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IEEE International Symposium on Information Theory (ISIT)

The Communication Value of a Quantum Channel

Eric Chitambar, Ian M. George, Brian Doolittle, Marius Junge

Published Jun 26, 2022• 1 citation
Publisher

Abstract

There are various ways to quantify the communication capabilities of a quantum channel. In this work we study the communication value (cv) of channel, which describes the optimal success probability of transmitting a randomly selected classical message over the channel. The cv also offers a dual interpretation as the classical communication cost for zero-error channel simulation using non-signaling resources. We first provide an entropic characterization of the cv as a generalized conditional min-entropy over the cone of separable operators. We evaluate the cv exactly for all qubit channels and the Werner-Holevo family of channels. The latter is shown to have non-multiplicative cv when d > 2. On the other hand, we prove that any pair of qubit channels have multiplicative cv when used in parallel. Even stronger, all entanglement-breaking channels and the partially depolarizing channel are shown to have multiplicative cv when used in parallel with any channel. We then turn to the entanglement-assisted cv and prove that it is equivalent to the conditional min-entropy of the Choi matrix of the channel. Combining with previous work on zero-error channel simulation, this implies that the entanglement-assisted cv is the classical communication cost for perfectly simulating a channel using quantum non-signaling resources. A final component of this work investigates relaxations of the channel cv to other cones such as the set of operators having a positive partial transpose (PPT).

