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On the Second-Order Asymptotics for Entanglement-Assisted Communication

Nilanjana Datta, Marco Tomamichel, Mark M. Wilde

arXiv:1405.1797v2quant-ph

TL;DR

Finite-use entanglement-assisted codes raise the question of how quickly their rates approach capacity as channel uses and error tolerance vary. The paper derives a second-order lower bound and establishes the conjectured expansion for covariant channels, with results also applying to entanglement-assisted quantum communication.

  • Problem

    Prior one-shot bounds converged to capacity in first order but did not characterize second-order behavior for finite channel uses.

    Source: §1 · pp. 2–3

  • Method

    The paper proves a second-order lower bound using a one-shot coding bound expressed through hypothesis testing relative entropy.

    Source: §3 · pp. 5–6

  • Results

    The direct part of the Gaussian approximation is established, and the conjectured converse is shown for covariant channels.

    Source: §1 · p. 2, §5 · p. 16

  • Takeaways & Limitations

    The results extend to entanglement-assisted quantum communication through the equivalence given by teleportation and super-dense coding.

    Source: §5 · p. 16

    The converse for general channels remains open; existing channel-simulation bounds are unsuitable for fixed error because post-selection scales the error polynomially in n.

    Source: §5 · p. 16

Abstract

from arXiv · show

The entanglement-assisted classical capacity of a quantum channel is known to provide the formal quantum generalization of Shannon's classical channel capacity theorem, in the sense that it admits a single-letter characterization in terms of the quantum mutual information and does not increase in the presence of a noiseless quantum feedback channel from receiver to sender. In this work, we investigate second-order asymptotics of the entanglement-assisted classical communication task. That is, we consider how quickly the rates of entanglement-assisted codes converge to the entanglement-assisted classical capacity of a channel as a function of the number of channel uses and the error tolerance. We define a quantum generalization of the mutual information variance of a channel in the entanglement-assisted setting. For covariant channels, we show that this quantity is equal to the channel dispersion, and thus completely characterize the convergence towards the entanglement-assisted classical capacity when the number of channel uses increases. Our results also apply to entanglement-assisted quantum communication, due to the equivalence between entanglement-assisted classical and quantum communication established by the teleportation and super-dense coding protocols.

1 Introduction

The introduction frames finite-use entanglement-assisted communication as a second-order problem: how the maximum message size approaches capacity with channel uses and error tolerance. It presents a lower bound, a conjectured expansion verified for covariant channels, and applicability to entanglement-assisted quantum communication.

1 Introduction

  • The capacity is single-letter in quantum mutual information and does not increase with noiseless quantum feedback from receiver to sender.Source: §1 · p. 2
  • The paper derives a lower bound on the maximum message size for large but finite n as a function of ε.Source: §1 · p. 2
  • The second-order coding rate b depends on the channel and allowed error threshold ε, with an explicit expression given in Theorem 3.Source: §1 · p. 2
  • The conjectured expansion is shown to hold for covariant channels [Hol02b].The conjecture is stated for all quantum channels, but this result establishes it for the covariant class.Source: §1 · p. 2
  • The results also apply to entanglement-assisted quantum communication through the equivalence established by teleportation and super-dense coding.Source: §1 · p. 3
  • Earlier one-shot bounds converged to the entanglement-assisted capacity in first order but did not characterize second-order asymptotics.Source: §1 · p. 3

2 Notations and Definitions

This section fixes the notation for quantum states and channels, specifies the entanglement-assisted coding task, and introduces mutual-information quantities used to analyze its capacity and second-order behavior.

2 Notations and Definitions

  • A quantum channel is a completely positive trace-preserving map, while partial trace gives a subsystem state and π_A denotes the completely mixed state.The section also defines finite-dimensional Hilbert-space notation and identifies pure states as rank-one states.Source: §2 · pp. 3–4

2.1 Entanglement-Assisted Codes

  • An entanglement-assisted code encodes each classical message into Alice’s system, sends it through the channel, and has Bob measure his systems to infer the message.Code size is the message-set size; the section defines the maximum size achievable with average error at most ε.Source: §2.1 · p. 4

2.2 Information Quantities

  • The section defines mutual information and its variance using the relative entropy and relative entropy variance of a bipartite state against the product of its marginals.It also introduces capacity-achieving resource states and the channel’s minimal and maximal mutual-information variances for lower-bound analysis.Source: §2.2 · p. 5

3 Results

The results establish a second-order lower bound for entanglement-assisted communication and conjecture a matching converse. The conjecture is proved for covariant channels.

