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Amortized Channel Divergence for Asymptotic Quantum Channel Discrimination

Mark M. Wilde, Mario Berta, Christoph Hirche, Eneet Kaur

arXiv:1808.01498v2quant-phcs.ITmath.IT

TL;DR

Adaptive strategies can help discriminate channels in finite settings, but their asymptotic value for quantum channels is less systematically understood. The paper uses amortized channel divergence to derive converse bounds and establishes that adaptivity does not improve the exponential error rate for classical-quantum channel discrimination.

  • Problem

    Finite-regime adaptive advantages are known, while asymptotic quantum channel discrimination has been studied less systematically and existing general-channel bounds have unclear quantitative performance.

    Source: §I · pp. 3–4

  • Method

    The paper introduces amortized channel divergence and analyzes it using data-processing inequalities to derive converse bounds for adaptive channel discrimination.

    Source: §I · p. 4, Appendix A · p. 47

  • Results

    For classical-quantum channels, adaptive strategies do not improve the asymptotic exponential error rate, and the paper also gives converse bounds for general quantum channel discrimination.

    Source: §I · p. 4, §VIII · p. 46

  • Takeaways & Limitations

    The established no-advantage conclusion applies to classical-quantum channels; the asymptotic value of adaptivity for quantum channels remains less systematically characterized.

    Source: §I · p. 4, §VIII · p. 46

    Asymptotic quantum channel discrimination has been studied less systematically, and the exact quantitative performance of existing general-channel bounds remains unclear.

    Source: §I · p. 4

Abstract

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It is well known that for the discrimination of classical and quantum channels in the finite, non-asymptotic regime, adaptive strategies can give an advantage over non-adaptive strategies. However, Hayashi [IEEE Trans. Inf. Theory 55(8), 3807 (2009)] showed that in the asymptotic regime, the exponential error rate for the discrimination of classical channels is not improved in the adaptive setting. We extend this result in several ways. First, we establish the strong Stein's lemma for classical-quantum channels by showing that asymptotically the exponential error rate for classical-quantum channel discrimination is not improved by adaptive strategies. Second, we recover many other classes of channels for which adaptive strategies do not lead to an asymptotic advantage. Third, we give various converse bounds on the power of adaptive protocols for general asymptotic quantum channel discrimination. Intriguingly, it remains open whether adaptive protocols can improve the exponential error rate for quantum channel discrimination in the asymmetric Stein setting. Our proofs are based on the concept of amortized distinguishability of quantum channels, which we analyse using data-processing inequalities.

I. INTRODUCTION … B. Quantum entropies

The paper develops amortized channel divergence to bound adaptive quantum-channel discrimination and establish asymptotic results. It proves the strong Stein’s lemma for classical-quantum channels while leaving open whether adaptation improves the quantum-channel Stein error exponent.

I. INTRODUCTION

  • Adaptive protocols can outperform non-adaptive ones at finite blocklength, but Hayashi [14] showed that this advantage disappears asymptotically for classical channels [10, Section 5].In the classical setting, optimal Stein and Chernoff exponential error rates are achieved using many copies of the best product-state input and discriminating the product outputs.Source: §I · pp. 3–4
  • Amortized channel divergence yields converse bounds on adaptive protocols and establishes the strong Stein’s lemma for classical-quantum channel discrimination.For classical-quantum channels, repeating the best product-state input and applying quantum Stein’s lemma to product outputs is asymptotically optimal; tight characterizations also cover unitary and isometry channels, projective measurements, replacer channels, and environment-seizable channels.Source: §I · p. 4
  • Whether adaptive protocols improve the quantum-channel error exponent in the asymmetric Stein setting remains open, despite many channel classes having no asymptotic adaptive advantage.The authors suspect a general gap may exist, possibly even for entanglement-breaking or quantum-classical channels.Source: §I · p. 4

II. NOTATION

  • Section II introduces the notation and relevant definitions used in the subsequent analysis.Source: §II · p. 5

A. Setup

  • The setup uses finite-dimensional quantum and classical systems, represents channels through Choi states, and models superchannels as physically realizable pre- and post-processing with memory.Source: A. Setup · p. 5

