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The quantum capacity of channels with arbitrarily correlated noise

Francesco Buscemi, Nilanjana Datta

arXiv:0902.0158v5quant-ph

TL;DR

The paper studies quantum communication rates beyond the asymptotic memoryless setting, including finite-use channels with arbitrarily correlated noise. It derives one-shot bounds via entanglement transmission and obtains capacity expressions for arbitrary channel sequences, reducing to regularized coherent information for memoryless channels.

  • Problem

    The paper addresses how to characterize quantum communication rates for finite uses and channels whose successive uses may have arbitrarily correlated noise.

  • Method

    It bounds one-shot entanglement-transmission capacity using decoupling, decoding fidelity, data processing, and smoothed 0-coherent informations.

  • Results

    The bounds give one-shot quantum-capacity bounds and an expression for the quantum capacity of an arbitrary infinite sequence of channels.

  • Takeaways & Limitations

    For memoryless channels, the asymptotic result is the regularized coherent-information expression, providing an alternative form of the LSD theorem.

Abstract

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We study optimal rates for quantum communication over a single use of a channel, which itself can correspond to a finite number of uses of a channel with arbitrarily correlated noise. The corresponding capacity is often referred to as the one-shot quantum capacity. In this paper, we prove bounds on the one-shot quantum capacity of an arbitrary channel. This allows us to compute the quantum capacity of a channel with arbitrarily correlated noise, in the limit of asymptotically many uses of the channel. In the memoryless case, we explicitly show that our results reduce to known expressions for the quantum capacity.

I. INTRODUCTION

The paper addresses quantum communication beyond the usual asymptotic, memoryless setting by studying finite-use capacities and channels with correlated noise. It formulates entanglement transmission with bounded error and derives one-shot bounds that recover familiar asymptotic results for memoryless channels.

  • Motivation: Quantum channels have distinct capacities because communication tasks differ in transmitted information, input states, measurements, auxiliary resources, and classical communication.The quantum capacity considered here concerns quantum information transmission without classical communication or additional resources.
  • Motivation: Existing capacity formulas were primarily evaluated asymptotically under the assumption that successive channel uses experience uncorrelated, memoryless noise.The paper questions both the memoryless assumption and the exclusive focus on asymptotically many uses.
  • Scope and contribution: The paper develops bounds for one-shot capacities and capacities of arbitrary channel sequences, including sequences with memory.A single channel use may itself represent finitely many uses of a channel with correlated noise.
  • Protocol: Because perfect transmission is generally impossible for finite uses, the capacities are defined subject to an error probability at most ε.The parameter ε satisfies ε ≥ 0.
  • Protocol: In entanglement transmission, Alice sends half of a maximally entangled state through the channel, and Bob decodes without classical communication to preserve overlap at least 1 − ε.The resulting one-shot entanglement-transmission capacity is denoted Qent(Φ; ε).
  • Proof strategy: The bounds use decoupling accuracy and decoding fidelity, while the upper bound generalizes arguments based on the quantum data-processing inequality.For asymptotically many uses of a memoryless channel, both bounds converge independently to the familiar quantum-capacity expression without explicitly invoking typicality.
  • Paper organization: The paper organizes the development around quasi-entropies, entanglement transmission, one-shot capacity, and the relation between one-shot entanglement transmission and quantum capacity.Its main result is Theorem 1, followed by asymptotic and sequence-of-channel applications.

B. Quasi-entropies and coherent information

The paper defines smoothed quasi-entropy quantities to characterize finite-error entanglement transmission and uses them to bound one-shot capacity. The central theorem connects these quantities to channel performance through a Stinespring-based construction and integer-rate correction.

