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Finite-resource teleportation stretching for continuous-variable systems

Riccardo Laurenza, Samuel L. Braunstein, Stefano Pirandola

arXiv:1706.06065v3quant-ph

TL;DR

The paper addresses how to simplify adaptive quantum and private communication protocols over bosonic Gaussian channels using finite-energy resources. It combines finite-resource Gaussian channel simulation with teleportation stretching and relative entropy of entanglement to derive weak-converse secret-key bounds. These bounds approximate infinite-energy limits and apply to both point-to-point and repeater-assisted private communication, with a pure-loss simulation requiring a separate construction.

Abstract

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We show how adaptive protocols of quantum and private communication through bosonic Gaussian channels can be simplified into much easier block versions that involve resource states with finite energy. This is achieved by combining the adaptive-to-block reduction technique devised earlier [S. Pirandola et al., Nat. Commun. 8, 15043 (2017)], based on teleportation stretching and relative entropy of entanglement, with an alternative simulation of Gaussian channels recently introduced by Liuzzo-Scorpo et al. [Phys. Rev. Lett. 119, 120503 (2017)]. In this way, we derive weak converse upper bounds for the secret-key capacity of phase-insensitive Gaussian channels, which approximate the optimal limit for infinite energy. Our results apply to both point-to-point and repeater-assisted private communications.

I. INTRODUCTION

The paper motivates finite-energy analyses of bosonic Gaussian communication by combining teleportation stretching with channel simulation, replacing adaptive protocols by simpler block forms. It targets secret-key capacity bounds for phase-insensitive Gaussian channels while connecting them to established rate-loss limits and infinite-energy simulations.

I. INTRODUCTION

  • Bosonic communication protocols are constrained by loss and noise, motivating ultimate-limit benchmarks for quantum and private communication.Source: §I · p. 1
  • Teleportation stretching combines channel simulation through LOCC and a resource state with relative entropy of entanglement to reduce adaptive protocols to block versions.Source: §I · p. 1

A. Preliminaries

  • Bosonic channel simulations are typically asymptotic because their Choi matrices are energy-unbounded, requiring controlled limits over resource-state sequences.Source: §I · p. 1, A. Preliminaries · p. 2

I. INTRODUCTION

  • The paper instead uses finite-energy two-mode Gaussian resource states to simulate bosonic Gaussian channels and simplify adaptive protocols.Source: §I · p. 1
  • The resulting secret-key capacity upper bounds for phase-insensitive Gaussian channels approximate the corresponding infinite-energy results and extend to repeater-assisted communication.Source: §I · p. 2

B. Simulation of Gaussian channels with finite-energy resource states

The finite-resource simulation represents phase-insensitive Gaussian channels using finite-energy two-mode Gaussian states and continuous-variable teleportation. Channel parameters are encoded through transmissivity and added-noise quantities, with the resource state's entanglement parameter linked to them.

B. Simulation of Gaussian channels with finite-energy resource states

III. FINITE-RESOURCE TELEPORTATION STRETCHING OF AN ADAPTIVE PROTOCOL

The paper inserts finite-resource Gaussian channel simulation into teleportation stretching to transform adaptive key-generation protocols into block protocols. This reduction prepares a single-letter relative-entropy bound for secret-key capacity.

III. FINITE-RESOURCE TELEPORTATION STRETCHING OF AN ADAPTIVE PROTOCOL

  • Finite-resource teleportation stretching replaces each channel use in an adaptive protocol with a Gaussian resource state and absorbs the associated teleportation operations into adaptive LOCCs.Source: §III · p. 2

A. Adaptive protocols and secret-key capacity

B. General upper bound

The general upper-bound argument uses relative entropy of entanglement, truncates continuous-variable outputs to discrete-variable private states, and then removes the truncation dependence. This yields a weak-converse route to bounds that remain applicable to continuous-variable channels.

B. General upper bound

  • The weak-converse bound upper-bounds secret-key capacity through the relative entropy of entanglement of the protocol's output state.Source: B. General upper bound · p. 3
  • Relative entropy of entanglement is defined by minimizing relative entropy over separable states.Source: B. General upper bound · p. 3
  • A continuous-variable protocol is truncated after its final LOCC, producing a finite-dimensional output that is close to a discrete-variable private state.Source: B. General upper bound · p. 3
  • The truncation-independent right-hand side allows the truncation to be removed from the final optimization over protocols.Source: B. General upper bound · p. 3
  • This truncation argument extends the discrete-variable relative-entropy bound to continuous-variable channels without retaining dependence on the truncated dimension.Source: B. General upper bound · p. 3

C. Simplification via teleportation stretching

Finite-resource channel simulation lets teleportation stretching replace each adaptive transmission with a Gaussian resource state, collapsing adaptive key generation into a block protocol and yielding a single-letter REE bound.

C. Simplification via teleportation stretching

  • Finite-resource simulation replaces each channel transmission with a two-mode Gaussian resource state while absorbing teleportation operations into one adaptive LOCC.The construction applies to phase-insensitive Gaussian channels and uses a Braunstein-Kimble protocol with gain √η.Source: C. Simplification via teleportation… · p. 4
  • The resulting output state is processed by a single trace-preserving LOCC, removing the explicit optimization over adaptive protocols from the REE bound.Monotonicity of relative entropy under this LOCC and regularized REE bounded by REE produce the simplification.Source: C. Simplification via teleportation… · p. 4
  • Theorem 2 gives a secret-key capacity bound for any phase-insensitive bosonic Gaussian channel stretchable into the specified finite-energy state σν.This is the single-letter bound derived from the finite-resource simulation.Source: C. Simplification via teleportation… · p. 4
  • The finite-resource bound cannot outperform the asymptotic bound of Ref. [11], but its performance is good and reasonably approximates that infinite-energy limit.The comparison establishes the finite-resource construction as an approximation rather than an improvement over the asymptotic result.Source: C. Simplification via teleportation… · p. 4

IV. FINITE-RESOURCE BOUNDS FOR PHASE INSENSITIVE GAUSSIAN CHANNELS

The paper evaluates the finite-resource REE bound for single-mode phase-insensitive Gaussian channels using Gaussian relative-entropy formulas and a constructed separable-state candidate.

