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IEEE International Symposium on Information Theory (ISIT)

Strong converse for the capacity of quantum Gaussian channels

Bhaskar Roy Bardhan, Raúl García−Patrón, Mark M. Wilde, Andreas Winter

Published Jun 1, 2014• 2 citations
Publisher

Abstract

We prove that a strong converse theorem holds for the classical capacity of all phase-insensitive bosonic Gaussian channels, when imposing a maximum photon number constraint on the inputs of the channel. This class is a natural extension of classical continuous Gaussian channels, and the well studied pure-loss, thermal, additive noise, and amplifier channels are all in this class of channels. The statement of the strong converse theorem is that the probability of correctly decoding a classical message rapidly converges to zero in the limit of many channel uses if the communication rate exceeds the classical capacity. We prove this theorem by relating the success probability of any code with its rate of data transmission, the effective dimension of the channel output space, and the purity of the channel as quantified by the minimum output entropy. Our result bolsters the understanding of the classical capacity of these channels by establishing it as a sharp dividing line between possible and impossible communication rates over them.

DOI
10.1109/isit.2014.6874928

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We prove that a strong converse theorem holds for the classical capacity of all phase-insensitive bosonic Gaussian channels, when imposing a maximum photon number constraint on the inputs of the channel. This class is a natural extension of classical continuous Gaussian channels, and the well studied pure-loss, thermal, additive noise, and amplifier channels are all in this class of channels. The statement of the strong converse theorem is that the probability of correctly decoding a classical message rapidly converges to zero in the limit of many channel uses if the communication rate exceeds the classical capacity. We prove this theorem by relating the success probability of any code with its rate of data transmission, the effective dimension of the channel output space, and the purity of the channel as quantified by the minimum output entropy. Our result bolsters the understanding of the classical capacity of these channels by establishing it as a sharp dividing line between possible and impossible communication rates over them.
We establish the classical capacity of optical quantum channels as a sharp transition between two regimes - one which is an error-free regime for communication rates below the capacity, and the other in which the probability of correctly decoding a classical message converges exponentially fast to zero if the communication rate exceeds the classical capacity. This result is obtained by proving a strong converse theorem for the classical capacity of all phase-insensitive bosonic Gaussian channels, a well-established model of optical quantum communication channels, such as lossy optical fibers, amplifier, and free-space communication. The theorem holds under a particular photon-number occupation constraint, which we describe in detail in this paper. Our result bolsters the understanding of the classical capacity of these channels and opens the path to applications, such as proving the security of noisy quantum storage models of cryptography with optical links.
We prove that several known upper bounds on the classical capacity of thermal and additive noise bosonic channels are actually strong converse rates. Our results strengthen the interpretation of these upper bounds, in the sense that we now know that the probability of correctly decoding a classical message rapidly converges to 0 in the limit of many channel uses if the communication rate exceeds these upper bounds. In order for these theorems to hold, we need to impose a maximum photon number constraint on the states input to the channel (the strong converse property need not hold if there is only a mean photon number constraint). Our first theorem demonstrates that Koenig and Smith's upper bound on the classical capacity of the thermal bosonic channel is a strong converse rate, and we prove this result by utilizing the structural decomposition of a thermal channel into a pure-loss channel followed by an amplifier channel. Our second theorem demonstrates that Giovannetti et al.'s upper bound on the classical capacity of a thermal bosonic channel corresponds to a strong converse rate, and we prove this result by relating the success probability to the rate, the effective dimension of the output space, and the purity of the channel as measured by the R\'enyi collision entropy. Finally, we use similar techniques to prove that similar previously known upper bounds on the classical capacity of an additive noise bosonic channel correspond to strong converse rates.
The classical capacity of a noisy optical channel is the maximum rate at which information can be transmitted reliably, given a constraint on the mean input energy. For the ubiquitous bosonic Gaussian channels, which include standard models of loss and noise in fibre and free-space optical links, determining this rate requires addressing whether entanglement between channel uses or non-Gaussian signals can outperform Gaussian encodings. More precisely, the question is whether the energy-constrained Holevo information is additive across arbitrarily many channel uses and coincides with its Gaussian counterpart. We answer these fundamental questions in the affirmative for every single-mode bosonic Gaussian channel at every input energy. Namely, we prove that independently prepared Gaussian states with Gaussian-distributed random displacements achieve the capacity, including in the low-energy regime where squeezing the signals competes with encoding information in their displacement. Our proof connects an entanglement inequality to a sharp, additive lower bound on output entropy. We then characterise communication above capacity: under a strict total photon-number cutoff, the probability of correct decoding decreases exponentially with the number of channel uses, and we derive explicit bounds on the decay exponent. When only the mean photon number is constrained, a simple energy-rescaling formula determines exactly the entire asymptotic rate--error trade-off: the capacity at mean photon number NNN and allowed average decoding error εεε equals the vanishing-error capacity at mean photon number N/(1−ε)N/(1-ε)N/(1−ε). Together, these results identify the ultimate limits of classical communication for some of the most fundamental models of optical communication.
We establish the classical capacity of optical quantum channels as a sharp transition between two regimes-one which is an error-free regime for communication rates below the capacity, and the other in which the probability of correctly decoding a classical message converges exponentially fast to zero if the communication rate exceeds the classical capacity. This result is obtained by proving a strong converse theorem for the classical capacity of all phase-insensitive bosonic Gaussian channels, a well-established model of optical quantum communication channels, such as lossy optical fibers, amplifier, and free-space communication. The theorem holds under a particular photon-number occupation constraint, which we describe in detail in this paper. Our result bolsters the understanding of the classical capacity of these channels and opens the path to applications, such as proving the security of noisy quantum storage models of cryptography with optical links.
This paper strengthens the interpretation and understanding of the classical capacity of the pure-loss bosonic channel, first established in [1]. In particular, we first prove that there exists a trade-off between communication rate and error probability if one imposes only a mean photon number constraint on the channel inputs. That is, if we demand that the mean number of photons at the channel input cannot be any larger than some positive number NS , then it is possible to respect this constraint with a code that operates at a rate g ( ηNS /(1- p )) where p is the code error probability, η is the channel transmissivity, and g ( x ) is the entropy of a bosonic thermal state with mean photon number x . Then we prove that a strong converse theorem holds for the classical capacity of this channel (that such a rate-error trade-off cannot occur) if one instead demands for a maximum photon number constraint, in such a way that mostly all of the “shadow” of the average density operator for a given code is required to be on a subspace with photon number no larger than nNS , so that the shadow outside this subspace vanishes as the number n of channel uses becomes large. Finally, we prove that a small modification of the well-known coherent-state coding scheme meets this more demanding constraint.
This paper reports the proof of a strong converse for the unconstrained quantum capacity of the pure-loss bosonic channel. At every fixed rate above capacity, the entanglement-generation fidelity of every code is bounded by a constant times the reciprocal of the number of channel uses. The bound holds without an energy constraint and for arbitrary encoded states, including states correlated across all input modes, and arbitrary joint decoders. The proof combines quantum Chebyshev and hockey-stick testing inequalities with a uniform relative-entropy-variance bound for the balanced pure-loss channel, corresponding to transmissivity η=1/2η=1/2η=1/2. The variance bound follows by expressing the balanced beam splitter in bright and dark modes: the dark modes are exactly in vacuum, and any state orthogonal to that vacuum contains at least one dark photon. For general transmissivity, dilating the degrading attenuator reduces the problem to this balanced-channel setting and bounds the decoder test by precisely the factor that produces the known quantum-capacity threshold. The resulting argument establishes the strong converse at the unconstrained quantum capacity for every pure-loss bosonic channel.