Glossary

Dirac's notation for quantum states: a ket $|\psi ⟩$ is a column vector in Hilbert space, a bra $⟨\varphi |$ its conjugate transpose, and $⟨\varphi |\psi ⟩$ their inner product. $|\psi ⟩⟨\varphi |$ is an operator (outer product). For qubits, $|0⟩$ and $|1⟩$ label the computational basis, and $|00⟩$ is shorthand for $|0⟩\otimes |0⟩$.

$|+⟩=(|0⟩+|1⟩)/\sqrt{2}$

In the folios: Quantum Foundations and Theory › The postulates; Quantum Information › States are density operators; Quantum Algorithms › Phase estimation and the QFT; Quantum AI › Variational quantum models; Building a Quantum Computer › The qubit and its coherence; Quantum Hardware Engineering › Reading out a qubitSee also Hilbert space, Probability amplitude, Born rule, Qubit, Tensor product

A complex vector space with an inner product, complete with respect to the norm that inner product defines. In quantum theory every system is assigned one, and its pure states are unit rays in it. Its dimension can be finite (2 for a qubit or a spin-½) or infinite (a particle in space). Von Neumann made it the mathematical setting of the theory in 1932.

In the folio: Quantum Foundations and Theory › The postulatesSee also Ket notation, Ray, Tensor product

The set of all vectors $e^{i\phi }|\psi ⟩$ that differ from a given unit vector only by an overall phase. A pure state is a ray rather than a single vector because the global phase has no observable consequence.

In the folio: Quantum Foundations and Theory › The postulatesSee also Global phase, Hilbert space

An operator with $P^2=P=P^{†}$, which maps any state onto a subspace. Each outcome of an ideal measurement is associated with one projector; together they sum to the identity, so the outcome probabilities sum to one.

In the folio: Quantum Foundations and Theory › The postulatesSee also Observable, Born rule, Gleason's theorem

The way quantum state spaces combine: the joint space of systems A and B is ${\mathcal{H}}_A\otimes {\mathcal{H}}_B$, whose dimension is the product of the parts' dimensions. Many states in this space are not products of states of the parts, and this is the origin of entanglement. ${\mathcal{N}}^{\otimes n}$ denotes $n$ parallel uses of a channel.

${\mathcal{H}}_{AB}={\mathcal{H}}_A\otimes {\mathcal{H}}_B$

In the folios: Quantum Foundations and Theory › The postulates; Quantum Information › States are density operatorsSee also Entanglement, Hilbert space, Exchange symmetry, Partial trace, Regularisation

A common factor $e^{i\phi }$ multiplying the whole state. It cancels in every probability and expectation value and is therefore unobservable, which is why states are rays rather than vectors.

In the folio: Quantum Foundations and Theory › The postulatesSee also Relative phase, Ray

The phase difference between components of a superposition, for example $\phi$ in $\alpha |0⟩+e^{i\phi }\beta |1⟩$. Unlike a global phase it is physical: it sets the position of interference fringes and, as the folio puts it, carries all interference.

In the folio: Quantum Foundations and Theory › The postulatesSee also Global phase, Interference term

The operator $[A,B]=AB-BA$. Observables whose operators commute can share eigenstates and be sharp together; a non-zero commutator is the mathematical source of uncertainty relations and of Heisenberg-picture dynamics.

$[A,B]=AB-BA$

In the folio: Quantum Foundations and Theory › Uncertainty from non-commutationSee also Canonical commutation relation, Robertson uncertainty relation, Heisenberg picture

The three $2\times 2$ Hermitian, unitary matrices ${\sigma }_x,{\sigma }_y,{\sigma }_z$, written $X$, $Y$, $Z$ in quantum information. Together with the identity they span all qubit operators. $X$ is a bit flip and $Z$ a phase flip, and these are the corrections Bob applies in teleportation and the encodings Alice uses in superdense coding.

${\sigma }_x=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),{\sigma }_y=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right),{\sigma }_z=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$

In the folios: Quantum Foundations and Theory › No-go theorems; Quantum Information › States are density operators; Building a Quantum Computer › Engineering the gatesSee also Peres–Mermin square, Tensor product, Bloch vector, Quantum teleportation, Superdense coding

The unit of molecular mass, one twelfth of the mass of a carbon-12 atom; also called the dalton. A C${}_{60}$ molecule is 720 u, about $1.2\times 10^{-24}$ kg.

$1\text{u}\approx 1.6605\times 10^{-27}\text{kg}$

In the folio: Quantum Foundations and Theory › Probing the quantum–classical boundarySee also Fullerene

Planck's constant $h$, introduced in 1900 to explain black-body radiation: energy is exchanged with radiation of frequency $\nu$ only in multiples of $h\nu$. The reduced constant $\hslash =h/2\pi$ sets the scale below which quantum effects matter.

$h=6.62607015\times 10^{-34}\text{J s}$ (exact, SI 2019)

In the folio: Quantum Foundations and Theory › MilestonesSee also Light quanta, Classical limit

A unit sphere on which every pure qubit state is a point, with $|0⟩$ at the north pole, $|1⟩$ at the south pole and superpositions in between; the polar angle $\theta$ sets the populations and the azimuth $\phi$ the relative phase. Single-qubit gates are rotations of the sphere, which makes it the natural picture for drive pulses and for dephasing (a spreading of $\phi$).

$|\psi ⟩=\cos \frac{\theta }{2}|0⟩+e^{i\phi }\sin \frac{\theta }{2}|1⟩$

In the folios: Quantum Information › States are density operators; Building a Quantum Computer › The qubit and its coherenceSee also Bloch vector, Pure state, Mixed state, Depolarising channel, Ket notation

The real three-vector that fixes a qubit's density operator through its Pauli components. Its length measures how pure the state is: $|\mathbf{r}|=1$ for pure states and $|\mathbf{r}|<1$ for mixed ones.

$\rho =\frac{1}{2}(I+\mathbf{r}\cdot \mathbf{\sigma }),|\mathbf{r}|\le 1$

In the folio: Quantum Information › States are density operatorsSee also Bloch sphere, Pauli matrices, Purity

A Hermitian operator with no negative eigenvalues, written $A⪰0$. This guarantees that every measurement gives non-negative probabilities. The same symbol appears in the Choi condition for complete positivity.

In the folio: Quantum Information › States are density operatorsSee also Density operator, Choi matrix

The unit of entanglement: one maximally entangled pair of qubits (one Bell pair). The rate plot gives the capacity of a link in "ebits or secret bits per channel use", meaning the number of Bell pairs or secret-key bits that can be produced per optical mode sent down the fibre.

In the folios: Quantum Information › Entanglement is a resource; Quantum Networking › Loss sets the rateSee also Bell pair, Resource inequality, Entanglement entropy, Secret bits per channel use, Repeaterless bound

Asymptotic notation for how a cost grows with problem size, ignoring constant factors: $O(n^2)$ means at most proportional to $n^2$ for large $n$. The folio's cost panel warns that the ignored constants matter, because a fault-tolerant quantum operation can be about $10^5$ times slower than a classical one.

In the folio: Quantum Algorithms › Phase estimation and the QFTSee also Speed-up, Break-even point

"The norm of the Hamiltonian's description": for $H={\sum }_j{c}_j{P}_j$ written as a sum of Pauli strings, typically $\lambda ={\sum }_j|c_j|$. It plays the role of the block-encoding subnormalisation $\alpha$, so simulation cost grows as $\lambda t$; reducing $\lambda$ (e.g. by tensor hypercontraction) was key to cheaper chemistry estimates.

$\lambda =\sum _j|c_j|$

In the folio: Quantum Algorithms › Simulating quantum systemsSee also Qubitisation, Subnormalisation, Block encoding

Information carried per received photon, $C/N$; also called photon information efficiency. It is the figure of merit for power-starved links such as deep-space optical communication. At low photon numbers the Holevo limit grows without bound, as ${\log }_2(1/N)$ bits per photon. Homodyne reception saturates at $2/\ln 2\approx 2.9$ and heterodyne at $1/\ln 2\approx 1.4$.

Holevo: $g(N)/N\approx {\log }_2(1/N)$ for $N\ll 1$

In the folio: Quantum Communications › Light as a channel: the Holevo limitSee also Holevo limit, Homodyne detection, Heterodyne detection, Pulse-position modulation

The conventional names in cryptography for the sender (Alice), the receiver (Bob) and the eavesdropper (Eve). In QKD security proofs Eve is allowed to control the channel completely, limited only by the laws of physics.

In the folios: Quantum Communications › QKD protocol families; Quantum Networking › What quantum networks enableSee also Trusted device, Quantum key distribution

A logarithmic unit for loss or gain: a loss of $L$ dB means a fraction $10^{-L/10}$ of the power survives. Thus 30 dB is a factor 1,000 and 120 dB a factor $10^{12}$. Because fibre loss in dB adds linearly with length, the transmitted fraction falls exponentially.

$\eta =10^{-L_{\text{dB}}/10}$

In the folios: Quantum Communications › Links: fibre, air and space; Quantum Networking › Loss sets the rate; Quantum Hardware Engineering › Inside a dilution refrigeratorSee also Transmissivity, Fibre attenuation, Cryogenic attenuation

The conventional telecom window, about 1530–1565 nm, where silica fibre has its lowest loss (0.18 dB/km at 1550 nm on the folio). It is also where erbium-doped amplifiers work, which helps classical traffic but not single-photon qubits.

In the folio: Quantum Networking › Light carries the qubitsSee also O band, Telecom band, Quantum frequency conversion

The "original" telecom window, about 1260–1360 nm, with near-zero chromatic dispersion and loss of about 0.33 dB/km at 1310 nm. It is a common target for converted photons from diamond nodes, for example 737 → 1350 nm.

In the folio: Quantum Networking › Light carries the qubitsSee also C band, Quantum frequency conversion

The near-infrared wavelengths used by optical-fibre communication, chiefly the O band near 1310 nm and the C band near 1550 nm, where fibre loss is lowest. Most matter qubits emit at shorter, much lossier wavelengths, so their photons must be converted to these bands for long links.

In the folio: Quantum Networking › Nodes: spin–photon interfacesSee also C band, O band, Quantum frequency conversion

The key rate counted per optical mode or pulse sent, independent of clock speed. Multiplying by the source repetition rate (e.g. 10 GHz) gives bits per second. It is the natural unit for comparing protocols against the PLOB bound.

In the folio: Quantum Networking › Loss sets the rateSee also Ebit, Repeaterless bound

The mean outcome of measuring an observable $O$ on a state; it is the output of a variational model. On hardware it is estimated by averaging over many repeated runs (shots), so a small value or a small difference needs many shots to resolve.

$⟨O⟩=⟨\psi |O|\psi ⟩$

In the folio: Quantum AI › Variational quantum modelsSee also Shot, Variational quantum model, Gradient variance

One run of a circuit ending in a measurement. Estimating a quantity to precision $ϵ$ takes roughly $1/{ϵ}^2$ shots, which is why exponentially small gradients mean "exponentially many measurements are needed to train".

In the folio: Quantum AI › Trainability: barren plateausSee also Expectation value, Barren plateau

One thousandth of a kelvin. Superconducting and spin qubits run at about 10 mK. At that temperature a 5 GHz mode holds only about $4\times 10^{-11}$ thermal photons, so it sits essentially in its ground state.

$1\text{mK}=10^{-3}\text{K}$; $k_B\times 10\text{mK}\approx 0.86\mu \text{eV}\approx h\times 0.21\text{GHz}$

In the folio: Quantum Hardware Engineering › Why millikelvinSee also Thermal photon number, Qubit energy quantum

A positive operator of unit trace that describes a quantum state in general, including statistical mixtures and the state of a subsystem of a larger entangled system. A pure state corresponds to $\rho =|\psi ⟩⟨\psi |$; for a mixture $\rho$ is a weighted sum of such projectors. Its diagonal elements in a basis are probabilities; its off-diagonal elements (coherences) carry the capacity to interfere.

$⟨A⟩=\mathrm{Tr}(\rho A)$, $\mathrm{Tr}\rho =1$

In the folios: Quantum Foundations and Theory › The postulates; Quantum Information › States are density operatorsSee also Populations and coherences, Reduced density matrix, Improper mixture, Pure state, Mixed state

A measurable quantity, represented by a self-adjoint operator $A$ on the Hilbert space. Its spectral decomposition $A={\sum }_{a}a{P}_a$ lists the possible outcomes (the eigenvalues $a$) and the projectors $P_a$ onto the corresponding eigenspaces, which enter the Born rule.

$A=\sum _{a}a{P}_a$

In the folio: Quantum Foundations and Theory › The postulatesSee also Projector, Born rule, Incompatible observables

The postulate that converts amplitudes into probabilities: the probability of outcome $a$ is the squared magnitude of the state's component in the corresponding eigenspace. Proposed by Born in 1926, it is the only link between the formalism and experimental frequencies. Gleason's theorem shows it is the only consistent rule in dimension three or more, but why it holds remains one of the folio's open problems.

