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Converse bounds for private communication over quantum channels

Mark M. Wilde, Marco Tomamichel, Mario Berta

arXiv:1602.08898v3quant-phcs.IT

TL;DR

Private quantum communication lacks fully general, practically relevant converse bounds beyond asymptotic settings, and the status of relative entropy of entanglement as a universal upper bound remains open. The paper develops a private-state-based meta-converse and applies it to strong-converse, second-order, and finite-blocklength bounds, obtaining precise characterizations for several channels and strong-converse results for important bosonic channels.

  • Problem

    Existing converse analyses are largely asymptotic, while it remains open whether relative entropy of entanglement upper-bounds two-way assisted private capacity for general quantum channels.

  • Method

    The paper uses private states and a privacy-test-based meta-converse, together with teleportation simulation for suitable channels, to derive multiple private-communication converse regimes.

  • Results

    The paper establishes strong-converse and second-order bounds, precise characterizations for several channels, and finite-blocklength converse bounds for all single-mode phase-insensitive bosonic channels.

  • Takeaways & Limitations

    Relative entropy of entanglement provides strong-converse rates in the stated settings, including unconstrained private and quantum capacities of pure-loss and quantum-limited amplifier channels.

  • Takeaways & Limitations

    The paper notes that second-order expansions of achievability results remain desirable and that the corresponding problem is generally open.

Abstract

from arXiv · show

This paper establishes several converse bounds on the private transmission capabilities of a quantum channel. The main conceptual development builds firmly on the notion of a private state, which is a powerful, uniquely quantum method for simplifying the tripartite picture of privacy involving local operations and public classical communication to a bipartite picture of quantum privacy involving local operations and classical communication. This approach has previously led to some of the strongest upper bounds on secret key rates, including the squashed entanglement and the relative entropy of entanglement. Here we use this approach along with a "privacy test" to establish a general meta-converse bound for private communication, which has a number of applications. The meta-converse allows for proving that any quantum channel's relative entropy of entanglement is a strong converse rate for private communication. For covariant channels, the meta-converse also leads to second-order expansions of relative entropy of entanglement bounds for private communication rates. For such channels, the bounds also apply to the private communication setting in which the sender and receiver are assisted by unlimited public classical communication, and as such, they are relevant for establishing various converse bounds for quantum key distribution protocols conducted over these channels. We find precise characterizations for several channels of interest and apply the methods to establish several converse bounds on the private transmission capabilities of all phase-insensitive bosonic channels.

1 Introduction

The paper frames private quantum communication through private states, which convert tripartite secrecy into bipartite quantum privacy, and develops converse bounds addressing limitations of asymptotic analyses.

  • Private-state framework: Private states recast tripartite secret-key states as bipartite states involving maximally entangled key systems, twisting, and an arbitrary shield state.Purification gives Alice and Bob purifying systems, allowing the eavesdropper to be eliminated from the analysis.
  • Private communication versus quantum communication: Quantum channels can have zero quantum capacity while still supporting private communication at a non-zero rate.This distinction highlights that private communication and entanglement distillation are fundamentally different capabilities.
  • Motivation: Existing theoretical bounds are primarily asymptotic, applying only when many independent and identical channel uses are available.The paper identifies limited channel uses as a practical setting not fully captured by those analyses.
  • Applications: The resulting bounds can assess practical non-asymptotic protocols and quantum-repeater performance for certain channels.This extends the usefulness of converse bounds beyond asymptotic quantum key-distribution analyses.
  • Contributions: The paper introduces a meta-converse that yields strong-converse bounds, second-order bounds, and finite-blocklength bounds for private communication.Applications include precise channel characterizations and converse bounds for phase-insensitive bosonic channels.

2 Preliminaries

The preliminaries define quantum channels, covariance, teleportation simulation, and LOCC-based private-state formulations that support the paper’s converse analysis.

