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Classical Capacity and Entanglement Cost of the Amplitude Damping Channel

Ziao Tang, Chengkai Zhu, Ge Bai, Xin Wang

arXiv:2609.28592v1quant-ph

TL;DR

Noisy quantum-channel capacity and entanglement cost generally require regularization because correlations across uses can change asymptotic rates. The paper proves a support criterion for strong superadditivity of entanglement of formation and uses it to remove both regularizations for qubit-to-qubit channels admitting a pure output, including amplitude damping.

  • Problem

    Classical capacity and parallel entanglement cost are regularized quantities, and correlations across channel uses can affect their asymptotic rates.

    Source: §1 · p. 2

  • Method

    The paper proves an arbitrary-finite-dimensional support criterion for strong superadditivity of entanglement of formation using a triangular block-matrix entropy inequality and decompositions preserving two expectations.

    Source: §1 · pp. 2–3

  • Results

    For qubit-to-qubit channels admitting a pure output, the classical capacity equals single-use Holevo information and parallel entanglement cost equals normalized-Choi entanglement of formation; these quantities are additive with arbitrary finite-dimensional partners.

    Source: §1 · p. 3, §7 · p. 18

  • Takeaways & Limitations

    The same pure-output class covers all qubit-to-qubit channels of Kraus rank at most two, while amplitude damping has cost h2((1 + √p)/2) ebits per use.

    Source: §1 · p. 3

    The entanglement-cost equalities are asymptotic vanishing-error statements and do not assert simulation from a single Choi copy or against adaptive tests.

    Source: §6.3 · p. 17

Abstract

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Determining a noisy quantum channel's classical capacity and entanglement cost generally requires regularization over many channel uses. We remove both regularizations for every qubit-to-qubit channel admitting a pure output. For each such channel, Holevo information, channel entanglement of formation, and parallel entanglement cost are additive with those of any finite-dimensional partner channel. This class includes all qubit-to-qubit channels of Kraus rank at most two. For the amplitude damping channel with damping probability $p$, the unassisted classical capacity equals the known single-use Holevo information, attained by a binary pure-state ensemble with collective decoding, and the entanglement cost is $h_2((1+\sqrt p)/2)$ ebits per use. The common mechanism is a support criterion for strong superadditivity of entanglement of formation: one marginal has no support on the sector in which both local systems are orthogonal to fixed distinguished vectors. We prove this criterion in arbitrary finite dimensions using a triangular block-matrix entropy inequality and decompositions preserving two expectations. For amplitude damping, we also derive an exact finite-block Holevo deficit, identify the unique optimal average input for $p<1$, and construct a binary Kraus representation attaining the uniform formation bound.

1 Introduction

Noisy-channel classical capacity and parallel entanglement cost generally require regularization because correlations across uses can change asymptotic rates. The paper removes both regularizations for qubit-to-qubit channels with a pure output and applies the result to amplitude damping.

1 Introduction

  • For amplitude damping, the unresolved capacity question was whether entangled encodings could exceed the known single-use binary-ensemble rate in the nontrivial regime 0 < p < 1.Prior additivity results for entanglement-breaking and unital qubit channels did not settle that regime.Source: §1 · p. 2
  • The common mechanism is a support constraint excluding the sector where both local systems are orthogonal to fixed distinguished vectors, which yields strong superadditivity of entanglement of formation.The criterion applies to output–environment states and reference–output Choi states.Source: §1 · pp. 2–3
  • In arbitrary finite dimensions, the proof uses a triangular coefficient matrix, a matrix entropy inequality, and decompositions preserving two expectations to establish the support criterion.The argument bounds marginal formation costs without evaluating either mixed-state convex roof.Source: §1 · p. 3
  • For every qubit-to-qubit channel admitting a pure output, Holevo information, channel entanglement of formation, and parallel entanglement cost are additive with arbitrary finite-dimensional partner channels.The class includes all qubit-to-qubit channels of Kraus rank at most two.Source: §1 · p. 3
  • For amplitude damping, the unassisted classical capacity equals the known single-use Holevo optimization, and the parallel simulation cost is h2((1 + √p)/2) ebits per use.The new operational conclusion is the matching many-use converse, not a new single-use capacity formula.Source: §1 · p. 3
  • The amplitude-damping results give an exact finite-block Holevo deficit, identify the unique optimal average input for p < 1, and provide a binary Kraus representation attaining a uniform formation bound.The unique average input need not determine a unique optimal ensemble or collective decoder.Source: §1 · p. 3

