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Theory of channel simulation and bounds for private communication

Stefano Pirandola, Samuel L. Braunstein, Riccardo Laurenza, Carlo Ottaviani, Thomas P. W. Cope, Gaetana Spedalieri, Leonardo Banchi

arXiv:1711.09909v3quant-phcond-mat.othermath-phphysics.optics

TL;DR

Quantum and private communication require converse bounds that remain valid for adaptive protocols and continuous-variable channels. This paper reviews channel simulation and teleportation stretching, rigorously controls Braunstein–Kimble asymptotics, and derives capacity bounds and related QKD conclusions. It also identifies maximum tolerable QKD excess noise as an open characterization problem.

  • Problem

    Fundamental limits for adaptive quantum and private communication, especially in continuous-variable settings, require rigorous converse analyses and a characterization of tolerable QKD excess noise.

  • Method

    The paper combines relative entropy of entanglement, channel simulation, teleportation stretching, and carefully controlled Braunstein–Kimble simulation for bosonic Gaussian channels.

  • Results

    The framework establishes two-way capacities for fundamental channels including the bosonic lossy channel and rigorously proves the relevant strong-converse claims for Gaussian private communication.

  • Takeaways & Limitations

    Teleportation stretching preserves the original task while converting adaptive protocols into block forms suitable for single-letter converse bounds.

  • Takeaways & Limitations

    The maximum excess noise tolerable by QKD remains an open problem.

Abstract

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We review recent results on the simulation of quantum channels, the reduction of adaptive protocols (teleportation stretching), and the derivation of converse bounds for quantum and private communication, as established in PLOB [Pirandola, Laurenza, Ottaviani, Banchi, arXiv:1510.08863]. We start by introducing a general weak converse bound for private communication based on the relative entropy of entanglement. We discuss how combining this bound with channel simulation and teleportation stretching, PLOB established the two-way quantum and private capacities of several fundamental channels, including the bosonic lossy channel. We then provide a rigorous proof of the strong converse property of these bounds by adopting a correct use of the Braunstein-Kimble teleportation protocol for the simulation of bosonic Gaussian channels. This analysis provides a full justification of claims presented in the follow-up paper WTB [Wilde, Tomamichel, Berta, arXiv:1602.08898] whose upper bounds for Gaussian channels would be otherwise infinitely large. Besides clarifying contributions in the area of channel simulation and protocol reduction, we also present some generalizations of the tools to other entanglement measures and novel results on the maximum excess noise which is tolerable in quantum key distribution.

I. INTRODUCTION

The paper reviews how channel simulation, teleportation stretching, and relative-entropy methods turn difficult adaptive communication problems into tractable converse bounds. It also addresses strong-converse technical issues for bosonic Gaussian channels and generalizes the framework.

  • I. INTRODUCTION: The paper introduces general channel simulation, replacing a channel by an LOCC acting on the input and a resource state, possibly through asymptotic sequences.This generalizes earlier teleportation-based reductions beyond particular discrete-variable channel classes.
  • I. INTRODUCTION: Teleportation stretching reduces arbitrary adaptive protocols to block protocols while preserving the original communication task.The reduction applies across channel dimensions and adaptive tasks, including key generation, metrology, discrimination, multipartite protocols, and networks.
  • I. INTRODUCTION: Combining stretching with channel relative entropy of entanglement yields single-letter upper bounds and establishes two-way capacities for several fundamental channels, including the bosonic lossy channel.The lossy-channel result sets the ultimate optical-communication limit without repeaters.
  • I. INTRODUCTION: The manuscript studies which entanglement-measure properties support single-letter bounds and asks for the maximum excess noise tolerable by QKD, which remains open.It also reviews adaptive two-way protocols whose channel uses are interleaved with LOCC operations and whose outputs approximate task-specific target states.
  • I. INTRODUCTION: The strong-converse analysis corrects the Braunstein–Kimble treatment for bosonic Gaussian channels, whose bounds otherwise become unbounded because simulation errors are not properly controlled.The simulation error must be propagated through the adaptive protocol and bounded using the correct asymptotic teleportation construction.