$p(a)=⟨\psi |P_a|\psi ⟩=|⟨a|\psi ⟩{|}^2$ (non-degenerate case)

In the folios: Quantum Foundations and Theory › The postulates; Quantum Algorithms › Where speed-ups come fromSee also Gleason's theorem, Probability amplitude, State update rule, Output readout

The postulate that, after an ideal measurement gives outcome $a$, the state is projected onto the matching eigenspace and renormalised. Unlike Schrödinger evolution it is non-linear, irreversible and random, which is the "second kind of change" whose relation to ordinary dynamics constitutes the measurement problem.

$|\psi ⟩\mapsto P_a|\psi ⟩/\sqrt{p(a)}$

In the folio: Quantum Foundations and Theory › The postulatesSee also Measurement problem, Born rule, Unitary evolution

A complex number, such as $⟨a|\psi ⟩$ or a wavefunction value $\psi (x)$, whose squared magnitude gives a probability. Amplitudes for indistinguishable alternatives add before squaring, which is the origin of all quantum interference; probabilities are never fixed by amplitudes alone but by $|⟨a|\psi ⟩{|}^2$.

$P(x)=|⟨x|\psi ⟩{|}^2$

In the folios: Quantum Foundations and Theory › Superposition and interference; Quantum Algorithms › Where speed-ups come fromSee also Superposition principle, Interference term, Born rule, Quantum interference, Ket notation

Any linear combination of allowed states is itself an allowed state, because the Schrödinger equation is linear. A superposition is not a statistical mixture of its components: its terms can interfere, and the cross term "has no classical counterpart".

In the folio: Quantum Foundations and Theory › Superposition and interferenceSee also Probability amplitude, Interference term, Density operator

The cross term that appears when two amplitudes are added and then squared. It produces the fringes in a double-slit pattern and is erased by any record, however inaccessible, of which path was taken.

$I=|{\psi }_1{|}^2+|{\psi }_2{|}^2+2\mathrm{Re}({\psi }_1^{\ast }{\psi }_2)$

In the folio: Quantum Foundations and Theory › Superposition and interferenceSee also Which-path information, Fringe visibility, Decoherence

The wavelength associated with a particle of momentum $p$, proposed by de Broglie in 1924. In a double slit of spacing $d$ observed at distance $L$ it sets the fringe spacing $\Delta y=\lambda L/d$; for heavy molecules it is far smaller than the molecule itself, which makes matter-wave interferometry so demanding.

$\lambda =h/p$, $\Delta y=\lambda L/d$

In the folio: Quantum Foundations and Theory › Superposition and interferenceSee also Matter-wave interferometry, Thermal de Broglie wavelength

Any record, in a detector, the environment or the particle's own internal state, that distinguishes which alternative (slit, arm, branch) a quantum object took. Once such information exists, even if nobody reads it, the corresponding interference disappears.

In the folio: Quantum Foundations and Theory › Superposition and interferenceSee also Path distinguishability, Duality relation, Decoherence

The contrast of an interference pattern, $V=(I_{\max }-I_{\min })/(I_{\max }+I_{\min })$, between 0 (no fringes) and 1 (perfect fringes). It is the standard measure of how much coherence survives in an interferometer.

$V=\frac{I_{\max }-I_{\min }}{I_{\max }+I_{\min }}$

In the folio: Quantum Foundations and Theory › Superposition and interferenceSee also Duality relation, Path distinguishability

A number between 0 and 1 measuring how well the path taken in an interferometer could in principle be inferred from the available which-path record; $D=1$ means the path is fully known.

In the folio: Quantum Foundations and Theory › Superposition and interferenceSee also Duality relation, Which-path information, Fringe visibility

The quantitative form of wave–particle complementarity: path distinguishability and fringe visibility cannot both be large. The folio cites Englert (1996); equality holds for pure states. It replaces the qualitative "either wave or particle" with a trade-off that can be measured.

$D^2+V^2\le 1$

In the folio: Quantum Foundations and Theory › Superposition and interferenceSee also Complementarity, Path distinguishability, Fringe visibility

Bohr's principle that some properties, such as wave-like interference and particle-like path, are revealed only by mutually exclusive experimental arrangements. The duality relation $D^2+V^2\le 1$ turns it into a quantitative statement.

In the folio: Quantum Foundations and Theory › Superposition and interferenceSee also Duality relation, Copenhagen interpretation

The defining relation between position and momentum operators along the same axis. It implies that position and momentum wavefunctions are Fourier transforms of each other and yields the Heisenberg bound ${\sigma }_x{\sigma }_p\ge \hslash /2$.

$[x,p]=i\hslash \Rightarrow {\sigma }_x{\sigma }_p\ge \hslash /2$

In the folio: Quantum Foundations and Theory › Uncertainty from non-commutationSee also Commutator, Robertson uncertainty relation, Minimum-uncertainty state

The general uncertainty relation for any two observables, where $\sigma$ is the standard deviation of outcomes in a given state. It is a property of the state itself, not of clumsy measurement: it bounds the spreads of separate, ideal measurements on identically prepared systems.

${\sigma }_A{\sigma }_B\ge \frac{1}{2}|⟨[A,B]⟩|$

In the folio: Quantum Foundations and Theory › Uncertainty from non-commutationSee also Commutator, Incompatible observables, Error–disturbance relation

Observables whose operators do not commute and so have no complete set of common eigenstates. They "cannot both be sharp in any state", and measuring one in general randomises the other.

In the folio: Quantum Foundations and Theory › Uncertainty from non-commutationSee also Commutator, Robertson uncertainty relation, Complementarity

A state that saturates the bound ${\sigma }_x{\sigma }_p=\hslash /2$. For position and momentum it is a Gaussian wave packet, whose Fourier transform is again Gaussian; squeezing it in $x$ necessarily broadens it in $p$.

In the folio: Quantum Foundations and Theory › Uncertainty from non-commutationSee also Canonical commutation relation, Robertson uncertainty relation

A relation of the form $\Delta t\Delta E\ge \hslash /2$ in which time is a parameter, not an operator, so it is not a Robertson relation. In the Mandelstam–Tamm version $\Delta t$ is the time a state needs to change appreciably (for instance to evolve to an orthogonal state), given its energy spread $\Delta E$.

$\Delta t\Delta E\ge \hslash /2$

In the folio: Quantum Foundations and Theory › Uncertainty from non-commutationSee also Robertson uncertainty relation, Unitary evolution

A distinct, measurement-based relation (Ozawa 2003) bounding how accurately one observable can be measured against how much that measurement disturbs an incompatible one. It corrects the naive "Heisenberg microscope" form $ϵ(A)\eta (B)\ge \frac{1}{2}|⟨[A,B]⟩|$, which can be violated.

In the folio: Quantum Foundations and Theory › Uncertainty from non-commutationSee also Robertson uncertainty relation, Incompatible observables

An uncertainty relation phrased with Shannon entropies $H$ of the outcome distributions of two measurements instead of standard deviations (Maassen–Uffink 1988). Here $c$ is the largest squared overlap $|⟨x|z⟩{|}^2$ between eigenvectors of the two observables; for two mutually unbiased qubit bases $c=1/2$, so the entropies must total at least one bit. It is the form used in security proofs for quantum cryptography.

$H(X)+H(Z)\ge {\log }_2(1/c)$, $c=\underset{x,z}{\max }|⟨x|z⟩{|}^2$

In the folio: Quantum Foundations and Theory › Uncertainty from non-commutationSee also Robertson uncertainty relation, Incompatible observables

The time evolution generated by the Schrödinger equation for a time-independent Hamiltonian $H$. A unitary operator preserves inner products, so probabilities are conserved and the evolution is reversible and linear. This is the "first kind of change" in the postulates.

$|\psi (t)⟩=e^{-iHt/\hslash }|\psi (0)⟩$

In the folio: Quantum Foundations and Theory › Dynamics: three equivalent picturesSee also Schrödinger picture, State update rule, Measurement problem

The formulation in which states evolve in time and operators representing observables are fixed. It is the most common starting point for wave mechanics and for quantum circuits.

In the folio: Quantum Foundations and Theory › Dynamics: three equivalent picturesSee also Heisenberg picture, Path integral, Unitary evolution

The formulation in which states are fixed and operators carry the time dependence, $A(t)=U^{†}AU$. It gives every prediction the Schrödinger picture does, and makes the analogy with classical equations of motion explicit.

$\frac{dA}{dt}=\frac{i}{\hslash }[H,A]$ (no explicit time dependence)

In the folio: Quantum Foundations and Theory › Dynamics: three equivalent picturesSee also Schrödinger picture, Ehrenfest theorem, Commutator

The statement that expectation values obey equations resembling Newton's laws: "Newton's law for averages". The match with classical motion is exact only when $⟨V^{\prime }(x)⟩\approx V^{\prime }(⟨x⟩)$, that is, when the wave packet is narrow on the scale over which the force varies.

$\frac{d}{dt}⟨p⟩=-⟨V^{\prime }(x)⟩$

In the folio: Quantum Foundations and Theory › Dynamics: three equivalent picturesSee also Heisenberg picture, Classical limit

Feynman's formulation (1948), in which the amplitude to go from $x_a$ to $x_b$ is a sum over every possible path, each weighted by a phase equal to its classical action in units of $\hslash$. It is equivalent to the Schrödinger and Heisenberg pictures and is the basis of quantum field theory and perturbation theory.

$K(x_b,x_a)=\int \mathcal{D}x{e}^{iS[x]/\hslash }$

In the folio: Quantum Foundations and Theory › Dynamics: three equivalent picturesSee also Propagator, Stationary-phase approximation, Classical limit

The amplitude for a particle at $x_a$ to be found at $x_b$ a given time later; equivalently the position-space matrix element of the evolution operator, $⟨x_b|U|x_a⟩$. Convolving it with an initial wavefunction gives the wavefunction at the later time.

In the folio: Quantum Foundations and Theory › Dynamics: three equivalent picturesSee also Path integral, Unitary evolution

An asymptotic method for integrals of rapidly oscillating phases: contributions cancel except near points where the phase is stationary. Applied to the path integral when the action is large compared with $\hslash$, it leaves only the neighbourhood of the classical path, where $\delta S=0$.

$\delta S=0$ (classical path)

In the folio: Quantum Foundations and Theory › Dynamics: three equivalent picturesSee also Classical limit, Path integral

The regime in which actions are large compared with $\hslash$, so phases of neighbouring paths cancel except near the path of stationary action and classical mechanics emerges as a stationary-phase approximation. This is why $\hslash$ sets the scale of quantum effects; decoherence explains why interference between macroscopically different alternatives is in practice not seen.

In the folio: Quantum Foundations and Theory › Dynamics: three equivalent picturesSee also Stationary-phase approximation, Ehrenfest theorem, Decoherence

A representation of a symmetry group that has no smaller invariant subspace. Wigner (1939) showed that elementary particles correspond to irreducible unitary representations of the Poincaré group (the symmetry group of special-relativistic spacetime), which are labelled by mass and spin.

In the folio: Quantum Foundations and Theory › Spin, symmetry, identical particlesSee also Spinor, Spin–statistics theorem

Particles of the same species, which in quantum theory are truly indistinguishable: no measurement can tell which is "particle 1". Consequently the many-particle state must be symmetric or antisymmetric under exchange, with consequences from Bose–Einstein condensation to the periodic table.

In the folio: Quantum Foundations and Theory › Spin, symmetry, identical particlesSee also Exchange symmetry, Boson, Fermion

The requirement that swapping two identical particles leaves the wavefunction unchanged (bosons, sign $+$) or flips its sign (fermions, sign $-$). In the composition postulate this is why states of identical particles are "(anti)symmetrised".

$\psi (x_2,x_1)=\pm \psi (x_1,x_2)$

In the folio: Quantum Foundations and Theory › Spin, symmetry, identical particlesSee also Boson, Fermion, Bunching

A particle of integer spin (0, 1, 2, …), whose many-particle wavefunction is symmetric under exchange. Photons, gluons, the Higgs boson and composite particles such as ${}^4$He atoms are bosons; any number can occupy the same state.

In the folio: Quantum Foundations and Theory › Spin, symmetry, identical particlesSee also Fermion, Exchange symmetry, Bunching

A particle of half-integer spin (½, 3/2, …), whose many-particle wavefunction is antisymmetric under exchange. Electrons, protons, neutrons and quarks are fermions; antisymmetry gives a node where two fermions would coincide and hence the exclusion principle.

In the folio: Quantum Foundations and Theory › Spin, symmetry, identical particlesSee also Boson, Pauli exclusion principle, Exchange symmetry

The enhanced probability of finding identical bosons close together, compared with distinguishable particles, arising from constructive interference of exchange amplitudes. Fermions show the opposite, a reduced probability (a node on the diagonal $x_1=x_2$ of the two-particle density).

In the folio: Quantum Foundations and Theory › Spin, symmetry, identical particlesSee also Boson, Fermion, Exchange symmetry

The theorem that particles of integer spin are bosons and those of half-integer spin are fermions. It cannot be derived in non-relativistic quantum mechanics, where it is put in by hand; it was proved in relativistic quantum field theory (Pauli 1940).

In the folio: Quantum Foundations and Theory › Spin, symmetry, identical particlesSee also Boson, Fermion, Irreducible representation

The kind of object that describes a spin-½ state. Under a full $2\pi$ rotation a spinor changes sign, returning to itself only after $4\pi$. The sign is invisible for a lone particle but appears as a relative phase in interference, and was seen in neutron interferometry in 1975.