B. Quantum entropies

  • The section defines Petz-Rényi divergences, specifies support conditions and endpoint limits, and introduces the Chernoff divergence.Source: B. Quantum entropies · p. 6
  • It defines sandwiched and log-Euclidean Rényi divergences, including their limits to max-relative entropy and quantum relative entropy.Source: B. Quantum entropies · pp. 6–7
  • The log-Euclidean form also yields a Chernoff distance and divergence-sphere optimization; inequalities compare the Rényi divergences, which recover classical versions on diagonal commuting states.Source: B. Quantum entropies · p. 7

III. SETTINGS FOR ASYMPTOTIC CHANNEL DISCRIMINATION … E. Symmetric setting – Chernoff

The section defines asymptotic channel discrimination through adaptive protocols and lays out asymmetric and symmetric error criteria, contrasting this focus with prior work on finite, non-asymptotic discrimination.

III. SETTINGS FOR ASYMPTOTIC CHANNEL DISCRIMINATION

  • The section frames its information-theoretic settings around asymptotic quantum channel discrimination, unlike most prior work focused on the finite, non-asymptotic regime.Source: §III · p. 7

A. Protocol for quantum channel discrimination

B. Asymmetric setting – Stein

  • In asymmetric Stein testing, type II error is minimized subject to type I error being at most a constant ε in (0, 1), with non-asymptotic and asymptotic quantities considered.Source: B. Asymmetric setting – Stein · p. 9

C. Strong converse exponent – Han-Kobayashi

D. Error exponent – Hoeffding

  • The Hoeffding error exponent characterizes type I error decay when type II error is constrained to decrease exponentially at rate r, and is non-trivial only when r is not too large.Source: D. Error exponent – Hoeffding · p. 10

E. Symmetric setting – Chernoff

  • Symmetric testing minimizes total error for a prior probability p, and its asymptotic exponent is unchanged by choosing priors other than 1/2; the section also relates it to the asymptotic Hoeffding exponent.Source: E. Symmetric setting – Chernoff · pp. 10–11

F. Energy-constrained channel discrimination … C. Properties of amortized channel divergence

The section develops an energy-constrained framework for channel discrimination and introduces amortized channel divergence as a tool for translating operational hypothesis-testing questions into mathematical properties of channels and distinguishability measures. It establishes structural properties of that divergence, including data processing under superchannels.

F. Energy-constrained channel discrimination

  • Without an energy constraint, two pure-loss bosonic Gaussian channels with different transmissivities can be perfectly distinguished in one call using a coherent state of arbitrarily large photon number [49, 50, Section 2].The output states become orthogonal in the limit of infinite photon number.Source: F. Energy-constrained channel… · p. 11
  • The framework constrains the average energy of reduced states at every channel input using a positive-semidefinite Hamiltonian and bound E; choosing HA = IA and E = 1 gives the unconstrained case.The constraint parameter satisfies E ∈ [0, ∞).Source: F. Energy-constrained channel… · p. 11

IV. AMORTIZED DISTINGUISHABILITY OF QUANTUM CHANNELS

  • Amortized distinguishability reduces channel hypothesis-testing questions to mathematical questions about channels, states, and their distinguishability measures.The section also develops properties of the measure as results of independent interest.Source: §IV · pp. 11–12

A. Generalized divergences

  • Generalized divergences obey data processing; a direct-sum property implies joint convexity, while generalized channel divergence optimizes output distinguishability over bipartite inputs and can be restricted to pure inputs with R isomorphic to A.For finite-dimensional channel inputs, this restriction makes the optimization bounded.Source: A. Generalized divergences · pp. 12–13

B. Amortized channel divergence

  • Amortized channel divergence subtracts the input states’ distinguishability from their output states’ distinguishability, and its optimization can be unbounded.In general, the optimizing states cannot be restricted to pure states, and no dimension bound on the reference system is guaranteed.Source: B. Amortized channel divergence · p. 13