  • Quasi-entropies: The quantum relative quasi-entropy is defined for positive operators ρ and σ, a test operator P, and order α ∈ (0, ∞)\{1}.When P is the identity, it reduces to the corresponding Rényi relative entropy.
  • Quasi-entropies: The order-0 quasi-entropy underlies two smoothed quantities used in the main theorem.These quantities are introduced to accommodate finite accuracy, or non-zero error, in the one-shot protocol.
  • Quasi-entropies: The smoothing sets include subnormalized states close in fidelity to ρ and operators P whose acceptance probability on ρ is at least 1 − δ.They are specified by b(ρ; δ) and p(ρ; δ), respectively.
  • Entanglement transmission: Entanglement transmission sends half of a rank-m maximally entangled state through Φ and applies a decoding CPTP map, with no classical communication.The target is a shared state whose overlap with the original maximally entangled state is at least 1 − ε.
  • Entanglement transmission: The paper defines entanglement-transmission fidelity, ε-achievable rates, and the one-shot capacity Qent(Φ; ε) from this protocol.The rate is represented as log m for a positive integer m satisfying the fidelity requirement.
  • Main result: For a chosen input subspace, the analysis constructs a maximally entangled state and a tripartite pure state using a Stinespring isometry of the channel.The resulting reduced states support the entropic quantities appearing in the main theorem.
  • Main result: Theorem 1 bounds one-shot entanglement-transmission capacity using smoothed 0-coherent informations.The lower bound includes a correction 0 ≤ Δ ≤ 1 so that it is the logarithm of a positive integer.
  • Main result: The correction Δ(x) = x − log ⌊2^x⌋ lies between 0 and 1 and decreases rapidly as x increases.This correction accounts for the requirement that achievable dimensions be integer-valued.

V. ONE-SHOT QUANTUM CAPACITY

This section relates minimum-output and entanglement-transmission fidelities, defines the one-shot quantum capacity, and uses their relationship to bound it. The analysis relies on quasi-entropic quantities and their associated conditional entropies.

  • Fidelity-based capacities: The minimum output fidelity optimizes over decoding operations and measures how well channel noise can be corrected.It differs from fidelities measuring a channel’s distance from the identity channel.
  • Fidelity-based capacities: The paper compares entanglement-transmission fidelity with minimum and average output fidelities through a pruning lemma and fidelity relations.The minimum output fidelity is bounded by the average fidelity, and these quantities are connected to entanglement transmission.
  • Fidelity-based capacities: The one-shot quantum capacity is defined using the minimum output fidelity, while entanglement transmission provides an alternative capacity formulation.The paper explicitly adopts the minimum-output-fidelity convention for quantum capacity.
  • Fidelity-based capacities: Corollary 1 relates one-shot entanglement-transmission capacity to one-shot quantum capacity.This relation lets bounds established for entanglement transmission yield bounds for quantum capacity.
  • Entropic tools: Quasi-entropies generate the relative Rényi entropies, conditional entropies, and coherent information used in the capacity analysis.The paper also introduces max-relative entropy, min-conditional entropy, quantum conditional entropy, and coherent information as supporting quantities.

B. Smoothed entropies

The paper introduces state-smoothed and operator-smoothed entropies to handle finite accuracy in one-shot protocols. Operator smoothing arises naturally from quasi-entropies, and the resulting quantities support monotonicity and coherent-information constructions.

  • B. Smoothed entropies: Finite-error one-shot protocols require smoothed entropies rather than unsmoothed quantities.The paper considers both state-smoothed and operator-smoothed classes.
  • B. Smoothed entropies: State-smoothed entropies originate with Renner, whereas operator-smoothed entropies arise naturally from quasi-entropies.The two smoothing approaches are treated as distinct classes.
  • B. Smoothed entropies: The paper defines smoothed conditional entropies and smoothed coherent information for bipartite states.These quantities are constructed from state- and operator-smoothed entropy expressions.
  • B. Smoothed entropies: The proof establishes monotonicity in α for the relevant quasi-entropy-derived quantity.The monotonicity follows from the stated derivative and convexity properties.

