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IEEE International Symposium on Information Theory (ISIT)

Quantum Capacity Amplification via Privacy

Peixue Wu, Yunkai Wang

Published Jun 22, 2025• 1 citation
Publisher

Abstract

Quantum capacity exhibits intriguing phenomena such as super-activation, amplification, and super-amplification, which can be explored using constructions based on private channels. In this work, we refine the connection between private channels and additivity questions in quantum channel capacity. Specifically, the shield system, which safeguards the key system's privacy by leaking quantum information, can recover some of this lost information when paired with an auxiliary channel like the erasure channel$\mathcal{E}_{p}$, thereby enhancing overall capacity. Building on this insight, we construct new and simple examples of super-amplification for the erasure channel$\mathcal{E}_{p}$with any erasure probability$p$. In addition, we provide a refined version of the Smith-Yard argument that also applies for arbitrary$p$. Finally, we revisit the fundamental question of whether regularization in channel capacity formulas requires infinitely many channel uses, showing that the necessary number may instead depend on the channel's dimension.

DOI
10.1109/isit63088.2025.11195355

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Quantum capacity exhibits intriguing phenomena such as super-activation, amplification, and super-amplification, which can be explored using constructions based on private channels. In this work, we refine the connection between private channels and additivity questions in quantum channel capacity. Specifically, the shield system, which safeguards the key system's privacy by leaking quantum information, can recover some of this lost information when paired with an auxiliary channel like the erasure channelEp\mathcal{E}_{p}Ep​, thereby enhancing overall capacity. Building on this insight, we construct new and simple examples of super-amplification for the erasure channelEp\mathcal{E}_{p}Ep​with any erasure probabilityppp. In addition, we provide a refined version of the Smith-Yard argument that also applies for arbitraryppp. Finally, we revisit the fundamental question of whether regularization in channel capacity formulas requires infinitely many channel uses, showing that the necessary number may instead depend on the channel's dimension.
We investigate superadditivity of quantum capacity through private channels whose Choi-Jamiolkowski operators are private states. This perspective links the security structure of private states to quantum capacity and clarifies the role of the shield system: information encoded in the shield system that would otherwise leak to the environment can be recycled when paired with an assisting channel, thereby boosting capacity. Our main contributions are threefold: Firstly, we develop a general framework that provides a sufficient condition for capacity amplification, which is formulated in terms of the assisting channel's Holevo information. As examples, we give explicit, dimension and parameter dependent amplification thresholds for erasure and depolarizing channels. Secondly, assuming the Spin alignment conjecture, we derive a single-letter expression for the quantum capacity of a family of private channels that are neither degradable, anti-degradable, nor PPT; as an application, we construct channels with vanishing quantum capacity yet unbounded private capacity. Thirdly, we further analyze approximate private channels: we give an alternative proof of superactivation that extends its validity to a broader parameter regime, and, by combining amplification bounds with continuity estimates, we establish a metric separation showing that channels exhibiting capacity amplification have nonzero diamond distance from the set of anti-degradable channels, indicating that existing approximate (anti-)degradability bounds are not tight. We also revisit the computability of the regularized quantum capacity and modestly suggest that this fundamental question still remains open.
Superadditivity of quantum capacity of communication channels is one of the most interesting findings of the field. Yard and Smith, finding a relation between the private capacity and the assisted quantum capacity, showed a striking example of superadditivity, i.e. two channels of zero quantum capacity could achieve a positive quantum capacity when used together [1]. The four dimensional channels they used are a 50% erasure channel (therefore zero quantum capacity, due to no-cloning theorem) and a Horodecki channel (again zero quantum capacity due to incapability of sharing free entanglement). In this work we present the more general cases of superadditivity. Directly calculating the lower bounds of joint quantum capacities without using the relation between private capacity and assisted quantum capacity, we examine scenarios considering erasure channels of arbitrary probabilities and different Horodecki channels, and discuss the roles of degradability and anti-degradability as well as the role of the private capacity in superadditivity. We also derive an upper bound for the joint quantum capacity for the superactivation case.
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.
The quantum capacity of a quantum channel is always smaller than the capacity of the channel for private communication. However, both quantities are given by the infinite regularization of respectively the coherent and the private information. Here, we construct a family of channels for which the private and coherent information can remain strictly superadditive for unbounded number of uses. We prove this by showing that the coherent information is strictly larger than the private information of a smaller number of uses of the channel. It turns out that even though the quantum capacity is upper bounded by the private capacity, the non-regularized quantities can be interleaved. From an operational point of view, the private capacity can be used for gauging the practical value of quantum channels for secure communication and, consequently, for key distribution. We thus show that in order to evaluate the interest a channel for this task it is necessary to optimize the private information over an unlimited number of uses of the channel.
Is there a meaningful quantum counterpart to public communication? We argue that it is the symmetric-side channel. This connection is partially motivated by recent work, where it was found that if a sender would like to communicate a secret message to a receiver through an insecure quantum channel using a shared quantum state as a key, then the insecure quantum channel is only ever used to simulate a symmetric-side channel. Here, we further show, in complete analogy to the role of public classical communication, that assistance by a symmetric-side channel makes equal the distillable entanglement, the recently introduced mutual independence, and a generalization of the latter, which quantifies the extent to which one of the parties can perform quantum privacy amplification. Symmetric-side channels, and the closely related erasure channel, have been recently harnessed to provide examples of superactivation of the quantum channel capacity. Our findings give new insight into this nonadditivity and its relation to quantum privacy. In particular, we show that single-copy superactivation protocols with the erasure channel, which encompasses all examples of nonadditivity of the quantum capacity found to date, can be understood as a conversion of mutual independence into distillable entanglement.