  • Quantum-information tools: A quantum channel is a completely positive, trace-preserving linear map between operator spaces, with isometries and instruments providing standard process descriptions.Quantum instruments output both classical and quantum systems, while LOCC protocols alternate conditioned operations by Alice and Bob.
  • Covariant channels: Covariant channels respect compatible group representations, including representations whose uniform average produces the maximally mixed state.This symmetry is central to the teleportation-simulation results used later.
  • Teleportation simulation: A teleportation-simulable channel can be replaced by shared state preparation followed by a teleportation protocol.This replacement lets Alice and Bob treat channel uses as shared resources within adaptive protocols.
  • Teleportation simulation: Every covariant channel is teleportation-simulable with associated state ωAB = N(ΦAA′).The proposition identifies the associated state as the channel output on a maximally entangled input.
  • Private states: Bipartite private states and tripartite key states are equivalent under purification, linking LOCC privacy protocols to bipartite state analysis.This correspondence is the operational basis for replacing tripartite key extraction with bipartite private-state generation.

3 Secret key transmission and generation over quantum channels

The paper formalizes secret-key transmission and generation over quantum channels, then purifies these protocols into bipartite private-state tasks. It also defines non-asymptotic performance boundaries and extends the framework to LOCC-assisted secret-key agreement.

  • Secret-key transmission codes: Secret-key transmission uses an encoder, n channel uses, and a decoder to produce an ε-approximate tripartite secret-key state.The protocol begins from a maximally classically correlated state and sends encoded systems through the channel.
  • Secret-key transmission codes: Purification converts the protocol into a bipartite private-state task by coherently replacing encoding and decoding operations and identifying key and shield systems.The purified output is compared with a private state using the correspondence between tripartite privacy and bipartite private states.
  • Secret-key generation codes: Secret-key generation starts from an initial state on the key and channel inputs, then applies the channel and a decoder to generate an ε-approximate tripartite secret-key state.Its purified formulation likewise has the goal of generating an ε-approximate bipartite private state.
  • Non-asymptotic achievable regions: The non-asymptotic achievable region tracks both the maximum rate at fixed error and the minimum error at fixed rate.The first boundary supports second-order coding-rate analysis, while the second supports strong-converse-exponent bounds.
  • Non-asymptotic achievable regions: LOCC-assisted secret-key agreement purifies tripartite rounds of local operations and public communication into a bipartite LOCC protocol, enabling private-state analysis.For teleportation-simulable channels, this adaptive structure can simplify substantially.
  • Secret-key generation codes: The transmission and generation problems have matching performance when classical communication is free, but transmission can directly realize generation without an additional resource.The reverse realization is not possible in general without assistance from another resource.
  • Secret-key transmission codes: Entanglement transmission at rate R and fidelity at least 1 − ε provides a secret-key transmission code with the same rate and fidelity.This follows because a maximally entangled state is a private state with trivial twisting and shield systems.

4 General (meta-converse) bounds

The paper develops a general meta-converse for private communication by testing whether output states are private and bounding how often separable states can pass the test. This framework yields channel converse bounds, including bounds for LOCC-assisted and teleportation-simulable settings.

  • General quantities: The general meta-converse uses hypothesis-testing quantities related to relative entropy of entanglement and their monotonicity under quantum and LOCC channels.The associated channel quantity is optimized over input states and supports upper bounds on private transmission capabilities.
  • Privacy test: A privacy test is a dichotomic measurement that checks whether a bipartite state is private with respect to a specified private state.Operationally, it untwists the candidate state and projects onto a maximally entangled state.
  • Privacy test: An ε-approximate private state passes its corresponding privacy test with probability at least 1 − ε.Thus, approximate privacy guarantees a high acceptance probability for the test.
  • Privacy test: Any separable state passes any γ-privacy test with probability no greater than 1/K, where K is the number of secret-key values.The proof establishes the bound first for pure product states and extends it to separable states by convex decomposition.
  • Privacy test: The two privacy-test bounds form the core inequalities underlying the paper’s converse bounds.They separate approximate private states from separable states through their different test-passing probabilities.
  • General converse bounds: For any quantum channel, the theorem converts the achievable-region analysis into a general channel converse bound.The construction applies the privacy test after channel transmission and LOCC decoding, comparing the result with separable-state acceptance limits.
  • General converse bounds: For teleportation-simulable channels, the resulting LOCC-assisted bound avoids dependence on shield-system dimension.This feature is beneficial for analyzing quantum Gaussian channels.

