arXiv:2408.17290 · Physical Review Letters

Fundamental Limit on the Power of Entanglement Assistance in Quantum Communication

Lasse Harboe Wolff, Paula Belzig, Matthias Christandl, Bergfinnur Durhuus, Marco Tomamichel

Published Jan 17, 2025• 9 citations

Abstract

The optimal rate of reliable communication over a quantum channel can be enhanced by preshared entanglement. Whereas the enhancement may be unbounded in infinite-dimensional settings even when the input power is constrained, a long-standing conjecture asserts that the ratio between the entanglement-assisted and unassisted classical capacities is bounded in finite-dimensional settings [Bennett et al., IEEE Trans. Inf. Theory 48, 2637 (2002)IETTAW0018-944810.1109/TIT.2002.802612]. In this Letter, we prove this conjecture by showing that their ratio is upper bounded by o(d^{2}), where d is the input dimension of the channel. An application to quantum communication with noisy encoders and decoders is given.

DOI: 10.1103/physrevlett.134.020802

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arXiv:2408.17290 · Physical Review Letters

Fundamental Limit on the Power of Entanglement Assistance in Quantum Communication

Lasse Harboe Wolff, Paula Belzig, Matthias Christandl, Bergfinnur Durhuus, Marco Tomamichel

Published Jan 17, 2025• 9 citations

arXiv HTML PDF

Abstract

DOI: 10.1103/physrevlett.134.020802

Related papers

Superadditivity for Entanglement-Assisted Communication

Hao-Chung Cheng, Mario Berta

The entanglement-assisted capacity of a quantum channel admits an additive single-letter characterization, implying that joint encodings across channel uses cannot increase the ultimate communication rate. Here, we show that this additive picture does not extend to communication reliability. Specifically, we prove that the Petz-Rényi channel information can be strictly superadditive for every $α\in[1/2,1)$ , yielding a genuine multi-copy enhancement of the entanglement-assisted random-coding error exponent, even though the entanglement-assisted capacity remains additive. We establish this phenomenon analytically already for measurement channels, which are entanglement-breaking and have additive unassisted capacity. Remarkably, this strict superadditivity is witnessed by a separable, classically correlated two-copy channel-input marginal, demonstrating that no entanglement between the transmitted systems is required. Our results show that, although correlations across channel uses cannot increase the ultimate rate of entanglement-assisted communication, they can enhance its reliability.The entanglement-assisted capacity of a quantum channel admits an additive single-letter characterization, implying that joint encodings across channel uses cannot increase the ultimate communication rate. Here, we show that this additive picture does not extend to communication reliability. Specifically, we prove that the Petz-Rényi channel information can be strictly superadditive for every $α\in[1/2,1)$, yielding a genuine multi-copy enhancement of the entanglement-assisted random-coding error exponent, even though the entanglement-assisted capacity remains additive. We establish this phenomenon analytically already for measurement channels, which are entanglement-breaking and have additive unassisted capacity. Remarkably, this strict superadditivity is witnessed by a separable, classically correlated two-copy channel-input marginal, demonstrating that no entanglement between the transmitted systems is required. Our results show that, although correlations across channel uses cannot increase the ultimate rate of entanglement-assisted communication, they can enhance its reliability.