2 Preliminaries

The preliminaries fix the channel and entropy notation, define unassisted classical communication, and specify the parallel simulation and entanglement-cost quantities used later.

2.1 Notation

  • A channel admits a pure output when some pure input is mapped to a pure state; Kraus rank is the minimum number of operators in a Kraus representation.All Hilbert spaces are finite dimensional, and entanglement is specified across a bipartition.Source: §2.1 · p. 3

2.2 Unassisted classical communication

  • Unassisted classical capacity is the supremum of vanishing-error rates, allowing encodings entangled across uses and collective decoding but no preshared entanglement or auxiliary classical communication.For an average input τ, χN(τ) is the largest Holevo information of an ensemble averaging to τ, and χ(N) = maxτ χN(τ).Source: §2.2 · p. 4

2.3 Parallel channel simulation

  • Parallel entanglement cost is measured in ebits per use for LOCC protocols approximating tensor-power channels in diamond norm, with classical communication free.The channel entanglement of formation is the maximum output-state entanglement of formation over pure inputs with a reference isomorphic to the input.Source: §2.3 · p. 5

3 Main results

The paper removes the many-use regularizations for qubit-to-qubit channels admitting a pure output, with additivity extending to arbitrary finite-dimensional partner channels. For amplitude damping, this yields the unassisted classical capacity and parallel entanglement cost from single-use quantities.

Main result 1: qubit channels admitting a pure output

  • For every qubit-to-qubit channel admitting a pure output, Holevo information, channel entanglement of formation, and parallel entanglement cost are additive with arbitrary finite-dimensional partner channels.The class includes every qubit-to-qubit channel of Kraus rank at most two.Source: §3 · p. 5

Main result 3: entanglement cost of amplitude damping

  • The amplitude-damping parallel simulation cost is h2((1 + √p)/2) ebits per use, equal to its Choi-state entanglement of formation and cost.The matching lower bound follows from strong Choi-state additivity; exact finite-copy channel formation and arbitrary-partner additivity are also established.Source: §3 · p. 6

Main result 2: classical capacity of amplitude damping

  • The amplitude-damping unassisted classical capacity equals its single-use Holevo information, attained by equiprobable binary pure states with collective decoding.For an optimizing q*, the states are √(1 − q*)|0⟩ ± √q*|1⟩; the converse covers codewords entangled across uses.Source: §3 · p. 6
  • For every blocklength and p < 1, the finite-block refinement determines the unique optimal average input.The result concerns the average input, in addition to the binary ensemble achieving the capacity.Source: §3 · p. 6

4 Strong superadditivity of entanglement of formation

A support constraint on one marginal—excluding the sector where both local systems are orthogonal to distinguished vectors—forces strong superadditivity of entanglement of formation. This yields additivity and formation-cost identities, including for arbitrary finite-dimensional auxiliary systems.

