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arXiv:2607.24693

Quantum Incapacity beyond No-Cloning and PPT Mechanisms

Chengkai Zhu, Xin Wang

quant-phSubmitted Jul 27, 2026• Announced Jul 28, 2026• 0 citations• Revised Jul 30, 2026 (v2)
v1 · Jul 27, 2026v2 · Jul 30, 2026v3 · Oct 5, 2026
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Abstract

We show an explicit qutrit channel whose private and quantum capacities both vanish, although it is neither antidegradable nor positive under partial transposition (PPT). This resolves two longstanding open problems in quantum information theory: whether zero quantum capacity can occur outside the PPT and antidegradable classes, and whether antidegradability is the only nontrivial mechanism that forces the private capacity to vanish. For qutrit systems $A$ and $B$, the channel is \[ Λ_{A\to B}(X)= \frac{1}{2}X+\frac{1}{4}\left(\operatorname{Tr}(X)\mathbb{1}_{B}-X^{\mathsf T} \right). \] We use the established Bogoliubov--Kubo--Mori/relative-entropy comparison results to show that its optimized coherent and private information vanish at every blocklength, and hence $P(Λ)=Q(Λ)=0$. Nevertheless, the Choi state of $Λ$ is not PPT, and $Λ$ is not antidegradable. The underlying mechanism is an observable-level relaxation of antidegradability. Specifically, for any input state and receiver observable, the complementary output can provide an unbiased expectation value of the receiver output with no larger variance. Our results show that this mechanism is strictly weaker than antidegradability and implies the complete less-noisy order of the complement. Therefore, the constructed channel establishes a new class of zero-capacity channels beyond the conventional PPT and no-cloning mechanisms.

Comments
23 pages. v2 adds results on observable-level relaxation of antidegradability

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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 χ2χ^2χ2 divergence. A new integral representation shows how measured χ2χ^2χ2 on a qubit extension recovers quantum χ2χ^2χ2. It lets us pass from complete measured-χ2χ^2χ2 ordering to complete quantum-χ2χ^2χ2, 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-χ2χ^2χ2 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≤p<11/2\le p<11/2≤p<1 and establish the same range for a new four-level family. Reference-assisted measured-χ2χ^2χ2 witnesses show why these activating constructions lack complete comparison.
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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.
A channel capacity quantifies the communication capability of a noisy physical process. In contrast to communication channels in the classical world, quantum theory makes this capability contextual. We show that two Pauli two-qubit channels, each with zero private classical capacity, can be used to transmit private information when used together. One is a two-qubit Pauli channel whose environment can reconstruct the receiver output up to matrix transposition; the other is an antidegradable channel. We obtain a similar result when the second channel is the 50% qubit erasure channel. A simple binary code built from rank-three mixtures of Bell states activates private communication. The main ingredient in our construction is a transpose-antidegradable channel that is not antidegradable.
Privacy is a fundamental feature of quantum mechanics. A coherently transmitted quantum state is inherently private. Remarkably, coherent quantum communication is not a prerequisite for privacy: there are quantum channels that are too noisy to transmit any quantum information reliably that can nevertheless send private classical information. Here, we ask how much private classical information a channel can transmit if it has little quantum capacity. We present a class of channels N(d) with input dimension d(2), quantum capacity Q(N(d)) ≤ 1, and private capacity P(N(d)) = log d. These channels asymptotically saturate an interesting inequality P(N) ≤ (1/2)[log d(A) + Q(N)] for any channel N with input dimension d(A) and capture the essence of privacy stripped of the confounding influence of coherence.
We show that informational completeness, while sufficient to have a bijection between ideal measurement probabilities and quantum states, does not guarantee statistically stable reconstruction from finite measurement data. To address this problem, we develop a general estimation theory for continuous-variable systems in which stable reconstructibility is characterized by the POVM effects forming a measurement frame. Informational completeness is therefore necessary, but not sufficient, for stable reconstruction. Our framework is based on measurement frames in a σσσ-regularized operator geometry, where the reference state σσσ encodes prior information about relevant features of the measured states. For any fixed measurement scheme, observables may be inaccessible, weakly reconstructible only through estimators with divergent variance, or stably reconstructible by finite-variance unbiased estimators. The relevant regime is determined by the range of the POVM synthesis operator. Our framework provides practical methods for constructing estimators and gives an operational interpretation of singular quasiprobability distributions, including the Glauber-Sudarshan PPP representation: quasiprobabilities act as unbiased estimators for associated observables, and their singularities reflect a pathological feature of the corresponding measurement: its lack of loewr frame bound. We furthermore show how this formalism naturally provides operational regularization procedures tied to prior information. Overall, our framework provides a unified view of continuous-variable tomography, quasiprobability representations, and classical-shadow estimation.