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Conditional contraction coefficients and their applications to quantum networks

Christoph Hirche, Ian George, Theshani Nuradha, Mark M. Wilde

arXiv:2608.27171v1quant-phcs.IT

TL;DR

The paper investigates contraction of quantum divergences when reference systems and quantum correlations are present. It develops a framework for these settings and shows correspondences and optimization properties that support applications to quantum networks, mixing times, and quantum memories.

  • Problem

    Existing contraction settings do not cover quantum correlations beyond a channel’s input system, despite their relevance to multi-user information-processing problems.

  • Method

    The paper develops a framework for contraction coefficients with quantum side information, including SDPI constants, expansion coefficients, relative contraction coefficients, and applications such as Doeblin-coefficient bounds.

  • Results

    For trace distance, the conditional contraction coefficient is optimized by orthogonal input states, while for relative entropy mutual-information contraction is identical to relative-entropy contraction.

  • Takeaways & Limitations

    The framework provides a basis for analyzing information loss in multi-user quantum settings and points toward compositional theories of information loss in quantum networks.

  • Takeaways & Limitations

    The entanglement-preservation applications acknowledge that non-entangling channels may be too powerful and that relaxing mutual information to twice the local dimension may be too loose.

Abstract

from arXiv · show

Contraction coefficients quantify the loss of distinguishability induced by a channel and provide a strong form of the data-processing inequality. While standard contraction coefficients ignore auxiliary quantum systems, existing extensions based on complete contraction coefficients require the compared states to have identical reference marginals. In this work, we introduce conditional contraction coefficients, a novel family that incorporates arbitrary quantum reference systems by subtracting the distinguishability already present in the reference system. We develop a general framework for contraction coefficients with such quantum side information, including the corresponding strong-data-processing-inequality (SDPI) constants, expansion coefficients, and relative contraction coefficients. For the trace distance, we show that the optimization can be restricted to orthogonal input states. For the quantum relative entropy, we prove that its conditional contraction coefficient is exactly equal to the contraction coefficient of the conditional mutual information, extending the classical correspondence between relative-entropy contraction and mutual-information contraction to the setting with quantum side information. More generally, we identify structural properties of divergences required for these results and discuss extensions beyond the relative entropy. These results establish a unified framework for analyzing information contraction in quantum network settings, where quantum side information and distributed correlations are intrinsic features of the information-processing task. Applications include an extension of the Polyanskiy-Wu bounds on mutual information contraction, new perspectives on mixing times, and fundamental limits on quantum memories.

1 Introduction

The paper introduces conditional contraction coefficients to analyze distinguishability loss with arbitrary quantum reference systems, overcoming the equal-marginal restriction of complete contraction coefficients. It develops this framework across divergences, SDPI constants, and applications to quantum networks, mixing, and quantum memories.

  • Motivation: Standard contraction coefficients quantify distinguishability loss under channels but do not account for correlations with auxiliary quantum reference systems.A naive extension can become trivial when distinguishability originates entirely in the reference system.
  • Motivation: Complete contraction coefficients handle reference systems by requiring identical reference marginals, but this restriction can be impractical and the reference optimization may be unbounded.For classical references and suitable divergences, complete and usual coefficients coincide; quantum references exhibit different behavior.
  • Conditional contraction coefficients: The conditional contraction coefficient incorporates arbitrary reference systems by subtracting the distinguishability already present in the reference.Restricting the optimization to equal marginals recovers the complete contraction coefficient.
  • Information-theoretic correspondences: For relative entropy, mutual-information contraction equals relative-entropy contraction, while conditional mutual-information contraction equals the conditional contraction coefficient even with quantum side information.The conditional mutual-information correspondence remains valid whether the auxiliary system is quantum, classical, or binary classical.
  • Structural results: For trace distance, conditional contraction remains optimized by orthogonal input states; for hockey-stick divergence, it can exceed complete contraction even classically.The framework also extends relations among divergence-based and mutual-information-based contraction coefficients beyond relative entropy.
  • Applications: Applications generalize contraction bounds to quantum networks, derive tensor-product and replacement-time results, and bound storage times in quantum memories.The paper also discusses SDPI constants, including approximate tensorization and exact tensorization for several special cases such as generalized depolarizing channels.