C. Properties of amortized channel divergence

  • The generalized channel divergence is no larger than its amortized version, and strong faithfulness of the underlying divergence makes the amortized divergence zero exactly for identical channels.When the reverse inequality also holds, the two channel divergences are equal, an amortization collapse.Source: C. Properties of amortized channel… · p. 14
  • Amortized channel divergence is monotone under a common superchannel and is jointly convex when the underlying generalized divergence has the direct-sum property.These results respectively establish data processing for amortized channel divergence and give a condition for its joint convexity.Source: C. Properties of amortized channel… · pp. 15–16
  • Amortized channel divergence is stable under adjoining an identity channel, and jointly covariant channels permit optimization over inputs whose reduced states on A satisfy the symmetry.The reference system in the stability property can be of arbitrary size.Source: C. Properties of amortized channel… · pp. 16–17

D. Amortization collapse for max-relative entropy

This section establishes that max-channel divergence does not increase under amortization and gives an efficiently computable Choi-state SDP form. It also identifies related collapses and limits on where general collapse has been proved.

D. Amortization collapse for max-relative entropy

  • For α ≥ 1, Proposition 10 establishes that max-channel divergence does not increase under amortization, complementing related max-relative-entropy collapses in quantum communication [61] [62].Source: D. Amortization collapse for… · pp. 17–18
  • The max-channel divergence can be expressed as the max-relative entropy of channel Choi states, yielding an efficiently computable SDP; the coherence-resource divergence D′max equals it.Source: D. Amortization collapse for… · pp. 18–19
  • The max-channel divergence is a special case of sandwiched Rényi channel divergences [15] and generalized channel divergences [24], and a coherence-resource divergence was proposed in [63].Source: D. Amortization collapse for… · p. 19
  • The section also establishes amortization collapse for Hilbert α-divergence [64], implying that the diamond norm of the channel difference does not increase under amortization, and notes nonincrease for certain fidelity-based metrics.Source: D. Amortization collapse for… · p. 20
  • For quantum relative entropy, Petz-Rényi and sandwiched divergences, and Chernoff distance, general collapse is unproved here, though collapses are established for special channels in Sections VI–VII.Source: D. Amortization collapse for… · p. 20

E. Meta-converse for quantum channel discrimination via amortized channel divergence

Lemma 14 is a meta-converse for quantum channel discrimination: for any protocol using the channels n times, the divergence between its final decision probabilities is limited by n times the channels’ amortized divergence. It can recover particular converse statements by choosing different faithful generalized divergences.

E. Meta-converse for quantum channel discrimination via amortized channel divergence

  • Lemma 14 bounds the divergence between final decision probabilities p and q by n times the amortized channel divergence for any protocol using the channels n times.The lemma functions as a meta-converse, similarly to, from which particular converse statements can be recovered by choosing different generalized divergences.Source: E. Meta-converse for quantum channel… · p. 20
  • The decision-probability divergence is defined by embedding p and q as binary states ζ(p) and ζ(q), and the lemma applies to faithful generalized divergences.ζ(p) = p|0⟩⟨0| + (1 − p)|1⟩⟨1|, with D(p∥q) := D(ζ(p)∥ζ(q)).Source: E. Meta-converse for quantum channel… · p. 20
  • The proof uses data processing at the final measurement and across channel uses, with faithfulness and equal initial states supporting the divergence bound.The argument then uses optimization and the definition of amortized channel divergence.Source: E. Meta-converse for quantum channel… · p. 21

F. Energy-constrained channel divergences and meta-converse for energy-constrained channel discrimination · V. CONVERSE BOUNDS FOR QUANTUM CHANNEL DISCRIMINATION · A. Stein bound

The section develops energy-constrained channel divergences and converse bounds for adaptive quantum channel discrimination. In the Stein setting, it gives bounds based on amortized relative entropy and max-relative entropy, alongside conditions for finite asymptotic error exponents.

