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Trade-off capacities of the quantum Hadamard channels

Kamil Bradler, Patrick Hayden, Dave Touchette, Mark M. Wilde

arXiv:1001.1732v2quant-phcs.ITmath.IT

TL;DR

Capacity formulas for quantum channels are often intractable because they require optimization over many channel uses. This paper proves single-letter formulas for the triple trade-off capacity region of Hadamard channels and computes regions for generalized dephasing, cloning, and Unruh channels. For all three examples, the classically-enhanced father protocol beats simple time-sharing.

  • Problem

    Quantum-channel capacity formulas often require optimization over infinitely many channel uses, making them intractable in the general case.

    Source: §I · p. 1

  • Method

    The paper proves single-letter formulas for the triple trade-off capacity region of Hadamard channels and computes example regions for dephasing, cloning, and Unruh channels.

    Source: §I · p. 2, §V · p. 11

  • Results

    The classically-enhanced father protocol beats a simple time-sharing strategy for generalized dephasing, cloning, and Unruh channels.

    Source: §VII · p. 17

  • Takeaways & Limitations

    The findings stress the need for non-trivial coding techniques when trading multiple resources.

    Source: §VII · p. 17

    Whether versions of the Unruh channel beyond its original definition have single-letterizable CQE capacity regions remains open.

    Source: §VII · p. 18

Abstract

from arXiv · show

Coding theorems in quantum Shannon theory express the ultimate rates at which a sender can transmit information over a noisy quantum channel. More often than not, the known formulas expressing these transmission rates are intractable, requiring an optimization over an infinite number of uses of the channel. Researchers have rarely found quantum channels with a tractable classical or quantum capacity, but when such a finding occurs, it demonstrates a complete understanding of that channel's capabilities for transmitting classical or quantum information. Here, we show that the three-dimensional capacity region for entanglement-assisted transmission of classical and quantum information is tractable for the Hadamard class of channels. Examples of Hadamard channels include generalized dephasing channels, cloning channels, and the Unruh channel. The generalized dephasing channels and the cloning channels are natural processes that occur in quantum systems through the loss of quantum coherence or stimulated emission, respectively. The Unruh channel is a noisy process that occurs in relativistic quantum information theory as a result of the Unruh effect and bears a strong relationship to the cloning channels. We give exact formulas for the entanglement-assisted classical and quantum communication capacity regions of these channels. The coding strategy for each of these examples is superior to a naive time-sharing strategy, and we introduce a measure to determine this improvement.

I. INTRODUCTION … B. Information-Theoretic Quantities

The paper establishes tractable, single-letter capacity regions for Hadamard channels, extending quantum Shannon theory’s study of trade-offs among noiseless communication resources. The sections also introduce the channel and information-theoretic notation used to state those results.

I. INTRODUCTION

  • Quantum Shannon theory studies limits on transmitting classical or quantum information and generating shared entanglement; these foundational tasks are captured by the HSW and LSD coding theorems [1, 2, 3–5, 11, 12].Source: §I · p. 1
  • Single-letter capacity formulas are rare in quantum Shannon theory: general classical and quantum capacities appear intractable, and single-letterization techniques for known examples are ad hoc.The introduction identifies channels with single-letter formulas for both classical and quantum capacity, including generalized dephasing, cloning, and Unruh channels.Source: §I · pp. 1–2
  • Prior work characterized triple trade-offs for entanglement-assisted classical and quantum communication, with single-letter regions for generalized dephasing and erasure channels [40] [41].The classically-enhanced father protocol was also reported to outperform time-sharing [40].Source: §I · p. 2
  • The Hadamard class has a single-letter triple trade-off capacity region, with generalized dephasing, cloning, and Unruh channels as examples; their coding strategies beat time-sharing.The paper also introduces a measure of the improvement over time-sharing.Source: §I · pp. 2–3

A. Quantum Channels and Classical-Quantum States

  • The channel notation covers CPTP maps, Kraus and isometric extensions, complementary channels and degradability, and classical–quantum ensembles transmitted through a channel.Source: A. Quantum Channels and… · p. 3

