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Bounding the energy-constrained quantum and private capacities of bosonic thermal channels.

Kunal Sharma, Mark M. Wilde, Sushovit Adhikari, Masahiro Takeoka

Published Aug 24, 2017• 7 citations
OpenAlex

Abstract

We establish three different upper bounds on the energy-constrained quantum and private capacities of bosonic thermal channels. The first upper which we call the bound, is obtained by decomposing a thermal channel as a pure-loss channel followed by a quantum-limited amplifier channel. We prove that the data-processing can be at most 1.45 bits larger than a known lower on these capacities of the thermal channel. The other two upper bounds, which we call the bound and the bound, are established using the notion of approximate degradability along with energy constraints. A comparison of these three bounds shows that the data-processing is near to a known lower for both low and high thermal noise and is very near to the $\varepsilon$-close-degradable only for low thermal noise. Moreover, for certain parameter regimes, we show that the $\varepsilon$-degradable is tighter than all other bounds. All three upper bounds are very near to a known lower for the case of low thermal noise and high transmissivity. We also find improved achievable rates of private communication through bosonic thermal channels, by employing coding schemes that make use of displaced thermal states. We end by proving that an optimal Gaussian input state for the energy-constrained, generalized channel divergence of two particular Gaussian channels is the two-mode squeezed vacuum state that saturates the energy constraint. What remains open for several interesting channel divergences, such as the diamond norm or the R\'enyi channel divergence, is to determine whether, among all input states, a Gaussian state is optimal.

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We establish three different upper bounds on the energy-constrained quantum and private capacities of bosonic thermal channels. The first upper which we call the bound, is obtained by decomposing a thermal channel as a pure-loss channel followed by a quantum-limited amplifier channel. We prove that the data-processing can be at most 1.45 bits larger than a known lower on these capacities of the thermal channel. The other two upper bounds, which we call the bound and the bound, are established using the notion of approximate degradability along with energy constraints. A comparison of these three bounds shows that the data-processing is near to a known lower for both low and high thermal noise and is very near to the ε\varepsilonε-close-degradable only for low thermal noise. Moreover, for certain parameter regimes, we show that the ε\varepsilonε-degradable is tighter than all other bounds. All three upper bounds are very near to a known lower for the case of low thermal noise and high transmissivity. We also find improved achievable rates of private communication through bosonic thermal channels, by employing coding schemes that make use of displaced thermal states. We end by proving that an optimal Gaussian input state for the energy-constrained, generalized channel divergence of two particular Gaussian channels is the two-mode squeezed vacuum state that saturates the energy constraint. What remains open for several interesting channel divergences, such as the diamond norm or the R'enyi channel divergence, is to determine whether, among all input states, a Gaussian state is optimal.
We establish several upper bounds on the energy-constrained quantum and private capacities of all single-mode phase-insensitive bosonic Gaussian channels. The first upper bound, which we call the 'data-processing bound,' is the simplest and is obtained by decomposing a phase-insensitive channel as a pure-loss channel followed by a quantum-limited amplifier channel. We prove that the data-processing bound can be at most 1.45 bits larger than a known lower bound on these capacities of the phase-insensitive Gaussian channel. We discuss another data-processing upper bound as well. Two other upper bounds, which we call the 'ϵ-degradable bound' and the 'ϵ-close-degradable bound,' are established using the notion of approximate degradability along with energy constraints. We find a strong limitation on any potential superadditivity of the coherent information of any phase-insensitive Gaussian channel in the low-noise regime, as the data-processing bound is very near to a known lower bound in such cases. We also find improved achievable rates of private communication through bosonic thermal channels, by employing coding schemes that make use of displaced thermal states. We end by proving that an optimal Gaussian input state for the energy-constrained, generalized channel divergence of two particular Gaussian channels is the two-mode squeezed vacuum state that saturates the energy constraint. What remains open for several interesting channel divergences, such as the diamond norm or the Rényi channel divergence, is to determine whether, among all input states, a Gaussian state is optimal.
We establish several upper bounds on the energy-constrained quantum and private capacities of all single-mode phase-insensitive bosonic Gaussian channels. The first upper bound, which we call the ‘data-processing bound,’ is the simplest and is obtained by decomposing a phase-insensitive channel as a pure-loss channel followed by a quantum-limited amplifier channel. We prove that the data-processing bound can be at most 1.45 bits larger than a known lower bound on these capacities of the phase-insensitive Gaussian channel. We discuss another data-processing upper bound as well. Two other upper bounds, which we call the ‘ ε -degradable bound’ and the ‘ ε -close-degradable bound,’ are established using the notion of approximate degradability along with energy constraints. We find a strong limitation on any potential superadditivity of the coherent information of any phase-insensitive Gaussian channel in the low-noise regime, as the data-processing bound is very near to a known lower bound in such cases. We also find improved achievable rates of private communication through bosonic thermal channels, by employing coding schemes that make use of displaced thermal states. We end by proving that an optimal Gaussian input state for the energy-constrained, generalized channel divergence of two particular Gaussian channels is the two-mode squeezed vacuum state that saturates the energy constraint. What remains open for several interesting channel divergences, such as the diamond norm or the Rényi channel divergence, is to determine whether, among all input states, a Gaussian state is optimal.
We determine the two-way quantum, entanglement-distribution, private, and secret-key capacities of the pure-loss bosonic channel under an unconditional mean transmitted-photon-number constraint. For transmissivity ηηη and mean photon number NNN, all four capacities coincide and are given by g(N)−g((1−η)N)g(N)-g((1-η)N)g(N)−g((1−η)N), where ggg is the entropy of a thermal mode. This resolves the longstanding energy-constrained capacity problem by establishing the optimality of the reverse-coherent-information rate introduced in 2009. Our central tool is sector teleportation simulation, which we develop specifically to transfer the input-energy constraint to the shared entanglement resource, thereby preserving the constraint throughout converse arguments for general adaptive protocols. For every N>0N>0N>0, we further show that the corresponding strong-converse thresholds coincide with the unconstrained value −log⁡2(1−η)-\log_2(1-η)−log2​(1−η). For parallel protocols with a hard total-photon-number cutoff, the same energy-constrained capacity formula instead satisfies an exponential strong converse. As an extension, we determine the energy-constrained two-way capacities of the quantum-limited amplifier, showing that classical assistance does not improve its unassisted quantum and private rates. Together, these results establish a unified finite-energy framework for adaptive bosonic communication and reveal how energy constraints fundamentally reshape both attainable rates and converse bounds.
There have been several upper bounds on the quantum capacity of the single-mode Gaussian channels with thermal noise, such as thermal attenuator and amplifier. We consider a class of attenuator and amplifier with more general noises, including squeezing or even non-Gaussian noises. We derive upper bounds the energy-constrained quantum capacity of those channels by using the quantum conditional entropy power inequality. Also, we obtain lower bounds for the same channels by means of Gaussian optimizer with fixed input entropy. They give narrow bounds when the transmissivity is near unity and the energy of input state is low.
There have been several upper bounds on the quantum capacity of the single-mode Gaussian channels with thermal noise, such as thermal attenuator and amplifier. We consider a class of attenuator and amplifier with more general noises, including squeezing or even non-Gaussian cases. We introduce new upper bounds on the energy-constrained quantum capacity of those channels by using the conditional quantum entropy power inequality. Also, we obtain lower bounds for the same channels by means of Gaussian optimizer with fixed input entropy. They give narrow bounds when the transmissivity is near unity and the energy of input state is low.