B. Extension to asymptotic states

This section extends relative-entropy and converse-bound reasoning to asymptotic continuous-variable states and channels. Truncation and LOCC monotonicity remove dimension dependence while preserving the resulting capacity bound.

  • B. Extension to asymptotic states: Asymptotic states are limits of finite-energy states, including the ideal EPR state as the limit of two-mode squeezed-vacuum states.The parameter µ quantifies both squeezing and local energy, making this formulation natural for continuous-variable systems.
  • B. Extension to asymptotic states: Lower semicontinuity of relative entropy extends the relative entropy of entanglement to asymptotic states and supports its use for bosonic-channel Choi matrices.The extension uses convergent sequences of bona-fide and separable states.
  • B. Extension to asymptotic states: Theorem 2 gives a weak-converse upper bound for the generic two-way capacity of any quantum channel, including finite- and infinite-dimensional channels.The proof begins with finite-dimensional systems and extends to continuous variables through truncation.
  • B. Extension to asymptotic states: For continuous-variable protocols, output truncation followed by trace-preserving LOCC and relative-entropy monotonicity produces a bound independent of the truncation dimension.This allows the truncation to be removed when optimizing over protocols.
  • B. Extension to asymptotic states: The shield-size lemma permits approximate protocols whose shield dimension grows at most exponentially with the number of channel uses.The same control is obtained for continuous variables after suitable Hilbert-space truncation.

D. Rebuttal of some unfounded claims

The section clarifies the channel-simulation and Braunstein–Kimble constructions used for bosonic channels. Teleportation covariance enables Choi-based simulation, while finite-energy convergence supplies the required asymptotic control.

  • D. Rebuttal of some unfounded claims: A teleportation-covariant channel is Choi-stretchable: it can be simulated by teleporting the input over its Choi matrix.This criterion covers Pauli and erasure channels in discrete variables and bosonic Gaussian channels in continuous variables.
  • D. Rebuttal of some unfounded claims: The Braunstein–Kimble protocol uses a two-mode squeezed-vacuum resource, Bell detection, and conditional phase-space displacements to teleport an input state.Its induced channel is an additive-noise Gaussian channel whose finite-energy behavior converges pointwise on energy-bounded inputs.
  • D. Rebuttal of some unfounded claims: For teleportation-covariant bosonic channels, modified displacement corrections commute the channel through the teleportation protocol and produce an energy-dependent simulation using a quasi-Choi resource state.The resulting simulated channel is expressed as the composition Eµ = E ◦ Iµ and converges pointwise in the simulation limit.
  • D. Rebuttal of some unfounded claims: Ancillary systems can be included by teleporting only the channel input subsystem while leaving the ancillary subsystem under the identity channel.The same asymptotic simulation limit then applies to bipartite inputs.

C. Considerations for bosonic Gaussian channels

The section analyzes how Braunstein–Kimble teleportation simulates bosonic Gaussian channels, emphasizing convergence failures for unbounded inputs and the need for carefully ordered or energy-constrained limits.

  • Gaussian-channel structure: The analysis characterizes phase-insensitive Gaussian channels through transmission and noise parameters, including thermal-loss, amplifier, and additive-noise channels.Pure loss is the zero-thermal-noise case, while quantum-limited amplification is the zero-thermal-noise amplifier case.
  • Convergence issues: Energy-unbounded input alphabets make the joint limit over input energy and simulation energy mathematically ambiguous.The issue arises because finite-energy BK simulations can have vanishing fidelity for some channels and increasingly squeezed inputs.
  • Resolution: The convergence problem can be resolved either by specifying the order of limits or by imposing an energy constraint on the input alphabet.The two approaches respectively use strong convergence or energy-constrained uniform convergence of BK teleportation.
  • Convergence issues: For any finite simulation energy, BK teleportation does not converge uniformly to the identity channel.This non-uniformity is reflected in the failure of uniform convergence in ordinary diamond distance.
  • Gaussian-channel convergence: Gaussian channels with full-rank noise matrices admit uniform teleportation-simulation convergence, whereas rank-deficient channels require bounded-uniform convergence.The bounded-uniform formulation uses an energy-constrained diamond distance.