B. Information-Theoretic Quantities

  • The information-theoretic notation defines von Neumann entropy, conditional entropy, mutual information, coherent information, and the Holevo quantity for quantum and classical–quantum states.It also notes the joint entropy relation when the conditioning system is classical [61].Source: B. Information-Theoretic Quantities · p. 4

C. Quantum Channels … 3. Generalized Dephasing Channels

The section defines Hadamard channels through their entanglement-breaking complements and explains why their structure supports single-letterization of CQ and CE capacity regions. Generalized dephasing channels model coherence loss without energy loss and belong to the Hadamard class.

1. Entanglement-breaking Channels

  • An entanglement-breaking channel can be viewed as a noisy channel, a complete von Neumann measurement, and another noisy channel, and it admits unit-rank Kraus operators.The Kraus operators need not arise from orthonormal sets of input and output states.Source: §1 · p. 4
  • The classical capacity of an entanglement-breaking channel single-letterizes, while its quantum capacity vanishes because the channel destroys entanglement.Source: §1 · p. 4

2. Hadamard Channels

  • Hadamard channels have entanglement-breaking complementary channels and can be represented by a Hadamard product with a Gram matrix.The product is taken with respect to the output basis.Source: §2 · pp. 4–5
  • Their degrading map measures the output and prepares an environment state, making it entanglement-breaking and crucial to single-letterizing both CQ and CE capacity regions.The map simulates the complementary channel by composing a von Neumann measurement with conditional state preparation.Source: §2 · p. 5

3. Generalized Dephasing Channels

  • Generalized dephasing channels model noise that loses quantum coherence without losing energy; they preserve states diagonal in the dephasing basis but mix coherent superpositions.The passage notes that this noise dominates, for example, in superconducting qubits.Source: §3 · p. 5
  • Generalized dephasing channels are Hadamard channels because their complementary channels are entanglement-breaking.Their isometric extension has the Hadamard-channel form, with the dephasing input states forming an orthonormal basis.Source: §3 · pp. 5–6
  • For a qubit dephasing channel, a bit flip on the input leaves the eigenvalues of the resulting environment state unchanged.Source: §3 · p. 6

4. Cloning Channels

The section characterizes universal 1 →N qubit cloning channels and establishes that every such channel is a Hadamard channel because it is entanglement-breaking. It also describes their isometric and Kraus representations.

4. Cloning Channels

  • A universal 1 →N cloner approximately copies an input with maximal copy fidelity independent of its state, consistent with the no-cloning theorem’s limit on exact cloning.The passage notes that approximate cloning is possible when fidelity is not too high.Source: §4 · p. 6
  • The section focuses on qubit channels that take one input qubit to N identical approximate copies and are unitarily covariant.An input unitary V acts on the output through an irreducible representation RV.Source: §4 · p. 6
  • The channel is specified through an isometric extension using symmetric N-qubit output states and includes Kraus representations for the cloning and complementary channels.The output and environment bases are described using normalized uniform superpositions of computational basis states.Source: §4 · p. 6
  • Every 1 →N cloning channel is entanglement-breaking and therefore belongs to the Hadamard class.For 1 →2 cloning, unit-rank Kraus operators of the complementary channel demonstrate this property; the paper states that other arguments establish it for every 1 →N channel.Source: §4 · p. 7

5. Unruh channels … B. Entanglement-Assisted Classical Capacity Region

The paper states that the entanglement-assisted classical and classically enhanced quantum capacity regions are single-letter for all Hadamard channels. It also describes the Unruh channel’s block structure and reviews capacity-region formulations for classical, quantum, and entanglement-assisted communication.

