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Potential capacities of quantum channels
Andreas Winter, Dong Yang
TL;DR
Quantum-channel capacities are difficult to characterize because non-additivity makes their formulas depend on arbitrarily many channel uses, while a channel’s utility can also depend on contextual channels. The paper defines potential capacities and develops bounds using additivity and Hadamard-channel lifting. Those bounds show that no noisy channel can be activated into a noiseless one by a contextual channel for the classical, quantum, or private capacity.
Problem
Non-additivity makes quantum-channel capacities difficult to compute, and contextual channels can affect a channel’s usefulness, motivating a measure of its capability in the best context.
Source: §I · p. 1Method
The paper defines potential capacity and bounds it by using strongly additive channels, including lifting a channel to a Hadamard channel to bound its quantum capacity.
Source: C. Potential quantum capacity · pp. 5–6Results
A noisy channel cannot be activated into a noiseless one by any contextual channel for the classical, quantum, or private capacity.
Source: §V · p. 8Takeaways & Limitations
Potential capacity captures a channel’s maximum transmission capability when combined with any contextual channel.
Source: §I · p. 1For most capacities, calculating the potential version is expected to be as prohibitive as calculating the plain version, so the paper relies on bounds.
Source: §V · p. 8
Abstract
from arXiv · showhide
We introduce potential capacities of quantum channels in an operational way and provide upper bounds for these quantities, which quantify the ultimate limit of usefulness of a channel for a given task in the best possible context. Unfortunately, except for a few isolated cases, potential capacities seem to be as hard to compute as their "plain" analogues. We thus study upper bounds on some potential capacities: For the classical capacity, we give an upper bound in terms of the entanglement of formation. To establish a bound for the quantum and private capacity, we first "lift" the channel to a Hadamard channel and then prove that the quantum and private capacity of a Hadamard channel is strongly additive, implying that for these channels, potential and plain capacity are equal. Employing these upper bounds we show that if a channel is noisy, however close it is to the noiseless channel, then it cannot be activated into the noiseless channel by any other contextual channel; this conclusion holds for all the three capacities. We also discuss the so-called environment-assisted quantum capacity, because we are able to characterize its "potential" version.
I. MOTIVATION
Quantum-channel capacities are difficult to characterize because non-additivity leads to regularization over arbitrarily many channel uses. The paper introduces potential capacity to account for contextual channels and uses bounds to address whether a noisy channel can become noiseless with assistance.
I. MOTIVATION
- Non-additivity makes capacity calculations difficult: the best-known formulas regularize over an unbounded number of channel uses, complicating even qualitative questions.Entangled inputs can transmit more information than product inputs.Source: §I · p. 1
- Superactivation occurs when two channels unable to transmit quantum information individually transmit at a positive rate when used together.This illustrates how entangled inputs across channels can change their usefulness.Source: §I · p. 1
- Potential capacity captures a channel’s maximum transmission capability when combined with any contextual channel.The motivation is that a channel’s utility can depend on the other channels available.Source: §I · p. 1
- The paper uses upper bounds to show that a noisy channel cannot become noiseless with a zero-capacity side channel, even when its quantum capacity is at most log d − δ.The motivating question is whether assistance could make such a channel perfectly noiseless.Source: §I · p. 1
II. NOTATION AND PRELIMINARIES
The preliminaries establish the channel and entropy notation and review the capacity formulas and channel properties used later. They also introduce entanglement-assisted and environment-assisted capacities.
II. NOTATION AND PRELIMINARIES
- A quantum channel is a completely positive, trace-preserving map from input system A to output system B, representable through an isometry and an environment.The complementary channel is obtained by tracing out B rather than the environment.Source: §II · p. 2
- The preliminaries define reduced states, von Neumann entropy, conditional entropy, mutual information, and conditional mutual information.The paper assumes finite-dimensional Hilbert spaces.Source: §II · pp. 1–2
- Classical, quantum, and private capacities are reviewed through regularized formulas because their single-letter quantities are non-additive.The classical capacity is the rate for reliably transmitting classical information through a quantum channel.Source: §II · p. 2
- A degradable channel can simulate its complementary channel, and degradable channels have additive single-letter quantum capacity with quantum and private capacities equal to single-letter coherent information.The review also defines entanglement-assisted classical capacity using shared prior entanglement and environment-assisted capacity using classical feed-forward from the environment.Source: §II · p. 2
III. POTENTIAL CAPACITIES
Potential capacity measures the best capability a channel can achieve with contextual assistance, while potential single-letter and regularized capacities have different additivity properties from their plain counterparts. Strong additivity identifies when a potential capacity equals the plain capacity.