E. Energy-constrained diamond distance

The section introduces energy-constrained diamond distance to control asymptotic teleportation-simulation errors and rigorously propagate them through adaptive protocols before taking the simulation limit.

  • Definition: For a bounded input alphabet, the energy-constrained diamond distance defines the simulation error between a bosonic channel and its finite-energy teleportation simulation.The energy constraint applies to the entire input space, including ancillas.
  • Convergence: Compactness of the energy-constrained state set upgrades point-wise convergence to uniform convergence for finite input energy.This allows asymptotic simulation to be treated as either point-wise or uniform convergence under an energy-constrained alphabet.
  • Teleportation stretching: An adaptive protocol with asymptotic channel simulation is reduced to a block protocol whose output is a single LOCC applied to n resource states.The same stretching procedure replaces each channel use and collapses the adaptive LOCCs into one trace-preserving LOCC.
  • Error control: Simulation errors must be propagated through every transmission to bound the trace distance between actual and simulated protocol outputs.The proof uses trace-distance properties, including monotonicity and the triangle inequality, together with the energy-constrained diamond distance.
  • Asymptotic limit: For finite energy and finitely many channel uses, taking the simulation energy to infinity yields the asymptotic stretching relation.This establishes the limiting block form needed for subsequent converse bounds.

VI. SINGLE-LETTER UPPER BOUNDS

The section combines channel relative entropy of entanglement with teleportation stretching to derive single-letter upper bounds and capacity formulas for several quantum channels.

  • General method: Combining channel REE with teleportation stretching yields simple single-letter upper bounds for all two-way capacities of a quantum channel.The method applies the weak converse bound after reducing adaptive outputs to LOCCs acting on tensor-product resource states.
  • Teleportation-covariant channels: For teleportation-covariant channels, the entanglement flux bounds all two-way assisted capacities and is computed from the channel’s Choi state.These channels are Choi-stretchable, making the bound especially direct.
  • Dephasing channel: For dephasing channels, the entanglement-flux upper bound coincides with a one-way distillability lower bound, so the two-way capacity is completely determined.The same result also establishes Q2(Pdeph) = Q(Pdeph).
  • Erasure channel: The erasure channel has secret-key capacity K(Eerase) = P2(Eerase) = 1 −p, extending the previously known two-way quantum-capacity result.The channel is both teleportation covariant and distillable, enabling the capacity calculation.
  • Asymptotic simulations: For asymptotically simulated channels, the energy constraint is released after the finite-energy analysis because the resulting upper bound is independent of the input-energy bound.Teleportation-covariant channels can use a Choi-state sequence as the simulation resource.

D. Formulas for Gaussian channels

The section develops Gaussian-channel formulas and uses teleportation simulation with relative-entropy methods to characterize capacities and bounds. It also treats amplitude damping through a dimension-independent LOCC simulation.

  • D. Formulas for Gaussian channels: A single-mode Gaussian channel is teleportation covariant and admits simulation through a Choi-state sequence.The section gives formulas for thermal-loss, amplifier, and additive-noise Gaussian channels.
  • D. Formulas for Gaussian channels: The Gaussian-state relative entropy is expressed directly through means and covariance matrices, avoiding symplectic diagonalization.This enables matrix-function implementations in numerical and symbolic software.
  • D. Formulas for Gaussian channels: PLOB established tightest-known upper bounds for two-way quantum and private capacities of all single-mode phase-insensitive Gaussian channels.The calculation uses a Gaussian relative-entropy formula within the REE framework.
  • D. Formulas for Gaussian channels: PLOB showed that the lossy channel’s upper bound equals its reverse-coherent-information lower bound, making all two-way capacities equal.For the lossy channel, D2 = Q2 = K = P2.
  • D. Formulas for Gaussian channels: At high loss, the lossy channel’s secret-key capacity scales as K ≃ 1.44η secret bits per channel use, establishing the repeaterless benchmark.A quantum repeater must surpass this point-to-point rate-loss benchmark to be effective.
  • D. Formulas for Gaussian channels: Quantum-limited amplifiers are distillable, so their two-way capacities coincide with one another and with the unassisted quantum capacity.The equality Q2(Eg) = Q(Eg) is stated explicitly.
  • E. Amplitude damping channel: For amplitude damping, a non-teleportation-covariant qubit channel is nevertheless LOCC-simulable by mapping through a single-rail bosonic lossy channel.The simulation combines DV-to-CV and CV-to-DV maps with the Braunstein-Kimble protocol, using η(p) = 1 − p.
  • E. Amplitude damping channel: The REE upper bound for amplitude damping performs well only at high damping, p > 0.9, and can be beaten by squashed entanglement.This example motivates a dimension-independent theory of channel simulation even for discrete-variable channels.