5 Relative entropy of entanglement as a strong converse rate

The section develops strong converse bounds for private communication using Rényi entanglement measures and a meta-converse. It shows that relative entropy of entanglement supplies strong converse rates generally, with stronger statements for teleportation-simulable channels and selected channel classes.

  • Proof ingredients: The proof strategy combines fidelity bounds, Rényi entanglement monotonicity, subadditivity on tensor-product states, and a postselection-based de Finetti argument.The postselection technique is used with permutation covariance of independent channel uses to obtain the needed finite-blocklength bound.
  • General strong converse framework: The meta-converse and Rényi entanglement bounds provide the tools for proving strong converse results for private communication.The relevant inequalities connect private rates to sandwiched Rényi relative entropy of entanglement and are used in the subsequent theorems.
  • Teleportation-simulable channels: For teleportation-simulable channels, the relative entropy of entanglement of the associated state is a strong converse rate for two-way assisted private communication.Rates exceeding this quantity force the optimal error to approach one exponentially fast, with exponent involving the gap to a Rényi entanglement quantity.
  • General channels: For any channel, its relative entropy of entanglement is a strong converse rate for CPPP-assisted private communication and therefore also for unassisted private communication.This conclusion follows from the corresponding bound for CPPP-assisted protocols and the relation between assisted and unassisted private communication.
  • Particular channels: Generalized dephasing and quantum erasure channels are identified as classes whose private capacities satisfy the strong converse property.The section gives propositions establishing strong converse characterizations for these channel families, with coherent information appearing for generalized dephasing channels.

6 Second-order expansions for private communication

The section derives second-order converse bounds for private communication from relative entropy variance and Gaussian approximations. For teleportation-simulable and sufficiently symmetric channels, these bounds can become tight characterizations through matching achievability results.

  • Second-order framework: Second-order analysis fixes the error probability and characterizes the maximum private communication rate as the number of channel uses grows.The resulting Gaussian approximation uses relative entropy variance and second-order expansions of quantum relative entropy.
  • Converse bounds: For teleportation-simulable channels, the meta-converse yields second-order expansions of relative entropy of entanglement upper bounds.The expansion uses a variance quantity defined over separable states that achieve the relative entropy of entanglement minimum.
  • Achievability connection: Entanglement transmission lower bounds transfer to secret-key transmission, enabling tight second-order private-communication characterizations for some channels.The connection relies on entanglement transmission achieving secret-key transmission and on one-way distillation or related achievability protocols.
  • Matching bounds: For sufficiently symmetric channels, coherent-information-based lower bounds can match the second-order converse, including covariant generalized dephasing channels.The cited characterization states equality between the achievable and lower-bound expressions for sufficiently large blocklength.
  • Scope of exact results: Covariant dephasing and quantum erasure channels have tight second-order characterizations, while qubit versions are characterized through the third order.The higher-order refinement is attributed to the symmetries of these channel classes.

7 Channels with higher-order characterizations

The section gives higher-order characterizations for qubit dephasing and erasure channels and discusses the negligible private-information capability of entanglement-breaking channels. For the qubit examples, private and quantum transmission capabilities coincide.

  • Qubit dephasing channel: For the qubit dephasing channel, private, CPPP-assisted, and two-way assisted boundaries coincide through a third-order expansion.The expansion includes the leading term 1 − h(γ), a Gaussian correction involving v(γ), and further terms specified in the proposition.
  • Qubit dephasing channel: The qubit dephasing channel has no difference between private and quantum transmission capabilities.The result follows from the matching higher-order characterization for the channel.
  • Qubit erasure channel: The qubit erasure channel admits a precise private-communication characterization with matching upper and lower bounds, including the CPPP-assisted setting.The channel is teleportation-simulable, allowing the converse bound to be paired with an achievability protocol.
  • Qubit erasure channel: For the qubit erasure channel, private and quantum transmission capabilities again coincide, refining the earlier two-way assisted capacity result of 1 − p.The stated characterization strengthens the previously established capacity result for this channel.
  • Entanglement-breaking channels: Entanglement-breaking channels have essentially no private-information transmission capability because their protocols can only produce separable final states.For these channels, the first-, second-, and third-order terms all vanish, and the same upper bound applies to two-way assisted quantum transmission.