III. POTENTIAL CAPACITIES
- A function is weakly-additive when f(N ⊗ n)=nf(N), and strongly-additive when f(N ⊗ M)=f(N)+f(M) for every channel pair.Strong additivity implies weak additivity, but not conversely.Source: §III · p. 2
- Potential capacity describes the maximum transmission capability achievable when a channel is combined with any contextual channel.The definition addresses the dependence of channel utility on context that arises from non-additivity.Source: §III · p. 3
- The potential capacity equals the plain capacity if and only if the channel is strongly additive.The paper also states that potential capacities are sub-additive and are upper-bounded by their potential single-letter forms.Source: §III · p. 3
- Unlike plain capacities, whose single-letter forms are lower bounds on their regularized forms, potential counterparts have the reverse relation.The potential and potential single-letter capacities are described as sub-additive.Source: §III · p. 3
IV. FIVE CONCRETE POTENTIAL CAPACITIES
The examples include an exactly characterized entanglement-assisted potential capacity and a non-additive environment-assisted capacity whose potential can also be evaluated. For the classical, quantum, and private capacities, the paper gives upper bounds instead.
IV. FIVE CONCRETE POTENTIAL CAPACITIES
- The entanglement-assisted classical capacity is strongly additive, so it equals its regularization and its potential capacity.This is presented as a trivial case among the concrete examples.Source: §IV · p. 3
- The environment-assisted quantum capacity is non-additive but has a single-letter formula, and its potential is evaluable in a simple single-letter formula.For the subsequent classical, private, and quantum capacities, the paper gives only upper bounds.Source: §IV · p. 3
- The paper distinguishes exact potential-capacity evaluations from upper bounds: the latter are used for the subsequent classical, private, and quantum capacities.The environment-assisted case is introduced as the case whose potential can be evaluated directly.Source: §IV · p. 3
A. Potential environment-assisted capacity
The section considers two types of zero-environment-assisted-capacity channels for use as context channels. Its proof uses them to vary input or output entropy independently.
A. Potential environment-assisted capacity
- The proposed context channels have either a one-dimensional input or a one-dimensional output.Both types have zero environment-assisted quantum capacity.Source: A. Potential environment-assisted… · p. 4
- The proof uses the first type, whose only input state has zero entropy, or the second, whose only output state has zero entropy.Source: A. Potential environment-assisted… · p. 4
- Tensoring with these channels can increase either the output entropy or the input entropy by an arbitrary amount without changing the other, respectively.Source: A. Potential environment-assisted… · p. 4
B. Potential classical capacity
The section bounds potential classical capacity using an entanglement measure and applies that bound to show that an imperfect channel cannot be made perfect by contextual assistance. The stated result also covers unequal input and output dimensions.
B. Potential classical capacity
- The paper identifies G as the particular entanglement measure used in its potential classical-capacity bound.G is faithful: it is zero exactly when the bipartite state is separable.Source: B. Potential classical capacity · pp. 4–5
- The section relates potential classical capacity to the single-letter Holevo capacity and establishes an upper bound using a specific entanglement measure.Source: B. Potential classical capacity · p. 4
- An imperfect channel for single-letter classical transmission cannot be activated into a perfect one by any contextual channel.Source: B. Potential classical capacity · p. 4
- The nonactivation result extends to the case din < dout and rules out activation by any contextual channel.The paper notes that covered only the case χ(N) < log dout implying C(N) < log dout.Source: B. Potential classical capacity · p. 5
C. Potential quantum capacity
The section bounds potential quantum capacity by lifting a channel to a Hadamard channel, whose single-letter quantum capacity is strongly additive. The bound is expressed using the channel’s entanglement of formation, and the section concludes that noisy channels cannot be activated into perfect ones by zero-quantum-capacity channels.
