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Lower and upper bounds on nonunital qubit channel capacities

Sergey N. Filippov

arXiv:1802.00646v2quant-ph

TL;DR

Classical capacity is well understood for unital qubit channels but not for general nonunital ones. The paper uses a decomposition relating nonunital and unital channels to derive capacity bounds. For mixtures of amplitude-damping and depolarizing channels, its upper bound is tighter than prior upper bounds over a stated parameter interval.

  • Problem

    Classical capacity is known for unital qubit channels, but general nonunital qubit-channel capacities remain difficult to determine and lack a closed-form χ-capacity.

    Source: §I · p. 1

  • Method

    The paper relates nonunital channels to unital ones through completely positive maps and applies a capacity inequality to obtain lower and upper bounds.

    Source: §II · p. 1, §III · pp. 2–3

  • Results

    For 0.29 < p < 0.33, the upper bound is tighter than Ref. for p < 0.33 and Ref. for p > 0.29.

    Source: §III · p. 4

  • Takeaways & Limitations

    The bounds hold for regularized Holevo capacity, and the lower bound is achievable with factorized coding for unital channels.

    Source: §IV · p. 4

    The decomposition does not directly order quantum capacities because its completely positive maps ΦA and ΦB are not trace preserving.

    Source: §IV · p. 4

Abstract

from arXiv · show

Classical capacity of unital qubit channels is well known, whereas that of nonunital qubit channels is not. We find lower and upper bounds on classical capacity of nonunital qubit channels by using a recently developed decomposition technique relating nonunital and unital positive qubit maps.

I. INTRODUCTION

The section frames classical information transmission through quantum channels and the difficulty of determining general nonunital qubit-channel capacity. It introduces lower and upper bounds as the paper’s response to that gap.

I. INTRODUCTION

  • Classical information is encoded into n-qubit density operators, transmitted through n uses of a channel, and decoded by a measurement followed by classical processing.The decoder’s measurement assigns operators to observed outcomes, which are then processed into a message.Source: §I · p. 1
  • Restricting inputs to product states yields C(1)(Φ) ≤ Cχ(Φ), with equality for pseudoclassical channels, including all unital and some nonunital qubit channels.The paper considers general nonunital qubit channels, not only those in the pseudoclassical class.Source: §I · p. 1
  • The paper seeks lower and upper bounds on qubit-channel capacity, whose estimation is difficult because the χ-capacity has no known closed formula for general qubit channels.The regularized capacity is harder to estimate because additivity has not been proved for a general qubit channel.Source: §I · p. 1
  • For some channels, the paper’s upper bound outperforms previously known upper bounds computed using semidefinite programming.The paper compares its bounds with known bounds that can be computed with semidefinite programming.Source: §I · p. 1

II. RELATION BETWEEN UNITAL AND NONUNITAL QUBIT CHANNELS

The section relates nonunital qubit channels to unital ones through a decomposition using positive maps. This connection lets the paper use known capacity results for unital qubit channels.

II. RELATION BETWEEN UNITAL AND NONUNITAL QUBIT CHANNELS

  • For positivity-improving Φ, the Sinkhorn normal form constructs a trace-preserving, unital map Υ using positive definite operators A and B.The relation is described as a quantum analogue of Sinkhorn’s theorem for strictly positive square matrices [25].Source: §II · pp. 1–2
  • When Φ is also completely positive and trace preserving, the resulting map Υ is completely positive and trace preserving as well.The construction applies to a given nonunital qubit channel using the particular operators A and B described in Refs..Source: §II · p. 2
  • For non-boundary nonunital qubit channels, the decomposition expresses the channel as a concatenation of completely positive maps around a unital channel.The factor maps are ΦB−1 and ΦA−1, and the middle map is unital.Source: §II · p. 2
  • The capacity of Υ can be evaluated using the unital-qubit formula because unital qubit channels satisfy additivity and their classical capacity equals their Holevo capacity.Unital channels can be represented in diagonal form in the conventional Pauli basis.Source: §II · p. 2

III. BOUNDS ON CLASSICAL CAPACITY OF NONUNITAL QUBIT CHANNELS

The section derives capacity bounds for nonunital qubit channels by relating them to unital channels, then illustrates the bounds on generalized amplitude-damping and mixed channels.

III. BOUNDS ON CLASSICAL CAPACITY OF NONUNITAL QUBIT CHANNELS

  • Theorem 1 bounds C(Φ) below by C(Ψ) minus 2 log2(∥A∥∥B∥) when Ψ = ΦA ◦ Φ ◦ ΦB is a channel.Applying the theorem to the relations between nonunital Φ and unital Υ yields upper and lower capacity bounds.Source: §III · pp. 2–3
  • −2 log2(∥A∥∥B∥) ≤ C(Φ)−C(Υ) ≤ 2 log2(∥A−1∥∥B−1∥) bounds the capacity difference for the unital map Υ = ΦA ◦ Φ ◦ ΦB.The result applies when Φ is a unital qubit channel in the interior of positive qubit maps and A, B make Υ unital.Source: §III · p. 3
  • For the four-parameter channel, complete positivity and |t3|+|λ3| < 1 ensure it lies inside the positive-map cone, enabling its decomposition into a unital map and filters.Substituting the diagonal-filter norms into Proposition 1 gives lower and upper capacity bounds.Source: §III · p. 3
  • For GAD channels with p ∈ (0, 1/2), C(1)(ΦGAD) < Cχ(ΦGAD) ≤ C(ΦGAD) ≤ C↑GAD; as p → 0, the derived bounds become trivial.At p = 0, the channel is conventional amplitude damping and lies outside the interior of the positive-map cone, so the filter-norm products diverge.Source: §III · p. 3
  • For the mixed channel, the upper bound is tighter than Ref. when p < 0.33 and Ref. when p > 0.29, outperforming both for 0.29 < p < 0.33.The authors report that the lower bound is not as precise as the χ-capacity.Source: §III · p. 4

IV. CONCLUSIONS

The paper gives lower and upper bounds on nonunital qubit-channel capacity, with factorized coding achieving the lower-bound rate. For some amplitude-damping and depolarizing mixtures, its upper bound outperforms prior bounds; extending the decomposition to quantum capacity remains a research direction.

IV. CONCLUSIONS

  • Factorized coding achieves the transmission rate corresponding to the lower bound, and the result holds for regularized Holevo capacity.The coding procedure is based on the known optimal factorized-state coding for unital qubit channels.Source: §IV · p. 4
  • Figure 2 plots the mixture-channel capacity bounds against p, distinguishing the derived bounds, χ-capacity lower bound, and prior upper bounds by line style.The derived lower and upper bounds are solid lines; prior upper bounds are plotted with dash-dotted and dashed lines.Source: Figure 2 · p. 4
  • For some mixtures of amplitude-damping and depolarizing qubit channels, the derived upper bound outperforms previously known upper bounds.The authors consider a four-parameter family that includes generalized amplitude-damping channels.Source: §IV · p. 4
  • The authors identify quantum capacity as a promising application of the decomposition, but its non-trace-preserving factors prevent an immediate capacity-ordering conclusion.At least one factor is not trace decreasing; studying these issues is left for future work.Source: §IV · p. 4
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