VII. MAXIMUM TOLERABLE NOISE IN QUANTUM KEY DISTRIBUTION

The section asks how much excess noise QKD can tolerate and compares upper and lower security thresholds. Trusted receiver noise improves the known lower bound, but a substantial gap remains.

  • VII. MAXIMUM TOLERABLE NOISE IN QUANTUM KEY DISTRIBUTION: The maximum tolerable excess noise in QKD is posed as an optimization over all protocols at fixed channel transmissivity.Security thresholds are obtained by setting the optimal key rate R(η, ε) to zero.
  • VII. MAXIMUM TOLERABLE NOISE IN QUANTUM KEY DISTRIBUTION: A squeezed-state protocol with trusted receiver noise outperforms the reverse-coherent-information lower bound when channel excess noise is nonzero.The protocol uses Gaussian-modulated squeezed states and reverse reconciliation.
  • VII. MAXIMUM TOLERABLE NOISE IN QUANTUM KEY DISTRIBUTION: Figure 4 compares security thresholds versus channel loss for the REE upper bound, reverse-coherent-information lower bound, trusted-noise protocol, and other one- and two-way protocols.Protocols are secure below their corresponding curves; the trusted-noise curve improves the lower bound but does not close the gap.
  • VII. MAXIMUM TOLERABLE NOISE IN QUANTUM KEY DISTRIBUTION: Trusted noise yields the best-known lower bound ε∞, yet it remains far below the universal upper bound εUB = 1.Determining the maximum tolerable excess noise therefore remains open.
  • VII. MAXIMUM TOLERABLE NOISE IN QUANTUM KEY DISTRIBUTION: A coherent implementation using quantum memories removes basis reconciliation and doubles the key rate of the trusted-noise protocol.It is equivalent to creating trusted noise with a beam splitter mixing the output with a thermal mode.
  • VII. MAXIMUM TOLERABLE NOISE IN QUANTUM KEY DISTRIBUTION: With zero trusted noise, the protocol rate is half the reference rate, while coherent implementation restores equality with that rate.This comparison is stated for ξ = 0.
  • VII. MAXIMUM TOLERABLE NOISE IN QUANTUM KEY DISTRIBUTION: The comparison also includes one-way coherent-state heterodyne QKD and two-way thermal-state homodyne protocols in coherent and largely-thermal limits.The thermal-state cases use V0 = 1/2 and V0 ≫ 1.

A. Main ingredients

The main ingredients are an entanglement-measure weak converse and teleportation stretching, which reduce adaptive communication protocols to resource-state bounds. The framework extends across capacities, simulations, and entanglement measures.

