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Minimax games for quantum channel discrimination

Kun Fang, Michael X. Cao, Hao-Chung Cheng, Li Gao, Masahito Hayashi

arXiv:2609.09839v1quant-phcs.ITmath.IT

TL;DR

Channel-discrimination models idealize inputs as fully tester-controlled or externally determined, although some scenarios include unpredictable disturbances. This paper models separate tester and jammer inputs across twelve games, finding that information-pattern effects vanish asymptotically for entangled jammers but can remain for IID jammers.

  • Problem

    Channel-discrimination models idealize inputs as fully tester-controlled or externally determined, while some relevant scenarios include unpredictable disturbances.

    Source: §1.1 · p. 3

  • Method

    The framework combines tester/jammer input structures and information patterns into twelve games, characterizes their Stein exponents, and develops an achievability-to-strong-converse principle.

    Source: §1.3 · p. 8, §9 · p. 42

  • Results

    Entangled jammers make visibility and hypothesis awareness irrelevant to the asymptotic exponent, whereas IID jammers can preserve information-pattern differences; replacer alternatives share one additive single-letter exponent.

    Source: §1.3 · p. 7, §9 · p. 42

  • Takeaways & Limitations

    The upgrade principle yields strong converses for selected games, resolves the composite-hypothesis question of Berta, Brandão, and Hirche [BBH21, Remark 2.2], and strengthens Lami’s Stein lemmas [Lam25b].

    Source: §1.3 · p. 8

    The framework treats parallel channel uses; extensions to adaptive or sequential strategies and strong converses for remaining vanishing-error results remain open.

    Source: §9 · p. 42

Abstract

from arXiv · show

Quantum channel discrimination is a primitive task for identifying, verifying, and benchmarking quantum dynamics. Previous studies have primarily considered either the best-case tester-input setting or the worst-case jammer-input setting. Here, we introduce a game-theoretic framework in which both the tester and jammer control separate inputs. Combining three input structures, characterized by whether the tester and jammer use entangled inputs or IID inputs across channel uses, with four information patterns, determined by the visibility of the jammer's strategy and its knowledge of the true hypothesis, yields twelve game models. We provide exact finite blocklength hypothesis testing characterizations of all twelve models in terms of nine minimax hypothesis testing divergences and derive their asymptotic Stein exponents. Notably, for entangled jammers, neither the visibility of the jammer's strategy nor its knowledge of the true hypothesis affects the asymptotic Stein exponent, whereas the information pattern remains consequential for IID jammers. As an example, we study the discrimination of a general channel from a replacer channel and show that all asymptotic Stein exponents coincide with the same additive, single letter quantity. We further develop a general argument that upgrades achievability results to strong converse results, thereby establishing strong converse properties for several game models, resolving an open problem in composite hypothesis testing posed by Berta et al. [Commun. Math. Phys. 385, 55 (2021)], and strengthening several recent results of Lami [arXiv:2510.06340]. The framework and techniques developed here may support future studies of quantum information tasks involving competing roles.

1 Introduction

The paper extends quantum channel discrimination beyond tester-controlled and jammer-controlled endpoints by giving a minimax framework in which both control separate inputs. It combines input structures and jammer information patterns to characterize the resulting games and their asymptotic behavior.

1.1 Background and motivation

  • The motivating gap is the contrast between tester-controlled best-case discrimination and jammer-controlled worst-case discrimination, while relevant settings can involve intermediate uncertainty.The paper’s minimax model gives the two parties separate input control.Source: §1.1 · pp. 3–4

1.2 Game models

  • The framework models tester and jammer control of separate inputs as an asymmetric hypothesis-testing game, with input structure and information pattern as independent ingredients.The framework recovers tester-input and jammer-input discrimination when either input is made trivial.Source: §1.2 · p. 4, §1.1 · p. 4
  • The twelve games combine three input structures with four information patterns, varying entanglement or IID restrictions and the jammer’s visibility and hypothesis awareness.The tester and jammer may each correlate their own systems across uses, while their inputs tensorize with one another.Source: §1.2 · p. 5