$R(2\pi )|\psi ⟩=-|\psi ⟩$ for spin ½

In the folio: Quantum Foundations and Theory › Spin, symmetry, identical particlesSee also Neutron interferometry, Relative phase

No two identical fermions can occupy the same single-particle quantum state, a direct consequence of antisymmetry. It underlies atomic shell structure, chemistry, the rigidity of solids and the stability of matter.

In the folio: Quantum Foundations and Theory › Spin, symmetry, identical particlesSee also Fermion, Spin–statistics theorem

The conflict between the two kinds of change in the postulates. Treated quantum mechanically, a measurement only entangles system and apparatus, so linear dynamics yields a superposition of records, never one definite outcome; yet experiments always show a single result. Every interpretation in the folio is a different response to it.

$(\alpha |\uparrow ⟩+\beta |\downarrow ⟩)|M_0⟩\to \alpha |\uparrow ⟩|M_{\uparrow }⟩+\beta |\downarrow ⟩|M_{\downarrow }⟩$

In the folio: Quantum Foundations and Theory › The measurement problemSee also Pre-measurement, Maudlin's trilemma, Decoherence

Von Neumann's model of measurement as an ordinary unitary interaction that correlates a pointer state $|M_a⟩$ of the apparatus with each eigenstate of the measured system. The result is an entangled state of system and apparatus, not a definite reading, which is the starting point of the measurement problem.

In the folio: Quantum Foundations and Theory › The measurement problemSee also Measurement problem, Entanglement, Pointer state

A correlation between quantum systems that has no classical counterpart. A pure joint state is entangled when it is not a product $|a⟩|b⟩$ (a product state). A mixed state is entangled when it is not separable. In quantum information it is treated as a resource, counted in ebits, that LOCC cannot create but can consume to teleport qubits, double classical capacity or distil secret keys.

In the folios: Quantum Foundations and Theory › The measurement problem; Quantum Information › Entanglement is a resourceSee also Tensor product, Bell's theorem, Decoherence, Schmidt decomposition, Ebit

Wigner's thought experiment in which a friend measures a system inside a closed laboratory while Wigner, outside, describes the whole laboratory, friend included, as being in superposition. It sharpens the measurement problem by asking whose description is correct.

In the folio: Quantum Foundations and Theory › The measurement problemSee also Frauchiger–Renner theorem, Local friendliness, Measurement problem

A 2018 extension of Wigner's friend with two friends and two outside observers. It shows that agents who all use quantum theory and reason about each other's conclusions can be forced into contradiction unless one natural assumption (universal validity of quantum theory, consistency between agents, or single outcomes) is dropped.

In the folio: Quantum Foundations and Theory › The measurement problemSee also Wigner's friend, Local friendliness

A set of assumptions (the friends' observations are absolute events, plus a locality condition and free choice of settings) that implies inequalities on Wigner's-friend-type experiments, stronger than Bell's. Experiments reported in 2020, with photonic "friends", violated these inequalities, showing that one assumption must fail.

In the folio: Quantum Foundations and Theory › The measurement problemSee also Wigner's friend, Frauchiger–Renner theorem, Bell's theorem

Maudlin's (1995) formulation of the measurement problem: the claims "$\psi$ is complete", "$\psi$ always evolves linearly" and "each measurement has one definite outcome" are mutually inconsistent, so at most two can hold. Many worlds keeps the first two, collapse models keep completeness and definite outcomes, and pilot wave and QBism keep linearity and definite outcomes.

In the folio: Quantum Foundations and Theory › The measurement problemSee also Measurement problem, Many-worlds interpretation, Objective collapse models, Pilot-wave theory

The rapid loss of observable interference in a system that becomes entangled with, and leaves records in, its environment. Tracing out the environment suppresses the off-diagonal elements of the system's density matrix between distinguishable alternatives. It is the main practical obstacle to building quantum computers and the main reason macroscopic objects look classical, but it does not by itself explain why one outcome occurs.

${\rho }_{\uparrow \downarrow }\propto ⟨E_{\downarrow }|E_{\uparrow }⟩\to 0$

In the folios: Quantum Foundations and Theory › Decoherence; Building a Quantum Computer › The qubit and its coherenceSee also Partial trace, Populations and coherences, Decoherence time, Improper mixture, Relaxation time

The operation that discards subsystem B and returns the reduced state of A, which is all that measurements on A alone can reveal. When the joint state is entangled, the reduced state is mixed even though the whole is pure. In the Stinespring picture, "traced out" means the environment is discarded in this way.

${\rho }_A={\mathrm{Tr}}_B{\rho }_{AB}$

In the folios: Quantum Foundations and Theory › Decoherence; Quantum Information › States are density operatorsSee also Reduced density matrix, Density operator, Tensor product, Mixed state, Stinespring dilation

The density operator of a subsystem, obtained by the partial trace of the joint state. If the subsystem is entangled with something else, its reduced state is mixed even though the global state is pure.

In the folio: Quantum Foundations and Theory › DecoherenceSee also Partial trace, Improper mixture, Density operator

The diagonal entries of a density matrix in a chosen basis (populations: the occupation probabilities) and its off-diagonal entries (coherences: what distinguishes a superposition from a classical mixture). For a qubit, coherences are the $x$ and $y$ components of the Bloch vector. Dephasing erases them while keeping the populations.

In the folios: Quantum Foundations and Theory › Decoherence; Quantum Information › Channels are CPTP mapsSee also Decoherence, Density operator, Fringe visibility, Dephasing channel, Bloch vector

The time over which the coherences between two alternatives decay. For a particle of mass $m$ in a superposition of positions separated by $\Delta x$, in contact with an environment at temperature $T$, it is shorter than the relaxation time by $({\lambda }_{\text{th}}/\Delta x{)}^2$. For 1 g at room temperature and $\Delta x=1$ cm, ${\tau }_D/{\tau }_R\sim 10^{-40}$ (Zurek), which is why decoherence acts at "extraordinary speed".

${\tau }_D\approx {\tau }_R({\lambda }_{\text{th}}/\Delta x{)}^2$

In the folio: Quantum Foundations and Theory › DecoherenceSee also Relaxation time, Thermal de Broglie wavelength, Decoherence

In Zurek's estimate of the decoherence time, the time scale on which the environment damps the system's motion classically (the inverse of the friction or damping rate). Decoherence of macroscopic superpositions is enormously faster than this relaxation.

In the folios: Quantum Foundations and Theory › Decoherence; Building a Quantum Computer › The qubit and its coherence; Quantum Hardware Engineering › Materials and fabricationSee also Decoherence time, Coherence time, Pure dephasing time, Decoherence, Two-level-system defect

A de Broglie wavelength corresponding to thermal momentum. In Zurek's decoherence estimate it is defined as $\hslash /\sqrt{2m{k}_{B}T}$; other fields often use $h/\sqrt{2\pi m{k}_{B}T}$, so factors of order unity depend on convention. Superpositions much wider than ${\lambda }_{\text{th}}$ decohere almost instantly.

${\lambda }_{\text{th}}=\hslash /\sqrt{2m{k}_{B}T}$ (Zurek's convention)

In the folio: Quantum Foundations and Theory › DecoherenceSee also Decoherence time, De Broglie wavelength

A state of a system (originally of a measuring apparatus's pointer) that is robust under its interaction with the environment, so it is recorded without being disturbed. For large objects, localised position states are pointer states, which is why we see objects in definite places.

In the folio: Quantum Foundations and Theory › DecoherenceSee also Einselection, Quantum Darwinism, Pre-measurement

Zurek's term for the process by which the system–environment coupling picks out a preferred set of pointer states: superpositions of pointer states decohere quickly, while pointer states themselves persist.

In the folio: Quantum Foundations and Theory › DecoherenceSee also Pointer state, Decoherence, Quantum Darwinism

Zurek's account of objectivity: information about pointer states is copied redundantly into many fragments of the environment (for example scattered photons), so many observers can independently read the same result without disturbing the system and therefore agree.

In the folio: Quantum Foundations and Theory › DecoherenceSee also Einselection, Pointer state

A mixed reduced state that arises from tracing out part of an entangled pure state, as opposed to a proper mixture, which describes ignorance of which pure state was actually prepared. The two give identical local statistics, but for an improper mixture there is no fact about which component is "really" there; this is why decoherence explains why interference is unobservable, not why one outcome occurs.

In the folio: Quantum Foundations and Theory › DecoherenceSee also Reduced density matrix, Decoherence, Measurement problem

An account of what the formalism says about reality, and in particular of how it answers the measurement problem. Each gives up something different. All but objective collapse models reproduce the standard predictions exactly, so they cannot be told apart by present experiments.

In the folio: Quantum Foundations and Theory › InterpretationsSee also Copenhagen interpretation, Many-worlds interpretation, Pilot-wave theory, Objective collapse models

The family of views associated with Bohr and Heisenberg from 1927 on, in which the wavefunction is a tool for predicting the results of classically described experiments. There is one outcome, at measurement, and measurement is treated as primitive rather than analysed. It makes no new predictions.

In the folio: Quantum Foundations and Theory › InterpretationsSee also Complementarity, Measurement problem, Interpretation of quantum mechanics

Everett's 1957 proposal that the wavefunction is physically real and complete and always evolves unitarily. Every outcome occurs, each in its own decohered branch, so the measurement problem is resolved by denying a single outcome. It makes no new predictions; deriving the Born rule within it is a central debate.

In the folio: Quantum Foundations and Theory › InterpretationsSee also Decoherence, Maudlin's trilemma, Born rule

A hidden-variable theory (de Broglie 1927, Bohm 1952) in which particles always have definite positions, guided by a real wavefunction that evolves by the Schrödinger equation. The outcome is fixed by the positions, and the guidance is explicitly non-local. When positions are distributed according to $|\psi {|}^2$ (quantum equilibrium) it reproduces every standard prediction.

In the folio: Quantum Foundations and Theory › InterpretationsSee also Quantum equilibrium, Hidden variables, Bell nonlocality

In pilot-wave theory, the condition that particle positions are distributed as $|\psi {|}^2$. It is preserved by the dynamics and guarantees agreement with the Born rule; departures from it ("quantum non-equilibrium") would give new predictions, but none have been observed.

In the folio: Quantum Foundations and Theory › InterpretationsSee also Pilot-wave theory, Born rule

Additional variables, beyond the quantum state, that would fix or influence measurement outcomes, so that quantum probabilities reflect ignorance of them. The no-go theorems of Bell, Kochen–Specker, PBR and Leggett–Garg each rule out whole classes of such models.

In the folio: Quantum Foundations and Theory › No-go theoremsSee also Bell's theorem, Kochen–Specker theorem, Psi-ontic and psi-epistemic models

The property of some entangled states that their measurement statistics cannot be reproduced by any local hidden-variable model. It is shown experimentally by violating a Bell inequality such as CHSH. Nonlocality does not allow signalling, but it can certify randomness and security without trusting the devices.

In the folios: Quantum Foundations and Theory › No-go theorems; Quantum Information › Bell nonlocalitySee also Bell's theorem, Pilot-wave theory, EPR paradox, CHSH inequality, Local hidden-variable model

Theories that modify the Schrödinger equation with small non-linear, stochastic terms so that superpositions collapse spontaneously, negligibly for microscopic systems but rapidly for macroscopic ones. The wavefunction is real and there is one outcome, by stochastic collapse. They are the only interpretations in the table with new predictions: spontaneous heating and loss of large-scale interference.

In the folio: Quantum Foundations and Theory › InterpretationsSee also GRW model, Continuous spontaneous localisation, Diósi–Penrose model

The first objective collapse model (Ghirardi, Rimini and Weber 1986). Each particle's wavefunction suffers, at random times, a spontaneous localisation to a width of about $10^{-7}$ m; the rate per particle is tiny (originally about $10^{-16}{\text{s}}^{-1}$), but in a body of $\sim 10^{23}$ particles collapse happens almost at once.

In the folio: Quantum Foundations and Theory › InterpretationsSee also Objective collapse models, Continuous spontaneous localisation

A refinement of GRW in which collapse acts continuously through a noise field coupled to mass density, with two parameters: a collapse rate $\lambda$ and a correlation length $r_C$. Matter-wave interferometry, mechanical cat states, heating measurements and gravitational-wave detectors all bound these parameters.

In the folio: Quantum Foundations and Theory › InterpretationsSee also GRW model, Objective collapse models, Matter-wave interferometry

A collapse model in which gravity triggers collapse: a superposition of different mass distributions decays in a time of order $\hslash /E_G$, where $E_G$ is the gravitational self-energy of the difference between them. The folio reports that an underground X-ray search excluded its parameter-free version (2020), by looking for the radiation its collapse noise would cause.

$\tau \sim \hslash /E_G$

In the folio: Quantum Foundations and Theory › InterpretationsSee also Objective collapse models, Gravity-induced entanglement

An interpretation (Caves, Fuchs and Schack 2002) in which the wavefunction represents an agent's degrees of belief about future experiences, not a physical object. Each agent has one outcome in their own experience, and collapse is simply updating belief. It makes no new predictions.