5. Unruh channels

  • The Unruh channel output is an infinite-dimensional block-diagonal state whose Nth block is an instance of a 1→N cloning channel.It represents the state detected by a uniformly accelerated observer when the input is a dualrail photonic qubit prepared by a Minkowski observer.Source: §5 · p. 7
  • The acceleration parameter z strictly increases with acceleration, and each Sl is the output of a 1→(l−1) cloning channel.The complementary channel has a similar structure, with Sl replaced by the complementary cloning-channel output Sc l.Source: §5 · p. 7

III. REVIEW OF CAPACITY REGIONS

A. Classically-Enhanced Quantum Capacity Region

  • A classically enhanced quantum protocol defines achievable rates C and Q for classical messages and quantum states transmitted over many uses of a noisy channel.The rates are measured in bits and qubits per channel use, respectively, with vanishing error and fidelity approaching unity.Source: A. Classically-Enhanced Quantum… · p. 7
  • The classically enhanced quantum capacity region also has a single-letter characterization for all Hadamard channels.The paper identifies this as a result shown in Section IV A.Source: A. Classically-Enhanced Quantum… · p. 7

B. Entanglement-Assisted Classical Capacity Region

  • An entanglement-assisted classical protocol defines achievable rates C and E for classical communication while consuming shared noiseless entanglement.The rates are measured in bits and ebits per channel use, respectively.Source: B. Entanglement-Assisted Classical… · pp. 7–8
  • For every Hadamard channel, the entanglement-assisted classical capacity region has a single-letter characterization, including generalized dephasing channels.The paper says the result is shown in Section IV B.Source: B. Entanglement-Assisted Classical… · p. 8

C. The Capacity Region for Entanglement-Assisted Transmission of Classical and Quantum Information · IV. SINGLE-LETTERIZATION OF THE CQ AND CE TRADE-OFF CURVES FOR HADAMARD CHANNELS · A. CQ Trade-off Curve

For Hadamard channels, the CQ trade-off curve single-letterizes, and together with single-letterization of the CE curve this yields a tractable full CQE capacity region. The section also characterizes the CQ curve through a parameterized optimization and explains how its boundary is achieved.

C. The Capacity Region for Entanglement-Assisted Transmission of Classical and Quantum Information

  • The CQE protocol tracks classical, quantum, and entanglement-consumption rates per channel use as an achievable triple when error vanishes and fidelity approaches unity.The rates are defined from message-set size M, quantum-state dimension K, and maximally entangled-state dimension D over n channel uses.Source: C. The Capacity Region for… · p. 8
  • Combining the CE curve with inverse super-dense coding and the CQ curve with inverse entanglement distribution gives two outer-bound surfaces for the CQE region.The surfaces are outer bounds because points beyond them would outperform the corresponding optimal CE or CQ trade-off curve.Source: C. The Capacity Region for… · pp. 8–9
  • The plane defined by the regularized h quantity has a single-letter characterization for any noisy quantum channel, making its boundary tractable to compute.Source: C. The Capacity Region for… · p. 9
  • The intersection of those surfaces has rates I(X; B), 1/2 I(A; B|X), and 1/2 I(A; E|X), achieved by the classically-enhanced father protocol.Combining that protocol with super-dense coding, entanglement distribution, and, when needed, wasting entanglement achieves the full CQE region.Source: C. The Capacity Region for… · p. 9
  • The full CQE capacity region single-letterizes whenever both the CQ and CE trade-off curves single-letterize; this condition holds for Hadamard channels.Source: C. The Capacity Region for… · p. 9

A. CQ Trade-off Curve

  • For each λ ≥ 1, a state maximizing the stated function yields a point on the CQ trade-off curve, and varying λ with the convex hull recovers the entire curve.Values 0 ≤ λ < 1 optimize only the classical capacity and do not extend the trade-off curve.Source: A. CQ Trade-off Curve · pp. 9–11
  • The one-shot CQ trade-off curve equals its regularized version for Hadamard channels, giving a single-letter characterization.The additivity argument applies when one channel is Hadamard and the other is arbitrary.Source: A. CQ Trade-off Curve · pp. 10–11

B. CE Trade-off Curve

The section characterizes the CE trade-off curve through points generated by a function maximized over states of form (15), for 0 ≤ λ ≤ 1. It describes the trade-off between classical communication and entanglement consumption.

