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Quantum fidelity measures for mixed states
Yeong-Cherng Liang, Yu-Hao Yeh, Paulo E. M. F. Mendonça, Run Yan Teh, Margaret D. Reid, Peter D. Drummond
TL;DR
Mixed-state fidelity is difficult to define because distance between density matrices is not unique, yet it is needed to assess noisy state preparation, communication, storage, and computation. This review compares candidate fidelity measures against required properties and finds that many candidates exist, while most do not fully satisfy the Josza axioms.
Problem
Mixed-state fidelity is needed to quantify how closely noisy or approximate quantum states match intended states, but distance between density matrices is not uniquely defined.
Method
The review analyzes candidate mixed- and pure-state fidelity measures against Josza's required properties, compares their mathematical behavior, and surveys applications.
Results
Most candidate fidelity measures do not fully comply with the Josza axioms, although some alternatives have useful properties; among the investigated Fp fidelities, F1 has the largest average values for random density matrices, universally for qubits and generally in higher dimensions.
Takeaways & Limitations
Fidelity choice matters because reduced or unmeasured Hilbert-space structure can bias estimates, while F2 is identified as particularly tractable for numerical Monte Carlo calculations.
Takeaways & Limitations
Which fidelity is the best unbiased estimator under partial-trace operations from a randomized extension of the measured Hilbert space remains an open problem.
Abstract
from arXiv · showhide
Applications of quantum technology often require fidelities to quantify performance. These provide a fundamental yardstick for the comparison of two quantum states. While this is straightforward in the case of pure states, it is much more subtle for the more general case of mixed quantum states often found in practice. A large number of different proposals exist. In this review, we summarize the required properties of a quantum fidelity measure, and compare them, to determine which properties each of the different measures has. We show that there are large classes of measures that satisfy all the required properties of a fidelity measure, just as there are many norms of Hilbert space operators, and many measures of entropy. We compare these fidelities, with detailed proofs of their properties. We also summarize briefly the applications of these measures in teleportation, quantum memories, quantum computers, quantum communications, and quantum phase-space simulations.
1. Introduction
Quantum fidelity quantifies similarity between quantum states, but mixed-state fidelity is non-unique and technically important because real experiments involve noise, environmental coupling, and incomplete state information. This review compares candidate measures, their properties, and applications across quantum technologies.
- Motivation: Fidelity quantifies the similarity between a produced quantum state and its intended target amid imperfections and noise.This is particularly relevant to quantum communications and quantum computing, where precisely defined states must be generated or transmitted.
- Applications: Mixed-state fidelity supports applications including entanglement quantification, optimal cloning, quantum memories, teleportation, and quantum phase-transition studies.These applications use fidelity to assess similarity, cloning quality, or the effects of noise and errors.
- The mixed-state problem: Mixed-state fidelity lacks a clearly unique definition, with multiple proposed approaches beyond the idealized pure-state case.Pure-state fidelity is typically non-scalable and does not reflect noisy real-world settings.
- The mixed-state problem: Comparing density matrices is a distance problem whose answer depends on additional choices about the appropriate Hilbert-space structure.The review frames fidelity selection as a non-unique notion of distance in a vector space.
- Review scope: The review examines fidelity measures for mixed and pure states, compares their mathematical properties and values, and surveys applications in quantum information protocols.It also discusses experimental fidelity measurements, relevant Hilbert-space choices, and phase-space fidelity for numerical evaluation.
2. Fidelity measures for mixed states
Mixed-state fidelity is needed because realistic quantum systems are noisy, coupled to environments, and difficult to characterize fully. The review compares proposed measures against Jozsa’s axioms and highlights alternatives, especially norm-based fidelities, that balance axiomatic properties with computational accessibility.
- Measured and relevant Hilbert spaces: Fidelity measurements must distinguish the experimentally relevant Hilbert space from irrelevant environmental degrees of freedom.Tracing over irrelevant parts can leave the relevant density matrix unchanged, but entanglement with those parts generically produces mixed states.
- Desirable properties of mixed state fidelities: The Hilbert-Schmidt inner product is an unsatisfactory mixed-state fidelity because it can equate maximally mixed and orthogonal-pure-state comparisons in a qubit system.Although normalization can resolve this specific problem, the example motivates explicit fidelity axioms.