In the folio: Quantum Foundations and Theory › InterpretationsSee also Interpretation of quantum mechanics, Relational quantum mechanics

Rovelli's 1996 interpretation in which states and measured values are relative to another physical system, not absolute. Outcomes are relative to each observing system, so the measurement problem is resolved by making facts observer-relative. It makes no new predictions.

In the folio: Quantum Foundations and Theory › InterpretationsSee also QBism, Wigner's friend

The 1964 theorem that no local hidden-variable theory can reproduce all the correlations of entangled quantum systems. It turns the EPR debate into a testable inequality; violations have been observed, without loopholes since 2015.

In the folios: Quantum Foundations and Theory › No-go theorems; Quantum Information › MilestonesSee also CHSH inequality, Local hidden-variable model, Loophole-free Bell test, EPR paradox, Bell nonlocality

A classical explanation of correlations in which each outcome is fixed by the local measurement setting and shared pre-existing variables, with no influence from the distant setting. All such models satisfy $|S|\le 2$ in the CHSH test, and the best one gives a correlation that is linear in the angle.

In the folios: Quantum Foundations and Theory › No-go theorems; Quantum Information › Bell nonlocalitySee also Bell's theorem, Hidden variables, CHSH inequality

The Bell inequality of Clauser, Horne, Shimony and Holt (1969), built from two measurement settings per side. $E(a,b)$ is the average product of the $\pm 1$ outcomes for settings $a$ and $b$. Local models obey $|S|\le 2$, quantum mechanics reaches up to $2\sqrt{2}$, and the algebraic maximum is 4.

$S=E(a,b)-E(a,b^{\prime })+E(a^{\prime },b)+E(a^{\prime },b^{\prime })$

$|S|\le 2$ (local), $|S|\le 2\sqrt{2}$ (quantum)

In the folios: Quantum Foundations and Theory › No-go theorems; Quantum Information › Bell nonlocalitySee also Tsirelson bound, Bell's theorem, Local hidden-variable model, Singlet

The maximum value of the CHSH quantity $|S|$ that quantum mechanics allows, $2\sqrt{2}$, reached with maximally entangled states and optimal settings. That nature stops there, rather than at the algebraic maximum of 4, bears on the open question of which principles single out quantum theory.

$|S{|}_{\text{QM}}\le 2\sqrt{2}$

In the folios: Quantum Foundations and Theory › No-go theorems; Quantum Information › Bell nonlocalitySee also CHSH inequality, Singlet

The 1967 theorem that, in Hilbert spaces of dimension three or more, it is impossible to assign definite values to all observables independently of which other compatible observables are measured with them. Hidden variables must therefore be contextual.

In the folio: Quantum Foundations and Theory › No-go theoremsSee also Contextuality, Peres–Mermin square, Gleason's theorem

The property that the value a hidden-variable model assigns to an observable must depend on the measurement context, that is, on which compatible observables are measured alongside it. Quantum theory is contextual by the Kochen–Specker theorem; contextuality is also considered a resource for quantum computational advantage.

In the folio: Quantum Foundations and Theory › No-go theoremsSee also Kochen–Specker theorem, Peres–Mermin square

A $3\times 3$ array of two-qubit Pauli observables in which the three operators in each row and column commute. The rows each multiply to $+I$ and the first two columns to $+I$, but the last column multiplies to $-I$, so no assignment of values $\pm 1$ can satisfy all six product constraints: a simple proof of the Kochen–Specker theorem.

In the folio: Quantum Foundations and Theory › No-go theoremsSee also Kochen–Specker theorem, Pauli matrices, Contextuality

The 1957 theorem that in Hilbert spaces of dimension three or more, any assignment of probabilities to projectors that is non-contextual (adds to one over every orthonormal frame regardless of the frame) must take the form $p(P)=\mathrm{Tr}(\rho P)$ for some density operator. The Born rule is thus forced; the Kochen–Specker theorem follows as a corollary.

$p(P)=\mathrm{Tr}(\rho P)$

In the folio: Quantum Foundations and Theory › No-go theoremsSee also Born rule, Kochen–Specker theorem, Projector

The 2012 theorem that, if independently prepared systems have independent hidden states, distinct pure quantum states cannot share hidden states. Within such models the wavefunction must be ontic (real) rather than merely knowledge.

In the folio: Quantum Foundations and Theory › No-go theoremsSee also Psi-ontic and psi-epistemic models, Hidden variables

Two classes of hidden-variable (ontological) models. In a $\psi$-ontic model each hidden state $\lambda$ determines the wavefunction, so the distributions over $\lambda$ for distinct states never overlap; in a $\psi$-epistemic model they can overlap, so $\psi$ is partly a state of knowledge. The PBR theorem rules out the epistemic class under its preparation-independence assumption.

In the folio: Quantum Foundations and Theory › No-go theoremsSee also PBR theorem, Hidden variables

A Bell-like inequality for correlations of one system at different times (Leggett and Garg 1985), with $C_{ij}$ the correlation between measurements at times $t_i$ and $t_j$. Macrorealism requires $K\le 1$; a precessing qubit, with $C=\cos \omega \tau$ between successive times, reaches 1.5 at $\omega \tau =\pi /3$.

$K=C_{12}+C_{23}-C_{13}\le 1$

In the folio: Quantum Foundations and Theory › No-go theoremsSee also Macrorealism, Bell's theorem

The view, formalised by Leggett and Garg, that a macroscopic system with two or more distinguishable states is always in one of them, and that this can be checked without disturbing its later behaviour (non-invasive measurability). It bounds correlations in time, $K\le 1$.

In the folio: Quantum Foundations and Theory › No-go theoremsSee also Leggett–Garg inequality, Schrödinger cat state

Interference experiments with massive particles, from electrons (diffraction, 1927) and neutrons (1974) to C${}_{60}$ fullerenes (1999) and molecules of more than 25,000 u and about 2,000 atoms (2019). Each heavier object tests whether superposition survives, and together they bound collapse models.

In the folio: Quantum Foundations and Theory › Probing the quantum–classical boundarySee also De Broglie wavelength, Neutron interferometry, Continuous spontaneous localisation

A closed carbon cage molecule; C${}_{60}$ ("buckyball") has 60 atoms and a mass of 720 u. Its interference in 1999 was the first for a molecule large and hot enough to emit thermal radiation, a landmark for testing decoherence.

In the folio: Quantum Foundations and Theory › Probing the quantum–classical boundarySee also Matter-wave interferometry, Atomic mass unit

Interferometry in which a neutron beam is split and recombined, typically by Bragg diffraction in a single silicon crystal, first done in 1974. In 1975 it showed that a $2\pi$ rotation multiplies a spin-½ state by $-1$; later it displayed the effect of gravity on a quantum phase.

In the folio: Quantum Foundations and Theory › Probing the quantum–classical boundarySee also Spinor, Matter-wave interferometry

A superposition of two macroscopically or mesoscopically distinct states, named after Schrödinger's 1935 thought experiment with a cat. In 2023 a cat state of motion was prepared in a 16 µg oscillator (Bild et al.), the heaviest object on the folio's mass scale.

In the folio: Quantum Foundations and Theory › Probing the quantum–classical boundarySee also Macrorealism, Objective collapse models, Measurement problem

A proposed test of whether gravity obeys quantum theory: if two masses, each in a spatial superposition, become entangled solely through their gravitational interaction, gravity must be able to carry quantum information. The open question "can it entangle two masses?" has not yet been answered experimentally.

In the folio: Quantum Foundations and Theory › Open problemsSee also Entanglement, Diósi–Penrose model

Einstein's 1905 hypothesis that light itself consists of quanta of energy $h\nu$, later called photons. It explained the photoelectric effect and was the first claim that the quantum is a property of radiation, not only of its emission.

In the folio: Quantum Foundations and Theory › MilestonesSee also Quantum of action

Heisenberg's 1925 formulation of quantum mechanics in terms of arrays (matrices) of transition quantities, developed with Born and Jordan. It was soon shown to be equivalent to Schrödinger's 1926 wave mechanics and survives as the Heisenberg picture.

In the folio: Quantum Foundations and Theory › MilestonesSee also Heisenberg picture, Schrödinger picture

The 1935 argument by Einstein, Podolsky and Rosen that the perfect correlations of an entangled pair imply quantum mechanics is incomplete. In the same year Schrödinger named the phenomenon entanglement. Bell's theorem later showed that the "elements of reality" EPR sought cannot be local.

In the folios: Quantum Foundations and Theory › Milestones; Quantum Information › MilestonesSee also Bell's theorem, Entanglement, Bell nonlocality

A state of maximal knowledge, described by a single state vector $|\psi ⟩$, or equivalently by the rank-one projector $\rho =|\psi ⟩⟨\psi |$. For a qubit, pure states lie on the surface of the Bloch ball ($|\mathbf{r}|=1$) and have purity 1.

In the folio: Quantum Information › States are density operatorsSee also Mixed state, Density operator, Bloch sphere, Purity

A state that cannot be written as a single state vector. It arises from classical ignorance about which pure state was prepared, or as the reduced state of one part of an entangled system. For a qubit, mixed states lie strictly inside the Bloch ball ($|\mathbf{r}|<1$) and have purity below 1.

In the folio: Quantum Information › States are density operatorsSee also Pure state, Maximally mixed state, Partial trace, Purity

A quantum two-level system used as the unit of quantum information; unlike a bit it can be in any superposition of its two basis states $|0⟩$ and $|1⟩$. Every hardware platform must isolate two levels from a larger spectrum, drive them coherently, couple pairs and read them out while keeping the environment from learning the state.

$|\psi ⟩=\alpha |0⟩+\beta |1⟩,|\alpha {|}^2+|\beta {|}^2=1$

In the folios: Quantum Information › States are density operators; Quantum Algorithms › Phase estimation and the QFT; Quantum AI › Variational quantum models; Building a Quantum Computer › The qubit and its coherenceSee also Bloch sphere, Ebit, Schumacher compression, Ket notation, Logical qubit

The qubit state $\rho =I/2$ ($I$ is the identity), at the centre of the Bloch ball. It gives completely random outcomes for every measurement and has the maximal entropy of 1 bit. Each half of a Bell pair, viewed alone, is maximally mixed, which is why Bob learns nothing in teleportation until the classical bits arrive.

In the folio: Quantum Information › States are density operatorsSee also Bloch sphere, Von Neumann entropy, Quantum teleportation

A scalar measure of how mixed a state is. It equals 1 for pure states and $1/d$ for the maximally mixed state in dimension $d$. For a qubit it depends only on the length of the Bloch vector.

$\mathrm{Tr}{\rho }^2=\frac{1}{2}\left(1+|\mathbf{r}{|}^2\right)$

In the folio: Quantum Information › States are density operatorsSee also Bloch vector, Pure state, Mixed state

The most general description of a quantum measurement: a set of positive operators $E_m$ summing to the identity, with outcome probabilities given by the Born rule. It includes projective (von Neumann) measurements as a special case. It also covers noisy detectors and measurements with more outcomes than the system has dimensions.

$p(m)=\mathrm{Tr}(E_m\rho ),\sum _m{E}_m=I$

In the folio: Quantum Information › States are density operatorsSee also Density operator, Accessible information

A collection of states ${\rho }_x$ prepared with probabilities $p_x$, written $\{p_x,{\rho }_x\}$. Different ensembles can average to the same density operator (for example equal mixtures of $|0⟩,|1⟩$ or of $|+⟩,|-⟩$ both give $I/2$), and then no measurement can tell them apart. Ensembles are also how classical messages are encoded into quantum states, which the Holevo quantity bounds.

In the folio: Quantum Information › States are density operatorsSee also Density operator, Holevo quantity

A canonical form of any pure bipartite state as a single sum over paired orthonormal bases, with non-negative weights. The state is a product exactly when only one Schmidt coefficient is non-zero. The coefficients alone fix how much entanglement it holds.

$|\psi {⟩}_{AB}=\sum _i\sqrt{{\lambda }_i}|a_i⟩|b_i⟩$

In the folio: Quantum Information › Entanglement is a resourceSee also Schmidt coefficients, Entanglement entropy, Entanglement

The weights ${\lambda }_i$ (summing to 1) in the Schmidt decomposition. They are also the eigenvalues of either reduced state. For two qubits, ${\lambda }_1=0$ or $1$ gives a product state and ${\lambda }_1=\frac{1}{2}$ gives a Bell state with 1 ebit.

In the folio: Quantum Information › Entanglement is a resourceSee also Schmidt decomposition, Entanglement entropy

The unique measure of entanglement for pure bipartite states: the von Neumann entropy of either reduced state. It counts how many Bell pairs the state is worth under LOCC in the many-copy limit. For two qubits it reaches its maximum of 1 ebit at the Bell states.

$E(\psi )=S({\rho }_A)=-\sum _i{\lambda }_i{\log }_2{\lambda }_i$

In the folio: Quantum Information › Entanglement is a resourceSee also Schmidt coefficients, Ebit, Von Neumann entropy, Entanglement concentration and dilution

Two qubits in one of the four maximally entangled Bell states. A Bell pair is the basic "service" a quantum network delivers. It is used up when it is consumed, for example when it is measured for a key bit or used to teleport a qubit, so the network must keep generating fresh pairs.