B. CE Trade-off Curve

  • The CE region trades off maximizing classical communication against minimizing entanglement consumption; both cannot be optimal simultaneously.Source: B. CE Trade-off Curve · p. 11
  • Accepting a small reduction in classical communication can lower the protocol’s entanglement consumption.Source: B. CE Trade-off Curve · p. 11
  • For 0 ≤ λ ≤ 1, maximizing the function in (23) over states ρ of form (15) yields points on the CE trade-off curve (Theorem 3).The function in (23) generates the curve’s points when ρ maximizes it.Source: B. CE Trade-off Curve · p. 11

V. PARAMETRIZATION OF THE TRADE-OFF CURVES · A. Qubit Dephasing Channel · B. 1 →N Cloning Channels

The section applies the single-letterization of Hadamard-channel CQE regions to characterize trade-off curves for qubit dephasing and 1 →N cloning channels. Both channels’ trade-off curves can outperform time-sharing under the stated conditions.

V. PARAMETRIZATION OF THE TRADE-OFF CURVES

  • The Hadamard-channel results make CQE capacity regions computable, and this section exactly characterizes trade-off curves for qubit dephasing and 1 →N cloning channels.It also introduces applying the cloning-channel results to the Unruh channel, which is treated later.Source: §V · p. 11

A. Qubit Dephasing Channel

  • The qubit-dephasing subsection recalls CQ and CE trade-off parametrizations previously given by Devetak and Shor and Hsieh and Wilde [40].The paper points to Figure 4 of Ref. [40] for the full CQE capacity region of the dephasing qubit channel rather than plotting it here.Source: A. Qubit Dephasing Channel · p. 12
  • For qubit dephasing, both classically-enhanced quantum coding and entanglement-assisted classical coding beat time-sharing whenever 0 < p < 1.The plotted curves use p values from 0 to 1 in increments of 1/10; the noiseless and classical-channel endpoints do not beat time-sharing.Source: A. Qubit Dephasing Channel · p. 12, Figure 1 · p. 12

B. 1 →N Cloning Channels

  • For a 1 →N cloning channel, the CQ and CE trade-off curves are parametrized by exact expressions, with H(B) = log(N + 1) and H(B|X) = H(λ_i(μ) / Δ_N).Here Δ_N = N(N + 1)/2, λ_i(μ) = (N − 2i)μ + i, and μ ∈ [0, 1/2].Source: B. 1 →N Cloning Channels · pp. 12, 14
  • For cloning channels with N > 1, both trade-off curves beat time-sharing, with a larger relative improvement as N increases.Figure 2 plots N = 1, 2, 3, 5, 8, 12, and 24.Source: Figure 2 · p. 14
  • The reported expression vanishes at μ = 0 and coincides with the quantum capacity of a 1 → N cloning channel at μ = 1/2 (Equation 30 in Ref. [33]).The expression coincides with the classical capacity of the 1 → N cloning channel at μ = 0 (Corollary 2 in Ref. [30]).Source: B. 1 →N Cloning Channels · pp. 14–15
  • The full CQE capacity region for a 1 →10 cloning channel beats time-sharing using the classically-enhanced father protocol, entanglement distribution, and super-dense coding.The paper attributes this to strict concavity of the first two surfaces described in Section III C for cloning channels when N > 1.Source: B. 1 →N Cloning Channels · p. 15

C. Unruh Channel

The section characterizes the Unruh channel’s CQ and CE trade-off curves and shows that their coding strategies outperform time-sharing for every nonzero acceleration parameter. It relates this tractable analysis to the cloning channel’s block structure.