IV. FINITE-RESOURCE BOUNDS FOR PHASE INSENSITIVE GAUSSIAN CHANNELS

  • The finite-resource bound is computed using the closed quantum relative entropy formula for Gaussian states derived in Ref. [11].The calculation uses the states' statistical moments and covariance matrices, together with the Gibbs representation.Source: §IV · p. 4
  • The REE optimization is upper-bounded by constructing a separable Gaussian state that preserves the diagonal covariance blocks and modifies the off-diagonal term.This candidate is built directly from the covariance matrix of the resource state.Source: §IV · p. 4
  • The resulting upper bound is evaluated for the different types of phase-insensitive Gaussian channels.Source: §IV · p. 4

A. Thermal-loss channel

For the thermal-loss channel, the finite-resource secret-key bound is obtained from its beam-splitter model and compared with the infinite-energy bound, with the two bounds remaining reasonably close.

A. Thermal-loss channel

  • The thermal-loss channel combines input signals with a thermal environment through a beam splitter of transmissivity η.Its thermal environment has n̄ photons, and the channel covariance description uses ν=(1−η)(n̄+1/2).Source: A. Thermal-loss channel · pp. 4–5
  • The finite-resource bound Ψ(Eη,n̄) is plotted against transmissivity η alongside the infinite-energy bound Φ(Eη,n̄) from Ref. [11].Figure 1 uses n̄=1 thermal photons and identifies Ψ as the red upper curve and Φ as the blue lower curve.Source: A. Thermal-loss channel · p. 5, Figure 1 · p. 5
  • The thermal-loss bound is zero when n̄ is not below η/(1−η).The expression uses h(x)=(x+1)log2(x+1)−x log2 x.Source: A. Thermal-loss channel · p. 5
  • The finite-resource and infinite-energy upper bounds are reasonably close for the thermal-loss channel.Source: A. Thermal-loss channel · p. 5

B. Noisy amplifier channel

For the noisy amplifier channel, the finite-resource secret-key bound is computed and compared with the infinite-energy bound as a function of gain.

B. Noisy amplifier channel

  • The noisy amplifier channel has gain η>1 and couples the input to a thermal environment containing n̄ photons.Its covariance description uses ν=(η−1)(n̄+1/2).Source: B. Noisy amplifier channel · p. 5
  • The finite-resource bound Ψ(Eη,n̄) is plotted against gain η together with the optimal infinite-energy bound Φ(Eη,n̄).Figure 2 uses n̄=1 thermal photons and shows Ψ as the red upper curve and Φ as the blue lower curve.Source: B. Noisy amplifier channel · p. 5, Figure 2 · p. 5
  • The noisy-amplifier bound is zero when n̄ is not below (η−1)^−1.Source: B. Noisy amplifier channel · p. 5

C. Additive-noise Gaussian channel

The finite-resource bound Ψ(Eξ) is evaluated for the additive-noise Gaussian channel and compared with the infinite-energy bound Φ(Eξ).

C. Additive-noise Gaussian channel

D. Pure-loss channel

For pure-loss channels, the infinite-energy upper bound equals the reverse-coherent-information lower bound, fully characterizing capacity, while finite-resource simulation yields a comparable bound. The framework also extends to repeater chains, where equidistant links produce a finite-resource bound that acceptably approximates the infinite-energy result.

D. Pure-loss channel

  • The pure-loss channel’s infinite-energy upper bound coincides with the reverse-coherent-information lower bound, fully characterizing its secret-key capacity and rate-loss scaling.This capacity is known as the Pirandola-Laurenza-Ottaviani-Banchi (PLOB) bound [11].Source: D. Pure-loss channel · p. 6
  • Finite-resource simulation of the pure-loss channel uses a two-mode squeezed resource state because the standard parametrization produces divergences.The resulting bound Ψ(Eη) is compared with K(η) in Fig. 4.Source: D. Pure-loss channel · p. 6, Figure 4 · p. 6

V. EXTENSION TO REPEATER-ASSISTED PRIVATE COMMUNICATION

  • For repeater-assisted communication, teleportation stretching and optimization over chain cuts reduce the adaptive protocol to a single-letter bound.The network reduction uses finite-resource simulations for the individual Gaussian channels.Source: §V · p. 6
  • With N equidistant repeaters, each additive-noise link has variance ξi = ξ/(N + 1), and Ψ({Ei}) = Ψ(Eξ/(N+1)).Equal link variances are described as the optimal repeater configuration in this setting.Source: §V · p. 6
  • The finite-resource repeater bound Ψ({Ei}) provides an acceptable approximation to the corresponding infinite-energy bound Φ({Ei}).The comparison is shown for different repeater numbers as a function of the chain’s overall noise ξ.Source: §V · p. 6, Figure 5 · p. 7

VI. CONCLUSION

The paper develops finite-energy channel simulations for teleportation stretching of single-mode bosonic Gaussian channels, replacing asymptotic energy-unbounded resource states in the core simulation.

VI. CONCLUSION

  • Teleportation stretching is implemented with a finite-energy two-mode Gaussian resource state processed by Braunstein-Kimble teleportation with suitable gains.The construction removes the need for an asymptotic simulation approaching an energy-unbounded Gaussian-channel Choi matrix.Source: §VI · p. 7
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