F. Energy-constrained channel divergences and meta-converse for energy-constrained channel discrimination

  • The energy-constrained generalized channel divergence includes the energy-constrained diamond norm and Bures distance as special cases, and its optimization can use pure states with R isomorphic to A.The generalized energy-constrained divergence was previously defined in [53].Source: F. Energy-constrained channel… · p. 21
  • The energy-constrained framework defines amortized channel divergence and extends the meta-converse to protocols satisfying an average energy constraint.The meta-converse assumes a faithful generalized divergence with the direct-sum property; the energy-constrained divergences retain many properties of their unconstrained counterparts.Source: F. Energy-constrained channel… · p. 22

V. CONVERSE BOUNDS FOR QUANTUM CHANNEL DISCRIMINATION

  • For channel pairs, n copies are perfectly distinguishable for some finite n if and only if I is not in the span of {N†i Mj}; under these conditions, the asymptotic quantities become trivial.The criterion is independent of the Kraus decompositions chosen.Source: §V · pp. 22–23

A. Stein bound

  • For adaptive protocols, Proposition 16 bounds the Stein error exponent using amortized quantum relative entropy, with a corresponding energy-constrained bound.The energy-constrained version applies for input Hamiltonian HA and energy constraint E.Source: A. Stein bound · p. 23
  • Proposition 17 gives a strong converse bound in terms of max-channel divergence, using the meta-converse and amortization collapse for max-relative entropy.The resulting bound applies uniformly to any channel discrimination protocol.Source: A. Stein bound · p. 24
  • Corollary 18 combines the Stein bounds to constrain the asymptotic Type II error exponent using amortized relative entropy and max-channel divergence.The amortized relative-entropy bound is an unbounded optimization in general, whereas the max-channel-divergence bound can be expressed as an efficiently computable semidefinite program.Source: A. Stein bound · p. 24
  • The asymptotic Type II error exponent is finite if and only if the stated support condition holds.When the condition fails, the relative-entropy lower bound makes the exponent infinite; when it holds, the max-divergence upper bound makes it finite.Source: A. Stein bound · p. 25
  • For generalized amplitude damping channels N(η1,p1) and M(η2,p2), the Stein quantity is bounded below by D(N∥M) and above by Dmax(N∥M); figures vary p1,p2 at η1=0.2, η2=0.3 and η1=η2=0.5, respectively.Source: A. Stein bound · p. 25, Figure 2 · p. 26, Figure 3 · p. 26

B. Strong converse exponent

This section bounds the strong-converse exponent using a product-state strategy and an amortized sandwiched Rényi channel divergence. Whether the product-state strategy is optimal is unclear, but the bounds match for channels where the amortized divergence collapses.

B. Strong converse exponent

  • A product-state discrimination strategy gives an achievable rate involving the sandwiched Rényi divergence, though its optimality is unclear.For channels where the amortized divergence collapses, the lower and upper bounds match.Source: B. Strong converse exponent · p. 27
  • For r > 0, Proposition 20 gives a lower bound on the strong-converse exponent involving the amortized sandwiched Rényi channel divergence.The excerpt does not include the proposition’s explicit bound.Source: B. Strong converse exponent · p. 27
  • The proof assumes β_n({Q, A}) ≤ 2^-nr and takes the supremum over α > 1 to obtain the sandwiched Rényi bound.It applies the meta-converse with the sandwiched Rényi divergence.Source: B. Strong converse exponent · p. 27
  • The proof also derives a max-relative-entropy bound using the meta-converse and the amortization collapse for max-relative entropy.The argument uses Dmax(p∥q) ≥ log(p/q) together with the assumed bound on β_n({Q, A}).Source: B. Strong converse exponent · pp. 27–28

C. Chernoff bound · VI. CLASSICAL-QUANTUM CHANNEL DISCRIMINATION

The Chernoff section establishes converse bounds for symmetric quantum-channel discrimination, while the classical-quantum section shows that best-input tensor-power strategies are optimal in several asymptotic settings, making adaptive strategies, quantum memories, and entangled inputs unnecessary there.