8 Quantum Gaussian channels

This section derives converse bounds for phase-insensitive bosonic channels, including strong converse rates and exact capacity characterizations for selected channels. It uses teleportation simulation and relative entropy methods to treat thermalizing, amplifier, and additive-noise channels.

  • Channel classes: The section studies thermalizing, amplifier, and additive-noise phase-insensitive bosonic channels and notes that other phase-insensitive Gaussian channels are entanglement-breaking.The channels are specified through Heisenberg input-output relations; entanglement-breaking cases have severely limited private and quantum communication abilities.
  • Channel classes: The bounds apply to both constrained and unconstrained capacities because private and quantum capacities remain bounded even with unlimited input energy.The energy constraint reflects practical resource limits, but the stated upper bounds cover both settings.
  • Converse bounds: The pure-loss and quantum-limited amplifier channels satisfy Q↔(Lη) = P↔(Lη) = −log(1 −η) and Q↔(AG) = P↔(AG) = log G.These equalities provide precise unconstrained capacity characterizations for the two channels.
  • Converse bounds: The bounds become strong converse rates for constrained and unconstrained private and quantum capacities, while pure-loss and quantum-limited amplifier channels have exact unconstrained characterizations.For these two channels, the strong converse property is established for unconstrained two-way assisted private and quantum capacities.
  • Converse bounds: Theorem 24 gives finite-blocklength converse bounds for all ε ∈(0, 1), n ≥1, and NB > 0, with corrections involving relative entropy variances and C(ε).The quantities include the unconstrained relative entropy variances of thermalizing, amplifier, and additive-noise channels, while C(ε) is an explicit error-dependent term.
  • Converse method: Teleportation simulation reduces arbitrary protocols to a simpler form whose infinite-energy limit preserves the original performance, enabling converse bounds.The simulation uses two-mode squeezed vacuum states, continuous-variable teleportation, and a limit as the squeezing energy grows.

9 Conclusion

The paper presents a private-state-based meta-converse framework that yields strong converse and second-order bounds, with precise characterizations for several channels. Its phase-insensitive bosonic results also constrain quantum key distribution, while further second-order achievability analysis remains open.

  • Conclusion: The general approach builds on private states and relative entropy of entanglement to derive meta-converse, strong converse, and second-order bounds.The paper applies methods developed in prior work and obtains precise characterizations for several channels.
  • Conclusion: The paper establishes the strong converse property for unconstrained private and quantum capacities of pure-loss and quantum-limited amplifier channels.It also gives converse bounds for more general phase-insensitive bosonic channels.
  • Conclusion: Several bounds are relevant to understanding limitations of quantum key distribution protocols over phase-insensitive bosonic channels.
  • Open questions: A second-order expansion of the achievability results remains desirable, and the paper notes that such an analysis might clarify channels with zero quantum capacity but non-zero private capacity.The cited discussion presents this as future work rather than a result established in the paper.
  • Open questions: Whether a quantum channel’s squashed entanglement upper-bounds its two-way assisted private capacity remains open.

A Covariant channels are teleportation simulable

The appendix proves that covariant quantum channels are teleportation-simulable. A shared resource state, a covariance-based measurement, classical communication, and a corrective unitary reproduce the channel output for any input.

  • Main result: The appendix establishes that every covariant channel is teleportation-simulable.The proof extends earlier teleportation-simulation developments.
  • Simulation protocol: Alice measures the input and resource systems using a covariance-based POVM, then sends the outcome g to Bob through a classical channel.The representation condition requires |A|^2 ≤ |G| for the POVM to be valid.
  • Simulation protocol: The construction starts with a shared resource state formed by applying the channel to one half of a maximally entangled state.The input state is held on a system isomorphic to the channel input.
  • Simulation protocol: Bob applies the corresponding unitary V^g, after which the protocol outputs N(ρ) independently of the input state.The final output is the channel applied to the original input state.

B Definitions of privacy and converse bounds

This appendix relates two definitions of secret-key privacy and transfers converse bounds between them. In particular, Type II security implies Type I security and preserves strong converse rates when the combined errors remain below one.

