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arXiv:1705.08878 · IEEE Transactions on Information Theory

Quantum Channel Capacities per Unit Cost

Dawei Ding, Dmitri S. Pavlichin, Mark M. Wilde

Published Jul 10, 2018• 13 citations
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Abstract

Communication over a noisy channel is often conducted in a setting in which different input symbols to the channel incur a certain cost. For example, for bosonic quantum channels, the cost associated with an input state is the number of photons, which is proportional to the energy consumed. In such a setting, it is often useful to know the maximum amount of information that can be reliably transmitted per cost incurred. This is known as the capacity per unit cost. In this paper, we generalize the capacity per unit cost to various communication tasks involving a quantum channel, such as classical communication, entanglement-assisted classical communication, private communication, and quantum communication. For each task, we define the corresponding capacity per unit cost and derive a formula for it analogous to that of the usual capacity. Furthermore, for the special and natural cases in which there is a zero-cost state, we obtain expressions in terms of an optimized relative entropy involving the zero-cost state. For each communication task, we construct an explicit pulse-position-modulation coding scheme that achieves the capacity per unit cost. Finally, we compute capacities per unit cost for various bosonic Gaussian channels and introduce the notion of a blocklength constraint as a proposed solution to the long-standing issue of infinite capacities per unit cost. This motivates the idea of a blocklength-cost duality on which we elaborate in depth.

DOI
10.1109/tit.2018.2854747

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How much classical information can a noisy optical channel carry when both signal energy and preshared entanglement are limited? This work gives an exact capacity frontier for this joint resource problem. It allows one to determine the best asymptotic communication rate, price the benefit of additional entanglement, and distinguish the effect of tolerating errors from that of changing how entanglement is supplied. Exact capacity under two resource constraints The setting is any fixed bosonic Gaussian channel \(\Phi\) with finitely many input and output modes. Signal energy is measured by a positive quadratic Hamiltonian \(H_G=\tfrac12 R^{\mathsf T}GR\), with \(G>0\) and ground energy \(E_0\); \(R\) collects the canonical quadratures. The budgets satisfy \(E\ge E_0\) and \(P\ge0\). For an \(n\)-use code, the mean total signal energy is at most \(nE\), while the same initially shared maximally entangled state has Schmidt rank \(K_n\) satisfying \(\log K_n\le nP\). Thus \(P\) is a strictly supplied entanglement rate. Encoders may be arbitrary joint quantum operations, and decoding may be collective. The classical capacity is \[ C(E,P)=\max_{\substack{U\ge V\;\mathrm{physical}\\ \frac12\operatorname{Tr}(GU)\le E,\;S_G(V)\le P}}\bigl[S_G(\Phi U)+S_G(V)-S_G(\Phi^c V)\bigr]. \] Here \(V\) is a Gaussian seed covariance, \(U\) is the average-input covariance, and \(U-V\) is the displacement-modulation covariance. A covariance is physical when \(V+i\Omega/2\ge0\), where \(\Omega\) is the canonical commutation matrix and the vacuum covariance is \(I/2\). The notation \(S_G\) denotes Gaussian von Neumann entropy, and \(\Phi^c\) is the full