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arXiv:2610.03706

Quantum estimation, channel orders, and private capacity

Christoph Hirche

quant-phcs.ITSubmitted Oct 2, 2026• Announced Oct 5, 2026• 0 citations
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Abstract

Recent examples have shown that zero private capacity need not imply antidegradability and that two channels with zero private capacity can nevertheless transmit private information together. Existing entropic channel orders compare what a receiver and its environment can learn and thereby bound capacities. We connect these orders to the recent constructions through binary estimation, whose minimum mean-square error is governed by measured $χ^2$ divergence. A new integral representation shows how measured $χ^2$ on a qubit extension recovers quantum $χ^2$. It lets us pass from complete measured-$χ^2$ ordering to complete quantum-$χ^2$, relative-entropy, and less-noisy ordering. Zhu and Wang used a signed lift to show that their qutrit channel has zero private capacity. We show that Hermitian-smoothed measured-$χ^2$ ordering characterizes when such lifts exist, even with a quantum reference. Environmental dominance in binary estimation at every blocklength yields a finite-code reliability--secrecy bound. These results explain why complete comparison prevents activation with antidegradable helpers whereas regularized comparison alone does not. Building on that qutrit example, we establish this stability for a range of noisy Werner--Holevo channels, determine sharp private-capacity and antidegradability thresholds, and compute exact complementary capacities in a nondegradable range. By contrast, we extend Pauli half-erasure activation to all $1/2\le p<1$ and establish the same range for a new four-level family. Reference-assisted measured-$χ^2$ witnesses show why these activating constructions lack complete comparison.

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40 pages

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Private communication over a noisy quantum channel requires reliable transmission to the receiver and secrecy from the environment. Whether two channels with zero private capacity can jointly enable private communication is a longstanding open problem in quantum information theory. Here we resolve this problem by exhibiting a four-level channel and a qubit erasure channel with half erasure probability, each with zero private capacity, whose joint use achieves more than 0.0001903 private bits per product use. The encoding gives the receiver a linear information gain with at most quadratic environmental leakage, enabling privacy through a fixed joint measurement and classical coding. This superactivation, impossible for independent classical memoryless wiretap channels, shows that a channel's private capacity alone does not determine its value for secure communication. The initial activation example was identified through interactions with large language models, and the result has been formalized in Lean 4.
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The quantum capacity of a noisy channel quantifies the maximum rate at which quantum information can be transmitted reliably. For general channels, its evaluation requires an optimization over arbitrarily many channel uses. Degradable channels form a central exception: their capacity is given by the single-letter coherent information, while antidegradable channels have zero capacity. Nevertheless, even for these fundamental classes, it has remained open whether communication above capacity becomes possible when a fixed non-maximal error is tolerated. Here we resolve this problem by proving an exponential strong converse for every finite-dimensional degradable and antidegradable channel: at any rate above capacity, the fidelity of every coding scheme decays exponentially with the number of channel uses. As an immediate consequence, we establish the first all-code exponential strong converse for the quantum erasure channel throughout its full parameter range, strengthening previous results that applied only to almost all codes. We also show that exponential strong-converse bounds are preserved under receiver post-processing. This yields efficiently computable semidefinite-programming bounds for arbitrary finite-dimensional channels, improved bounds for Pauli channels, and an exact exponential strong converse for a nondegradable multilevel amplitude-damping family.
When does a difference in information become a way to communicate? Imagine two quantum channels as competing routes through noise. One might preserve a distinction that the other loses. That difference is interesting on its own; the question here is whether it can be turned into something operational: a channel that can carry private or quantum information at a positive rate. This paper presents a construction intended to make that step. Starting from a failure of complete quantum-channel information order, it constructs a channel claimed to have strictly positive ordinary private and quantum capacity. “Complete” matters because the comparison allows an input to be correlated with a finite quantum reference: information may be visible in the joint system even when it is hidden from the channel output alone. The construction accounts for the resources it uses, including transmission of the reference and uses of an activation helper. Its conclusion concerns strict positivity: a rate greater than zero, however small. It does not claim a uniform positive gap or fixed-error capacity inapproximability. The paper also develops reductions connecting the capacity questions to one-way distillable entanglement and secret key. The state constructions are designed to retain any prescribed lower bound on two-way distillation. On this basis, the manuscript claims exact-zero undecidability classifications for finite rational-complex memoryless channels and corresponding classifications for the one-way quantities. The information-order equivalences and undecidability premises used in these conclusions are developed in the companion paper, Complete information orders of quantum channels: equivalence and undecidability. A further introduction and citation details are available on the manuscript page.
Quantum communication is ruled out when an eavesdropper can reconstruct the receiver's state. Yet states are operationally specified by measurement statistics, raising a sharper question: can access to every expectation value destroy communication? We show that it can. We call unbiased, variance-nonincreasing reconstruction under arbitrary reference extensions \emph{complete observable shadowing} and prove that it forces unassisted private and quantum capacities to vanish. An explicit qutrit channel realizes this mechanism while being neither antidegradable nor PPT, resolving the long-standing questions of whether quantum capacity can vanish outside those two classes and whether antidegradability is the only route to zero private capacity. Notably, it superactivates every finite-dimensional channel with zero quantum but positive private capacity. Statistical access can therefore preclude standalone communication without erasing latent coherent utility.