C. Chernoff bound

  • For non-adaptive product inputs with a quantum reference, the symmetric error exponent is the Chernoff channel divergence C(N∥M), which is also achievable adaptively but does not generally quantify the quantum-channel exponent [9] [10] [18].The product-input result follows from the quantum-state Chernoff bound [3] [4] [6].Source: C. Chernoff bound · p. 28
  • Proposition 21 gives an all-n converse bound on the non-asymptotic symmetric error exponent for adaptive discrimination of quantum channels.The proof uses a meta-converse applied to the final states of an adaptive protocol.Source: C. Chernoff bound · pp. 28–29
  • The converse proof also uses the inequality eD1/2(ρ∥σ) ≤ Dmax(ρ∥σ) and amortization collapse for max-relative entropy.These are combined with the meta-converse to obtain a bound in terms of max-channel divergence.Source: C. Chernoff bound · p. 29
  • The symmetric fidelity-based bound also holds with N and M interchanged because quantum fidelity is symmetric.Source: C. Chernoff bound · p. 29
  • The fidelity divergence defined in [59] equals eDA1/2(N∥M) after taking a negative logarithm; restricting its state optimization to pure states does not bound their dimension.Source: C. Chernoff bound · p. 29
  • Proposition 21 implies an upper bound that is provably finite when the channels cannot be perfectly distinguished by an adaptive protocol, making the symmetric error exponent meaningful in that case.The cited prior result [18, Lemma 10] supplied a non-explicit upper bound.Source: C. Chernoff bound · p. 29

VI. CLASSICAL-QUANTUM CHANNEL DISCRIMINATION

  • In several classical-quantum discrimination settings, choosing the best input and using a tensor-power strategy is optimal, so adaptivity, quantum memories, and entangled inputs provide no advantage.This also extends the classical setting by resolving whether quantum memories could help there.Source: §VI · p. 30

A. Stein bound · B. Strong converse exponent

For classical-quantum channels, the section establishes a Stein bound using amortization collapse and shows that the strong converse exponent matches what product-state strategies achieve. The strong converse exponent may require optimizing over input distributions rather than choosing a single input letter.

A. Stein bound

  • For classical-quantum channels, Theorem 24 establishes a Stein bound whose achievability uses product-state discrimination and whose converse relies on amortized relative entropy.The proof reduces the converse to evaluating the amortized quantity via Lemma 25.Source: A. Stein bound · p. 30
  • Lemma 25 establishes amortization collapse for quantum relative entropy on classical-quantum channels, supplying the quantity needed for the Stein-bound converse.The proof invokes data processing and partial trace to bound the divergence for arbitrary states.Source: A. Stein bound · pp. 30–31

B. Strong converse exponent

  • For classical-quantum channels, the strong converse exponent from Proposition 20 matches the exponent achievable with product states.The proof combines Proposition 20 with Lemma 26; its product-state strategy inputs a share of a classical state at every channel use.Source: B. Strong converse exponent · pp. 31, 33, 34
  • The strong-converse proof establishes its minimax exchange using convexity, continuity under the support condition, and compactness of the input-distribution space.The cited proof applies the Kneser-Fan minimax theorem.Source: B. Strong converse exponent · pp. 34–35
  • Theorem 27 implies a strong variant of Stein’s lemma, and when H(r, N, M) > 0 the Type I error probability tends to one exponentially fast.The corollary states its result for all ε ∈ (0, 1).Source: B. Strong converse exponent · p. 35
  • Unlike the weak and strong Stein lemmas, the strong converse exponent cannot generally be obtained by selecting only the best input letter; it requires optimizing over input distributions.The passage notes that the classical case has an example where one input is insufficient, while distributions positive on two letters suffice there [14, Section IV] [14, Theorem 3].Source: B. Strong converse exponent · p. 35

C. Error exponent

This section derives Hoeffding-exponent bounds for classical–quantum channel discrimination using a log-Euclidean Rényi divergence and an environment-parametrized representation. The first approach is not generally tight, while the second applies state-discrimination converse bounds to adaptive protocols.

C. Error exponent

  • Proposition 30 gives a log-Euclidean Rényi-divergence upper bound, but unlike the classical case this approach is not generally tight.The authors suggest that closing the gap may require an improved proof strategy that would also be novel for the classical case; the upper-bound technique is related to [14, Equation (16)].Source: C. Error exponent · pp. 35–36
  • The lower bound in Proposition 30 follows by repeatedly sending the optimizing input letter and applying the state Hoeffding bound [74].This uses a non-adaptive strategy.Source: C. Error exponent · p. 36
  • Proposition 31 bounds the Hoeffding exponent by applying the state converse to an environment-parametrized representation of adaptive channel discrimination.The common interaction channel makes the adaptive channel task equivalent to parallel state discrimination, so the converse Hoeffding bound for states applies.Source: C. Error exponent · pp. 37–38
  • The environment-parametrized bound can approach the lower bound when one same-letter output-state pair is highly distinguishable and the other pairs are not.This describes the condition under which the bound can be close; the supplied passage does not expose the bound’s full formula.Source: C. Error exponent · p. 38

D. Chernoff bound · VII. OTHER SINGLE-LETTER EXAMPLES · A. Zero-error examples

For classical-quantum channels, the Chernoff upper bound lies within a factor of two of a tensor-power, non-adaptive lower bound, and perfect distinguishability has an orthogonality criterion. Further examples show where single-letter characterizations and zero-error discrimination apply.