VII. PROOF OF THEOREM 1

The proof of the lower bound uses random coding to connect decoupling-based estimates with entanglement-transmission fidelity. It then bounds the resulting group-averaged fidelity using smoothed entropic quantities.

  • VII. Proof of Theorem 1: The lower bound begins with a random-coding lower bound on entanglement-transmission fidelity.The proof seeks a subspace whose transmission fidelity meets the random-coding estimate.
  • VII. Proof of Theorem 1: Lemma 11 bounds the fidelity achievable by restricting a channel to an s-dimensional input subspace.The restriction is defined by projecting inputs onto the subspace before applying the channel.
  • VII. Proof of Theorem 1: For a channel acting noiselessly on the selected subspace, the entanglement-transmission fidelity equals 1 for every m ≤ s.The conclusion follows from the factorization conditions and entropy evaluations for the environment.
  • VII. Proof of Theorem 1: The proof uses group averaging over subspaces and evaluates the resulting fidelity through decoupling estimates.The construction starts from a purified state and a maximally entangled state on the selected subspace.
  • VII. Proof of Theorem 1: A decoupling-to-decoding-fidelity inequality and trace-distance estimates convert environmental decoupling into a lower bound on transmission fidelity.The argument combines Lemma 4 with the fidelity inequality F^2(ρ,σ) ≥ 1 − ||ρ − σ||_1.
  • VII. Proof of Theorem 1: The proof optimizes over an environmental state and expresses the resulting bound using smoothed conditional or coherent information.The α = 2 specialization supplies the relevant operator-smoothed quantity.

2. Proof of the lower bound in (18)

This subsection converts the fidelity lower bound into achievable-rate and one-shot-capacity lower bounds. The resulting conditions apply first to a restricted channel and then to the full channel.

  • 2. Proof of the lower bound in (18): Lemma 11 yields a sufficient condition for R = log m to be an ε-achievable entanglement-transmission rate through the restricted channel.The condition holds for δ ∈ [0, ε/4].
  • 2. Proof of the lower bound in (18): Because the restricted subspace dimension satisfies s ≤ d = dim HA, the sufficient condition can be stated using the full input dimension.This produces a dimension-based rate criterion.
  • 2. Proof of the lower bound in (18): The achievable-rate condition implies a lower bound on the one-shot entanglement-transmission capacity.The bound is obtained for any δ ∈ [0, ε/4].
  • 2. Proof of the lower bound in (18): The capacity bound extends from the restricted channel to the original channel, with a correction term ensuring the rate is the logarithm of a positive integer.The correction quantity satisfies Δ ≤ 1.
  • 2. Proof of the lower bound in (18): The proof permits the particular choice δ = ε/8 within the stated admissible range.This choice is recorded as a special case of the general δ condition.

B. Proof of the upper bound in Theorem 1

The upper bound in Theorem 1 is derived using a quantum data-processing inequality and a sequence of standard reductions from achievable entanglement transmission rates to coherent-information expressions.

  • Conclusion: The proof concludes that the standard data-processing argument yields the upper bound in Theorem 1.The theorem concerns the one-shot entanglement-transmission capacity of an arbitrary quantum channel.
  • Data-processing inequality: The upper-bound proof begins from a quantum data-processing inequality for a bipartite state, a channel, and a smoothing parameter.The inequality is established through an operator construction involving the adjoint channel and the Gentle Measurement Lemma.
  • Application to achievable rates: The proof applies the inequality to an s-dimensional input subspace and an associated decoding operation achieving the ε-achievable rate.Here s is defined from the maximum ε-achievable entanglement-transmission rate.
  • Identification of the bound: The resulting upper-bound expression is identified with the coherent-information quantity defined earlier in the paper.The final equality follows by recognizing the last line as the relevant coherent-information expression.

VIII. QUANTUM CAPACITY OF A SEQUENCE OF CHANNELS

The paper extends one-shot capacity bounds to sequences of channels, defining capacity per use and asymptotic capacity, and shows that the memoryless case recovers the known quantum-capacity expression.

