arXiv:2610.16270

Quantitative upper bounds for certain quantum channel divergences

Morgan Jones

quant-phSubmitted Oct 12, 2026• Announced Oct 13, 2026• 0 citations

Abstract

We show that completely positive maps can be decomposed into sums of submaps which are close in completely bounded trace norm but as far apart as possible in terms of geometric channel fidelity. We then show that this kind of behavior extends to a large class of quantum channel divergences. We construct quantitative norm estimates for quantum channel divergences in terms of the Arveson-Radon-Nikodym theorem.

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arXiv:2610.16270

Quantitative upper bounds for certain quantum channel divergences

Morgan Jones

quant-phSubmitted Oct 12, 2026• Announced Oct 13, 2026• 0 citations

arXiv HTML PDF

Abstract

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We prove that learning an unknown quantum channel with input dimension

d_A

, output dimension

d_B

, and Choi rank

r

to diamond distance

\varepsilon

requires $ Ω\!\left( \frac{d_A d_B r}{\varepsilon \log(d_B r / \varepsilon)} \right)$ channel queries when

d_A= rd_B

, and

Ω\!\left( \frac{d_A d_B r}{\varepsilon^2 \log(d_B r / \varepsilon)} \right)

channel queries when

d_A\le rd_B/2

. These lower bounds improve upon the best previous

Ω(d_A d_B r)

bound by introducing explicit, near-optimal

\varepsilon

-dependence. Moreover, when

d_A\le rd_B/2

, the lower bound is optimal up to a logarithmic factor. The proof constructs ensembles of channels that are well separated in diamond norm yet admit Stinespring isometries that are close in operator norm.

We establish the quantum channel Stein lemma at every fixed type-I error tolerance in

(0,1)

and prove an exponential strong converse under arbitrary adaptive strategies. For any finite-dimensional channel pair, the optimal type-II exponent equals the regularized channel relative entropy and is attained by parallel strategies. When this divergence is finite, every strictly larger type-II rate forces the probability of accepting the null hypothesis to decay exponentially, uniformly over strategies and quantum memories. The proof rests on a uniform Stinespring approximation: tensor powers of the null dilation are approximated by tensor powers of the alternative dilation after applying an auxiliary linear map to the environment. The operator-norm error is exponentially small under a squared-norm bound at any rate above the regularized relative entropy. We construct the approximation from a weak testing bound by iterative rate reduction and a fixed-block tensor expansion with an exact residual. It implies right continuity at order one of the regularized sandwiched Rényi channel divergence, which yields the adaptive converse through the channel chain rule. We also obtain a sharp hockey-stick threshold and a subchannel smoothing asymptotic equipartition property for fixed and subexponentially vanishing diamond-norm errors.

We show that the regularized channel Rényi divergence is continuous at

α= 1

. As consequences, we establish an exponentially strong converse for channel discrimination, a subchannel-smoothed asymptotic equipartition property, a relative entropy accumulation theorem, and a new proof of the generalized quantum Stein's lemma. Our result further shows that the regularized channel relative entropy is computable. The continuity proof relies on a variational formula that expresses the regularized channel divergence as a single-letter optimization over variables of unrestricted finite dimension. We then establish continuity using a novel spectral confinement technique.