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Quantum Channel Stein Theorem beyond Definite Causal Order
arxiv.orgpaper

Quantum Channel Stein Theorem beyond Definite Causal Order

This arXiv paper proves that for finite-dimensional memoryless quantum channels, parallel, adaptive, and general (including indefinite causal order) hypothesis-testing strategies all achieve the same Stein exponent — the regularized channel relative entropy — showing adaptivity and exotic causal structures give no asymptotic advantage in quantum channel discrimination. The key technical result is continuity of the regularized sandwiched Rényi channel divergence at order one, plus an exact strong-converse exponent for general testers.

TL;DR Proves adaptive and indefinite-causal-order testing strategies offer no asymptotic advantage over parallel strategies for distinguishing quantum channels, extending the quantum Stein's lemma.
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What it claims

Beyond the equal-exponent result the summary states, the page establishes the architecture that gets there and two things a reader building on it would need. The main theorem (Thm 2.4) is stated with an explicit exponential strong converse: if the rate $D^\infty(N\|M)$ is finite, then for every $r>D^\infty$ there exist $c>0$ and $n_0$ (depending only on the channels and $r$) such that any general tester with Type-II error $b\le e^{-nr}$ has null-acceptance error $a\le e^{-cn}$ for $n\ge n_0$. The result holds for all three strategy classes (parallel, adaptive, general/indefinite-causal-order), and also when the rate is infinite.

How it works

The converse is reduced to a scalar quantity. The paper defines a weighted testing score $\Delta_n(\lambda)=\max_{\text{parallel testers}}(a-\lambda b)$ and proves (Lemma 4.1) it equals the least normalized positive Choi slack $h$ such that $N_n\le\lambda M_n+hQ$ with $Q$ a channel Choi operator. A fixed-marginal de Finetti reduction [12, Cor 1.1] turns the general-tester slack into a mixture of IID channel Choi operators with only polynomial loss ($\log g_n=O(\log n)$), which vanishes on exponential rates. An explicit general-to-parallel conversion is given in Appendix A. The order-one continuity of the regularized sandwiched Rényi channel divergence (Thm 5.1) is proved via a supported-inverse factorization lifting the slack to a single amplitude approximation, a tensor-moment estimate for entangled probes, and combining two-cost approximations into three amplitudes.

The page's other substantive contribution is an exact one-shot SDP duality (Thm 3.1): parallel testers are dual to smoothing over all Choi operators ($K_{\text{all}}$), general testers to the positive part of the affine hull of product channel Choi operators ($K_{\text{aff}}$). Four nested Choi sets $K_{\text{iid}}\subseteq K_{\text{prod}}\subseteq K_{\text{aff}}\subseteq K_{\text{all}}$ all share the asymptotic limit $D^\infty(N\|M)$ (Cor 5.4). This connects channel testing to the finite-blocklength smooth-entropy framework.

Limits

  • $\varepsilon>0$ is necessary. The equal-exponent claim holds only for fixed $\varepsilon\in(0,1)$. At zero Type-I error the paper gives a counterexample: replacer channels with distinct full-rank commuting outputs $\rho,\sigma$ force $D^{0,\text{par}}_H=0$ despite $D(\rho\|\sigma)>0$.
  • All systems are finite-dimensional and channels memoryless; nothing is claimed for infinite dimensions or channels with memory.
  • The general-tester class is a mathematical set (positive process-testers with product-channel normalization); the authors explicitly do not claim every feasible tester has a circuit realization — the value is that the converse then covers any physically realizable subclass.
  • This is v1 of a preprint (dated Sept 2026); no peer review.

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