1.3 Main results

  • When the jammer may use entangled inputs, all four information patterns have the same asymptotic Stein exponent.Neither visibility of the jammer’s input nor its hypothesis awareness changes the rate.Source: §1.3 · p. 7
  • With both players restricted to IID inputs, the four information patterns do not all collapse, although the two secret-jammer exponents coincide.The IID tester constraint prevents use of the same minimax argument as in the entangled-input case; the paper instead constructs a common jammer input uniformly effective across tester choices.Source: §1.3 · pp. 7–8
  • For replacer alternatives, all twelve vanishing-error Stein exponents reduce to the same additive, single-letter value.The result is a structural specialization of the game framework.Source: §1.3 · p. 8
  • An achievability-to-strong-converse principle yields strong converses for selected minimax games and composite hypothesis testing.The results resolve the question posed by Berta, Brandão, and Hirche [BBH21, Remark 2.2] and strengthen Lami’s Stein lemmas [Lam25b].Source: §1.3 · p. 8

1.4 Organization of the paper

  • The paper proceeds from notation and divergences to operational characterizations, analyses of the three input structures, replacer alternatives, strong converses, and concluding directions.The conclusion identifies adaptive or sequential strategies and further single-letter and converse results as open directions.Source: §1.4 · p. 8

2 Preliminaries

The preliminaries establish notation and recall quantum state divergences and hypothesis-testing bounds used in the paper. Figure 4 previews how the game models relate to finite-block and asymptotic divergences.

2 Preliminaries

  • The section collects notation, state-divergence properties, and one-shot hypothesis-testing bounds used throughout the paper.It defines the Umegaki, sandwiched Rényi, measured Rényi, max-relative, and hypothesis-testing divergences.Source: §2 · p. 8, §2.2 · pp. 10–11
  • Figure 4 summarizes twelve game models as nine finite-block divergences and five asymptotic Umegaki relative entropy rates.Its solid and dashed connectors indicate strong- and weak-converse results; for a replacer alternative, all Stein exponents share one single-letter value.Source: Figure 4 · p. 9

2.1 Notation and conventions

  • The notation defines tensor-power states as ξ^⊗n and permutation-invariant states separately, alongside operators, density operators, and test effects.All Hilbert spaces are finite dimensional, and logarithms use base two.Source: §2.1 · p. 9

2.2 Quantum state divergences

  • A quantum divergence is monotone under CPTP maps, and the section records common properties and bounds linking hypothesis-testing divergence with Umegaki and Rényi divergences.Recorded properties include direct-sum equality, normalization, operator dominance, joint convexity, boundedness, and lower semicontinuity.Source: §2.2 · p. 10

3 Minimax quantum channel divergences

The section develops channel divergences for tester and jammer inputs, then introduces dual-input maximin and minimax divergences. These divergences exactly characterize the finite-blocklength exponents of the twelve discrimination games.

3 Minimax quantum channel divergences

  • The section establishes finite-block operational correspondences between its divergences and the twelve games, building on tester-input and jammer-input divergences from prior studies.The public correspondences follow by fixing the announced jammer states and applying the state hypothesis-testing formulation; secret cases also use convexification for IID jammer inputs.Source: §3 · p. 11, §3.3 · pp. 15–16

3.1 Two extreme channel divergences

  • Tester-input and jammer-input channel divergences represent the two extreme input-control settings, and the jammer-input versions need no auxiliary reference system.The two jammer-input variants can differ in general, as illustrated for quantum relative entropy.Source: §3.1 · pp. 11–12

3.2 Maximin and minimax channel divergences

  • The maximin and minimax divergences unify optimizations over the tester’s and jammer’s separate inputs, with arrow notation specifying optimization domains and shared or distinct jammer variables.Maximin divergences interpolate between the tester-input and jammer-input divergences; a finiteness lemma supplies bounds used repeatedly for these quantities.Source: §3.2 · pp. 12–13

3.3 Operational correspondence

  • The twelve channel-discrimination games have finite-blocklength Stein exponents exactly characterized by maximin or minimax divergences induced by quantum hypothesis-testing divergence.The twelve games yield nine potentially distinct divergence expressions; public-game correspondences follow directly from definitions, while secret-game correspondences require additional steps.Source: §3.3 · pp. 14–15
  • For secret games, the tester uses one test with a uniform type-I constraint over jammer inputs and worst-case type-II performance, so jammer hypothesis awareness does not change the value.The secret-game identities require double-down-arrow divergences: at finite blocklength, a single- and double-down-arrow hypothesis-testing divergence can differ strictly.Source: §3.3 · pp. 15–16

4 Entangled tester against entangled jammer

With entangled inputs for both players, all four information patterns have the same asymptotic Stein exponent. The proof establishes minimax and asymptotic-equivalence properties, then develops supporting results for the theorem.