2 Contraction with reference systems

The paper develops generalized contraction, SDPI, expansion, and relative contraction coefficients that incorporate quantum reference systems. Conditional versions account for distinguishability already present in the reference, while the framework establishes general properties and divergence-dependent bounds.

  • Definitions: Conditional coefficients extend contraction and SDPI constants to arbitrary reference systems by accounting for reference-system distinguishability.The framework also includes product-reference and complete variants, plus corresponding expansion coefficients.
  • Definitions: The conditional coefficient optimization requires a finite, strictly positive conditional divergence denominator.If no admissible state satisfies this condition, the coefficient is defined as zero.
  • General properties: Additivity gives tensor-product inequalities for SDPI and contraction coefficients, although these inequalities are usually strict for contraction coefficients.SDPI achievability depends on the exact coefficient definition.
  • General properties: Nonnegative and jointly convex divergences yield general bounds, including convexity of contraction coefficients and concatenation inequalities.The framework also derives Doeblin-coefficient bounds, with the Doeblin coefficient expressible as a semidefinite program.
  • Expansion coefficients: Generalized expansion coefficients introduce product-reference, complete, and conditional versions for channels and divergences.The usefulness of expansion coefficients depends strongly on the divergence because standard expansion coefficients can be trivial for many divergences.

3 Relative entropy-like divergences

This section connects relative-entropy contraction with mutual-information and conditional-mutual-information contraction in the presence of quantum reference systems. It also identifies divergence properties supporting analogous results beyond relative entropy.

  • Conditional mutual information: Conditional mutual-information contraction is identified as the information counterpart of the conditional contraction coefficient.This extends the relative-entropy–mutual-information correspondence to quantum side information.
  • Optimization reductions: Theorem 1 and Theorem 2 show that testing binary classical systems suffices for the stated conditional and complete contraction equivalences.The binary restriction is compared with optimization over arbitrary classical systems.
  • Comparison of coefficients: A numerical search found no mutual-information state attaining a comparable value to the complete contraction coefficient, but this does not establish nonexistence.The paper therefore treats the observed separation as a conjectural indication rather than a proof.
  • Mutual information with reference systems: The paper introduces MIR contraction coefficients to represent mutual-information contraction with quantum reference systems and gives equivalent formulations.One formulation uses conditional mutual information, while another uses tripartite mutual-information combinations.
  • Beyond relative entropy: The framework extends alternative contraction-coefficient expressions to divergences satisfying the two structural properties in (3.147) and (3.148).Analogous generalizations are stated for expansion and relative contraction coefficients.

4 On tensorization of SDPI constants

The section studies tensorization of SDPI constants with and without reference systems. Relative-entropy and stabilized conditional quantities admit positive tensorization results in specified settings, whereas complete and conditional SDPIs can fail to tensorize exactly.

  • Motivation and setting: Tensorization prevents the exponential contraction rate from deteriorating with system size when identical noise acts independently on multiple systems.This motivates studying SDPI constants on tensor products of channels and states.
  • Without reference systems: The relative-entropy SDPI constant tensorizes in special cases including an erasure channel paired with an arbitrary channel.Other listed cases include identical generalized depolarizing channels and a quantum-to-classical channel under an attainment condition.
  • Stabilization: Mutual-information SDPI constants tensorize, whereas the conditional mutual-information SDPI constant does not tensorize according to the cited counterexample.The distinction reflects the need for stabilization in the conditional setting.
  • Reference systems: Approximate tensorization results extend to settings with reference systems, but exact tensorization does not hold generally for complete and conditional relative-entropy SDPIs.The failure persists even when the reference system is one-dimensional.
  • Reference systems: The failure of exact tensorization follows because fixing the reference state can allow the reference system to be trivial, reproducing a known counterexample.The argument applies the counterexample to channels and their tensor product.
  • Stabilization: Stabilized conditional SDPI constants tensorize, and equivalently conditional-mutual-information SDPI constants tensorize.The result is established for product channels and corresponding states.