  • A. Main ingredients: The weak converse begins with an entanglement measure satisfying exponential-size control for the target state, including maximally entangled and private states.The target-state dimension may grow at most exponentially with the number of channel uses.
  • A. Main ingredients: The framework applies to REE and squashed entanglement because both satisfy the required monotonicity, subadditivity, and continuity properties.The REE uses the binary entropy in its continuity correction.
  • A. Main ingredients: Taking the large-block and small-error limits yields a weak converse, which is then optimized over the admissible protocol class.The resulting capacity may describe a two-way channel capacity or an end-to-end repeater or network capacity.
  • A. Main ingredients: Teleportation stretching reorganizes an adaptive protocol into a block form whose output is a trace-preserving LOCC applied to tensor powers of a resource state.For teleportation-covariant channels, the resource state is the channel’s Choi state.
  • A. Main ingredients: Combining stretching with monotonicity and subadditivity converts a regularized converse into a single-letter bound based on the resource state.This is the central PLOB simplification for two-way assisted capacities.
  • A. Main ingredients: For an approximating simulable channel, protocol outputs are compared through diamond-distance error propagation and stretching of the simulated channel.The resulting upper bound is uniform over protocols and therefore bounds the n-use two-way secret-key capacity.
  • A. Main ingredients: The approximation argument requires nδ to remain small; this is automatic for one-shot capacity but can fail at large n when the error accumulates.The one-shot bound is stated explicitly, while the large-block problem is identified separately.

C. Sequences of channels

Sequences of simulable channels resolve accumulated simulation errors by taking limits after controlling finite-energy or finite-dimensional approximations. The method extends the weak converse to continuous-variable channels and asymptotic resource states.

  • C. Sequences of channels: A diamond-norm-convergent sequence of simulable channels removes the large-block error problem and yields a weak converse after taking the simulation limit.The sequence approach is particularly useful for continuous-variable systems.
  • C. Sequences of channels: For continuous-variable channels, truncation is applied to output states and energy-constrained diamond distance controls the approximation at finite truncation.The truncation is released only after the relevant limits are taken.
  • C. Sequences of channels: The finite-dimensional truncated protocol and its simulated counterpart are compared with triangle inequalities before the simulation limit is taken.This preserves control of the target-state approximation for each truncation dimension.
  • C. Sequences of channels: The limiting resource-state bound uses an inferior limit because the simulating states may form an unbounded sequence.The construction therefore accommodates asymptotic states rather than only finite-energy resource states.
  • C. Sequences of channels: After optimization, the truncation and energy constraint can be relaxed because the final upper bound no longer depends on the constraint N.Specifying the entanglement measure as REE recovers an alternate proof of the earlier REE result.

SIMULATION AND PROTOCOL REDUCTION

PLOB generalized channel simulation and teleportation stretching to arbitrary dimensions and adaptive tasks, combining these tools with REE to derive tight two-way capacity upper bounds. The paper also revisits earlier reductions and corrects WTB’s strong-converse treatment for bosonic Gaussian channels.

  • Extensions: PLOB’s framework extends beyond earlier reductions to applications including quantum metrology, channel discrimination, multipartite protocols, and quantum networks.The review contrasts these advances with previous literature in Table I.
  • Prior approaches: Earlier work simulated Pauli channels through teleportation and their Choi matrices, enabling reductions from quantum communication to entanglement distillation.Subsequent approaches considered probabilistic or deterministic teleportation-covariant simulations, but finite-dimensional non-asymptotic methods covered narrower channel classes.
  • PLOB advances: PLOB simulates arbitrary quantum channels using an LOCC and resource state, including asymptotic constructions in finite or infinite dimensions.This general framework covers continuous-variable channels and deterministic asymptotic simulation of channels such as amplitude damping.
  • PLOB advances: Teleportation stretching converts arbitrary adaptive protocols into block protocols while preserving the original communication task.The output is expressed through tensor products of resource states followed by a single LOCC, rather than being reduced to entanglement distillation.
  • Capacity bounds: Combining REE with teleportation stretching yields tight known upper bounds for secret-key and other two-way assisted capacities at any dimension.PLOB introduced the adaptive-to-block reduction for private communication and combined it with entanglement-measure properties.
  • Strong converse: WTB’s Gaussian-channel strong-converse derivation is technically invalid because its Braunstein–Kimble simulation treatment makes the resulting bounds unbounded.The issue concerns an imprecise interpretation of continuous-variable teleportation, whose simulation error requires rigorous control.