C. Potential quantum capacity
- The lifting strategy uses simulation: if N is simulated by a degradable channel, its single-letter capacity supplies an upper bound on Q(N).The section motivates lifting to a strongly additive channel to bound potential capacity.Source: C. Potential quantum capacity · p. 5
- Hadamard channels are a special class of degradable channels whose complementary channel can be simulated by measuring and preparing states.Source: C. Potential quantum capacity · p. 5
- A Hadamard channel has strongly additive single-letter quantum capacity, so its potential quantum capacity equals its single-letter quantum capacity.Source: C. Potential quantum capacity · pp. 5–6
- The lifting gives the receiver a coherent copy of the environment, making the complementary channel entanglement-breaking and the lift Hadamard.Source: C. Potential quantum capacity · p. 6
- The canonical lifting gives an upper bound on potential quantum capacity in terms of the original channel’s entanglement of formation.The optimal canonical lifting’s quantum capacity equals the channel’s entanglement of formation.Source: C. Potential quantum capacity · p. 6
- A channel that is not perfect for quantum transmission has nonmaximal potential single-letter quantum capacity and cannot be activated to a perfect one by zero-quantum-capacity channels.Source: C. Potential quantum capacity · p. 7
D. Potential private capacity
The section establishes strong additivity of private capacity for Hadamard channels, making their potential private capacity equal to their private capacity. It also gives an entanglement-of-formation upper bound and rules out activation of imperfect private channels into perfect ones.
D. Potential private capacity
- The single-letter private capacity of a Hadamard channel is strongly additive under tensoring with any contextual channel.Source: D. Potential private capacity · p. 7
- For Hadamard channels, the potential private capacity equals the private capacity, which in turn equals Q(1)(N).Source: D. Potential private capacity · p. 8
- For any channel, potential private capacity is bounded above by potential single-letter private capacity, which is bounded above by channel entanglement of formation.Source: D. Potential private capacity · p. 8
- A channel that is imperfect for private information cannot be activated into a perfect one by any contextual channel.Source: D. Potential private capacity · p. 8
V. DISCUSSION AND OPEN QUESTIONS
Potential capacities quantify a channel’s nonadditivity in its most favorable context. The discussion connects these quantities to bounds, fair pricing, and open questions about their scope and characterization.
V. DISCUSSION AND OPEN QUESTIONS
- A noisy channel cannot be activated into a noiseless one by any contextual channel for classical, quantum, or private capacity.The paper presents this as its central result and an improvement on previous statements.Source: §V · p. 8
- Hadamard-channel lifting yields an outer bound on the potential region of achievable triples (q, c, e), using strong additivity of the qubitcbit-ebit tradeoff region.Source: §V · p. 8
- The study treats basic capacities and one exactly characterized potential capacity, QA; for most capacities, exact computation is expected to be prohibitive, leaving bounds as the likely recourse.The discussion notes exact characterizations for some zero-error potential capacities [38].Source: §V · p. 8
- For quantum and private capacity, symmetric side-channel-assisted capacities serve as fair prices, with equality asymptotically attained by symmetric contextual channels.The discussion identifies Qss as a fair price for quantum capacity and extends the reasoning to Pss.Source: §V · pp. 8–9
- In zero-error capacities, fair pricing with K = log α and V = log ϑ is established, while a corresponding value for Shannon-theoretic capacities remains an open question.Source: §V · p. 8
- Whether Qss or Pss are unique fair prices, whether alternative fair pricing schemes exist, and whether analogous statements hold for C or χ remain open.Source: §V · p. 9
APPENDIX
The appendix relates conditional entropy, squashed entanglement, and entanglement of formation, then characterizes states attaining key equalities through subsystem structure.
APPENDIX
- For mixed ρBE, S(B) − S(BE) ≤ C←(ρBE); equality is obtained when a unitary on B yields the stated factorization.The proof uses a purification and equality EF(R : B) = S(R) to establish that ρRE is a product state.Source: APPENDIX · p. 9
- In the equality case, the state factors as ρBL ⊗ φBRE, with HB = HBL ⊗ HBR and φBRE pure.Source: APPENDIX · p. 9
- For a state ρBE, S(B) − S(BE) ≤ G(B : E) ≤ EF(B : E), and equality with G implies equality with EF.Source: APPENDIX · p. 9
- Equality S(B) − S(BE) = EF(B : E) holds exactly for states with the stated direct-sum tensor-product structure and pure component states.The proof iterates the decomposition until the component states are pure; each iteration reduces the dimension of B.Source: APPENDIX · pp. 9–10