DOI
10.1109/isit50566.2022.9834380

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There are various ways to quantify the communication capabilities of a quantum channel. In this work we study the communication value (cv) of channel, which describes the optimal success probability of transmitting a randomly selected classical message over the channel. The cv also offers a dual interpretation as the classical communication cost for zero-error channel simulation using non-signaling resources. We first provide an entropic characterization of the cv as a generalized conditional min-entropy over the cone of separable operators. We evaluate the cv exactly for all qubit channels and the Werner-Holevo family of channels. The latter is shown to have non-multiplicative cv when d > 2. On the other hand, we prove that any pair of qubit channels have multiplicative cv when used in parallel. Even stronger, all entanglement-breaking channels and the partially depolarizing channel are shown to have multiplicative cv when used in parallel with any channel. We then turn to the entanglement-assisted cv and prove that it is equivalent to the conditional min-entropy of the Choi matrix of the channel. Combining with previous work on zero-error channel simulation, this implies that the entanglement-assisted cv is the classical communication cost for perfectly simulating a channel using quantum non-signaling resources. A final component of this work investigates relaxations of the channel cv to other cones such as the set of operators having a positive partial transpose (PPT).
There are various ways to quantify the communication capabilities of a quantum channel. In this work we introduce the communication value (cv) of quantum channel, which describes the optimal probability of guessing the channel input from its output. By connecting to prior work on zero-error channel simulation, we show that the cv and its entanglement-assisted variant also offer dual interpretations as the classical communication cost for perfectly simulating different aspects of a channel using non-signaling resources. Our study involves characterizing the communication value as a generalized conditional min-entropy over the cone of separable operators. Using this characterization, we evaluate the cv for all qubit channels and higher-dimensional channels with certain symmetries. We find that the any entanglement-breaking channel has multiplicative cv when used in parallel with any other channel; the same is shown to hold for Pauli channels and partially depolarizing channels. In contrast, the cv is found to be non-multiplicative for a subset of the well-known Werner-Holevo channels. A final component of this work investigates relaxations of the channel cv to other cones such as the set of operators having a positive partial transpose (PPT).
Understanding the classical communication cost of simulating a quantum channel is a fundamental problem in quantum information theory, which becomes even more intriguing when considering the role of non-locality in quantum information processing. This paper investigates the bidirectional classical communication cost of simulating a bipartite quantum channel assisted by non-signalling correlations, which are permitted across the spatial dimension between the two parties. By introducing non-signalling superchannels, we present lower and upper bounds on the one-shot εεε-error one-way classical communication cost of a bipartite channel via its smooth max-relative entropy of one-way classical communication, and establish that the asymptotic exact cost is given by its max-relative entropy of one-way classical communication. For the bidirectional scenario, we derive semidefinite programming (SDP) formulations for the one-shot exact bidirectional classical communication cost via non-signalling bipartite superchannels. We further introduce a channel's bipartite conditional min-entropy as an efficiently computable lower bound on the asymptotic cost of bidirectional classical communication. Our results in both one-shot and asymptotic settings provide lower bounds on the entanglement-assisted simulation cost in scenarios where entanglement is available to the two parties. Moreover, we propose a seesaw-based algorithm to compute an upper bound on the minimum simulation error via local operations and shared entanglement, which provides valuable insights into the relationship between non-signalling bipartite superchannels and more physically realizable protocols. Numerical experiments demonstrate the effectiveness of our bounds in estimating communication costs for various quantum channels, showing that our bounds can be tight in different scenarios.
This paper establishes single-letter formulas for the exact entanglement cost of simulating quantum channels under free quantum operations that completely preserve positivity of the partial transpose (PPT). First, we introduce the $\ensuremath{\kappa}$-entanglement measure for point-to-point quantum channels, based on the idea of the $\ensuremath{\kappa}$ entanglement of bipartite states, and we establish several fundamental properties for it, including amortization collapse, monotonicity under PPT superchannels, additivity, normalization, faithfulness, and nonconvexity. Second, we introduce and solve the exact entanglement cost for simulating quantum channels in both the parallel and sequential settings, along with the assistance of free PPT-preserving operations. In particular, we establish that the entanglement cost in both cases is given by the same single-letter formula, the $\ensuremath{\kappa}$-entanglement measure of a quantum channel. We further show that this cost is equal to the largest $\ensuremath{\kappa}$ entanglement that can be shared or generated by the sender and receiver of the channel. This formula is calculable by a semidefinite program, thus allowing for an efficiently computable solution for general quantum channels. Noting that the sequential regime is more powerful than the parallel regime, another notable implication of our result is that both regimes have the same power for exact quantum channel simulation, when PPT superchannels are free. For several basic Gaussian quantum channels, we show that the exact entanglement cost is given by the Holevo-Werner formula [Holevo and Werner, Phys. Rev. A 63, 032312 (2001)], giving an operational meaning of the Holevo-Werner quantity for these channels.
In one-shot and zero-error information theory, the conditional min-entropy is a fundamental tool. It may be expressed as a conic program over the positive semidefinite cone. Recently, Chitambar et al. showed that the same conic program altered to be over the separable cone is a measure of transmitting classical communication over a quantum channel called the ‘communication value.’ In this work, we extend this idea to a broad class of convex cones to induce new families of entropic quantities. We show this methodology has operational relevance by characterizing a generalized notion of communication value and relating a class of cone-restricted entropies to a partial ordering on converting quantum channels via bistochastic preprocessing. We also show regularized smooth versions of these entropic quantities do not in general converge to the von Neumann entropy, which shows tasks characterized by these quantities are not equivalent even in an asymptotic i.i.d. fashion.
We study the quantum umlaut information, a correlation measure defined for bipartite quantum states ρABρ_{AB}ρAB​ as a reversed variant of the quantum mutual information: U(A;B)ρ=min⁡σBD(ρA⊗σB∥ρAB)U(A;B)_ρ= \min_{σ_B} D(ρ_A\otimes σ_B\|ρ_{AB})U(A;B)ρ​=minσB​​D(ρA​⊗σB​∥ρAB​) in terms of the quantum relative entropy DDD. As in the classical case [Girardi et al., arXiv:2503.18910], this definition allows for a closed-form expression and has an operational interpretation as the asymptotic error exponent in the hypothesis testing task of deciding whether a given bipartite state is product or not. We generalise the umlaut information to quantum channels, where it also extends the notion of `oveloh information' [Nuradha et al., arXiv:2404.16101]. We prove that channel umlaut information is additive for classical-quantum channels, while we observe additivity violations for fully quantum channels. Inspired by recent results in entanglement theory, we then show as our main result that the regularised umlaut information constitutes a fundamental measure of the quality of classical information transmission over a quantum channel -- as opposed to the capacity, which quantifies the quantity of information that can be sent. This interpretation applies to coding assisted by activated non-signalling correlations, and the channel umlaut information is in general larger than the corresponding expression for unassisted communication as obtained by Dalai for the classical-quantum case [IEEE Trans. Inf. Theory 59, 8027 (2013)]. Combined with prior works on non-signalling--assisted zero-error channel capacities, our findings imply a dichotomy between the settings of zero-rate error exponents and zero-error communication. While our results are single-letter only for classical-quantum channels, we also give a single-letter bound for fully quantum channels in terms of the `geometric' version of umlaut information.
A fundamental objective in quantum information science is to determine the cost in classical resources of simulating a particular quantum system. The classical simulation cost is quantified by the signaling dimension which specifies the minimum amount of classical communication needed to perfectly simulate a channel's input-output correlations when unlimited shared randomness is held between encoder and decoder. This paper provides a collection of device-independent tests that place lower and upper bounds on the signaling dimension of a channel. Among them, a single family of tests is shown to determine when a noisy classical channel can be simulated using an amount of communication strictly less than either its input or its output alphabet size. In addition, a family of eight signaling dimension witnesses is presented that completely characterize when any four-outcome measurement channel, such as a Bell measurement, can be simulated using one communication bit and shared randomness. Finally, we bound the signaling dimension for all partial replacer channels in ddd dimensions. The bounds are found to be tight for the special case of the erasure channel.