C. Unruh Channel

  • The Unruh channel’s output is an infinite-dimensional block-diagonal density matrix whose blocks each contain a 1 → N cloning channel, so maximizing transmission is equivalent to maximizing it in each block.The paper links the Unruh channel’s capacity questions to those of the cloning channel through this structure.Source: C. Unruh Channel · p. 15
  • Theorem 6 gives CQ trade-off points as (CCQ (z, µ), QCQ (z, µ)) and CE trade-off points as (CCE (z, µ), ECE (µ)).H2 is the binary entropy function.Source: C. Unruh Channel · pp. 15–16
  • For z > 0, both classically-enhanced quantum coding and entanglement-assisted classical coding beat a time-sharing strategy for the Unruh channel.This holds for the CQ and CE trade-off curves across the acceleration values plotted.Source: C. Unruh Channel · p. 16
  • For z > 0, the classically-enhanced father protocol combined with entanglement distribution and super-dense coding beats time-sharing because the first two surfaces are strictly concave.The full CQE capacity region is plotted for z = 0.95.Source: C. Unruh Channel · p. 16

VI. MEASURING THE GAIN OVER A TIME-SHARING STRATEGY · VII. CONCLUSION

The paper establishes tractable CQE capacity regions for Hadamard channels and introduces an area-based relative-gain measure to compare specialized coding with time-sharing. It finds that the classically-enhanced father protocol beats time-sharing for the considered channel examples.

VI. MEASURING THE GAIN OVER A TIME-SHARING STRATEGY

  • For a cloning channel with N = 1,000,000, at 0.165 bits/channel use of classical data, quantum transmission rises from 5.9 × 10^-7 with time-sharing to 7.2 × 10^-7 qubits/channel use with specialized coding.The channel’s quantum capacity is approximately 1.5 × 10^-6 qubits/channel use, while its classical capacity is approximately 0.27 bits/channel use, illustrating substantial relative improvement despite a small absolute increase.Source: §VI · p. 16
  • The proposed relative gain compares areas under the CQ and CE trade-off curves with areas under their respective time-sharing lines.The CQ gain is ACQ/ATSCQ; the passage defines the CE gain as ATSCQ/ACE.Source: §VI · p. 16
  • The relative-gain measures remain nontrivial when a capacity vanishes, average gains across the trade-off curve, and can extend to CQE regions using volumes.A necessary-seeming condition for a large CQ relative gain is a quantum capacity much smaller than the classical capacity.Source: §VI · p. 16

VII. CONCLUSION

  • The CQE capacity region for every Hadamard channel has a single-letter characterization, with exact formulas for generalized dephasing and cloning channels and expressions for the Unruh channel.The proof uses the channels’ degradability and entanglement-breaking complementary channels to generalize the Devetak–Shor technique.Source: §VII · pp. 16–18
  • The classically-enhanced father protocol beats time-sharing for generalized dephasing, cloning, and Unruh channels; relative CQ gains are strongest for cloning and Unruh channels.Dephasing channels transmit classical data noiselessly, yet their CQ trade-off is nontrivial; cloning channels with N > 1 and Unruh channels with z > 0 lack noiseless classical transmission and require error correction.Source: §VII · p. 17
  • Future work includes finding other single-letter channels, studying static Hadamard-state capacities, and examining alternative or qudit Unruh channels and trade-offs involving private quantum information, private or public classical information, and secret key.The proposed single-letterization approach relies on Hadamard-channel degradability and entanglement-breaking complementary channels.Source: §VII · pp. 17–18

Appendix A: Single-letterization of the Plane in (17) · Appendix B: Proof of the Single-letterization of the CE Trade-off Curve for Hadamard channels · Appendix C: On the parametrization of the trade-off curve

The appendices establish additivity tools that yield single-letterization of the entanglement-assisted trade-off curve for Hadamard channels, and clarify how a parameter traces trade-off curves except on constant-slope segments.

Appendix A: Single-letterization of the Plane in (17)

  • The function h is additive across any two channels, with the nontrivial upper bound proved using strong subadditivity and subadditivity.The proof reduces the joint-channel quantity to terms bounded by h(N1) + h(N2).Source: Appendix A · p. 19

Appendix B: Proof of the Single-letterization of the CE Trade-off Curve for Hadamard channels