Theorem 4.1: Strong superadditivity of entanglement of formation

  • E_F(ρ_AB) + E_F(σ_A′B′) ≤ E_F(Ω_AA′:BB′) whenever ρ_AB is supported on the allowed subspace.The auxiliary systems may be arbitrary finite dimensional, and the extension Ω may be mixed and correlated.Source: §4 · p. 7

Additivity and state cost

  • E_F(ρ ⊗ σ) = E_F(ρ) + E_F(σ), and E_C(ρ) = E_F(ρ) under the support criterion.Iteration gives E_F(ρ^⊗n) = nE_F(ρ), removing the state-cost regularization.Source: §4 · pp. 7, 10

Proof mechanism

  • For pure extensions, the excluded joint sector makes the coefficient matrix triangular, separating entanglement into a scalar term and weighted block entropies.The scalar and block terms bound the two marginal formation costs; convexity extends the argument to mixed extensions.Source: §4 · pp. 7, 9
  • The scalar-term bound uses pure-state decompositions preserving two support-projector expectations, so the components share the same sector weights.The application needs only two expectations; preserving three can fail even for a mixed qubit state.Source: §4 · pp. 8–9

Scope and applications

  • For two qubits, a product vector in ker ρ guarantees the criterion; every two-qubit state of rank at most two has such a kernel vector.In higher dimensions, the excluded sector is the whole subspace A1 ⊗ B1, while the allowed subspace has dimension dA + dB − 1.Source: §4 · p. 10

Channel applications

  • For qubit-to-qubit channels, admitting a pure output is equivalent to the Stinespring support condition, and the class includes every channel of Kraus rank at most two.The communication and simulation applications use the criterion across output–environment and reference–output bipartitions, respectively.Source: §4 · p. 10

5 Classical capacity and finite-block optimality

The paper converts a Stinespring support condition into strong Holevo additivity, then evaluates the amplitude-damping channel’s single-use optimum. It also gives a finite-block converse and proves uniqueness of the optimal average input when p < 1.

5.1 Strong Holevo additivity from Stinespring support

  • χ(N ⊗ M) = χ(N) + χ(M) and C(N) = χ(N) when a Stinespring image lies in the specified support subspace.The identities hold with any finite-dimensional partner M, and the channel capacity is additive with that partner.Source: §5.1 · p. 11
  • Every qubit-to-qubit channel admitting a pure output satisfies the Stinespring criterion; this includes channels of Kraus rank at most two.For these channels, the support condition is equivalent to admitting a pure output.Source: §5.1 · pp. 11–12

5.2 Classical capacity of amplitude damping

  • The fixed-input output-entropy convex roof depends only on excitation probability q, and an explicit fixed-excitation decomposition attains the bound g_p(q).The binary signals have individual output entropy g_p(q) and average-output entropy h2((1 − p)q).Source: §5.2 · pp. 12–13
  • The amplitude-damping channel’s unassisted classical capacity equals its single-use Holevo information, attained by an optimal binary pure-state ensemble with collective decoding.The cited ensemble achieves the fixed-q value F_p(q); maximizing over q gives the capacity, and product codewords achieve every smaller rate.Source: §5.2 · pp. 12–13

5.3 Finite-block converse and the optimal average input

  • For p < 1, F_p is strictly concave and has a unique interior maximizer q*; the finite-block converse has an exact decomposition into nonnegative terms.The optimal average input is unique, though optimal ensembles and collective decoders need not be.Source: §5.3 · pp. 13–14
  • The finite-block identity does not by itself establish a strong converse for codes with nonvanishing error.The optimal product input is an information-radius center at every blocklength in the relative-entropy characterization of Holevo information.Source: §5.3 · p. 14

6 Entanglement cost

The section derives parallel simulation-cost bounds from a Choi-state criterion, then applies them to low-rank and pure-output qubit channels. For amplitude damping, it evaluates the cost and gives a uniform formation bound.