$|{\Phi }^{\pm }⟩=(|00⟩\pm |11⟩)/\sqrt{2}$, $|{\Psi }^{\pm }⟩=(|01⟩\pm |10⟩)/\sqrt{2}$

In the folios: Quantum Information › Entanglement is a resource; Quantum Networking › Why quantum networks differSee also Ebit, Singlet, Bell measurement, Fidelity

The antisymmetric Bell state $\frac{1}{\sqrt{2}}(|01⟩-|10⟩)$, the spin-0 state of two spin-½ particles. Measured along directions at angle $\theta$, its spins are anticorrelated with correlation $E(\theta )=-\cos \theta$. With optimal measurement angles this reaches the Tsirelson bound.

In the folio: Quantum Information › Bell nonlocalitySee also Bell pair, CHSH inequality, Tsirelson bound

The class of protocols in which separated parties act only on their own systems and talk over classical channels. LOCC cannot create entanglement, so entanglement is the resource whose value is defined by what LOCC can do with it.

In the folio: Quantum Information › Entanglement is a resourceSee also Entanglement, Entanglement concentration and dilution, Separable state

The two LOCC procedures that make pure-state entanglement interchangeable. Concentration (distillation) turns $n$ copies of a partly entangled pure state into about $nE$ Bell pairs. Dilution turns about $nE$ Bell pairs back into $n$ copies. Both are asymptotically lossless, and this is what makes $E(\psi )$ the unique pure-state measure.

In the folio: Quantum Information › Entanglement is a resourceSee also LOCC, Entanglement entropy, Entanglement purification

A mixed state of two systems that can be prepared by LOCC: a probabilistic mixture of product states. A state that is not separable is entangled. Deciding whether a given state is separable is NP-hard in general (Gurvits, 2003).

$\rho =\sum _k{p}_k{\rho }_A^k\otimes {\rho }_B^k$

In the folio: Quantum Information › Entanglement is a resourceSee also Peres–Horodecki criterion, Entanglement, LOCC

An entanglement test based on the partial transpose: transposing only subsystem B's indices of ${\rho }_{AB}$. If the result has a negative eigenvalue (negative partial transpose, NPT), the state is certainly entangled. The test is decisive for $2\times 2$ and $2\times 3$ systems; in larger dimensions some entangled states have a positive partial transpose (PPT).

In the folio: Quantum Information › Entanglement is a resourceSee also Separable state, Bound entanglement

Entanglement that cannot be distilled into Bell pairs by LOCC, even though it took entanglement to create the state. All PPT entangled states are bound. Whether bound-entangled states with a negative partial transpose exist is a long-standing open problem named in the folio.

In the folio: Quantum Information › Open problemsSee also Peres–Horodecki criterion, Entanglement purification

The general description of a physical process acting on a quantum system, including noise, transmission and storage. It is a linear map from input density operators to output density operators. Kraus operators, a Stinespring dilation and a Choi matrix are three equivalent ways to describe one.

In the folio: Quantum Information › Channels are CPTP mapsSee also CPTP map, Kraus operators, Stinespring dilation, Choi matrix

The mathematical requirement on a quantum channel. Trace-preserving means probabilities still sum to 1. Completely positive means the map gives a valid state even when it acts on one half of an entangled system. Every physical process, noise included, is of this form.

In the folio: Quantum Information › Channels are CPTP mapsSee also Complete positivity, Quantum channel, Kraus operators

A map $\mathcal{E}$ is completely positive if $\mathcal{E}\otimes \mathrm{id}$ maps positive operators to positive operators for every size of ancillary system. Plain positivity is not enough: transposition maps states to states but fails on half of an entangled pair. This is why the partial transpose can detect entanglement.

In the folio: Quantum Information › Channels are CPTP mapsSee also CPTP map, Choi matrix, Peres–Horodecki criterion

The operators of the operator-sum (Kraus) form of a channel. The completeness condition makes the map trace-preserving, and $†$ denotes the Hermitian adjoint. Each $K_k$ can be read as one possible "error" that happens with a state-dependent probability.

$\mathcal{E}(\rho )=\sum _k{K}_k\rho K_k^{†},\sum _k{K}_k^{†}{K}_k=I$

In the folio: Quantum Information › Channels are CPTP mapsSee also CPTP map, Stinespring dilation

The theorem (Stinespring, 1955) that every channel can be realised as a unitary interaction $U$ of the system with an environment starting in a fixed state $|0{⟩}_E$, after which the environment is traced out. Noise is therefore information leaking into an environment, an idea used throughout capacity and security proofs.

$\mathcal{E}(\rho )={\mathrm{Tr}}_E\left[U(\rho \otimes |0⟩⟨0{|}_E)U^{†}\right]$

In the folio: Quantum Information › Channels are CPTP mapsSee also Kraus operators, Partial trace, Quantum channel

The operator obtained by applying a channel to one half of an (unnormalised) maximally entangled state. It encodes the channel completely. Choi's theorem says the map is completely positive exactly when this matrix is positive semidefinite, which gives a practical test.

$J=\sum _{ij}|i⟩⟨j|\otimes \mathcal{E}(|i⟩⟨j|)⪰0$

In the folio: Quantum Information › Channels are CPTP mapsSee also Complete positivity, Positive semidefinite operator

The noise model that, with some probability, replaces the state by the maximally mixed state. It shrinks the Bloch ball uniformly towards the centre. It is the standard symmetric noise model for qubits, yet its quantum capacity is still unknown.

$\mathcal{E}(\rho )=(1-p)\rho +pI/2$

In the folio: Quantum Information › Channels are CPTP mapsSee also Bloch sphere, Quantum capacity, Maximally mixed state

Noise that destroys superpositions between $|0⟩$ and $|1⟩$ but leaves their probabilities untouched. It squeezes the Bloch ball into a cigar along the $z$ axis. It models pure dephasing, the $T_2$-type decay of qubit coherence.

In the folio: Quantum Information › Channels are CPTP mapsSee also Populations and coherences, Amplitude damping channel, Bloch sphere

Noise that models energy loss, such as spontaneous emission or photon loss, taking $|1⟩$ to $|0⟩$ with some probability. It shrinks the Bloch ball and pulls every state towards $|0⟩$. In hardware it corresponds to energy relaxation with time constant $T_1$.

In the folio: Quantum Information › Channels are CPTP mapsSee also Dephasing channel, Depolarising channel, Bloch sphere

The quantum generalisation of Shannon entropy: the Shannon entropy of the eigenvalues of $\rho$, measured in bits when the logarithm is base 2. It is 0 for pure states and ${\log }_{2}d$ for the maximally mixed state of dimension $d$. Operationally it is the number of qubits per symbol needed to compress a quantum source.

$S(\rho )=-\mathrm{Tr}\rho {\log }_2\rho ,0\le S\le {\log }_{2}d$

In the folio: Quantum Information › Entropy measures quantum informationSee also Shannon entropy, Schumacher compression, Conditional entropy

The classical measure of uncertainty, or information content, of a random variable: $H(X)=-{\sum }_{x}p(x){\log }_{2}p(x)$ bits. Shannon's 1948 theory shows it is the optimal compression rate. $H(X|B)$ in the uncertainty relation is the entropy of a measurement outcome conditioned on a quantum memory.

In the folio: Quantum Information › Entropy measures quantum informationSee also Von Neumann entropy, Binary entropy, Channel capacity

The Shannon entropy of a coin with bias $p$. In QKD, $h(Q)$ is both the error-correction cost and the information an eavesdropper may have gained, per sifted bit at error rate $Q$. Since $h(0.11)\approx 0.5$, the BB84 key rate $1-2h(Q)$ falls to zero near $Q\approx 11\%$.

$h(p)=-p{\log }_{2}p-(1-p){\log }_2(1-p)$

In the folios: Quantum Information › Protocols: trading ebits, bits and qubits; Quantum Communications › From raw clicks to a secret keySee also Key rate, Quantum bit error rate, Shannon entropy, Decoy-state key rate

The total correlation, classical and quantum, between systems A and B. For a Bell pair it equals 2 bits, twice the classical maximum for two bits. It sets the entanglement-assisted capacity of a channel.

$I(A:B)=S(A)+S(B)-S(AB)$

In the folio: Quantum Information › Entropy measures quantum informationSee also Conditional entropy, Entanglement-assisted capacity, Von Neumann entropy

The entropy of A given access to B. Classically it is never negative; quantum mechanically it can be. A negative value is a signature of entanglement: for a Bell pair $S(A|B)=-1$. Its negative is the number of ebits left over in state merging.

$S(A|B)=S(AB)-S(B)$

In the folio: Quantum Information › Entropy measures quantum informationSee also State merging, Quantum mutual information, Entropic uncertainty relation with quantum memory, Coherent information

A measure of how distinguishable state $\rho$ is from state $\sigma$ (Umegaki's quantum Kullback–Leibler divergence). It is non-negative, zero only when the states are equal, and not symmetric, so it is not a distance. Most other entropic quantities can be written as relative entropies.

$D(\rho \Vert \sigma )=\mathrm{Tr}\rho (\log \rho -\log \sigma )$

In the folio: Quantum Information › Entropy measures quantum informationSee also Data-processing inequality, Von Neumann entropy

For an ensemble of states ${\rho }_x$ sent with probabilities $p_x$, the entropy of the average state minus the average entropy. It upper-bounds the classical information any measurement can extract (the Holevo bound), and its maximum over ensembles gives a channel's one-shot classical capacity.

$\chi =S\left(\sum _x{p}_x{\rho }_x\right)-\sum _x{p}_{x}S({\rho }_x)$

In the folio: Quantum Information › Entropy measures quantum informationSee also Holevo bound, Accessible information, HSW theorem, Ensemble

The fundamental inequality of quantum entropy, proved by Lieb and Ruskai in 1973. It is equivalent to the statement that conditioning cannot increase entropy, $S(A|BC)\le S(A|B)$. Almost every coding theorem in quantum information depends on it.

$S(AB)+S(BC)\ge S(ABC)+S(B)$

In the folio: Quantum Information › Entropy measures quantum informationSee also Data-processing inequality, Conditional entropy

The principle that no channel can make two states more distinguishable: relative entropy can only decrease under any CPTP map. It formalises "post-processing cannot create information" and is the source of many converse (upper-bound) proofs.

$D(\mathcal{E}(\rho )\Vert \mathcal{E}(\sigma ))\le D(\rho \Vert \sigma )$

In the folio: Quantum Information › Entropy measures quantum informationSee also Relative entropy, CPTP map, Strong subadditivity

The quantum analogue of Shannon's source coding theorem (Schumacher, 1995). Many copies of a source with state $\rho$ can be faithfully compressed to $S(\rho )$ qubits per symbol, and no further. This gave von Neumann entropy its operational meaning and the qubit its role as a unit of information.

In the folio: Quantum Information › Entropy measures quantum informationSee also Von Neumann entropy, Qubit

The task of transferring A's share of a joint state to B, who already holds a correlated system, using LOCC and shared entanglement (Horodecki, Oppenheim and Winter, 2005). The entanglement cost is $S(A|B)$ ebits. When that is negative, the transfer leaves $-S(A|B)$ ebits for later use, which gives negative conditional entropy a physical meaning.

In the folio: Quantum Information › Entropy measures quantum informationSee also Conditional entropy, Ebit, LOCC

Key distribution (and related tasks) whose security follows from an observed Bell violation alone. It needs no trusted model of the sources or detectors, which may even come from an adversary. It requires loophole-free conditions and is currently far slower than standard QKD.

In the folios: Quantum Information › Bell nonlocality; Quantum Networking › Stages of a quantum internetSee also Bell nonlocality, Quantum key distribution, Certified randomness, E91, Device-independent QKD

Random numbers whose unpredictability is guaranteed by a Bell violation rather than by trust in the hardware. Because no local model can reproduce the correlations, the outcomes cannot have been fixed in advance.

In the folio: Quantum Information › Bell nonlocalitySee also Bell nonlocality, Device-independent cryptography

A bookkeeping notation for what a protocol consumes and produces. $[qq]$ is one shared ebit, $[c\to c]$ one classical bit sent and $[q\to q]$ one qubit sent. "$\ge$" means the resources on the left can simulate those on the right. Teleportation and superdense coding are the two basic trades.

teleportation: $[qq]+2[c\to c]\ge [q\to q]$

superdense coding: $[qq]+[q\to q]\ge 2[c\to c]$

In the folio: Quantum Information › Protocols: trading ebits, bits and qubitsSee also Ebit, Quantum teleportation, Superdense coding

A protocol that transfers an unknown qubit state using one shared Bell pair and two classical bits (Bennett et al., 1993). Alice makes a Bell measurement on her qubit and her half of the pair. Bob then applies $X^{m_2}$ and $Z^{m_1}$ according to her two bits. Nothing travels faster than light and the original is destroyed, as no-cloning requires. First demonstrated with photons in 1997.

In the folios: Quantum Information › Protocols: trading ebits, bits and qubits; Quantum Networking › A protocol stack for entanglementSee also Bell measurement, Resource inequality, No-cloning theorem, Pauli matrices, Bell pair

A joint measurement of two qubits in the Bell basis, with four outcomes (two bits). In circuits it is a CNOT followed by a Hadamard and two computational-basis measurements. It is the key step of teleportation, superdense decoding and entanglement swapping. With linear optics alone, only two of the four Bell states can be told apart.