- Alternative fidelities: Other fidelity measures besides Uhlmann-Jozsa satisfy Jozsa’s axioms, including an infinite class of norm-based fidelities.The review frames this multiplicity as analogous to the existence of many operator norms and entropy-like measures.
- Alternative fidelities: Among previously proposed generalized formulas, only FQ and F1 fully comply with Jozsa’s axioms.FQ remains computationally challenging because it uses fractional powers and optimization over a continuous set of candidate measures.
- Hilbert-Schmidt fidelities: F2 uses the Hilbert-Schmidt norm and is easier to calculate than F1, while all even-order p-fidelities reduce to an easily evaluated form.F2 is based on expectation values of Hermitian operators, which standard quantum-mechanical techniques can compute and measure.
- Hilbert-Schmidt fidelities: Norm-based measures based on operator expectation values are more accessible in large or infinite-dimensional Hilbert spaces than measures involving nested operator square roots.This accessibility supports considering alternatives even when the Uhlmann-Jozsa fidelity has desirable axiomatic properties.
3. Auxiliary fidelity properties
The review compares auxiliary properties of mixed-state fidelity measures, including concavity, multiplicativity, monotonicity under quantum operations, and metric construction. These properties distinguish candidate measures across ensembles, tensor products, quantum operations, and distance-based applications.
- The review focuses predominantly on F1, F2, and FQ because they satisfy all Jozsa axioms.
- Concavity properties: Separate concavity concerns one argument, while joint concavity is stronger and implies separate concavity.Joint concavity reduces to separate concavity when the second-state mixture is constant.
- Concavity properties: F2, FGM, and FAM provide counterexamples to separate concavity, while Table 3 summarizes concavity across candidate measures.
- Multiplicativity under tensor products: F2 and FQ are supermultiplicative, while FAM is neither multiplicative nor supermultiplicative in general.The review refers to Appendix A for proofs of the positive results and a counterexample for FAM.
- Multiplicativity under tensor products: Multiplicativity tests fidelity under uncorrelated ancillary states, tensor powers, and general tensor-product combinations.Appending the same uncorrelated state leaves fidelity unchanged when F(τ,τ)=1.
- Monotonicity under quantum operations: Monotonicity is summarized separately for partial trace, projective measurement, and general quantum operations, with counterexamples for several measures.F1 and FQ can increase under partial trace, creating bias when smaller-Hilbert-space measurements estimate fidelity on an enlarged space; F2 does not share this property.
- Metrics: Functionals including 1 − F(ρ, σ) are assessed for metric properties, with results summarized in Table 6 and additional metric constructions reported for FN and F1.
4. Comparisons, bounds, and relations between measures
The review compares fidelity measures through rigorous bounds, graphical examples, and random-state studies, showing that their relationships depend on dimension, rank, and the states being compared.
- Bounds: The review adds the result that F2(ρ, σ) provides a lower bound to FN(ρ, σ).Theorem 1 applies to arbitrary Hermitian matrices satisfying tr(ρ^2) ≤ 1 and tr(σ^2) ≤ 1.
- Bounds: The review finds that proposed conjectured bounds can fail for qutrit density matrices, although nonlinear functionals may still provide bounds.The text specifically identifies counterexamples to all three conjectures discussed in this comparison.
- Interpolated qubit states: For the interpolated qubit states, several bounds are saturated: F1 = FN = FC, F2 = FAM = FGM, and √FA = FQ.The measures are plotted against the states’ purity P, with r ranging from maximally mixed states at r = 0 to distinct pure states at r = 1.
- Random density matrices: For qubit density matrices, F1 ≥ F2, whereas for qutrits both F1 > F2 and F1 < F2 are possible.The qubit relation follows from the stated comparison, while qutrit behavior admits both orderings.
- Random density matrices: For qutrit random matrices, the average fidelity satisfies ⟨F1⟩ > ⟨F2⟩ despite individual pairs allowing either ordering.The F1 < F2 case is illustrated by diagonal qutrit density matrices, but is described as atypical.
- Random density matrices: For higher-dimensional random density matrices, F1 < F2 remains possible, while the fraction with F1 > F2 increases substantially and average fidelity decreases.In dimension d = 10, purity obeys P ≥ 0.1, and the larger Hilbert space reduces the probability that random matrices are similar.