6.1 Channel-simulation bounds and a Choi-state criterion

  • The lower simulation-cost bound follows by simulating the channel on halves of local Bell pairs to prepare copies of its Choi state.Diamond-norm convergence gives vanishing trace-distance error.Source: §6.1 · p. 14
  • A product vector in the Choi state’s kernel makes its entanglement cost and entanglement of formation coincide, so the simulation bounds meet.The matching bounds support the parallel LOCC simulation result.Source: §6.1 · pp. 14–15

6.2 Qubit channels of Kraus rank at most two

  • For every qubit-to-qubit channel of Kraus rank at most two, the communication and simulation criteria yield strong additivity with any finite-dimensional partner channel.The pure-output class also satisfies these conclusions, even beyond Kraus rank two.Source: §6.2 · pp. 15–16

6.3 Entanglement cost of amplitude damping

  • Theorem 6.3 gives an entanglement-cost result for amplitude damping for every p ∈ [0, 1], with endpoint costs of one ebit per use at p = 0 and zero at p = 1.The cost equalities are asymptotic vanishing-error statements; (6.13) is an exact finite-copy formation identity.Source: §6.3 · pp. 16–17
  • A two-outcome Kraus representation gives a formation upper bound uniformly over all block inputs.Its symmetry under τ → I − τ identifies I/2 as a maximizer of the average conditional entropy.Source: §6.3 · pp. 17–18

7 Discussion

The discussion connects a shared support criterion to classical communication and channel simulation. It also identifies limits on the theorem’s scope and on the operational conclusions.

7 Discussion

  • Excluding the joint orthogonal sector A1 ⊗ B1 suffices for strong superadditivity of entanglement of formation in arbitrary finite dimensions.The proof uses a triangular block entropy inequality and a fixed-expectation decomposition.Source: §7 · p. 18
  • For qubit-to-qubit channels, the communication and simulation support conditions coincide with admitting a pure output.The resulting class includes channels of Kraus rank at most two.Source: §7 · p. 18
  • The support hypothesis is sufficient, but a single product vector in the kernel does not exclude the full orthogonal sector in higher dimensions.Generalized amplitude damping at finite temperature does not in general satisfy the hypothesis.Source: §7 · p. 18
  • The operational results cover vanishing-error classical communication and parallel LOCC simulation, not strong-converse thresholds or simulation against adaptive tests.The paper leaves reliability functions and adaptive-test simulation unresolved.Source: §7 · p. 18

A Proof of the triangular block entropy inequality

The appendix proves the triangular block entropy inequality through an integral representation and a sequence of convexity reductions.

A Proof of the triangular block entropy inequality

  • The proof expresses the entropy difference as an integral of log-determinants.Convexity in two matrix arguments, with the third fixed, permits dephasing and scalar reduction.Source: A Proof of the triangular block… · p. 21
  • Polar decomposition identifies the entropy difference with the triangular block expression.The proof treats square blocks first and extends the result to rectangular blocks by isometric embedding.Source: A Proof of the triangular block… · p. 21
  • The appendix uses natural logarithms in its entropy calculations and converts to bits at the end.This is a convention of the proof rather than a change in the inequality.Source: A Proof of the triangular block… · p. 21

A.1 Convexity of a log-determinant functional

The appendix establishes the triangular block entropy inequality by proving convexity of a log-determinant functional, reducing the entropy difference to scalars, and normalizing block matrices.

A.1 Convexity of a log-determinant functional

  • The log-determinant functional is convex on positive definite Hermitian matrices, and strictly convex when J is invertible.The proof bounds its Hessian using the operator norm of (I + A0)^−1.Source: §A.1 · p. 22

A.2 Convexity of the entropy difference

  • The entropy difference is convex in its first and third matrix arguments when the middle argument is fixed.It is strictly convex in those arguments when the fixed middle matrix is positive definite.Source: §A.2 · p. 23

A.3 Dephasing and scalar reduction

  • Dephasing in an eigenbasis of the fixed middle matrix reduces the matrix inequality to a scalar inequality.Homogeneity and Jensen’s inequality complete the scalar reduction, with continuity extending it to singular arguments.Source: §A.3 · pp. 23–24

A.4 Polar decomposition and normalization

  • Polar decomposition and normalization convert the scalar inequality into the triangular block entropy inequality for square blocks.Isometric embedding extends the result to rectangular blocks, and continuity covers vanishing weights.Source: §A.4 · p. 24

A.5 Equality conditions

For invertible middle block B, equality requires proportionality relations among the blocks, equivalently a tensor-product form. The appendix also establishes convexity of a binary-spectrum entropy function and evaluates concurrence when a product vector lies in the kernel.