complementary channel, including any environment purification. All logarithms are natural, so rates are in nats per channel use. This formula replaces optimization over arbitrary quantum block codes by a finite-dimensional covariance optimization. It supplies both an achievable rate and a converse against non-Gaussian, correlated encodings. At \(P=0\), it also determines the unassisted capacity of general fixed finite-mode Gaussian channels, including channels without a passive decomposition into independent modes. The reduction specifies the exact capacity value; it does not require a closed-form optimizer for every device. Energy and strict entanglement supply respond differently to errors For a fixed allowed average error \(0\le\varepsilon<1\), the exact capacity under the mean-energy constraint is \[ C_\varepsilon(E,P)=C\!\left(E_0+\frac{E-E_0}{1-\varepsilon},\,P\right). \] Allowing failure rescales the mean energy above the ground state, but leaves the strict initial entanglement budget unchanged. The proof uses an exponential strong converse for codes whose inputs obey a hard total-energy cutoff. In a different model, where a readable flag permits random entanglement supply constrained only on average, the work establishes an achievable bound and a strict small-supply advantage for thermal attenuation at positive energy and fixed nonzero error. That comparison identifies the operational importance of the supply rule; it is not a complete capacity characterization of the flagged model. The communication value of entanglement For a thermal attenuator, let \(0<\eta<1\) be the transmissivity, \(b>0\) the output mean photon number for vacuum input, and \(N\) the input mean photon number. Write its capacity as \(C_N(P)\), and define \(g(N)=(N+1)\log(N+1)-N\log N\). The gain from entanglement obeys the sharp bound \[ 0\le C_N(P)-C_N(0)\le\frac{\eta}{1+b}\,P. \] For every fixed \(N>0\), the coefficient \(\eta/(1+b)\) is attained as the initial right derivative at \(P=0\). In the low-photon limit, the stronger resource law is \[ C_N(P)-C_N(0)=\frac{\eta}{1+b}\min\{P,g(N)\}+O(N),\qquad N\downarrow0. \] The remainder is uniform over \(P\ge0\) at fixed \(\eta\) and positive \(b\). Noise therefore determines the conversion efficiency, while \(g(N)\sim N\log(1/N)\) sets the entanglement scale sufficient for saturation. This uniform limit concerns the entanglement budget, not a simultaneous limit of vanishing thermal noise. Multimode resource allocation and scope A coupled two-mode example demonstrates how the frontier can guide resource allocation. A global resource-price certificate proves that a legal covariance pair exceeds the entire common-passive-frame Gaussian profile hull by at least \(10^{-5}\) nat per two-mode use at specified budgets. The comparison includes all common passive frames, local squeezing and average-resource mixtures. The mechanism is the freedom to choose different mode allocations for the seed and average input; the benchmark is this precisely defined Gaussian class, rather than all unentangled encodings. The universal scope is the stated family of fixed finite-mode Gaussian channels, positive quadratic signal costs and strict initial maximally entangled Schmidt-rank supply, with unrestricted joint encoding and collective decoding. The model has no feedback or credit for returned entanglement. The capacity frontier and its resource consequences provide a common way to assess optical communication designs under these operational constraints.