5 Trace distance and hockey-stick divergences

The section shows that trace-distance conditional contraction remains optimizable over orthogonal states, while hockey-stick divergences reveal sharper distinctions between conditional and complete coefficients. It also establishes tensorization results and identifies open limitations for reference-system settings.

  • Hockey-stick divergence: For hockey-stick divergence, orthogonal pure states suffice without reference systems, but the corresponding reference-system approach does not generally extend to every γ ≥ 1.The paper states that the general-γ approach fails in the reference-system setting, while γ = 1 remains useful.
  • Tensorization: The tensorization analysis reports exact tensorization for some fixed-product-state quantities, failure in general for red entries and stabilized complete SDPI, and approximate tensorization for all quantities on the left.The ordinary SDPI still tensorizes under sufficient conditions established in Corollary 9.
  • Orthogonal-state reductions: Theorem 6 constructs orthogonal state pairs that preserve the relevant contraction and expansion comparisons, with equal reference marginals when the original pair has them.The construction underlies the simplified trace-distance formulas and also applies to relative contraction and expansion coefficients.
  • Trace distance: For trace distance, the conditional contraction coefficient can be optimized over orthogonal states, including mixed orthogonal states.This extends the favorable orthogonal-state optimization property to settings with reference systems, although reduction to pure orthogonal states remains unresolved.
  • Open questions: Monotonicity in γ for contraction coefficients with reference systems remains open, with preliminary numerical evidence suggesting it might fail.The absence of the orthogonal-state simplification available without reference systems prevents the same immediate monotonicity argument.
  • Separating contraction coefficients: For γ > 1, a classical channel can have conditional hockey-stick contraction equal to zero even with quantum reference systems, separating conditional from complete contraction behavior.The example uses a binary symmetric channel with flipping probability p satisfying (1 − p)/p = γ, and exploits the non-faithfulness of the hockey-stick divergence.

6 Applications to quantum networks

The applications extend contraction-coefficient techniques to quantum networks, tensor-product channels, quantum memories, and secret-sharing schemes. They derive network bounds and use information contraction to limit resource preservation and recoverability under noise.

  • 6 Applications to quantum networks: The framework generalizes a classical mutual-information contraction bound to quantum network channels composed as N = R ◦ M.The input reference system tracks mutual information with different network outputs.
  • 6.1 Contraction of mutual information in networks: 1 + (1 − p)^2 ≤ ηD(D_p^⊗2) ≤ 1 − 2p^2 + p^3 ≤ 1 − p^2 gives bounds for two-qubit depolarizing channels.These tensor-product bounds extend results previously known only for special cases, including one erasure channel.
  • 6.2 Limits on quantum memories: Quantum-memory limits are obtained by applying contraction coefficients to resource-distillation rates after noisy, time-inhomogeneous Markov evolution.The bounds hold independently of the initial shared state and identify times after which post-processing cannot recover the resource.
  • 6.2 Limits on quantum memories: Entanglement distillation with LOCC is controlled by ηCMI, whereas non-entangling post-processing is controlled by ηMI, with ηMI ≤ ηCMI.The relevant constants in the resulting theorems differ despite the inclusion relation between the operation classes.
  • 6.2 Limits on quantum memories: After sufficiently long storage, no state with singlet fraction greater than 1 − ε can be distilled using either non-entangling or LOCC post-processing.The stated no-go results apply to every input state, with separate bounds for the two post-processing classes.
  • 6.3 Limits on quantum memories for quantum secret sharing schemes: For quantum secret sharing, sufficiently noisy memories can prevent any ε-recoverable (t, m)-sharing encoding, including under homogeneous or inhomogeneous memory channels.The analysis uses contraction bounds and explains that error-correcting encodings can preserve distinguishability across product channels, limiting tensorization-based guarantees.