D. Chernoff bound

  • The classical-quantum Chernoff upper bound does not match the product-state lower bound, but the gap is less than a factor of two and another upper bound is tighter in some cases.The authors suspect an improved proof could close the gap.Source: D. Chernoff bound · pp. 38–39
  • The lower bound uses a tensor-power non-adaptive input, while the upper bounds follow from amortized-divergence collapse and a Hoeffding-exponent argument.The proof derives the lower bound by repeatedly sending the letter maximizing the state Chernoff divergence, using the quantum-state Chernoff bound [3] [4]; the upper-bound arguments also use Corollary 23, Lemma 26, and Proposition 35.Source: D. Chernoff bound · pp. 38–39
  • Perfect distinguishability of classical-quantum channels is possible exactly when some letter's two outputs are orthogonal in the Hilbert-Schmidt inner product; this fails already for entanglement-breaking channels [10].Source: D. Chernoff bound · p. 39

VII. OTHER SINGLE-LETTER EXAMPLES

  • The paper extends classical single-letter results [14] to classical-quantum channels, while noting that unamortized divergences fail to characterize Chernoff exponents [9] [10] and have no known counterexample for Stein exponents.This motivates the additional single-letter examples collected in the section.Source: §VII · p. 39

A. Zero-error examples

  • Two isometries or unitaries can be perfectly distinguished with some finite number of uses, without a reference system or entangled inputs [9] [25] [76].Consequently, all asymptotic quantities introduced in Section III A are trivial in this setting.Source: A. Zero-error examples · p. 39
  • Two projective measurements have the same finite-use perfect distinguishability, trivial-reference, and no-entanglement properties [26].All asymptotic quantities introduced in Section III A are therefore trivial for this setting as well.Source: A. Zero-error examples · p. 39

B. Environment-parametrized channels

For environment-parametrized channels, adaptive distinguishability is bounded by that of the underlying environment states. When the channels are environment-seizable, discrimination reduces to state discrimination and the resulting bounds are achievable.

B. Environment-parametrized channels

  • For any sub-additive generalized divergence, Proposition 33 bounds the amortized distinguishability of environment-parametrized channels by the distinguishability of their environment states.This collapse underlies the later converse bounds.Source: B. Environment-parametrized channels · p. 40
  • The cited passages state bounds for environment-parametrized channels: Proposition 34 covers ε∈(0,1), α>1, and Proposition 35 covers fixed ε∈(0,1), r>0, p∈(0,1), for all n≥1.Source: B. Environment-parametrized channels · pp. 40–41
  • Any adaptive protocol for environment-parametrized channels can be viewed as a measurement on n copies of the corresponding environment states.This reformulation expresses channel type-I and type-II errors as state-discrimination errors, enabling bounds from asymptotic quantum state discrimination.Source: B. Environment-parametrized channels · pp. 40–41, Figure 4 · p. 41
  • For environment-seizable channels, applying a common input and post-processing to seize the environment states reduces channel discrimination to state discrimination and achieves the preceding upper bounds.Erasure and dephasing channel pairs are given as examples; for them, the environment can be seized using a simple input and output processing.Source: B. Environment-parametrized channels · pp. 41–42
  • Jointly teleportation-simulable channels are a special case of environment-parametrized channels; when their resource states equal their Choi states, they are also environment-seizable by recovering the Choi state.Jointly covariant channels are jointly teleportation-simulable with Choi states as their associated resource states.Source: B. Environment-parametrized channels · p. 43
  • For multiple environment-seizable channels, discrimination reduces to state discrimination, so the multiple Chernoff bound applies [83].Source: B. Environment-parametrized channels · p. 43

C. Replacer channels and quantum illumination · VIII. CONCLUSION & OUTLOOK

Amortized channel divergence yields results for energy-constrained discrimination involving replacer channels and supports the strong Stein’s lemma for classical-quantum channels. The conclusion also identifies unresolved questions, including whether adaptive protocols improve the asymmetric Stein exponent for quantum channels.