4 Entangled tester against entangled jammer

  • All four information patterns yield the same optimal type-II error exponent when both tester and jammer use entangled inputs.The theorem assumes Dmax(N∥M) < ∞.Source: §4 · p. 16

Minimax identity

  • Under the stated regularity conditions, maximin and minimax channel divergences coincide, permitting the tester and jammer optimization order to be exchanged.The result applies to Umegaki relative entropy, measured relative entropy, and specified ranges of measured and sandwiched Rényi divergences.Source: Minimax identity · p. 17

Asymptotic equivalence

  • A suitable mixture of hypothesis-dependent jammer inputs makes their common-input and separate-input maximin divergences asymptotically equivalent.The mixture incurs a vanishing penalty after regularization, and the four regularized minimax and maximin divergences coincide under the stated assumptions.Source: Asymptotic equivalence · pp. 18–19

Continuity in the Rényi order

Superadditivity and existence of regularized rates

  • The maximin divergence induced by measured Rényi divergence is superadditive, so its regularized channel divergence exists.This superadditivity does not extend to the variant that minimizes over a common input; it can fail even for the underlying jammer-input channel divergence.Source: Superadditivity and existence of… · pp. 20–21

Permutation-symmetry reduction

  • The n-fold maximin channel divergences can be optimized over permutation-invariant states without changing their values.The symmetry reduction also holds for the common-input variant described in the lemma.Source: Permutation-symmetry reduction · p. 21

Measured-to-quantum asymptotic equivalence

  • After regularization, the maximin divergence induced by measured divergence coincides with that induced by its fully quantum counterpart.The result extends a prior asymptotic-equivalence lemma to the maximin setting.Source: Measured-to-quantum asymptotic… · p. 22

4.3 Proof of Theorem 18

  • The proof establishes achievability using a one-shot bound and measured-to-sandwiched Rényi ordering, then takes a limit inferior in blocklength.It suffices to establish achievability for one divergence setting; the other operational settings follow by sandwiching.Source: §4.3 · p. 23

5 IID tester against entangled jammer

For an IID tester facing an entangled jammer, the four information patterns have the same asymptotic Stein exponent. Because the IID input family is nonconvex, the proof constructs a jammer strategy directly at the operational level.

5 IID tester against entangled jammer

  • All four information patterns have the same asymptotic Stein exponent for an IID tester against an entangled jammer.The theorem assumes Dmax(N∥M) < ∞.Source: §5 · p. 24
  • The IID tester input family is nonconvex, so the minimax argument used for entangled testers does not apply.The converse instead averages optimal jammer input pairs over a finite subcover to construct an input independent of the tester's choice.Source: §5 · p. 24
  • For each fixed number of channel uses, the relevant divergence is finite and continuous in the IID tester input under the theorem's assumptions.The proof uses this continuity to support its finite-subcover construction.Source: §5 · pp. 24, 25, 27
  • Subadditivity of the jammer-input divergences gives subadditivity of the optimized divergence and indicates existence of its regularized limit.The proof separately establishes achievability and a converse bound to identify the operational exponent.Source: §5 · p. 25

6 IID tester against IID jammer

For an IID tester and IID jammer, the asymptotic Stein exponent can depend substantially on the information pattern. The secret-jammer exponents are characterized by regularized channel divergences, and the section’s exponents need not coincide universally.

6 IID tester against IID jammer

  • The asymptotic Stein exponent can differ significantly across information patterns for an IID tester against an IID jammer.Theorem 32 gives exponents for public IID jammers, while Theorem 33 treats secret IID jammers under a finite Dmax condition.Source: §6 · pp. 28–29
  • The secret-IID-jammer Stein exponents are characterized by regularized channel divergences.Theorem 33 assumes Dmax(N∥M) < ∞.Source: §6 · p. 29
  • The three asymptotic Stein exponents in this section are not universally equal; examples exhibit strict differences for particular channel pairs.The examples are given in Appendix A.Source: §6 · p. 31

7 Replacer alternatives

For a replacer alternative, the alternative-side jammer input is irrelevant, and all twelve vanishing-error Stein exponents coincide at one single-letter value. The finite-block Umegaki divergences are additive under the stated finite-Dmax condition.