  • A maximizing state for the weighted objective lies on the one-shot CE trade-off curve, since an equal-classical-rate state using less entanglement would contradict its optimality.The argument compares states with the same I(AX; B) but lower H(A|X).Source: Appendix B · p. 19
  • The trade-off curve is traced by the maximization parameter for 0 ≤ λ ≤ 1; λ > 1 optimizes only classical capacity.For λ > 1, the maximized quantity is at most the classical capacity.Source: Appendix B · pp. 19–20
  • For Hadamard channels, the one-shot CE trade-off curve equals the regularized curve, using additivity of g_λ when one tensor-factor channel is Hadamard.The proof bounds g_λ(N1 ⊗ N2) above by g_λ(N1) + g_λ(N2) for λ ≤ 1.Source: Appendix B · p. 20

Appendix C: On the parametrization of the trade-off curve

  • The parameter λ traces all points on the CQ and CE trade-off curves except possibly points on constant-slope segments.At nondifferentiable points, choosing −λ within the gap between one-sided slopes satisfies the supporting-line condition.Source: Appendix C · p. 21
  • Points on differentiable constant-slope segments may not be parametrized by λ, but lie in the convex hull of points that it does parametrize.The proof uses the convexity and monotonicity of the trade-off curve.Source: Appendix C · p. 21

Appendix D: Form of the CQ Trade-off Curve for Qubit Dephasing Channels

Appendix D derives the CQ trade-off curve for qubit dephasing channels by reducing the optimization to diagonal conditional states and a specific parametrized ensemble. It then evaluates the ensemble’s information quantities, including its coherent information.

Appendix D: Form of the CQ Trade-off Curve for Qubit Dephasing Channels

  • For generalized dephasing channels, it suffices to optimize the CQ trade-off over states with diagonal conditional density operators.The proof uses the fact that complete dephasing can only increase the relevant output entropies and commutes with the channel.Source: Appendix D · p. 21
  • A specific ensemble parametrizes all points on the CQ trade-off curve for a qubit dephasing channel.The proof shows that bit-flip symmetrization yields an ensemble sufficient to attain a point on the curve.Source: Appendix D · pp. 21–22
  • For the parametrized ensemble, the coherent information is I(A⟩BX) = H2(µ) − H2(λ+).The appendix computes the Holevo information and coherent information for this ensemble to establish the CQ trade-off in Theorem 4.Source: Appendix D · pp. 22–23

Appendix E: Form of the CE Trade-off Curve for Qubit Dephasing Channels

The appendix derives the exact CE trade-off curve for qubit dephasing channels using lemmas that restrict the optimization and parametrize the curve, then evaluates the quantities in Theorem 4.

Appendix E

  • For generalized dephasing channels, the CE optimization can be restricted to classical-quantum states with diagonal conditional density operators.The proof uses that complete dephasing can only increase the relevant entropies and that the channel and its complement satisfy stated dephasing relations.Source: Appendix E · p. 23
  • An ensemble specified in Lemma 15 parametrizes every point on the CE trade-off curve for a qubit dephasing channel.The ensemble uses the states ψ0 and ψ1 from Lemma 13.Source: Appendix E · p. 23
  • Theorem 4’s CE trade-off follows from I(AX; B) = H(A|X) + H(B) − H(E|X), with H(A|X) = H2(µ) and H(B), H(E|X) computed earlier in its proof.Source: Appendix E · p. 24

Appendix F: CQ and CE Trade-off Curves for the Unruh Channel

Appendix F applies the cloning-channel input state to the Unruh channel’s CQ curve and presents calculations for its Holevo and coherent information. The supplied passages report two endpoint properties of the Holevo information.

Appendix F: CQ and CE Trade-off Curves for the Unruh Channel

  • The input state in (24) used to trace the cloning channels’ CQ curve also traces the Unruh channel’s CQ curve, and the calculation uses its purification.Source: Appendix F · p. 24
  • At µ = 0, the Holevo information coincides with the Unruh channel’s classical-capacity expression in Ref. [30]; it vanishes at µ = 1/2.The cited work uses a slightly different definition of ∆l.Source: Appendix F · p. 24
  • For coherent information, the appendix uses outputs of the Unruh channel’s complementary channel for the two computational-basis inputs.Source: Appendix F · p. 24
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