  • Capacity per use: A sequence channel Φ_n may represent n uses of a channel with arbitrary memory, so Q_x(Φ_n; ε)/n measures capacity per channel use.This finite-n normalization is motivated by practical transmission over a large but finite number of uses.
  • Finite-blocklength bounds: Theorem 1 supplies bounds on this per-use quantity, with correction terms that decrease rapidly as n increases.The bounds become sharp for entanglement transmission even at finite n.
  • Asymptotic capacity: The asymptotic capacity is defined for an infinite channel sequence, and equivalent fidelity formulations yield the same asymptotic quantum capacity.The relevant limit exists in the formulation used for the asymptotic capacity.
  • Memoryless channels: For memoryless channels, the sequence is {Φ^⊗n}, and the paper presents an alternative formulation of the LSD theorem.The memoryless expression is equivalent to the entanglement-transmission formulation through the stated relation.

1. Direct part of Theorem 2

The direct part of Theorem 2 evaluates the asymptotic lower bound by relating smoothed quantities to ordinary coherent information and controlling their continuity errors.

  • Lower-bound reduction: The proof evaluates the asymptotic behavior of the lower bound from Theorem 1 after the first two terms vanish.The remaining task is the evaluation of the third term.
  • Supporting lemmas: Two supporting lemmas relate operator-smoothed quantities to quantum relative entropy and operator-smoothed coherent information to ordinary coherent information.These relations are identified as essential for evaluating the bound.
  • Relative-entropy comparison: Lemma 13 compares an operator-smoothed relative-entropy quantity with quantum relative entropy using monotonicity, matrix convexity, and Fannes’ inequality.The proof constructs a modified operator and controls its smoothing parameter before applying continuity arguments.
  • Coherent-information comparison: Lemma 14 bounds the smoothed coherent-information expression by applying entropy continuity estimates to the marginal and joint states.The proof uses dimension-dependent entropy bounds and Fannes’ continuity property.
  • Completion of the direct part: Combining these lemmas with the lower-bound expression yields the direct part of the memoryless-channel capacity theorem.The argument uses rescaling identities for relative entropy and the completely mixed state in intermediate steps.

B. Multiple uses of an arbitrary channel

For arbitrary channel sequences, the paper uses quantum information-spectrum divergence and coherent-information rates to characterize quantum capacity through direct and weak-converse arguments.

  • Information-spectrum framework: The quantum information-spectrum method introduces spectral sup- and inf-divergence rates for sequences of quantum states.These rates are defined using difference operators Π_n(γ)=ρ_n−2^nγσ_n.
  • Spectral coherent information: The corresponding spectral sup- and inf-coherent information rates are defined by analogy with ordinary coherent information.The paper notes that some infima can be replaced by minima under the stated conditions.
  • Capacity characterization: Theorem 3 states the quantum capacity of an arbitrary channel sequence in terms of these spectral quantities.Its proof follows from Theorem 1 together with the direct and weak-converse lemmas.
  • Direct and converse arguments: The direct part establishes achievability for sequences of bipartite states, while the weak converse rules out rates exceeding the spectral bound.The weak-converse proof proceeds by contradiction using sequences of operators satisfying a fixed acceptance condition.

IX. DISCUSSION

The paper establishes one-shot entanglement-transmission bounds for arbitrary channels and derives quantum-capacity expressions for memoryless and arbitrary channel sequences.

  • The bounds apply to arbitrary quantum channels, including finite uses with arbitrarily correlated noise.
  • For multiple uses of a memoryless channel, the results yield quantum capacity in terms of regularized coherent information.This provides an alternative form of the LSD theorem, known to be equivalent to it.
  • The Quantum Information Spectrum Method gives an expression for the quantum capacity of an arbitrary infinite sequence of channels.
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