5. Applications
The review applies mixed-state fidelity measures to quantum processes, teleportation, cloning, phase-space simulations, and experimental measurements, emphasizing that different measures can agree only under special conditions.
- Quantum processes: Under orthogonal error-state assumptions, F1, FQ, and FA satisfy Eq. (5.4) without additional conditions.The assumption is that the error state is orthogonal to every signal state.
- Quantum processes: When a completely absorbing channel always outputs an orthogonal pure vacuum state, Fave = 0 for F1, FQ, and F2.Every input then produces an incorrect output, and the result follows for fidelity measures satisfying the generalized Jozsa axioms.
- Quantum processes: Average fidelity equals mixed-state fidelity only in special circumstances because orthogonal error states and saturation of joint concavity are not generally valid.This distinction applies even to the simpler F2 measure.
- Experimental measurements: Experimental fidelity measurements must account for the relevant Hilbert space, including degrees of freedom such as center-of-mass position and mode structure.Photonic fidelity estimates may be conditional on detection and may therefore differ from the true fidelity.
- Phase-space simulations: Phase-space simulations can compute F2 from sampled probability distributions, making it suitable for estimating quantum-technology or memory performance in large Hilbert spaces.F2 is singled out because it uses easily computable Hilbert-Schmidt norms while satisfying all Jozsa axioms.
6. Summary
The review finds that many candidate mixed-state fidelities satisfy the required axioms, including an infinite family of compliant measures. It highlights three physically interpretable measures and notes circumstances where alternatives may be preferable.
- Most candidate measures do not fully satisfy the Josza axioms, although some alternatives retain useful properties.
- An infinite number of compliant fidelities exist, including the Uhlmann-Josza fidelity F1, non-logarithmic quantum Chernoff fidelity FQ, and Hilbert-Schmidt fidelity F2.
- The review focuses on F1, F2, and FQ, while properties of other integer-p norm-based measures and several candidate properties remain unresolved.
- For the investigated norm-based fidelities, F1 has the largest average values for random density-matrix comparisons, universally for qubits and generally in higher dimensions.
- F1 is defined as a maximum over purifications, so subspace measurements may be biased high when errors occur in unmeasured parts of the relevant Hilbert space.
- Alternatives can be preferable when they are simpler to compute or better suited to an application; F2 also applies to unnormalized density matrices and appears less biased toward high values.
Appendix A. Detailed proofs
The appendix supplies detailed proofs for fidelity results established earlier in the review.
- The appendix provides detailed proofs for fidelity results from earlier sections when those proofs were not already included.
Norm based fidelity properties
All norm-based fidelities Fp satisfy the Josza axioms for p ≥ 1. The proofs establish bounds, equality conditions, symmetry, pure-state reduction, and unitary invariance.
- All norm-based fidelities Fp obey the Josza axioms for p ≥ 1.
- Fp lies in [0, 1], with the lower and upper bounds established using norm positivity and Hölder's inequality.
- Fp(ρ, σ) = 1 if and only if ρ = σ, while Fp(ρ, σ) = 0 if and only if the states have orthogonal support.
- Fp is symmetric under exchanging its two state arguments and reduces to tr(ρσ) when either state is pure.
- Fp is invariant under simultaneous unitary transformations of both density matrices.
Normalization
The arithmetic- and geometric-mean fidelity measures FAM and FGM are normalized between zero and one, attaining one exactly for identical density matrices.
- FAM and FGM lie in [0, 1], with the upper bound attained if and only if ρ = σ.
- The ordering FAM(ρ, σ) ≤ FGM(ρ, σ) ≤ 1 follows from the arithmetic-geometric mean relation and the Cauchy-Schwarz inequality.
- Equality at one requires saturation of Cauchy-Schwarz, which for unit-trace density matrices occurs only when ρ = σ.
Multiplicativity
The appendix examines how fidelity measures behave under tensor products, identifying general supermultiplicativity and special cases of exact multiplicativity. The measures differ: F2, FQ, and FC have supermultiplicative behavior, while FAM can violate even supermultiplicativity.
- F2: F2 is generally supermultiplicative, with multiplicativity in a specified appended-state case.The supplied theorem statement truncates the full condition.