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Sending quantum information reliably over long distances is a central challenge in quantum technology in general, and in quantum optics in particular, since most quantum communication relies on optical fibres or free-space links. Here, we address this problem by shifting the focus from the quantity of information sent to the quality of the transmission, i.e. the rate of decay of the transmission error with respect to the number of channel uses. For the general class of teleportation-simulable channels, which includes all channels arising in quantum optical communication, we prove that the single-letter reverse relative entropy of entanglement of the Choi state upper bounds the error exponent of two-way assisted quantum communication - paralleling the celebrated capacity bound of [Pirandola et al., Nat. Comm. (2017)] in terms of the regularised relative entropy of entanglement. Remarkably, for Gaussian channels our bound can be computed efficiently through a convex program with simple constraints involving only finite-dimensional covariance matrices. As a prototypical application, we derive closed-form analytical expressions of our upper bound as well as random-coding-based lower bounds for several one-mode Gaussian channels. Extending recent work [Lami et al., arXiv:2408.07067 (2024)] to infinite-dimensional systems, we further endow the reverse relative entropy of entanglement with an exact operational interpretation in entanglement testing, and show that it characterises the rate of entanglement distillation under non-entangling operations. These findings offer a new perspective on entanglement as a resource and sharpen the theoretical benchmarks for future quantum optical networks.Sending quantum information reliably over long distances is a central challenge in quantum technology in general, and in quantum optics in particular, since most quantum communication relies on optical fibres or free-space links. Here, we address this problem by shifting the focus from the quantity of information sent to the quality of the transmission, i.e. the rate of decay of the transmission error with respect to the number of channel uses. For the general class of teleportation-simulable channels, which includes all channels arising in quantum optical communication, we prove that the single-letter reverse relative entropy of entanglement of the Choi state upper bounds the error exponent of two-way assisted quantum communication - paralleling the celebrated capacity bound of [Pirandola et al., Nat. Comm. (2017)] in terms of the regularised relative entropy of entanglement. Remarkably, for Gaussian channels our bound can be computed efficiently through a convex program with simple constraints involving only finite-dimensional covariance matrices. As a prototypical application, we derive closed-form analytical expressions of our upper bound as well as random-coding-based lower bounds for several one-mode Gaussian channels. Extending recent work [Lami et al., arXiv:2408.07067 (2024)] to infinite-dimensional systems, we further endow the reverse relative entropy of entanglement with an exact operational interpretation in entanglement testing, and show that it characterises the rate of entanglement distillation under non-entangling operations. These findings offer a new perspective on entanglement as a resource and sharpen the theoretical benchmarks for future quantum optical networks.

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Quantum communications promises reliable transmission of quantum information, efficient distribution of entanglement and generation of completely secure keys. For all these tasks, we need to determine the optimal point-to-point rates that are achievable by two remote parties at the ends of a quantum channel, without restrictions on their local operations and classical communication, which can be unlimited and two-way. These two-way assisted capacities represent the ultimate rates that are reachable without quantum repeaters. Here, by constructing an upper bound based on the relative entropy of entanglement and devising a dimension-independent technique dubbed 'teleportation stretching', we establish these capacities for many fundamental channels, namely bosonic lossy channels, quantum-limited amplifiers, dephasing and erasure channels in arbitrary dimension. In particular, we exactly determine the fundamental rate-loss tradeoff affecting any protocol of quantum key distribution. Our findings set the limits of point-to-point quantum communications and provide precise and general benchmarks for quantum repeaters.Quantum communications promises reliable transmission of quantum information, efficient distribution of entanglement and generation of completely secure keys. For all these tasks, we need to determine the optimal point-to-point rates that are achievable by two remote parties at the ends of a quantum channel, without restrictions on their local operations and classical communication, which can be unlimited and two-way. These two-way assisted capacities represent the ultimate rates that are reachable without quantum repeaters. Here, by constructing an upper bound based on the relative entropy of entanglement and devising a dimension-independent technique dubbed 'teleportation stretching', we establish these capacities for many fundamental channels, namely bosonic lossy channels, quantum-limited amplifiers, dephasing and erasure channels in arbitrary dimension. In particular, we exactly determine the fundamental rate-loss tradeoff affecting any protocol of quantum key distribution. Our findings set the limits of point-to-point quantum communications and provide precise and general benchmarks for quantum repeaters.

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A universal energy–entanglement capacity frontier for Gaussian optical channels

Zixuan He

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.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.

Oct 19, 20260 citations

α\in[1/2,1)

, yielding a genuine multi-copy enhancement of the entanglement-assisted random-coding error exponent, even though the entanglement-assisted capacity remains additive. We establish this phenomenon analytically already for measurement channels, which are entanglement-breaking and have additive unassisted capacity. Remarkably, this strict superadditivity is witnessed by a separable, classically correlated two-copy channel-input marginal, demonstrating that no entanglement between the transmitted systems is required. Our results show that, although correlations across channel uses cannot increase the ultimate rate of entanglement-assisted communication, they can enhance its reliability.

d

that, at each use, sample a pair of classical random variables

(F,Z)

, apply one of the

d^2

Heisenberg--Weyl operators indexed by

Z

to the input, and deliver the transformed quantum system together with

F

to the receiver. If

E

is the available entanglement rate in bits per channel use, then the capacity is

\min\{H(Z|F),\log_2d+E\}

. This class of channels encompasses quantum erasure and depolarizing channels, as well as families of Pauli channels. Additionally, for interactive protocols, we show that the commitment rate cannot exceed

\log_2d+E

bits per channel use, so that when

H(Z|F)\geq\log_2d+E

, interactive communication does not increase the capacity. As a consequence, for interactive protocols, we determine the capacity of the quantum erasure 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.