Single-Letterization of the Simultaneous Classical-Quantum Capacity Region of a Quantum Channel and Its Conjectured Analogues

Devetak and Shor (2005) characterized the set of rate pairs at which classical and quantum information can be sent simultaneously over a quantum channel, but their expression is a regularized (multi-letter) limit over arbitrarily many channel uses. The unresolved problem, as posed in that paper, is to determine when this regularization is genuinely necessary and to obtain a single-letter, computable form of the simultaneous classical-quantum capacity region and of its conjectured structural analogues: the simultaneous public-private capacity of a channel, the static trade-off between distillable common randomness and distillable entanglement of a bipartite state (including the classical-communication cost, giving a three-dimensional region), and the optimal joint source-channel coding rate for transmitting a visible pure-state ensemble with classical side information over an arbitrary channel. Later work has extended the regularized theory, added new single-letter special classes (Hadamard channels, the erasure channel), constructed channels whose classical-quantum capacity region is strictly superadditive even though their classical and quantum capacities are additive, and tightened bounds on the depolarizing channel, but the general single-letter question, the quantum capacity of the depolarizing channel, and the general joint source-channel coding problem remain open.

Background

A quantum channel is a completely positive, trace-preserving linear map $\mathcal{N}$ describing the noisy evolution of a quantum system. Two of its fundamental capacities are the classical capacity $C(\mathcal{N})$, the largest rate at which classical bits can be sent reliably with arbitrarily many channel uses, and the quantum capacity $Q(\mathcal{N})$, the largest rate at which qubits (equivalently, entanglement between sender and receiver) can be sent. The quantum capacity is given by a regularized formula, the maximized coherent information (the entropy difference $I(A\rangle B)=H(B)-H(AB)$ of the channel output), optimized over arbitrarily many copies of the channel, because coherent information is superadditive for some channels.

When both kinds of information must be sent over the same noisy channel, the two capacities compete. Devetak and Shor (2005) introduced this problem and derived the capacity region $S(\mathcal{N})$ of admissible pairs $(r,R)$, where $r$ is the classical rate and $R$ the quantum rate. The achievability strategy is trade-off coding: the quantum code is chosen randomly depending on the classical message, so the classical information is 'piggy-backed' on top of the quantum transmission, beating the naive time-sharing strategy (interleaving dedicated classical and quantum codes) for channels such as the dephasing channel, whose only effect is to randomize relative phases. Their formula is regularized, $S(\mathcal{N})=\bigcup_{l\ge 1}\tfrac{1}{l}S^{(1)}(\mathcal{N}^{\otimes l})$, where $S^{(1)}$ is a single-letter region built from the entropic quantities $I(X;B)$ and the conditional coherent information $I(A\rangle BX)$, optimized over an auxiliary classical variable $X$ and input states. The regularization is a serious defect: it makes the region uncomputable in general, since evaluating it requires optimizing over unboundedly many channel uses.

Some structure is known. For the noiseless qubit channel and the erasure channel (which with some probability discards the input and otherwise transmits it perfectly), time-sharing is optimal and the single-letter region is exact. Devetak and Shor proved that for generalized dephasing channels a single copy ($l=1$) suffices, giving a strictly concave trade-off curve. For the depolarizing channel, which applies one of the three Pauli errors with total probability $p$, DiVincenzo, Shor and Smolin (1998) had shown that degenerate block codes (codes whose distinct codewords can suffer identical errors, so that errors on some codewords act trivially) beat the single-letter hashing bound (the largest coherent information achievable with a single channel use, the best one-use lower bound on $Q(\mathcal{N})$) near $p\approx 0.189$, the noise level at which the single-letter quantum rate vanishes; hence the regularization is known to be needed there. Separately, comparing the single-copy ($l=1$) trade-off curve with time-sharing, Devetak and Shor report that trade-off coding beats time-sharing only for small noise (roughly $p<0.04$), that time-sharing is optimal at $l=1$ for larger noise (roughly $p>0.05$), and that for these larger noise levels away from the near-threshold point $p\approx 0.189$ it is not known whether using many channel copies strictly enlarges $Q(\mathcal{N})$ or the joint trade-off.

The same paper raised three structural extensions. First, a cryptographic analogue in which the classical rate is private classical communication (kept secret from the channel's environment, as in a classical wire-tap channel), conjectured to obey the same single-letter formula with the entropic quantities replaced by privacy quantities (a difference of conditional mutual informations with the environment). Second, a static analogue in which the parties share many copies of a bipartite mixed state $\rho_{AB}$ instead of using a channel: trade off distillable common randomness (perfectly correlated classical bits extractable by one-way communication) against one-way distillable entanglement, conjectured to be given by the same theorem with the channel replaced by general quantum instruments (measurement maps with quantum outputs) applied to the $A$ system; including the forward classical-communication cost as a third dimension was posed as a more ambitious open goal. Third, a joint source-channel coding problem: a visible ensemble $E$ of pure states (visible meaning the encoder is told which state of the ensemble was emitted) can be compressed by storing partial classical side information about the state identities at rate $C$ and the quantum remainder at rate $Q(C)$; given such a source and a channel, what is the optimal rate of transmitted ensemble copies per channel use? The separable strategy (compress the source, then send the classical and quantum parts through the channel using the capacity region) is known to be suboptimal for 'well-matched' pairs: in Smolin's trine-ensemble example (the trine ensemble is three equiprobable qubit pure states arranged symmetrically), feeding the identity of each state directly to a matched channel transmits one copy per use, while the separable strategy achieves only $1/\log_2 3$ copies per use.