3 Results

  • Theorem 3 gives a second-order lower bound on bits sent through n independent channel uses at error threshold ε, with K(n; N, ε) = O(log n).The theorem concerns the maximum number of classical message bits transmitted by an entanglement-assisted protocol.Source: §3 · p. 5
  • The proof first derives a one-shot coding lower bound using hypothesis testing relative entropy, then applies its product-state asymptotic expansion.The expansion used in the second step was derived independently by Tomamichel and Hayashi and by Li.Source: §3 · p. 6
  • The authors conjecture that the lower bound is an equality with K(n; N, ε) = o(√n), and define an entanglement-assisted ε-channel dispersion.The conjecture concerns general quantum channels; the equality is established in the paper only for covariant channels.Source: §3 · p. 6
  • For covariant quantum channels, the paper proves the conjectured equality for the second-order expansion.The argument recapitulates an analysis by Matthews and Wehner and uses the asymptotic expansion of hypothesis testing relative entropy.Source: §3 · p. 6

4 Proofs

The proofs develop one-shot entanglement-assisted coding bounds and use them to establish a second-order achievability result, then prove the matching Gaussian approximation for covariant channels with irreducible input representations.

4.1 Technical Preliminaries

  • The preliminaries define maximally entangled states and invoke the transpose trick for operators on isomorphic Hilbert spaces.The proofs also recall types and relative-entropy tools used in the coding and asymptotic arguments.Source: §4.1 · pp. 6–7

4.1.1 Distance Measures

  • The distance-measure preliminaries state definitions of trace distance and fidelity and give their relation for pairs of pure states.The supplied passages do not state the relation's formula.Source: §4.1.1 · p. 7

4.1.2 Relative Entropies for One-Shot Analysis

  • The text defines hypothesis-testing relative entropy [WR12] and identifies collision and information-spectrum relative entropies as relevant to its analysis [TH13, Def. 8]; [BG14, Thm. 4] relates the latter quantities.Source: §4.1.2 · pp. 7–8

4.2 One-Shot Achievability

  • The one-shot achievability proof randomizes message codewords with unitary encodings, uses pretty-good measurements, and derives a lower bound on expected decoding success.The expected-success bound implies that a deterministic code achieves success probability at least 1 − ε under the stated condition.Source: §4.2 · pp. 9–11

4.3 Second-Order Analysis for Achievability

  • The second-order achievability analysis restricts a tensor-power resource state to types close to its distribution and applies a product-state hypothesis-testing expansion.The resulting bound has a remainder of O(log n).Source: §4.3 · pp. 12–14

4.4 Second-Order Converse for Covariant Quantum Channels

  • For covariant channels with irreducible input representations, averaging over local unitaries reduces the converse to the tensor-power maximally mixed input and yields the Gaussian approximation.The argument uses covariance, convexity, Schur's lemma, and an asymptotic expansion.Source: §4.4 · pp. 14–16

5 Discussion

The discussion distinguishes the established direct part from the conjectured general converse. It explains why existing converse approaches do not settle the Gaussian approximation and identifies the general-channel converse as an open problem.

Discussion

  • The paper establishes the direct part of the Gaussian approximation but leaves the converse conjectural for general channels.The authors say proving the conjecture requires new techniques.Source: §5 · p. 16

Arimoto Converse

  • Arimoto-style converses give exponentially vanishing success above capacity but become trivial in the Gaussian-approximation regime.The cited approach therefore does not provide the needed second-order converse.Source: §5 · p. 16

De Finetti Theorems

  • A de Finetti approach would sacrifice at least order √n systems, significantly affecting the second-order term.The text presents this as a problem with approximating the state by a convex combination of product states.Source: §5 · p. 16

Relation to Channel Simulation

  • The finite-channel simulation bounds in [BCR11] have error scaled by a polynomial in n after post-selection, making them unsuitable for fixed-error statements.The authors say adapting that proof to fixed error is unclear.Source: §5 · p. 16

A Heisenberg-Weyl Operators

The appendix introduces the Heisenberg-Weyl operators through their actions on computational-basis vectors.

A Heisenberg-Weyl Operators

  • The operators X(x) and Z(z) are defined by their actions on qudit computational-basis vectors; when d = 1, both are identity operators.The index shift j ⊕ x is addition modulo d.Source: A Heisenberg-Weyl Operators · p. 17
  • For d = 1, the shift and phase operators both reduce to the identity.This is the stated one-dimensional special case.Source: A Heisenberg-Weyl Operators · p. 17

B The Method of Types

The appendix reviews the method of types used in the proofs, defining empirical distributions and the sequence classes associated with each type.

B The Method of Types

  • A sequence's type is its empirical probability distribution, formed from the frequencies of its letters.The set of all types for sequences of length n is denoted Pn.Source: B The Method of Types · p. 17
  • The type class T^t consists of all sequences with type t, and the appendix states bounds on the number of types and the size of a type class.The supplied passages do not give complete expressions for those bounds.Source: B The Method of Types · p. 17
  • The appendix also introduces the Shannon entropy of a type and the Kullback-Leibler divergence between a type and a distribution.These quantities are used in the listed type-related properties.Source: B The Method of Types · pp. 17–18
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