  • Relations between definitions: A converse bound for the paper’s privacy definition also applies to the modified quantum privacy definition used in this appendix.The modification is needed to connect the framework with other privacy notions in the literature.
  • Privacy definitions: Type I secret-key security requires both a bounded decoding error and small trace distance between the key-eavesdropper state and an ideal product state.The definition uses parameters ε and δ for correctness and secrecy.
  • Privacy definitions: Type II security uses closeness to a maximally classically correlated key state together with the eavesdropper marginal fixed to the actual state.This differs from the main-text definition, where the ideal eavesdropper state may be arbitrary.
  • Relations between definitions: As η →1, a Type II secret-key state becomes an (ε, δ) Type I state with ε + δ → ξ ≥1.Consequently, the approach gives strong converse rates for ε + δ < 1.
  • Relations between definitions: A second relation states that an η Type II secret-key state is an (√η, √η) Type I secret-key state.

C One-shot distillable entanglement lower bound

The appendix proves a one-shot distillable-entanglement lower bound using non-smooth decoupling, smooth-entropy conversion, and isometric embedding constructions. The argument uses quantum instruments, classical communication, fidelity monotonicity, and Uhlmann’s theorem.

  • The proof begins with a purification ρABE of ρAE and a quantum channel corresponding to a quantum instrument.
  • The construction uses projectors and an isometric channel that embeds the projected subspace into HA1, assuming |A1| divides |A|.
  • A non-smooth decoupling theorem based on conditional collision entropy provides the initial bound before conversion to a smooth-entropy statement.
  • Choosing |A1| appropriately yields the desired entropy bound, with Hmin(A|E)ρ related to −Hmax(A|B)ρ by entropy duality.
  • Uhlmann’s theorem, fidelity monotonicity, and purified-distance conversions establish channels and unitary operations that complete the entanglement-distillation construction.
  • A final application of Uhlmann’s theorem and conversion from purified distance to fidelity concludes the proof.

D Variance of the relative entropy of entanglement for phase-insensitive Gaussian channels

The appendix derives relative-entropy-variance formulas for phase-insensitive Gaussian channels by comparing channel outputs with carefully selected separable Gaussian states. The resulting expansions characterize the large-energy behavior, including vanishing variance for pure-loss and quantum-limited amplifier channels.

  • For two-mode Gaussian states in standard form, separability is determined using the covariance-matrix conditions reviewed from prior work.
  • For fixed parameters a and b, the separable comparison state uses maximal correlations with c set to csep, providing a guiding choice for minimizing relative-entropy distance.
  • The comparison input is a two-mode squeezed vacuum whose single-mode energy μ is directly related to the input entanglement.
  • Sending one mode through thermal, amplifier, and additive-noise channels produces corresponding two-mode Gaussian output states with specified covariance matrices.
  • The relative entropy variances are expanded about μ = ∞ using a formula from prior work and computer algebra for the algebraic manipulations.
  • V(ρµG,ω=0) = O(µ−1), so the relative entropy variance vanishes in the infinite-energy limit for the pure-loss and quantum-limited amplifier channels.

E Upper bound for the hypothesis testing relative entropy

The appendix derives an upper bound for hypothesis testing relative entropy from a proposition involving a random variable with relative-entropy mean and variance. Applying the resulting Chebyshev-like inequality to tensor-power states yields the stated bound with an explicit ε-dependent constant.

  • The derivation relies on prior results for hypothesis testing and takes relative entropy, relative entropy variance, and hypothesis-testing quantities with natural logarithms.
  • Proposition 30 bounds measurement-test behavior using a random variable whose mean is D(ρ∥σ) and variance is V(ρ∥σ).
  • Combining Proposition 30 with the reasoning behind a prior lemma gives a Chebyshev-like bound valid for ε ∈ (0, 1) and n ≥ 1.
  • The bound uses the explicit constant C(ε) ≡ ln 6 + 2 ln([1 + ε] / [1 − ε]).
  • For tensor-power states, the associated random variable has mean nD(ρ∥σ) and variance nV(ρ∥σ), enabling the asymptotic test bound.
  • Choosing θn through a constant C0(ε) and applying the resulting probability inequality yields the desired upper bound.
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