In the folios: Quantum Information › Protocols: trading ebits, bits and qubits; Quantum Networking › Heralded entanglementSee also Bell pair, CNOT gate, Hadamard gate, Entanglement swapping, Heralded entanglement

The single-qubit gate $H=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}1& 1\\ 1& -1\end{array}\right)$. It maps $|0⟩,|1⟩$ to the equal superpositions $|\pm ⟩$ and swaps the $Z$ and $X$ bases. In the teleportation and superdense circuits it converts a Bell measurement into a computational-basis one. (Not to be confused with the Shannon entropy $H$.)

$H=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}1& 1\\ 1& -1\end{array}\right)$

In the folios: Quantum Information › Protocols: trading ebits, bits and qubits; Quantum Algorithms › Phase estimation and the QFT; Building a Quantum Computer › Engineering the gatesSee also CNOT gate, Bell measurement, Uniform superposition, Quantum circuit, Universal gate set

The two-qubit gate that flips the target qubit ($\oplus$) when the control qubit (dot) is $|1⟩$. With the Hadamard gate it creates and analyses Bell states, and with single-qubit gates it is universal for quantum computation.

In the folios: Quantum Information › Protocols: trading ebits, bits and qubits; Building a Quantum Computer › Engineering the gatesSee also Hadamard gate, Bell measurement, Controlled-Z gate, Universal gate set, Cirac–Zoller gate

A protocol that sends two classical bits by transmitting one qubit, using a Bell pair shared in advance (Bennett and Wiesner, 1992). Alice applies one of four Pauli operations to her half, selecting one of four orthogonal Bell states, and Bob's Bell measurement reads off both bits. Without prior entanglement the Holevo bound allows only one bit per qubit.

In the folio: Quantum Information › Protocols: trading ebits, bits and qubitsSee also Resource inequality, Holevo bound, Bell pair, Quantum teleportation

A method by which two parties, Alice and Bob, grow a shared secret key by exchanging quantum states of light and then post-processing the results over an authenticated classical channel. Any eavesdropping disturbs the states and shows up as errors, so security follows from quantum physics rather than computational hardness. Per the folio, it needs dedicated optical links, reaches about 100 km per link and relies on trusted nodes beyond that.

In the folios: Quantum Information › Protocols: trading ebits, bits and qubits; Quantum Communications › The quantum threat, two answers; Quantum Networking › What quantum networks enableSee also BB84, E91, Quantum bit error rate, Key rate, Key growing

The first QKD protocol (Bennett and Brassard, 1984). Alice sends single photons prepared in one of two conjugate bases, for example rectilinear or diagonal polarisation. Bob measures in a randomly chosen basis, and the two keep only the results where their bases agree. Eavesdropping causes errors because the bases are mutually non-orthogonal. In practice it is run with weak laser pulses and decoy states.

In the folios: Quantum Information › Protocols: trading ebits, bits and qubits; Quantum Communications › QKD protocol families; Quantum Networking › MilestonesSee also Quantum key distribution, Sifting, Key rate, E91, Decoy states

The entanglement-based QKD protocol of Artur Ekert (Phys. Rev. Lett. 67, 661, 1991). Alice and Bob measure shared entangled pairs and use a violation of a Bell (CHSH) inequality to certify that no eavesdropper holds information. It is the conceptual ancestor of device-independent QKD.

In the folios: Quantum Information › Protocols: trading ebits, bits and qubits; Quantum Communications › QKD protocol families; Quantum Networking › MilestonesSee also BB84, CHSH inequality, Device-independent cryptography, Entanglement-based QKD, Bell violation

The QKD step in which Alice and Bob publicly compare their basis choices (not their bits) and discard the rounds where the bases differ. On average half the bits survive, forming the sifted key.

In the folios: Quantum Information › Protocols: trading ebits, bits and qubits; Quantum Communications › From raw clicks to a secret keySee also BB84, Quantum bit error rate, Raw key, Parameter estimation

The fraction of sifted-key bits on which Alice and Bob disagree, from channel noise, detector imperfections and possible eavesdropping. All errors are conservatively attributed to Eve. For BB84 with ideal one-way error correction, secure key is possible only below about 11%.

BB84 threshold $Q\approx 11\%$

In the folios: Quantum Information › Protocols: trading ebits, bits and qubits; Quantum Communications › From raw clicks to a secret keySee also Key rate, Sifting, Binary entropy, Secret fraction, Error correction

Also called information reconciliation: the classical step that makes Alice's and Bob's keys identical using public discussion, for example with LDPC codes or the Cascade protocol. Every bit disclosed counts as leaked to Eve, about $fh(Q)$ bits per key bit.

In the folios: Quantum Information › Protocols: trading ebits, bits and qubits; Quantum Communications › From raw clicks to a secret keySee also Privacy amplification, Binary entropy, Quantum key distribution, LDPC code, Cascade

The final step: Alice and Bob compress the corrected key with a randomly chosen universal hash function, so that Eve's partial information about it is reduced to a negligible amount. The output is shorter by roughly the amount Eve could know.

In the folios: Quantum Information › Protocols: trading ebits, bits and qubits; Quantum Communications › From raw clicks to a secret keySee also Error correction, Key rate, Universal hashing, Composable security

The number of final secret-key bits produced per pulse sent (or per second). It falls with channel loss, as $\eta$ for most protocols and $\sqrt{\eta }$ for twin-field QKD. At the longest record distances, keys accumulate at bits per second or less.

$r=1-2h(Q),Q\lesssim 11\%$

In the folios: Quantum Information › Protocols: trading ebits, bits and qubits; Quantum Communications › QKD protocol familiesSee also Binary entropy, Quantum bit error rate, Repeaterless bound, Decoy-state key rate, Secret fraction

A Bell-state measurement on one qubit from each of two pairs, A–R and R–B, leaves A and B entangled even though they never interacted. It is the quantum counterpart of store-and-forward and the basic step of every repeater. Proposed in 1993 (Żukowski, Zeilinger, Horne and Ekert).

In the folios: Quantum Information › Protocols: trading ebits, bits and qubits; Quantum Networking › Quantum repeatersSee also Quantum repeater, Bell measurement, Quantum teleportation, Swap fidelity

A chain of stations ($R_1,R_2,\dots$) that divides a long link into short elementary links of length $L_0$, entangles each one by heralding, then joins them by entanglement swapping and protects fidelity by purification or error correction. With memories, it turns the exponential loss of direct transmission into roughly polynomial scaling. Proposed by Briegel, Dür, Cirac and Zoller in 1998.

In the folios: Quantum Information › Protocols: trading ebits, bits and qubits; Quantum Networking › Quantum repeatersSee also Entanglement swapping, Quantum memory, Entanglement purification, Repeaterless bound, Quantum repeater generations

A device that stores a quantum state, ideally one received from or entangled with a photon, and returns it on demand. Its key figures are efficiency, storage (coherence) time, fidelity, bandwidth and multimode capacity. In the stages of a quantum internet, adding memory enables stored entanglement, blind computing and clock networks.

In the folios: Quantum Information › Protocols: trading ebits, bits and qubits; Quantum Networking › Stages of a quantum internet; Quantum AI › Where advantage is provenSee also Quantum repeater, Entanglement swapping, Memory qubit, Coherence time, Multiplexing

A procedure that uses local operations and classical communication to turn several noisy entangled pairs into fewer pairs of higher fidelity. It is probabilistic and needs two-way classical exchange, and it replaces classical forward error correction in first-generation repeaters.

In the folios: Quantum Information › Protocols: trading ebits, bits and qubits; Quantum Networking › Quantum repeatersSee also Quantum repeater, Entanglement concentration and dilution, Bound entanglement, BBPSSW protocol, Fidelity

The fraction of photons (or optical power) that survives a channel, between 0 and 1. In QKD, key rates are quoted as scaling with $\eta$ or $\sqrt{\eta }$. In dB, the loss is $-10{\log }_{10}\eta$, so 0.2 dB/km of fibre gives $\eta =10^{-0.02L}$ for length $L$ in km.

loss (dB) $=-10{\log }_{10}\eta$

In the folios: Quantum Information › Protocols: trading ebits, bits and qubits; Quantum Communications › Light as a channel: the Holevo limit; Quantum Networking › Light carries the qubitsSee also Repeaterless bound, Quantum repeater, Decibel, Pure-loss channel, Fibre attenuation

The maximum secret-key rate of any point-to-point QKD protocol over a lossy channel without quantum repeaters (Pirandola, Laurenza, Ottaviani and Banchi, 2017). It is about $1.44\eta$ bits per mode at high loss, so ordinary protocols such as BB84 scale at best linearly in $\eta$. Twin-field QKD beats it by scaling as $\sqrt{\eta }$. The folio's "1.44 η scaling" label refers to this bound.

$-{\log }_2(1-\eta )\approx 1.44\eta$ for $\eta \ll 1$

In the folios: Quantum Information › Protocols: trading ebits, bits and qubits; Quantum Communications › QKD protocol families; Quantum Networking › Loss sets the rateSee also Transmissivity, Quantum repeater, Key rate, Twin-field QKD

No physical process can make an independent copy of an unknown quantum state (Wootters and Zurek; Dieks, 1982). For networking this means a lost or corrupted photonic qubit cannot be amplified, regenerated, buffered as a copy or resent. Copies are "free" in a classical network but "impossible" in a quantum one.

$U|\psi ⟩|0⟩=|\psi ⟩|\psi ⟩\forall \psi$ is impossible

In the folios: Quantum Information › What quantum information forbids; Quantum Networking › Why quantum networks differSee also Quantum teleportation, Quantum key distribution, Erasure channel, Heralding, Entanglement purification

The principle that local measurements on one half of an entangled pair cannot change the statistics seen at the other half, so entanglement on its own cannot carry a message or beat light speed. Every use of entanglement in a network therefore also needs classical communication.

${\mathrm{Tr}}_A[({\mathcal{E}}_A\otimes \mathrm{id}){\rho }_{AB}]={\rho }_B$

In the folios: Quantum Information › What quantum information forbids; Quantum Networking › Why quantum networks differSee also Bell nonlocality, Quantum teleportation, Partial trace, Classical channel

Holevo's 1973 theorem that the classical information extractable from an ensemble of quantum states is at most the Holevo quantity $\chi$. Since $\chi \le n$ for $n$ qubits, at most $n$ bits can be read out of $n$ qubits, even though describing the state takes exponentially many amplitudes.

$I_{\text{acc}}\le \chi \le n$

In the folio: Quantum Information › What quantum information forbidsSee also Holevo quantity, Accessible information, Superdense coding

The maximum classical mutual information between the label $x$ of an ensemble and the outcome of any measurement (POVM) on the received state. It is generally hard to compute, and the Holevo quantity bounds it from above.

In the folio: Quantum Information › What quantum information forbidsSee also Holevo bound, Holevo quantity, POVM

The fact that entanglement cannot be freely shared. If A is maximally entangled with B, it can have no entanglement, or even classical correlation, with C. For qubits this is the Coffman–Kundu–Wootters inequality (2000). Monogamy is why entanglement-based keys are private.

$C_{AB}^2+C_{AC}^2\le C_{A(BC)}^2$

In the folio: Quantum Information › What quantum information forbidsSee also Concurrence, Quantum key distribution

An entanglement measure for two qubits introduced by Wootters (1998), ranging from 0 (separable) to 1 (Bell state). It has a closed form for any two-qubit mixed state. $C_{A(BC)}$ is its generalisation to the split between qubit A and the pair BC.

In the folio: Quantum Information › What quantum information forbidsSee also Monogamy of entanglement, Entanglement entropy

An uncertainty principle for an observer B who holds a quantum memory entangled with the measured system A (Berta et al., 2010). The bound on the entropies of the outcomes of incompatible measurements $X$ and $Z$ is lowered by the conditional entropy $S(A|B)$. With a maximally entangled memory $S(A|B)=-1$ and Bob can predict both outcomes. The relation underpins modern QKD security proofs.

$H(X|B)+H(Z|B)\ge {\log }_2\frac{1}{c}+S(A|B)$

In the folio: Quantum Information › What quantum information forbidsSee also Mutually unbiased bases, Conditional entropy, Shannon entropy

Two bases in which every state of one gives uniformly random outcomes in the other, like the $Z$ and $X$ (rectilinear and diagonal) bases of a qubit. The overlap constant $c=\max |⟨x|z⟩{|}^2$ is then $1/d$, which is $1/2$ for a qubit, making the uncertainty bound as large as possible. BB84 uses this pair.

In the folio: Quantum Information › What quantum information forbidsSee also Entropic uncertainty relation with quantum memory, BB84

The highest rate, in bits or qubits per channel use, at which information can be sent with vanishing error over many uses. A quantum channel has several capacities depending on what is sent (classical or quantum data) and what is shared beforehand (for example entanglement). Unlike Shannon's formula, most require regularisation.

In the folio: Quantum Information › Channel capacitiesSee also Classical capacity, Quantum capacity, Entanglement-assisted capacity, Regularisation

A capacity expression computed from one use of the channel, like Shannon's $C={\max }_{p(x)}I(X:Y)$. Single-letter formulas are computable; the entanglement-assisted capacity has one, but the classical and quantum capacities of a general quantum channel do not.