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Communication over a noisy channel is often conducted in a setting in which different input symbols to the channel incur a certain cost. For example, for bosonic quantum channels, the cost associated with an input state is the number of photons, which is proportional to the energy consumed. In such a setting, it is often useful to know the maximum amount of information that can be reliably transmitted per cost incurred. This is known as the capacity per unit cost. In this paper, we generalize the capacity per unit cost to various communication tasks involving a quantum channel, such as classical communication, entanglement-assisted classical communication, private communication, and quantum communication. For each task, we define the corresponding capacity per unit cost and derive a formula for it analogous to that of the usual capacity. Furthermore, for the special and natural cases in which there is a zero-cost state, we obtain expressions in terms of an optimized relative entropy involving the zero-cost state. For each communication task, we construct an explicit pulse-position-modulation coding scheme that achieves the capacity per unit cost. Finally, we compute capacities per unit cost for various bosonic Gaussian channels and introduce the notion of a blocklength constraint as a proposed solution to the long-standing issue of infinite capacities per unit cost. This motivates the idea of a blocklength-cost duality on which we elaborate in depth.
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.
In most communication scenarios, sending a symbol encoded in a quantum state requires spending resources such as energy, which can be quantified by a cost of communication. A standard approach in this context is to quantify the performance of communication protocol by classical capacity, quantifying the maximal amount of information that can be transmitted through a quantum channel per single use of the channel. However, different figures of merit are also possible and a particularly well-suited one is the classical capacity per unit cost, which quantifies the maximal amount of information that can be transmitted per unit cost. I generalize this concept to account for the quantum nature of the information carriers and communication channels and show that if there exists a state with cost equal to zero, e.g., a vacuum state, the capacity per unit cost can be expressed by a simple formula containing maximization of the relative entropy between two quantum states. This enables me to analyze the behavior of photon information efficiency for general communication tasks and show simple bounds on the capacity per unit cost in terms of quantities familiar from quantum estimation theory. I calculate also the capacity per unit cost for general Gaussian quantum channels.
This paper establishes a general theory of energy-constrained quantum and private capacities of quantum channels. We begin by defining various energy-constrained communication tasks, including quantum communication with a uniform energy constraint, entanglement transmission with an average energy constraint, private communication with a uniform energy constraint, and secret key transmission with an average energy constraint. We develop several code conversions, which allow us to conclude non-trivial relations between the capacities corresponding to the above tasks. We then show how the regularized, energy-constrained coherent information is equal to the capacity for the first two tasks and is an achievable rate for the latter two tasks, whenever the energy observable satisfies the Gibbs condition of having a well-defined thermal state for all temperatures and the channel satisfies a finite output-entropy condition. For degradable channels satisfying these conditions, we find that the single-letter energy-constrained coherent information is equal to all of the capacities. We finally apply our results to degradable quantum Gaussian channels and recover several results already established in the literature (in some cases, we prove new results in this domain). Contrary to what may appear from some statements made in the literature recently, proofs of these results do not require the solution of any kind of minimum output entropy conjecture or entropy photon-number inequality.
Quantum information processing leverages the principles of quantum mechanics, utilizing qubits, to improve computational and communicative tasks. In this realm, the quantum channel’s capacity is pivotal in determining the efficiency and accuracy of quantum information handling, with its performance being significantly influenced by channel noise. Our study aims to establish a holistic hybrid quantum noise model to determine the quantum channel capacity. In this paper, we formulated a mathematical expression for this capacity and conducted simulations for both Gaussian and non-Gaussian inputs. A hybrid noise model is constructed by convolution of Poisson-distributed quantum noise with classical additive white Gaussian noise. We characterized the quantum-classical noise and the received signal using Gaussian Mixture Models. The maximum amount of quantum information that can be reliably transmitted over a quantum channel (per use of the channel) is determined by its capacity, and entropy and related quantities like mutual information play a role in calculating this capacity. Our formulation of quantum channel capacity is derived from the mutual information shared between the transmitter and receiver, encompassing the entropies of the signals. The quantum channel presents a higher capacity-to-signal-to-noise ratio for Gaussian inputs than non-Gaussian ones.