7 Replacer alternatives

  • All twelve vanishing-error Stein exponents against a replacer alternative coincide at one single-letter value.Theorem 35 states the finite-block and asymptotic identities when Dmax(N∥Rτ) < ∞.Source: §7 · p. 31
  • The alternative-side jammer input is irrelevant because the replacer output is fixed independently of channel inputs.The replacer discards its input and prepares τB.Source: §7 · p. 31
  • The eleven Umegaki divergences are additive against replacer alternatives, yielding a finite-block identity and its regularized consequence.The proof combines the finite-block identity with division by n and regularization.Source: §7 · pp. 31, 33
  • A full quantum-illumination application remains future work because it requires analysis of bosonic modes under energy constraints.The paper relates quantum illumination to general-channel versus replacer-channel discrimination.Source: §7 · p. 33

8 Strong-converse upgrades

The section develops a general route from suitable vanishing-error achievability results to strong converses, first for states and then for sets of states. It applies these upgrades to channel games and composite hypotheses, including a result resolving an open question posed in [BBH21, Remark 2.2].

8 Strong-converse upgrades

  • The upgrade technique is applied to an IID tester against a secret entangled jammer and to composite hypotheses, including the trivial-tester game.The composite IID specialization resolves the strong-converse question in [BBH21, Remark 2.2] and strengthens results in [Lam25b].Source: §8 · p. 33

8.1 General principles

  • The general principles proceed from classical distributions to quantum state sequences and then to sets of quantum states.These steps are established in Lemmas 37 and 38 and Theorems 39 and 40.Source: §8.1 · p. 33

Upgrade for classical sequences

  • The classical-sequence formulation allows any positive scale ck tending to infinity, including blocklength subsequences.The usual blocklength setting takes ck = k; a subsequence uses ck = nk.Source: Upgrade for classical sequences · pp. 33–34

If the vanishing error exponent achieves the relative entropy rate

Upgrade for quantum state sequences

  • For quantum state sequences, a correction depending only on the error parameters vanishes after scale normalization, enabling the classical upgrade to lift to states.The Nussbaum–Szkoła construction connects quantum state pairs to classical measures while preserving relative entropy.Source: Upgrade for quantum state sequences · p. 35

Upgrade for sets of quantum states

  • The state-sequence upgrade extends to nonempty compact sets of states when the vanishing-error exponent reaches the upper relative-entropy rate.The extension selects a relative-entropy-minimizing pair at each index.Source: Upgrade for sets of quantum states · pp. 36–37

8.2 Upgrade for the minimax game in Section 5

For the minimax game considered here, the section establishes strong-converse results for an IID tester against a secret entangled jammer. Its key step is to obtain an achievability result using a tester-state sequence independent of the vanishing-error parameter.

8.2 Upgrade for the minimax game in Section 5

  • The upgrade requires an achievability sequence independent of the vanishing-error parameter, which the earlier achievability result does not guarantee.The authors construct a more suitable achievability result to apply Theorem 39.Source: §8.2 · p. 37
  • Theorem 41 establishes strong-converse results for an IID tester against a secret entangled jammer under a finite-Dmax condition.The theorem assumes Dmax(N∥M) < ∞.Source: §8.2 · pp. 37, 39
  • The construction fixes a tester-state subsequence and then chooses jammer states to achieve the relevant divergence comparison.The subsequence is fixed before the auxiliary error parameter is introduced.Source: §8.2 · p. 38

8.3 Upgrade for composite state hypotheses

The section develops a general upgrade from vanishing-error achievability to fixed-error strong-converse results for composite hypotheses. It applies this framework to composite IID, correlated, and arbitrarily varying hypotheses, including a trivial-tester specialization.