In the folio: Quantum Information › Channel capacitiesSee also Regularisation, Entanglement-assisted capacity

Taking the limit ${\lim }_n\frac{1}{n}f({\mathcal{N}}^{\otimes n})$ of a one-shot quantity over $n$ parallel channel uses, needed when entangled inputs across uses can beat independent ones. Regularised formulas are generally not computable. Whether a computable formula for quantum capacity exists is an open problem.

In the folio: Quantum Information › Channel capacitiesSee also Single-letter formula, Classical capacity, Quantum capacity

The best rate of sending classical bits through a quantum channel without prior entanglement. By the HSW theorem it is the regularised Holevo quantity of the channel. Hastings (2009) showed that $\chi$ is not additive, so the regularisation cannot be dropped in general.

$C(\mathcal{N})=\underset{n}{\lim }\frac{1}{n}\chi ({\mathcal{N}}^{\otimes n})$

In the folio: Quantum Information › Channel capacitiesSee also HSW theorem, Holevo quantity, Regularisation

The coding theorem (Holevo, 1998; Schumacher and Westmoreland, 1997) showing that the Holevo quantity maximised over input ensembles is an achievable rate for classical communication over a quantum channel with product-state inputs. Regularised, it gives the classical capacity.

In the folio: Quantum Information › Channel capacitiesSee also Classical capacity, Holevo quantity

The best rate, in qubits per use, at which quantum states can be sent faithfully, equivalent to how fast entanglement can be generated. The LSD theorem gives it as regularised coherent information. It can show superactivation, and it is unknown even for the qubit depolarising channel.

$Q(\mathcal{N})=\underset{n}{\lim }\frac{1}{n}{I}_c({\mathcal{N}}^{\otimes n})$

In the folio: Quantum Information › Channel capacitiesSee also Coherent information, LSD theorem, Superactivation, Depolarising channel

The quantum analogue of mutual information for sending quantum data: $I_c=S(B)-S(E)$, the output entropy minus the entropy leaked to the environment, maximised over inputs. It equals $-S(A|B)$ for a purified input and can be negative.

In the folio: Quantum Information › Channel capacitiesSee also Quantum capacity, Conditional entropy, Stinespring dilation

The quantum channel coding theorem (Lloyd, 1997; Shor, 2002; Devetak, 2005): the regularised coherent information is the quantum capacity.

In the folio: Quantum Information › Channel capacitiesSee also Quantum capacity, Coherent information

The effect, found by Smith and Yard (2008), that two channels each with zero quantum capacity can have positive capacity when used together. Their example paired a PPT entangling channel with a 50% erasure channel. Quantum capacity is therefore not a simple property of single channels.

In the folio: Quantum Information › Channel capacitiesSee also Quantum capacity, Erasure channel, Bound entanglement

The classical capacity when sender and receiver share unlimited prior entanglement (Bennett, Shor, Smolin and Thapliyal, 2002). It has a single-letter formula, the maximal quantum mutual information between a purifying reference and the channel output, making it the quantum analogue of Shannon's formula.

$C_E(\mathcal{N})=\underset{\rho }{\max }I(A:B)$

In the folio: Quantum Information › Channel capacitiesSee also Quantum mutual information, Superdense coding, Single-letter formula

A channel that delivers the input intact with probability $1-\epsilon$, or else replaces it by a flag telling the receiver it was lost. All three capacities are known exactly. $Q$ vanishes at $\epsilon =\frac{1}{2}$, since otherwise receiver and environment would share a clone.

$C_E=2(1-\epsilon ),C=1-\epsilon ,Q=\max (0,1-2\epsilon )$

In the folio: Quantum Information › Channel capacitiesSee also Channel capacity, No-cloning theorem, Superactivation

Entanglement among three or more parties. Unlike the two-party case it has no single measure or ordering: for three qubits there are already two inequivalent classes, GHZ and W, that cannot be converted into each other even probabilistically. How to classify and quantify it in general is open.

In the folio: Quantum Information › Open problemsSee also Entanglement, Monogamy of entanglement

A prospective network that distributes entanglement between distant nodes on demand. It would enable QKD, distributed and blind quantum computing, and networked sensing. Its achievable rates across continents depend on repeaters, memories and satellites.

In the folios: Quantum Information › Open problems; Quantum Networking › Stages of a quantum internetSee also Quantum repeater, Entanglement swapping, Micius satellite, Stages of a quantum internet, Quantum network

The puzzle, raised by Hawking, of whether information that falls into a black hole is lost when the hole evaporates by thermal radiation, which would violate the unitarity of quantum mechanics. Quantum-information ideas such as entanglement entropy and error-correcting codes, together with holography, now drive the proposals that spacetime geometry itself is built from entanglement.

In the folio: Quantum Information › Open problemsSee also Entanglement entropy, Von Neumann entropy

A step-by-step procedure for a quantum computer: prepare qubits, apply a sequence of unitary gates, measure, and usually finish with classical post-processing. Its power comes from arranging interference so that wrong answers cancel and right answers reinforce, not from "trying all answers at once". A quantum algorithm is useful only if its speed-up survives every step from problem formulation to verified answer.

In the folio: Quantum Algorithms › Where speed-ups come fromSee also Quantum interference, Quantum parallelism, Quantum circuit, Speed-up

The addition of probability amplitudes from different computational paths leading to the same outcome. Paths "with scattered phases" sum to nearly zero (cancelled), while paths "in phase" add to a large amplitude and a likely outcome. Every quantum speed-up is, at bottom, a choreography of interference.

In the folio: Quantum Algorithms › Where speed-ups come fromSee also Probability amplitude, Quantum parallelism, Quantum algorithm

The fact that a unitary acting on a superposition of $N$ inputs evaluates a function on all of them in one step. On its own it gives no advantage, because a measurement returns only one outcome at random; the folio stresses that "parallelism alone is not enough" without interference to concentrate amplitude on useful answers.

In the folio: Quantum Algorithms › Where speed-ups come fromSee also Quantum interference, Born rule, Output readout

The ratio, as problem size grows, between the cost of the best known (or provably best) classical method and that of the quantum algorithm. The folio grades speed-ups as exponential, super-polynomial, quadratic or heuristic; the grade matters because slow fault-tolerant gates mean only large speed-ups pay off in practice.

In the folio: Quantum Algorithms › The algorithm mapSee also Exponential speed-up, Super-polynomial speed-up, Quadratic speed-up, Break-even point

A quantum algorithm whose cost grows only polynomially (sometimes polylogarithmically) in the problem size where the best known classical algorithm's cost grows exponentially. Many early quantum ML claims of this kind hid costs in data loading, readout or conditioning (Aaronson 2015), or were later dequantised.

In the folios: Quantum Algorithms › Where speed-ups come from; Quantum AI › The fine printSee also Hidden structure, Hamiltonian simulation, Super-polynomial speed-up, Quadratic speed-up, Dequantisation

A speed-up larger than any polynomial but not necessarily exponential. Shor's algorithm is labelled this way because the best classical factoring method, the number field sieve, is sub-exponential, running in roughly $\exp (c(\log N{)}^{1/3}(\log \log N{)}^{2/3})$ time.

In the folio: Quantum Algorithms › The algorithm mapSee also Shor's algorithm, Exponential speed-up, Speed-up

A quantum cost proportional to the square root of the classical cost, $\sqrt{N}$ versus $N$, as in Grover search and amplitude estimation. It is provably the best possible for unstructured search, but in the folio's illustration it breaks even only after about 3 hours, so quadratic gains are easily eroded by slow logical gates.

quantum runtime $\propto \sqrt{N}$

In the folios: Quantum Algorithms › Speed-ups and the real cost; Quantum AI › The fine printSee also Grover's algorithm, Amplitude estimation, Break-even point, Super-quadratic speed-up, Exponential speed-up

Any speed-up larger than quadratic, such as quartic (quantum cost $\propto N^{1/4}$), polynomial of higher degree or exponential. Babbush et al. (2021) and Hoefler, Häner and Troyer (2023) argue that practical advantage on early fault-tolerant machines needs super-quadratic, ideally exponential, speed-ups; in the folio's illustration a quartic speed-up breaks even after about 5 s.

quartic: quantum runtime $\propto N^{1/4}$

In the folio: Quantum Algorithms › Speed-ups and the real costSee also Quadratic speed-up, Break-even point, Exponential speed-up

Algebraic regularity in a problem, such as a period of a function or a symmetry group, that a quantum Fourier transform can expose. Formally many such problems are instances of the hidden subgroup problem; factoring, discrete logarithms and Simon's problem all belong here and are the source of exponential speed-ups.

In the folio: Quantum Algorithms › Where speed-ups come fromSee also Quantum Fourier transform, Period finding, Simon's algorithm, Exponential speed-up

Finding a marked item among $N$ when the only access is a yes/no oracle and nothing is known about where solutions lie. Classically this needs about $N$ queries; Grover's algorithm needs about $\sqrt{N}$, which the BBBV bound shows is optimal.

In the folio: Quantum Algorithms › Where speed-ups come fromSee also Grover's algorithm, Oracle, BBBV lower bound

A black-box subroutine that evaluates a function, used reversibly on quantum states; typically it flips a phase or a target bit when the input is a solution. Algorithm costs are often counted in oracle calls ("queries"), and in practice each call must itself be compiled into gates, which the folio's cost chart prices at 0.1 s per fault-tolerant call against 1 µs classically.

$O_f|x⟩=(-1{)}^{f(x)}|x⟩$ (phase oracle)

In the folio: Quantum Algorithms › From problem to answerSee also Query complexity, Unstructured search, Oracle separation

The number of oracle calls an algorithm needs, ignoring other work. It is the setting in which quantum lower bounds can actually be proven, such as the $\sqrt{N}$ optimality of search, and in which the first quantum–classical separations were found.

In the folio: Quantum Algorithms › Where speed-ups come fromSee also Oracle, BBBV lower bound, Oracle separation

The theorem of Bennett, Bernstein, Brassard and Vazirani (1997) that any quantum algorithm for unstructured search over $N$ items needs of order $\sqrt{N}$ oracle queries. It proves Grover's algorithm optimal and shows that quantum computers cannot solve NP-complete problems by brute-force search exponentially faster.

$\Omega (\sqrt{N})$ queries

In the folio: Quantum Algorithms › Where speed-ups come fromSee also Unstructured search, Grover's algorithm, NP-complete

A proof that, relative to some black-box oracle, one model of computation needs far fewer queries than another. Deutsch–Jozsa (1992) separated quantum from deterministic classical computing, and Simon (1994) gave an exponential separation from randomised classical computing, which directly inspired Shor.

In the folio: Quantum Algorithms › MilestonesSee also Deutsch–Jozsa algorithm, Simon's algorithm, Query complexity

The standard way of writing a quantum algorithm: horizontal wires are qubits, boxes are gates applied left to right, dots joined by vertical lines mark controls, and meter symbols are measurements. Double lines in the folio's phase-estimation diagram mark the eigenstate register $|u⟩$.

In the folios: Quantum Algorithms › Phase estimation and the QFT; Quantum AI › Variational quantum modelsSee also Hadamard gate, Controlled unitary, Circuit depth, Parameterised quantum circuit, Qubit

A gate that applies $U$ to a target register only when a control qubit is $|1⟩$, and acts coherently on superpositions of the control. Phase estimation uses controlled powers $U^{2^k}$ (drawn as $U^4$, $U^2$, $U^1$), so the number of $U$ applications doubles with each extra bit of precision.

In the folio: Quantum Algorithms › Phase estimation and the QFTSee also Quantum phase estimation, Phase kickback

The discrete Fourier transform applied to the amplitudes of an $n$-qubit state, $N=2^n$. It "turns phases into measurable bits" and needs only $O(n^2)$ gates, against $O(n{2}^n)$ for the classical FFT on the same vector; the catch is that its output is a quantum state, not a readable list of numbers. Its inverse, ${\text{QFT}}^{†}$, ends phase estimation.

$\text{QFT}|x⟩=\frac{1}{\sqrt{N}}\sum _{y=0}^{N-1}{e}^{2\pi ixy/N}|y⟩$

In the folio: Quantum Algorithms › Phase estimation and the QFTSee also Quantum phase estimation, Fast Fourier transform, Hidden structure, Output readout

The classical algorithm computing a discrete Fourier transform of length $N$ in $O(N\log N)$ operations, i.e. $O(n{2}^n)$ for $N=2^n$. It is the benchmark against which the folio compares the QFT's $O(n^2)$ gates.

In the folio: Quantum Algorithms › Phase estimation and the QFTSee also Quantum Fourier transform, Big-O notation

The algorithm (Kitaev, 1995) that reads out the eigenphase $\phi$ of a unitary $U$ given its eigenstate $|u⟩$: Hadamards prepare control qubits, controlled powers of $U$ imprint the phase by kickback, and an inverse QFT converts it to binary digits. It is "the engine inside Shor's algorithm and quantum chemistry", where $U$ encodes time evolution and $\phi$ gives an energy.