How much classical information can a noisy optical channel carry when both signal energy and preshared entanglement are limited? This work gives an exact capacity frontier for this joint resource problem. It allows one to determine the best asymptotic communication rate, price the benefit of additional entanglement, and distinguish the effect of tolerating errors from that of changing how entanglement is supplied. Exact capacity under two resource constraints The setting is any fixed bosonic Gaussian channel Φ\PhiΦ with finitely many input and output modes. Signal energy is measured by a positive quadratic Hamiltonian HG=12RTGRH_G=\tfrac12 R^{\mathsf T}GRHG​=21​RTGR, with G>0G>0G>0 and ground energy E0E_0E0​; RRR collects the canonical quadratures. The budgets satisfy E≥E0E\ge E_0E≥E0​ and P≥0P\ge0P≥0. For an nnn-use code, the mean total signal energy is at most nEnEnE, while the same initially shared maximally entangled state has Schmidt rank KnK_nKn​ satisfying log⁡Kn≤nP\log K_n\le nPlogKn​≤nP. Thus PPP is a strictly supplied entanglement rate. Encoders may be arbitrary joint quantum operations, and decoding may be collective. The classical capacity is C(E,P)=max⁡U≥Vphysical12Tr⁡(GU)≤E,SG(V)≤P[SG(ΦU)+SG(V)−SG(ΦcV)]. C(E,P)=\max_{\substack{U\ge V\;\mathrm{physical}\\ \frac12\operatorname{Tr}(GU)\le E,\;S_G(V)\le P}}\bigl[S_G(\Phi U)+S_G(V)-S_G(\Phi^c V)\bigr]. C(E,P)=maxU≥Vphysical21​Tr(GU)≤E,SG​(V)≤P​​[SG​(ΦU)+SG​(V)−SG​(ΦcV)]. Here VVV is a Gaussian seed covariance, UUU is the average-input covariance, and U−VU-VU−V is the displacement-modulation covariance. A covariance is physical when V+iΩ/2≥0V+i\Omega/2\ge0V+iΩ/2≥0, where Ω\OmegaΩ is the canonical commutation matrix and the vacuum covariance is I/2I/2I/2. The notation SGS_GSG​ denotes Gaussian von Neumann entropy, and Φc\Phi^cΦc is the full complementary channel, including any environment purification. All logarithms are natural, so rates are in nats per channel use. This formula replaces optimization over arbitrary quantum block codes by a finite-dimensional covariance optimization. It supplies both an achievable rate and a converse against non-Gaussian, correlated encodings. At P=0P=0P=0, it also determines the unassisted capacity of general fixed finite-mode Gaussian channels, including channels without a passive decomposition into independent modes. The reduction specifies the exact capacity value; it does not require a closed-form optimizer for every device. Energy and strict entanglement supply respond differently to errors For a fixed allowed average error 0≤ε<10\le\varepsilon<10≤ε<1, the exact capacity under the mean-energy constraint is Cε(E,P)=C ⁣(E0+E−E01−ε, P). C_\varepsilon(E,P)=C\!\left(E_0+\frac{E-E_0}{1-\varepsilon},\,P\right). Cε​(E,P)=C(E0​+1−εE−E0​​,P). Allowing failure rescales the mean energy above the ground state, but leaves the strict initial entanglement budget unchanged. The proof uses an exponential strong converse for codes whose inputs obey a hard total-energy cutoff. In a different model, where a readable flag permits random entanglement supply constrained only on average, the work establishes an achievable bound and a strict small-supply advantage for thermal attenuation at positive energy and fixed nonzero error. That comparison identifies the operational importance of the supply rule; it is not a complete capacity characterization of the flagged model. The communication value of entanglement For a thermal attenuator, let 0<η<10<\eta<10<η<1 be the transmissivity, b>0b>0b>0 the output mean photon number for vacuum input, and NNN the input mean photon number. Write its capacity as CN(P)C_N(P)CN​(P), and define g(N)=(N+1)log⁡(N+1)−Nlog⁡Ng(N)=(N+1)\log(N+1)-N\log Ng(N)=(N+1)log(N+1)−NlogN. The gain from entanglement obeys the sharp bound 0≤CN(P)−CN(0)≤η1+b P. 0\le C_N(P)-C_N(0)\le\frac{\eta}{1+b}\,P. 0≤CN​(P)−CN​(0)≤1+bη​P. For every fixed N>0N>0N>0, the coefficient η/(1+b)\eta/(1+b)η/(1+b) is attained as the initial right derivative at P=0P=0P=0. In the low-photon limit, the stronger resource law is CN(P)−CN(0)=η1+bmin⁡{P,g(N)}+O(N),N↓0. C_N(P)-C_N(0)=\frac{\eta}{1+b}\min\{P,g(N)\}+O(N),\qquad N\downarrow0. CN​(P)−CN​(0)=1+bη​min{P,g(N)}+O(N),N↓0. The remainder is uniform over P≥0P\ge0P≥0 at fixed η\etaη and positive bbb. Noise therefore determines the conversion efficiency, while g(N)∼Nlog⁡(1/N)g(N)\sim N\log(1/N)g(N)∼Nlog(1/N) sets the entanglement scale sufficient for saturation. This uniform limit concerns the entanglement budget, not a simultaneous limit of vanishing thermal noise. Multimode resource allocation and scope A coupled two-mode example demonstrates how the frontier can guide resource allocation. A global resource-price certificate proves that a legal covariance pair exceeds the entire common-passive-frame Gaussian profile hull by at least 10−510^{-5}10−5 nat per two-mode use at specified budgets. The comparison includes all common passive frames, local squeezing and average-resource mixtures. The mechanism is the freedom to choose different mode allocations for the seed and average input; the benchmark is this precisely defined Gaussian class, rather than all unentangled encodings. The universal scope is the stated family of fixed finite-mode Gaussian channels, positive quadratic signal costs and strict initial maximally entangled Schmidt-rank supply, with unrestricted joint encoding and collective decoding. The model has no feedback or credit for returned entanglement. The capacity frontier and its resource consequences provide a common way to assess optical communication designs under these operational constraints.