Quantum Information Glossary

Quantum-information terminology often compresses a resource model or an experimental assumption into a short phrase. This glossary expands that phrase just far enough to identify the object being discussed. Definitions of states and channels follow the finite-dimensional framework of Watrous; circuit and coding language follows Nielsen and Chuang.

States, operations, and resources

Qubit. A quantum system with a specified two-dimensional state space. A pure state is a ray represented by the normalized vector $\alpha\lvert0\rangle+\beta\lvert1\rangle$, with $|\alpha|^2+|\beta|^2=1$. A physical device may have additional levels; occupation outside the intended two-dimensional space is leakage. See Bits, Qubits, Qudits, and Modes.

Physical qubit and logical qubit. A physical qubit is a controlled carrier used in an implementation. A logical qubit is information encoded in a code space or protected subsystem. Its resource cost includes data carriers, ancillas (extra carriers used, for example, to learn about errors), control, and time; counting encoded data carriers alone is not a complete overhead estimate. See Resource Estimation.

Quantum channel. A completely positive trace-preserving linear map on density operators. Trace preservation describes an unconditional operation; an individual heralded branch is generally trace nonincreasing. See Quantum Channels for QI.

Instrument. A collection of completely positive maps indexed by a classical outcome, whose sum is a channel. It specifies both outcome probabilities and conditional state changes. A POVM alone specifies only the former. See Measurement in Circuits.

Entanglement. Nonseparability relative to a specified subsystem split. It is a resource only relative to a task and allowed operations; neither correlation nor a mixed reduced state alone certifies mixed-state entanglement. See Entanglement Measures.

Magic state. A resource state outside the stabilizer-state convex set used to supply operations beyond a chosen stabilizer toolbox. Preparation quality, injection, and distillation overhead belong in the computation cost. See Magic-State Distillation.

Distances and information quantities

Trace distance. For states $\rho,\sigma$,

$$D(\rho,\sigma)=\frac12\lVert\rho-\sigma\rVert_1.$$

For two known states with equal prior probabilities, the best measurement identifies the input correctly with probability $(1+D)/2$. For example, $D=0.4$ gives success probability $0.7$. This is an operational probability statement, not a Euclidean distance between amplitude vectors.

Fidelity. This collection uses the squared convention

$$F(\rho,\sigma)=\left\lVert\sqrt\rho\sqrt\sigma\right\rVert_1^2.$$

For a pure target, $F(\rho,\lvert\psi\rangle)=\langle\psi|\rho|\psi\rangle$. Some texts use its square root, so the defining equation must accompany numerical comparisons. State fidelity, entanglement fidelity, and average gate fidelity answer different questions; see Device Characterization.

Quantum Fisher information (QFI). A local statistical sensitivity associated with a parameterized state family and a chosen parameter. For the symmetric logarithmic derivative $L_\theta$, defined by $\partial_\theta\rho=(L_\theta\rho+\rho L_\theta)/2$, it is $F_Q=\operatorname{Tr}(\rho L_\theta^2)$. Its inverse bounds variance only under the conditions of the relevant estimation theorem. See Classical and Quantum Fisher Information.

Computation, protection, and communication

Circuit depth. Number of sequential layers after fixing a gate set, connectivity, and scheduling rule. Gates on disjoint carriers may share a layer; pulse duration and classical feedforward can make equal-depth circuits take different times. See Circuit Model.

Oracle. A declared access operation, such as $U_f\lvert x,y\rangle=\lvert x,y\oplus f(x)\rangle$. Query complexity counts uses of that access and does not automatically include building the oracle. See Quantum Oracles.

Fault tolerance. A design property that prevents faults in allowed locations from spreading into uncorrectable errors faster than the correction procedure can handle. It concerns noisy preparation, gates, measurements, and recovery, not only an encoding. See Fault-Tolerant Gates.

Threshold. A boundary in a specified noise and architecture model below which increasing protection can suppress logical failure. It is neither a universal device fidelity nor necessarily the point where the smallest encoded circuit beats an unencoded one. See Threshold Theorem.

Quantum key distribution (QKD). Generation of shared secret key using quantum communication and authenticated classical communication, with security stated against an adversary model and a finite failure parameter. It does not remove the need for authentication or secure endpoints. See Quantum Key Distribution.

Quantum advantage. A demonstrated benefit over a specified classical comparison for a specified task and cost. Reproducing a sampling distribution, proving an asymptotic speedup, reducing elapsed runtime, and improving an application are distinct achievements. See Verification of Quantum Advantage.

A terminology check

Suppose an experiment reports “a logical qubit with 99% fidelity.” Before interpreting this as fault-tolerant memory, identify the input ensemble, storage duration, recovery procedure, acceptance probability, and fidelity convention. A conditional state overlap after discarding unsuccessful shots cannot by itself determine the unconditional logical failure rate. The Hardware Metrics Table and Benchmark Index make these distinctions explicit.

References

• Nielsen, Michael A., and Isaac L. Chuang, Quantum Computation and Quantum Information, 10th anniversary edition, Cambridge University Press, 2010, doi:10.1017/CBO9780511976667 — circuit, coding, and communication terminology.
• Watrous, John, The Theory of Quantum Information, Cambridge University Press, 2018, author manuscript and errata — operational definitions of states, channels, distance, and fidelity.