$U|u⟩=e^{2\pi i\phi }|u⟩$

precision $\epsilon$ costs $O(1/\epsilon )$ uses of $U$

In the folio: Quantum Algorithms › Phase estimation and the QFTSee also Eigenphase, Phase kickback, Quantum Fourier transform, Heisenberg limit

The number $\phi \in [0,1)$ defining an eigenvalue $e^{2\pi i\phi }$ of a unitary; every eigenvalue of a unitary has unit modulus, so the phase carries all the information. Phase estimation outputs the "bits of $\phi$"; for $U=e^{-iHt}$ the phase gives an energy eigenvalue of $H$.

In the folio: Quantum Algorithms › Phase estimation and the QFTSee also Quantum phase estimation, Ground-state problem

The effect whereby a controlled-$U$ acting on an eigenstate of $U$ leaves the target unchanged ("eigenstate, untouched") and transfers the eigenvalue's phase to the control qubit instead. In phase estimation the $k$-th control picks up $2\pi \cdot 2^k\phi$, writing successive binary digits of $\phi$ into relative phases.

In the folio: Quantum Algorithms › Phase estimation and the QFTSee also Controlled unitary, Quantum phase estimation

The target additive error of an estimated quantity (a phase, an energy, an expectation value). The folio contrasts $O(1/\epsilon )$ cost for phase and amplitude estimation with the $O(1/{\epsilon }^2)$ samples of classical Monte Carlo or repeated measurement.

In the folio: Quantum Algorithms › Phase estimation and the QFTSee also Heisenberg limit, Amplitude estimation

The fundamental scaling in which estimation error falls as $1/T$ in the total number $T$ of uses of the probed process, rather than the $1/\sqrt{T}$ "standard quantum limit" of independent repetitions. Phase estimation reaches it: precision $\epsilon$ costs $O(1/\epsilon )$ uses of $U$.

$\epsilon \propto 1/T$ (vs. $1/\sqrt{T}$)

In the folios: Quantum Algorithms › Phase estimation and the QFT; Quantum Networking › What quantum networks enableSee also Quantum phase estimation, Precision, Amplitude estimation, Clock network

Peter Shor's 1994 algorithm that factors integers and computes discrete logarithms in polynomial time by reducing them to period finding, solved with the QFT. It "breaks RSA and elliptic-curve cryptography" but needs full fault tolerance; the latest estimate for RSA-2048 is under 1 million physical qubits, under a week (Gidney 2025).

In the folios: Quantum Algorithms › The algorithm map; Quantum Communications › The quantum threat, two answers; Building a Quantum Computer › Overhead: how many qubits?See also Period finding, RSA-2048, Discrete logarithm, Fault tolerance, RSA

The task of finding the period $r$ of a function such as $f(x)=a^x\mathrm{mod}N$. Superposing many $x$, evaluating $f$, and applying a QFT produces peaks at multiples of $1/r$; factoring reduces to it because $r$ yields a factor of $N$ via $\gcd (a^{r/2}\pm 1,N)$.

In the folio: Quantum Algorithms › Phase estimation and the QFTSee also Shor's algorithm, Quantum Fourier transform, Continued fractions

The classical expansion of a real number as $a_0+1/(a_1+1/(a_2+\dots ))$, whose truncations give its best rational approximations. In Shor's algorithm this post-processing step turns the measured approximation of $s/r$ into the period $r$.

In the folio: Quantum Algorithms › From problem to answerSee also Period finding, Shor's algorithm

Lov Grover's 1996 search algorithm, which finds one of $M$ marked items among $N$ in about $(\pi /4)\sqrt{N/M}$ oracle calls by rotating the state "step by step, from the uniform superposition towards the marked solutions". Its quadratic speed-up is provably optimal, but slow logical gates erode the gain in practice.

$k\approx \frac{\pi }{4}\sqrt{N/M}$ steps

In the folios: Quantum Algorithms › Amplitude amplification; Quantum Networking › What quantum networks enableSee also Amplitude amplification, Grover iteration, Unstructured search, BBBV lower bound, Distributed quantum computing

The generalisation of Grover's algorithm (Brassard, Høyer, Mosca and Tapp, 2000) to any quantum subroutine that succeeds with probability $p$: about $1/\sqrt{p}$ repetitions of a reflection pair boost the success probability to near 1, instead of the $1/p$ classical repetitions. It "speeds up any probabilistic subroutine" quadratically.

In the folio: Quantum Algorithms › Amplitude amplificationSee also Grover's algorithm, Grover iteration, Amplitude estimation

The equal-weight superposition of all $N$ basis states, prepared by Hadamards on every qubit; it is Grover's starting state. Its small angle $\theta$ to the non-solution axis is what each Grover step increases.

$|s⟩=\frac{1}{\sqrt{N}}\sum _x|x⟩$

In the folio: Quantum Algorithms › Amplitude amplificationSee also Hadamard gate, Grover angle, Marked state

The normalised superposition of all $M$ solutions ("solutions $|w⟩$") that the oracle recognises. Grover's algorithm acts entirely in the two-dimensional plane spanned by $|w⟩$ and the normalised non-solution state $|s^{\prime }⟩$.

In the folio: Quantum Algorithms › Amplitude amplificationSee also Uniform superposition, Oracle, Grover iteration

One step of Grover's algorithm: a reflection about the non-solution axis (the oracle, flipping the sign of solutions) followed by a reflection about the starting state (the "diffusion" operator). Two reflections make a rotation by $2\theta$, so after $k$ steps the state sits at angle $(2k+1)\theta$.

$G=(2|s⟩⟨s|-I)O_f$

In the folio: Quantum Algorithms › Amplitude amplificationSee also Grover angle, Success probability, Amplitude amplification

The angle between the starting state and the non-solution axis, set by the fraction of solutions. Because $\theta \approx \sqrt{M/N}$ is small, about $(\pi /4)\sqrt{N/M}$ rotations by $2\theta$ reach the solution axis.

$\sin \theta =\sqrt{M/N}$

In the folio: Quantum Algorithms › Amplitude amplificationSee also Grover iteration, Success probability

The probability that measuring after $k$ Grover steps returns a solution. It oscillates with $k$, so overshooting the optimal number of steps lowers it again; the optimum is near $k\approx (\pi /4)\sqrt{N/M}$.

$P_k={\sin }^2((2k+1)\theta )$

In the folio: Quantum Algorithms › Amplitude amplificationSee also Grover angle, Grover's algorithm

An algorithm (Brassard, Høyer, Mosca and Tapp, 2000) that estimates a success probability or mean value by applying phase estimation to the Grover operator. It gives Monte Carlo averages, e.g. in risk analysis, with $O(1/\epsilon )$ rather than $O(1/{\epsilon }^2)$ samples, a quadratic speed-up "useful only for very costly samples".

In the folio: Quantum Algorithms › Amplitude amplificationSee also Amplitude amplification, Quantum phase estimation, Heisenberg limit, Quadratic speed-up

Loading the input of a quantum algorithm: an initial state, an oracle circuit or a block encoding (step 2, "Prepare", of the folio's pipeline). For chemistry it means preparing a state close to the ground state; for data problems, encoding classical numbers into amplitudes, which can be costly.

In the folio: Quantum Algorithms › From problem to answerSee also Data-loading problem, Block encoding, Overlap

The bottleneck that writing $N$ classical numbers into a quantum state generally takes about $N$ steps ("loading $N$ classical numbers can cost $N$ steps"), which can wipe out a speed-up that is polylogarithmic in $N$. Hypothetical quantum random-access memory (QRAM) would ease it but has not been built at scale; simulation avoids it because "the problem is already quantum".

In the folio: Quantum Algorithms › From problem to answerSee also State preparation, HHL algorithm, Hamiltonian simulation, Output readout

The constraint that measurement yields only a few bits or expectation values per run, never the full state ("reading out a whole vector erases the gain"). Reconstructing all $N$ amplitudes would need of order $N$ repetitions, so useful algorithms must output a compact answer.

In the folio: Quantum Algorithms › From problem to answerSee also Born rule, Data-loading problem, HHL algorithm

The number of sequential layers of gates in a circuit, which sets its runtime. The folio warns that the core algorithm's depth "must fit the error-corrected budget": the number of logical operations a fault-tolerant machine can perform before logical errors become likely.

In the folios: Quantum Algorithms › From problem to answer; Quantum AI › Trainability: barren plateausSee also Fault tolerance, Logical qubit, Quantum circuit, Expressivity, Barren plateau

Classical post-processing techniques (e.g. zero-noise extrapolation, probabilistic error cancellation) that reduce the bias that noise causes in measured expectation values, without full error correction. They typically cost extra repetitions that grow exponentially with circuit size, so they suit only modest circuits.

In the folio: Quantum Algorithms › From problem to answerSee also Quantum error correction, Noisy intermediate-scale quantum

Checking that a quantum computer's output is correct. Some outputs are easy to check classically, such as factors (multiply them), but most, such as a simulated energy or sampled distribution, are not; the folio insists "verification is part of the application", and lists verifying classically uncheckable results as an open problem.

In the folio: Quantum Algorithms › From problem to answerSee also Shor's algorithm, Quantum echoes, Beyond-classical computation

Computing the time evolution $e^{-iHt}|\psi ⟩$ under a Hamiltonian $H$, Feynman's original motivation and "still the most natural application". Lloyd (1996) showed that local Hamiltonians can be simulated efficiently on a universal quantum computer, giving exponential speed-ups over classical methods; no data-loading penalty applies because the input is a compact description of $H$.

In the folio: Quantum Algorithms › Simulating quantum systemsSee also Product formula, Qubitisation, Local Hamiltonian, Quantum phase estimation

A Hamiltonian that is a sum of terms each acting on only a few particles or qubits, as in physical systems with short-range interactions. Its description grows only polynomially with system size, which makes it "the natural input" for quantum simulation.

$H=\sum _j{H}_j$, each $H_j$ acting on $O(1)$ qubits

In the folio: Quantum Algorithms › The algorithm mapSee also Hamiltonian simulation, Product formula

A simulation method that splits $H=A+B$ and approximates $e^{-iHt}$ by $r$ repetitions of short evolutions under each part separately. It is "simple, hardware-friendly", and higher-order (Suzuki) formulas reduce the error; for the first-order (Lie–Trotter) formula the error scales with the commutator $[A,B]$.

$e^{-i(A+B)t}=(e^{-iAt/r}{e}^{-iBt/r}{)}^r+O(\frac{t^2}{r}\Vert [A,B]\Vert )$

In the folio: Quantum Algorithms › Simulating quantum systemsSee also Trotter error, Hamiltonian simulation, Qubitisation

The error of a product formula, arising because $A$ and $B$ do not commute. For first order it falls as $t^2\Vert [A,B]\Vert /r$, so the number of steps $r$ needed for fixed accuracy grows with the simulation time and the size of the commutator.

In the folio: Quantum Algorithms › Simulating quantum systemsSee also Product formula

A technique (Low and Chuang) that turns a block encoding of $H$ into a quantum walk whose eigenphases are functions of $H$'s eigenvalues, enabling simulation and phase estimation with optimal query cost. It underlies the FeMoco estimate in the folio's resource table.

cost $O(\lambda t+\log (1/\epsilon ))$ (folio; optimal is $O(\lambda t+\frac{\log (1/\epsilon )}{\log \log (1/\epsilon )})$)

In the folio: Quantum Algorithms › Simulating quantum systemsSee also Block encoding, Normalisation, Quantum signal processing, FeMoco

Finding the lowest-energy eigenstate and energy of a Hamiltonian, central to chemistry and materials. Phase estimation solves it efficiently only if one can prepare a state with substantial overlap with the true ground state; in general (the local Hamiltonian problem) it is believed hard even for quantum computers.

In the folio: Quantum Algorithms › Simulating quantum systemsSee also Overlap, Quantum phase estimation, Variational quantum eigensolver

The squared inner product between the trial state $|\varphi ⟩$ fed into phase estimation and the true ground state $|{\psi }_0⟩$; it is the probability that phase estimation returns the ground-state energy. A small overlap means many repetitions, so "a good initial guess is essential", and preparing one is listed as an open problem.

success probability $=|⟨{\psi }_0|\varphi ⟩{|}^2$

In the folios: Quantum Algorithms › Simulating quantum systems; Quantum AI › Quantum kernelsSee also Ground-state problem, State preparation, Quantum phase estimation, Quantum kernel, Ket notation

The iron–molybdenum cofactor, the active site of the nitrogenase enzyme by which bacteria fix atmospheric nitrogen into ammonia at room temperature. Its strongly correlated electronic structure defeats classical methods, making it a benchmark for quantum chemistry: about 4 million physical qubits, about 4 days, with phase estimation and qubitisation (Lee et al. 2021).

In the folio: Quantum Algorithms › Resource estimatesSee also Qubitisation, Resource estimate, Ground-state problem

A way of representing a general (non-unitary) matrix $A$ inside a larger unitary $U$, placing $A/\alpha$ in its top-left block ("$A$ sits in a corner of $U$"). It is the common input format of modern quantum algorithms: once $A$ is block-encoded, QSVT can apply polynomial functions to it.

$U=\left(\begin{array}{cc}A/\alpha & \cdot \\ \cdot & \cdot \end{array}\right)$