7 Applications to discrete-time quantum Markov chains

The paper extends contraction-based analysis to discrete-time quantum Markov chains with reference systems, relating replacing behavior to mixing and deriving replacement-time bounds. It also connects these ideas to decoupling, tensorization, and exponential information decay.

  • 7.1 On hierarchies of operational times: Replacing and mixing are equivalent for time-homogeneous quantum Markov chains, and any intermediate asymptotic notion such as decoupling is equivalent as well.Both characterize convergence of channel powers toward a replacer channel.
  • 7.1 On hierarchies of operational times: A replacing chain is characterized by the convergence of N^n to a replacer channel, preserving only the distinguishability already present in the reference system.The defining trace-distance difference converges to zero for all bipartite input states.
  • 7.1.2 Finite-time bounds: The replacement time is bounded using a conditional SDPI constant, extending analogous finite-time bounds for other operational times.The construction assumes a homogeneous chain with a fixed point and derives the bound by iterating the relevant contraction inequality.
  • 7.1.2 Finite-time bounds: For relative entropy, decoupling and replacing times become equivalent, while replacement time is controlled by the product SDPI constant.The relative-entropy SDPI constant also tensorizes in the stated setting, implying stability under tensor powers.
  • 7.1 On hierarchies of operational times: A finite iterate has positive quantum Doeblin coefficient if and only if the channel is mixing.This gives a coefficient-based characterization of mixing.
  • 7.2 Other applications: If all channels in a sequence have Choi operators with minimum eigenvalue bounded strictly above zero, conditional mutual information decays exponentially with circuit depth.The result does not require the input state to be maximally entangled.

8 Applications to quantum machine learning

The paper applies conditional contraction tools to noisy quantum learning and circuits, relating channel contraction to stability, generalization, information decay, and scrambling. It also distinguishes globally preserved information from information hidden from restricted subsystems.

  • 8.1 Stability and generalization: For a fixed channel learning algorithm, contraction coefficients quantify the noisy model's impact, while mutual information terms capture dataset and encoding dependencies.This separates noise effects from correlations introduced by the data and encoding protocol.
  • 8.1 Stability and generalization: For layered channels and unitaries, the stability bound includes a product of factors 1 − α+(P_i), which decreases with increasing circuit depth.The bound multiplies the mutual information term by the contraction contributions of the noisy channels.
  • 8.1 Stability and generalization: The considered quantum-learning generalization error scales as √γ′ through its connection to the defined stability parameter.The connection is stated for the fixed-distribution setup under discussion.
  • 8.2 Noisy quantum circuits: For noisy circuits with encoding or pretrained channels that correlate reference systems, the paper's tools analyze information loss where earlier noiseless-reference analyses do not apply.The setting includes noisy encoders or circuits involving decoherence, randomness, and correlations.
  • 8.2 Noisy quantum circuits: When the relevant contraction quantity is small, the circuit can require exponentially many samples for error mitigation and exhibit noisy barren plateaus.The discussion also relates contraction behavior to choices of circuit depth and encoder.
  • 8.3 Information scrambling: Scrambling is analyzed through reduced channels describing information accessible to restricted observers, distinguishing delocalization from destruction.The framework studies information about C available to collections of output systems B_i given prior correlations R.
  • 8.3 Information scrambling: Small local contraction together with a global quantity near one indicates information that survives globally but is hidden from small regions, whereas both small quantities indicate destruction.The paper gives corresponding interpretations for selected subsystem collections and for all small regions.

9 Conclusion

The conclusion presents conditional contraction as a framework for quantum side information, highlights distinct tensorization behavior across SDPI formulations, and identifies applications and open problems. It emphasizes the unresolved question of when local contraction properties yield global network guarantees.

