arXiv:2610.16270

Continuity of certain quantum channel divergences

Morgan Jones

quant-phSubmitted Oct 12, 2026• Announced Oct 13, 2026• 0 citations• Revised Oct 21, 2026 (v2)

Abstract

The geometric channel fidelity, the geometric Rényi divergences and the Belavkin--Staszewski relative entropy give converse bounds for capacities, discrimination and estimation. Channels are known only approximately, yet two pairs of channels with equal sums can be arbitrarily close in the diamond norm and have fidelities $0$ and $1$. The remedy is the Arveson--Radon--Nikodym derivative $T$ of one map with respect to the sum $Ω$, whose transpose is the environment effect selecting that map in a Stinespring dilation of $Ω$. In $T$, these quantities and all maximal $f$-divergences of operator convex $f$ are continuous, with explicit dimension-free moduli; for two pairs of channels with the same sum the fidelities differ by at most $2\|T_1-T_2\|_\infty^{1/2}$, which gives Hölder continuity in the diamond norm on such splittings. The fidelity is continuous at a pair of completely positive maps exactly when the Choi matrix of the sum is invertible or the fidelity vanishes: noisy channels of full Kraus rank are continuity points, unitary gates are not.

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arXiv:2610.16270

Continuity of certain quantum channel divergences

Morgan Jones

quant-phSubmitted Oct 12, 2026• Announced Oct 13, 2026• 0 citations• Revised Oct 21, 2026 (v2)

arXiv HTML PDF

Abstract

0

and

1

. The remedy is the Arveson--Radon--Nikodym derivative

T

of one map with respect to the sum

Ω

, whose transpose is the environment effect selecting that map in a Stinespring dilation of

Ω

. In

T

, these quantities and all maximal

f

-divergences of operator convex

f

are continuous, with explicit dimension-free moduli; for two pairs of channels with the same sum the fidelities differ by at most

2\|T_1-T_2\|_\infty^{1/2}

, which gives Hölder continuity in the diamond norm on such splittings. The fidelity is continuous at a pair of completely positive maps exactly when the Choi matrix of the sum is invertible or the fidelity vanishes: noisy channels of full Kraus rank are continuity points, unitary gates are not.

Related papers

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quant-pharXiv:2609.29885Sep 25, 20260 citations

0

and

1

. The remedy is the Arveson--Radon--Nikodym derivative

T

of one map with respect to the sum

Ω

, whose transpose is the environment effect selecting that map in a Stinespring dilation of

Ω

. In

T

, these quantities and all maximal

f

-divergences of operator convex

f

are continuous, with explicit dimension-free moduli; for two pairs of channels with the same sum the fidelities differ by at most

2\|T_1-T_2\|_\infty^{1/2}

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.