8.3 Upgrade for composite state hypotheses

  • The strong-converse upgrade identifies the operational exponent with the pairwise divergence of the convex hulls of the composite hypothesis sets.The section applies the general upgrade to sequences of nonempty compact sets in finite-dimensional spaces.Source: §8.3 · p. 40

Composite IID hypotheses and the trivial-tester specialization

  • The composite IID result establishes an ordinary relative entropy rate and a strong converse, resolving the question posed in [BBH21, Remark 2.2].The proof does not require the pairwise support assumption imposed in [BBH21, Theorem 1.1].Source: Composite IID hypotheses and the… · p. 41
  • A trivial tester against a secret IID jammer is treated as a specialization, under the stated channel finiteness condition.The passage states the result for a channel N and a completely positive map M with Dmax(N∥M) < ∞.Source: §8.3 · p. 40, Composite IID hypotheses and the… · p. 41

Correlated and arbitrarily varying hypotheses

  • The framework also upgrades composite Stein exponents for correlated and IID/AV hypotheses to fixed-error limits.The stated results cover correlated alternatives and IID or arbitrarily varying composite hypotheses.Source: Correlated and arbitrarily varying… · p. 42

9 Conclusion

The conclusion summarizes a framework for binary quantum channel discrimination with independently controlled tester and jammer inputs. It reports how jammer structure affects Stein exponents, gives the replacer-channel result, and identifies open extensions.

9 Conclusion

  • Combining three input structures with four information patterns yields twelve games whose asymptotic Stein exponents are characterized by maximin and minimax channel divergences.The games interpolate between tester-input and jammer-input discrimination.Source: §9 · p. 42
  • For entangled jammers, visibility and hypothesis-awareness do not affect the asymptotic exponent, while information patterns can matter for IID jammers.Source: §9 · p. 42
  • For replacer alternatives, all twelve vanishing-error Stein exponents coincide at the same additive, single-letter value.Source: §9 · p. 42
  • The work upgrades achievability results to strong converses, resolving the open problem in [BBH21, Remark 2.2] and strengthening results in [Lam25b].Source: §9 · p. 42
  • The framework treats parallel channel uses; adaptive or sequential strategies and single-letter forms beyond replacer alternatives remain open directions.The conclusion also leaves fixed-error or strong-converse upgrades for remaining vanishing-error results and quantum illumination for future study.Source: §9 · p. 42

A Strict separations between IID jammer models

The appendix introduces two channel-level separation examples using qubit channels.

A Strict separations between IID jammer models

  • Both separation examples use qubit channels and satisfy the main text’s max-relative-entropy finiteness assumption.The first example considers measure-and-prepare channels.Source: A Strict separations between IID… · p. 46

They satisfy

The supplied passages describe a strict separation between public IID-jammer exponents and note a bound used in establishing it.

They satisfy

  • The common-input and separate-input minima differ, yielding a strict separation of the corresponding public IID-jammer exponents.The common-input minimum is at least 1/(8 ln 2), and the lower bound is independent of the common input.Source: They satisfy · p. 47
  • Strictly positive Pauli weights for M imply a positive definite Choi operator and finite Dmax(N∥M).Source: They satisfy · p. 47
  • The construction provides an upper bound on the regularized divergence; the argument does not require that bound to be tight.Source: They satisfy · p. 49

B Some useful identities

This appendix relates operational exponents to hypothesis-testing relative entropy for arbitrary diverging scales, including the one-sided limits needed at fixed error. It also records endpoint identities and proof arguments for the testing characterizations.

B Some useful identities

  • For arbitrary positive scales c_k→∞, the appendix relates operational exponents to hypothesis-testing relative entropy, including the one-sided limits required at fixed error.The one-sided limits exist because h_k is nondecreasing.Source: B Some useful identities · pp. 49–50
  • The vanishing-error and strong-converse exponents are defined operationally, and classical distributions and finite measures are included without changing testing values.The classical setting is represented by diagonal operators, with tests restricted to diagonal effects.Source: B Some useful identities · p. 49
  • Optimal tests exist because the feasible set is compact and the type-II objective is continuous.This supports the test constructions used in the identities.Source: B Some useful identities · p. 50
  • The reverse bounds construct tests from optimal fixed-error tests, using increasing blocks for Eq. (340) and selected indices for the other bound.The appendix also expresses the endpoint cases of [NH07, Theorem 1] through Lemma 48.Source: B Some useful identities · p. 50
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