A.5 Equality conditions

  • For invertible B and square blocks of the same size, equality holds exactly under the stated proportionality relations, equivalently a block-unitary tensor-product form.For singular B, the tensor-product form remains sufficient, but necessity is not asserted.Source: §A.5 · p. 25

B.1 Convexity of the binary-spectrum entropy √

  • The binary-spectrum entropy function e is increasing and convex on [0, 1], and the associated function is convex in q for every p ∈ [0, 1].It vanishes identically at p = 0 and p = 1.Source: §B.1 · p. 25

B.2 Concurrence for a product vector in the kernel

  • For a two-qubit state with |01⟩ in its kernel, Wootters’ formula yields the concurrence expression, including singular states and arbitrary allowed x, y.The calculation follows from two zero eigenvalues of ρeρ and the remaining two eigenvalues.Source: §B.2 · pp. 25–26

C Uniform Kraus representations

A uniform Kraus representation bounds channel entanglement of formation across all inputs, supporting subadditivity and, with the matching lower bound, additivity for the channels considered. The appendix also records established operational rates and reductions used for amplitude damping.

C Uniform Kraus representations

  • For every finite-dimensional channel, one finite Kraus representation can approximate the channel entanglement-of-formation bound uniformly over all inputs.For amplitude damping, an explicit representation attains the bound; the matching lower bound for channels considered follows from strong Choi-state additivity.Source: C Uniform Kraus representations · pp. 26–27
  • The uniform Kraus bounds yield subadditivity of channel entanglement of formation for any finite-dimensional channel pair.The proof applies product instruments to joint inputs and then uses entropy subadditivity and strong subadditivity.Source: C Uniform Kraus representations · p. 27

D.1 Quantum and entanglement-assisted capacities

  • The appendix distinguishes unassisted quantum capacity, measured in qubits per use, from entanglement-assisted classical capacity, measured in bits per use.The quantum-capacity formulas rely on degradability and antidegradability, while the entanglement-assisted formula is evaluated for diagonal inputs.Source: §D.1 · pp. 27–28
  • For diagonal input τq, the dilation entropies are S(R) = h2(q), S(B) = h2((1 − p)q), and S(E) = h2(pq).These give coherent information and reference–output mutual information for the established capacity formulas.Source: §D.1 · pp. 27–28

D.2 Relations at a fixed input excitation

At fixed input excitation, the appendix relates the quantum, classical, and entanglement-assisted rates while cautioning that their equalities do not generally persist under separate optimization. It also gives high-damping asymptotics and identifies comparisons not settled by the analysis.

D.2 Relations at a fixed input excitation

  • At a common excitation, the rates satisfy Q(Ap) ≤ C(Ap) ≤ CE(Ap), but the stated equalities do not generally survive separate optimization.The bound uses data processing on the environment Holevo information; at p = 1, all four rates vanish.Source: §D.2 · p. 28

D.3 Asymptotic expansion in the high-damping limit

  • As p approaches 1, the appendix derives high-damping expansions for the classical capacity, entanglement cost, and entanglement-assisted capacity.The capacity and cost expansions follow from uniform entropy estimates and the expansion of (1 − √(1 − ε))/2.Source: §D.3 · p. 29
  • For 1/2 ≤ p < 1, quantum capacity vanishes while parallel entanglement cost remains positive; two-way-assisted distillation and adaptive simulation are not determined.The comparison alone does not establish irreversibility under a common class of free operations.Source: §D.3 · p. 29

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