  • Conclusion: Quantum reference systems create a richer contraction structure than the classical setting, even though many quantities reduce to the usual coefficient classically.The paper frames this difference as central for network and multi-user problems.
  • Conclusion: Classical-quantum SDPI constants do not tensorize in general, whereas fully quantum mutual-information and conditional-mutual-information SDPI constants recover tensorization.The conclusion attributes this contrast to how quantum side information is incorporated.
  • Conclusion: The framework is intended to support a compositional theory in which local noisy-component contractions control information propagation and lifetime in larger quantum networks.The paper identifies the validity of such local-to-global principles as a central future question.
  • Open problems: An open problem is determining when relative-entropy SDPI tensorizes or fails, and which divergence and side-information properties govern that behavior.The conclusion also notes unresolved questions for the hockey-stick divergence and unbounded reference systems.
  • Open problems: The paper leaves nonlinear strong-data-processing inequalities with reference systems as a natural future direction.This topic is explicitly outside the paper's scope.

Statement on AI usage

The paper acknowledges using ChatGPT in developing Lemma 12 and Proposition 26 and in figure polishing, typo correction, and error detection.

  • ChatGPT was used for conceiving Lemma 12 and Proposition 26, polishing figures, correcting typos, and detecting an earlier-draft error.

A Contraction coefficients with classical reference systems

Classical reference systems do not require complete or conditional contraction coefficients when the divergence has the extended direct-sum property. Under that condition, the classical-reference coefficients equal the standard contraction coefficient.

  • A Contraction coefficients with classical reference systems: Classical reference systems need not be considered for complete or conditional contraction coefficients when the divergence has the extended direct-sum property.The appendix defines these coefficients for classical–quantum states and proves equality with the standard coefficient.
  • A Contraction coefficients with classical reference systems: The proof applies the extended direct-sum property to classical–quantum states and uses D(p∥q) = D(ρR∥σR).
  • A Contraction coefficients with classical reference systems: The resulting classical-reference contraction coefficient is bounded by the usual coefficient, and combining inequalities gives equality.
  • A Contraction coefficients with classical reference systems: The extended direct-sum property includes Umegaki and Belavkin–Staszewski quantum relative entropy and many quantum α-Hellinger divergences.
  • A Contraction coefficients with classical reference systems: The property does not include the trace distance or hockey stick divergences.

B Limit property of relative entropy and generalized divergences

This appendix establishes a limit-property framework for relative entropy and generalized divergences. It relates two manifestations of the relevant condition through a proof using full-rank perturbations and a vanishing first derivative.

  • B Limit property of relative entropy and generalized divergences: A full-rank state is introduced through a parameterized construction involving λ.
  • B Limit property of relative entropy and generalized divergences: The resulting expression includes Tr[(ln ρ(λ) −ln σ) (σ −ρ)] divided by (1 −λ)2.
  • B Limit property of relative entropy and generalized divergences: The appendix states that one of two manifestations of the condition implies the other, even for general divergences.
  • B Limit property of relative entropy and generalized divergences: The argument differentiates a parameterized expression using the chain rule and the conditions µ(0) = 0 and µ′(0) = 1/ε.

C Failure of tensorization for the stabilized complete SDPI

The stabilized complete relative-entropy SDPI constant does not tensorize in general. The counterexample uses a qubit reference, a four-state input construction, and a channel recording whether the input equals 2.

  • C Failure of tensorization for the stabilized complete SDPI: The stabilized complete SDPI constant is shown not to tensorize in general.
  • C Failure of tensorization for the stabilized complete SDPI: The counterexample takes R to be a qubit and defines four input states ω0, ω1, ω2, and ω3.
  • C Failure of tensorization for the stabilized complete SDPI: The channel N records whether the input is equal to 2, while the reference marginal remains unchanged in the construction.
  • C Failure of tensorization for the stabilized complete SDPI: The proof reduces one-copy analysis to dephased states because dephasing leaves the numerator unchanged and cannot increase the denominator.
  • C Failure of tensorization for the stabilized complete SDPI: The output states are decomposed into mutually orthogonal sectors, enabling the relative-entropy calculation sector by sector.
  • C Failure of tensorization for the stabilized complete SDPI: The construction attains the value 1/2 in the relevant one-copy calculation before establishing the two-copy failure.
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