C. Technical gap and exploding bound

The paper identifies that WTB’s strong-converse proof does not control simulation errors through adaptive protocols or unbounded input alphabets. Consequently, the derived Gaussian-channel bound can explode to infinity rather than establish the claimed finite rate.

  • Error propagation: WTB does not prove how single-use Gaussian-channel simulation error propagates to the n-use output state of an adaptive protocol.The missing propagation step concerns the dependence of output infidelity on the number of channel uses.
  • Error propagation: A peeling argument using triangle inequality and relative-entropy data processing is required to quantify this propagation through adaptive protocols.Its absence means WTB’s proof does not actually apply to adaptive protocols.
  • Unbounded inputs: The problem already appears for one channel use when input energy competes with the Braunstein–Kimble resource energy.Such asymptotic inputs must be included for unconstrained quantum and private capacities because the input alphabet is energy-unbounded.
  • Unbounded inputs: WTB’s limit expression leaves the order of simulation-energy and input-energy limits unspecified, so it cannot justify convergence for an unbounded alphabet.The paper distinguishes the resulting ambiguity from the uniform-convergence assumption implicitly used in WTB.
  • Exploding bound: The resulting joint limit is undefined and can yield Δ(n, µ) →∞, producing only a trivial upper bound.Extending the single-use reasoning to arbitrary n through the peeling argument gives the exploding bound.

D. Fixing the mathematical issues

The paper fixes bosonic Gaussian-channel proofs by imposing energy constraints, controlling Braunstein–Kimble simulation error, and then removing the constraint to establish strong converse bounds.

  • D. Fixing the mathematical issues: The corrected proof propagates simulation error through arbitrary protocols using trace-distance and fidelity bounds before deriving an energy-constrained converse.This repairs the ambiguity caused by asymptotic input states with unbounded energy.
  • D. Fixing the mathematical issues: Energy-constrained diamond distance makes Gaussian-channel simulation asymptotically perfect for every fixed input-energy bound.The simulation error is defined between the channel and its finite-energy Braunstein–Kimble simulation and vanishes in the simulation limit.
  • D. Fixing the mathematical issues: The resulting upper bound is independent of the energy constraint, allowing the constraint to be relaxed and proving the strong converse for noisy Gaussian channels.The same reasoning applies to the pure-loss channel and quantum-limited amplifier.
  • D. Fixing the mathematical issues: Rigorous continuous-variable simulation must also approximate Bell detection and the associated LOCC as finite-energy sequences.The omitted finite-squeezing measurement issue affects teleportation stretching and related Gaussian-channel derivations.
  • D. Fixing the mathematical issues: The broader framework combines relative-entropy weak converses, general LOCC channel simulation, and teleportation stretching to obtain capacity bounds and exact results for fundamental channels.For some channels, these upper bounds coincide with lower bounds, including the bosonic lossy channel.
  • D. Fixing the mathematical issues: Two-way capacities remain open for depolarizing and amplitude-damping channels and for noisy single-mode phase-insensitive Gaussian channels such as thermal loss.The thermal-loss case is highlighted for its importance in quantum key distribution.

Appendix A: Fidelity limits in the BK teleportation protocol

Appendix A analyzes fidelity limits for Braunstein–Kimble teleportation applied to Gaussian inputs and relates the finite-energy protocol to additive-noise simulation.

  • Appendix A: Fidelity limits in the BK teleportation protocol: The Braunstein–Kimble protocol applies an ideal continuous-variable Bell measurement and classical communication to realize a finite-energy teleportation channel.The protocol acts on an input mode and a resource state supplied by a two-mode squeezed vacuum.
  • Appendix A: Fidelity limits in the BK teleportation protocol: The resulting channel is locally equivalent to an additive-noise Gaussian channel whose noise depends on the resource energy.The passage identifies the added noise as ξ = 2µ − …, with the expression truncated in the supplied text.
  • Appendix A: Fidelity limits in the BK teleportation protocol: Gaussian-state fidelity formulas and asymptotic expansions determine the limiting behavior of the teleportation simulation.These limits imply opposite behaviors for the relevant limits when the simulated channel is the identity.
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