Subsequent work enlarged the regularized theory but did not remove the regularization. Hsieh and Wilde extended the region to the three-resource 'dynamic capacity region' trading classical communication, quantum communication, and entanglement, for both channels and (statically) bipartite states, always with multi-letter converses; they and later authors proved additivity of the single-letter formula for Hadamard channels (those whose complementary channel, which sends the channel input to the environment, is entanglement-breaking, meaning it destroys all entanglement between its input and any reference system) and for the erasure channel, and determined a regularized public-private-secret-key region. Channel classes organized by what the environment can see also structure the field: a channel is degradable if the receiver's output can be transformed by a further channel into everything the environment receives, antidegradable if the environment's output can be transformed into the receiver's output, in which case the no-cloning theorem forces $Q(\mathcal{N})=0$, and non-degradable if neither holds. A decisive structural obstruction is now known: Zhu, Zhuang, Hsieh and Shor constructed channels whose classical and quantum capacities are each additive but whose classical-quantum capacity region is strictly superadditive, so no universal single-letter formula can exist. What remains undetermined is for which channels the regularization is unnecessary, the exact region for the depolarizing channel, the single-letter status of the public-private and static conjectures, and the general joint source-channel coding rate.

Problem Statement

Determine the simultaneous classical-quantum transmission capability of a general quantum channel in computable form, resolving the regularization left open by Devetak and Shor. For a finite-dimensional channel $\mathcal{N}$ with simultaneous capacity region $S(\mathcal{N})=\bigcup_{l\ge 1}\tfrac{1}{l}S^{(1)}(\mathcal{N}^{\otimes l})$, where $S^{(1)}(\mathcal{N}^{\otimes l})$ is the achievable single-block region obtained by optimizing the entropic trade-off quantities (the mutual information $I(X;B)$ and the conditional coherent information $I(A\rangle BX)$) over an auxiliary classical variable $X$ and input states for $l$ channel uses, the linked questions are: (1) characterize the channels for which the regularization is unnecessary, that is, $S(\mathcal{N})=S^{(1)}(\mathcal{N})$, beyond the noiseless, erasure, dephasing, and Hadamard (complementary channel entanglement-breaking) classes; equivalently, determine for which channels $l>1$ copies strictly enlarge the region, the open crux being the non-degradable channels (channels that are neither degradable nor antidegradable, so that neither the receiver's nor the environment's output can simulate the other's), where strict superadditivity of the classical-quantum region is known to occur but no classification exists); (2) as the central open instance, determine the trade-off curve of the qubit depolarizing channel (from which the quantum capacity $Q(\mathcal{N})$ follows as the quantum rate at vanishing classical rate) in the regime between the hashing-bound zero at $p\approx 0.189$ (the noise level at which the best single-letter lower bound on $Q$ vanishes) and the noise level at which the channel becomes antidegradable (its output reconstructible from the environment) and $Q$ is known to vanish, and correspondingly for the $d$-dimensional depolarizing channel in its non-antidegradable regime; (3) prove or refute the conjectured single-letter structural forms of the two analogues: the simultaneous public-private capacity region of a channel, and the static trade-off between simultaneously distillable common randomness and one-way distillable entanglement of a bipartite state, with the three-dimensional extension that adds the forward classical-communication cost as an optional, more ambitious goal; (4) determine the optimal joint source-channel coding rate, in copies of a visible pure-state ensemble per channel use, for an arbitrary pair $(E,\mathcal{N})$ without entanglement assistance, beyond the separable source-then-channel strategy whose suboptimality is shown by the trine-ensemble example. Items (1)-(4) are the source question together with its author-stated extensions and analogues, and they share one objective: replacing the regularized Devetak-Shor region and its conjectured analogues by a complete, single-letter or otherwise computable characterization, together with the resulting optimal source-channel coding rate. They are four heterogeneous open sub-problems rather than four formulations of a single proposition: (1) is the general classification of when the regularization is unnecessary, (2) is its central depolarizing-channel instance, (3) is the single-letter status of the public-private and static conjectures, and (4) is the joint source-channel coding rate built on the region. Each is independently refutable, and no single item settles the others on its own. The items also overlap: the classification in (1), applied to the depolarizing channel, decides only whether that channel's region single-letterizes, giving its exact region or trade-off curve only in the single-letter case.

References

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