C. Replacer channels and quantum illumination

  • For an arbitrary channel and a replacer channel, energy-constrained quantum relative entropy collapses under amortization.This result underlies a weak Stein’s lemma showing that adaptive strategies do not help in energy-constrained discrimination in this setting.Source: C. Replacer channels and quantum… · pp. 43–44
  • In quantum illumination [85], the energy-constrained relative-entropy optimizer is the two-mode squeezed vacuum, completing the Stein-exponent characterization [84].The optimizer saturates the energy constraint; second-order characterizations are discussed in [86].Source: C. Replacer channels and quantum… · p. 44
  • For α > 1, the sandwiched Rényi divergence between an arbitrary channel and a replacer channel also collapses under amortization.The paper recasts a main result of [15] in this form.Source: C. Replacer channels and quantum… · p. 45

VIII. CONCLUSION & OUTLOOK

  • The framework establishes the strong Stein’s lemma for classical-quantum channels: adaptive strategies do not improve their asymptotic exponential error rate.The paper also derives converse bounds for general quantum channel discrimination.Source: §VIII · p. 46
  • Tight Chernoff and Hoeffding characterizations remain open for classical-quantum channels and replacer channels; the classical case provides a comparison [14].The replacer-channel questions are connected to open questions about quantum-feedback-assisted communication [15].Source: §VIII · p. 46
  • Whether adaptive protocols improve the asymmetric Stein exponent for quantum channel discrimination remains open, as does an asymptotic advantage for parallel strategies over tensor-power inputs.The authors suspect adaptive improvement; the parallel setting prepares an entangled input across channel uses and performs a joint measurement.Source: §VIII · p. 46
  • With trivial quantum memory, Hastings counterexamples yield a separation from the tensor-product strategy for a replacer channel; nontrivial memory raises entropic additivity questions.The authors expect the framework to find plentiful applications in quantum information theory.Source: §VIII · p. 46

Appendix A: Amortization does not increase the Hilbert α-channel divergences · Appendix B: Generalized Fuchs-van-de-Graaf inequality

Appendix A shows that data-processed triangle inequalities yield amortization collapse for Hilbert α-channel divergences and fidelity-based channel distances. Appendix B states a generalized Fuchs-van-de-Graaf inequality for positive semidefinite operators.

Appendix A: Amortization does not increase the Hilbert α-channel divergences

  • For every α ≥1, the Hilbert α-divergence obeys a data-processed triangle inequality under positive trace-preserving maps.The proof uses the trace-distance triangle inequality and monotonicity at α=1, and establishes the inequality for α>1 using the map’s positive unital adjoint.Source: Appendix A · p. 47
  • Hilbert α-channel divergences collapse under amortization for all α ≥1, and the diamond norm of the difference of two channels also does not increase under amortization.The collapse follows from strong faithfulness of the Hilbert α-divergence [64, Theorem 1(i)].Source: Appendix A · p. 48
  • The c-distance and Bures distance satisfy data-processed triangle inequalities, yielding amortization collapse for their induced channel divergences.These state distances are based on quantum fidelity, whose monotonicity under positive trace-preserving maps supports the inequalities [7, Corollary A.5].Source: Appendix A · p. 48

Appendix B: Generalized Fuchs-van-de-Graaf inequality

  • Lemma 46 generalizes the Fuchs-van-de-Graaf relation to positive semidefinite, trace-class operators on a separable Hilbert space, with a tighter bound than [6, Theorem 7].The generalized relation is attributed to [70].Source: Appendix B · p. 49
  • The generalized inequality’s proof chooses a unitary from the polar decomposition of B^1/2A^1/2 after first allowing an arbitrary unitary.The authors give a complete proof that follows [6, Theorem 7] closely but changes the reasoning slightly.Source: Appendix B · pp. 49–50
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