- FC: FC is generally supermultiplicative under tensor products.The proof parameterizes subsystem dimensions and establishes the relevant inequality by maximizing an auxiliary function over [0,1]^2.
- FQ: FQ is generally supermultiplicative but multiplicative for uncorrelated ancillary states and tensor powers of identical states.For tensor powers, the single-copy minimizer remains optimal; an uncorrelated ancillary state cancels through the definition and axiom J1b.
- FAM: FAM is multiplicative when states are appended by an uncorrelated ancilla, with additional equality conditions involving equal purities or tensor powers.When none of the stated conditions holds, the appendix reports supermultiplicative rather than multiplicative behavior.
Proofs of average fidelity properties
The appendix proves equivalences involving average state-by-state fidelity under stated orthogonality and purity assumptions. Pairwise orthogonal signal states make the relevant density matrices jointly diagonalizable, yielding the result for several measures.
- Assumptions: Assuming the error state is orthogonal to every signal state, the appendix derives equivalences for average-fidelity expressions.This orthogonality assumption is used throughout the stated proofs.
- Orthogonal signals: When all signal states are pairwise orthogonal, F1, FQ, and FA satisfy Eq. (5.4) independently of signal-state purity.Pairwise orthogonality implies pairwise commutation and simultaneous diagonalizability.
- FQ: Under the relevant premise, the equivalence in Eq. (5.4) is verified for FQ.The proof substitutes the derived expression into the average-fidelity relation.
- Pure-state case: If the state is pure, the mixture reduces to one term, so FN equals the average state-by-state fidelity.The appendix states this when tr(ρ^2) = 1.
Counterexamples
The appendix supplies counterexamples showing that some proposed fidelity measures fail desired properties in general. It specifically exhibits a violation of supermultiplicativity for FAM and tests contractivity under partial trace for other measures.
- Multiplicativity: FAM can violate both multiplicativity and supermultiplicativity.For the supplied example, FAM(ρ⊗ρ, σ⊗σ) ≈ 0.702, whereas [FAM(ρ, σ)]^2 ≈ 0.718.
- Partial trace: The appendix examines whether FC, FGM, and FAM are contractive under partial trace.The test uses a pair of two-qubit density matrices and traces out the first qubit.
Appendix B. Metric properties
Appendix B develops metric properties for functionals derived from fidelity measures. It defines the metric requirements, invokes a simplified Schoenberg theorem, and notes a flaw in an earlier proof for two F1-based functionals.
- Setup: The appendix studies metric properties of functionals constructed from a fidelity measure.The functionals are defined before proving their properties for the measures discussed in Section 3.3.
- Metric definition: A metric must satisfy nonnegativity, identity of indiscernibles, symmetry, and the triangle inequality.These are listed as conditions M1–M4.
- Proof strategy: The appendix uses a simplified Schoenberg theorem as its main proof tool.The theorem links a symmetric nonnegative kernel with a metric under a specified implication.
- Related work: An earlier proof that B[F1(ρ, σ)] and C[F1(ρ, σ)] are metrics is described as flawed because it misapplied Schoenberg’s theorem.The review contrasts this with the prior result for C[FN].
F2 metric properties
The section proves that several fidelity-derived functions define metrics on spaces of density matrices. The arguments establish triangle inequalities using vector expansions, angular bounds, lemmas, and known metric results.
- F2 metric properties: C[F2(ρ, σ)] is established as a metric by verifying non-negativity, identity, symmetry, and the triangle inequality.The first three properties follow from F2 satisfying Jozsa’s axioms; the proof therefore focuses on the triangle inequality.
- F2 metric properties: The triangle-inequality proof for C[F2] reduces the problem to three inequalities ordered by the purities of ρ, σ, and τ.The proof represents states with expansion-coefficient vectors and uses angular relations plus bounds on cos θrt.
- F2 metric properties: 1 − FC(ρ, σ) is shown to be a metric for density matrices of fixed Hilbert-space dimension.The fixed-dimension condition makes the parameter r constant in the proof.
- F2 metric properties: C[FGM(ρ, σ)] is established as a metric, while explicit proofs are also provided for B[FA(ρ, σ)] and C[FA(ρ, σ)].The FA metric properties had previously been suggested numerically or mentioned via Schoenberg’s theorem; this section supplies explicit proofs.