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Physical Review A

Leveraging magic resources for quantum channel capacity enhancement under stabilizer convolution

Wenlong Sun, Yuanfeng Jin

Published Nov 6, 2025• 2 citations
Publisher

Abstract

In this paper, we focus on discrete-variable quantum systems of odd prime dimension. We investigate the quantum capacity of stabilizer convolution channels under a variety of environmental states. Inspired by the recent result of Bu and Jaffe [Phys. Rev. Lett. 134, 050202 (2025)], our main contribution is to generalize the analysis beyond the discrete beam splitter---which is only applicable for odd prime dimensions $d\ensuremath{\ge}7$---to broader classes of stabilizer convolutions, allowing us to explore the behavior of magic resources in lower dimensions such as $d=3$ and $d=5$, where their role in enhancing quantum capacity has remained unclear. In fact, many of our mathematical results are derived under the broader setting of positive convolutions, not limited strictly to stabilizer convolutions.

DOI
10.1103/qnz3-4fsc

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In this paper, we focus on discrete-variable quantum systems of odd prime dimension. We investigate the quantum capacity of stabilizer convolution channels under a variety of environmental states. Inspired by the recent result of Bu and Jaffe [Phys. Rev. Lett. 134, 050202 (2025)], our main contribution is to generalize the analysis beyond the discrete beam splitter---which is only applicable for odd prime dimensions d\ensuremath≥7d\ensuremath{\ge}7d\ensuremath≥7---to broader classes of stabilizer convolutions, allowing us to explore the behavior of magic resources in lower dimensions such as d=3d=3d=3 and d=5d=5d=5, where their role in enhancing quantum capacity has remained unclear. In fact, many of our mathematical results are derived under the broader setting of positive convolutions, not limited strictly to stabilizer convolutions.
We investigate the private capacity of quantum channels using the recently proposed quantum convolution theory for discrete-variable quantum systems. We focus on the role of the magic resource played in this framework. Firstly, for a large class of convolutional channels, we find that the private capacity is zero if the fixed environmental state is a stabilizer state. Moreover, we show that the private capacity can be nonzero for some magic environmental states. Furthermore, we show that the private capacity of a discrete beam splitter unitary is upper-bounded by the amount of magic of the environmental state. In addition, if the environmental state exhibits a certain symmetric structure, even if it is magic, the corresponding private capacity will also vanish for a class of convolution. These results emphasize the role of magic resources in quantum communication
We investigate the role of magic resource in the quantum capacity of channels. We consider the quantum channel of the recently proposed discrete beam splitter with the fixed environmental state. We find that if the fixed environmental state is a stabilizer state, then the quantum capacity is zero. Moreover, we find that the quantum capacity is nonzero for some magic states, and the quantum capacity increases linearly with respect to the number of single-qudit magic states in the environment. We also bound the maximal quantum capacity of the discrete beam splitter in terms of the amount of magic resource in the environmental states. These results suggest that magic resource can increase the quantum capacity of channels; it sheds new insight into the role of stabilizer and magic states in quantum communication.
We study how Clifford interactions with a fixed environment determine quantum transmission, in a setting that extends positive discrete quantum convolution . We characterize when Clifford operations on the environment can be replaced by operations on the signal input and the two outputs, and classify the interactions under local Clifford transformations . For pure stabilizer environments, we determine the noiseless, dephasing, and completely depolarizing parts of the channel directly from the interaction and environmental stabilizers, obtaining its quantum and private capacities . This reveals a distinction between interactions: those locally equivalent to positive convolution produce entanglement-breaking channels for every pure stabilizer environment, whereas every other interaction in the class admits noiseless quantum transmission with a suitable pure stabilizer environment . For positive convolution, environmental Clifford transport also allows us to separate the stabilized coordinates of an arbitrary mixed environment . These coordinates carry only a classical label shared by the receiver and environment, so removing them preserves quantum and private capacities . The resulting channel reduction yields a private-capacity bound by the modified relative entropy of magic of the environmental state .
We introduce a quantum convolution and a conceptual framework to study states in discrete-variable (DV) quantum systems. All our results suggest that stabilizer states play a role in DV quantum systems similar to the role that Gaussian states play in continuous-variable systems. Hence we suggest the name ``discrete quantum Gaussians'' for stabilizer states. For example, we prove that the convolution of two stabilizer states is another stabilizer state, and that stabilizer states extremize both quantum entropy and Fisher information. We establish a ``maximal entropy principle,'' a ``second law of thermodynamics for quantum convolution,'' and a ``quantum central limit theorem.'' The latter is based on iterating the convolution of a zero-mean quantum state, which we prove converges to a stabilizer state. We bound the exponential rate of convergence of the quantum central limit theorem by the ``magic gap,'' defined in terms of the support of the characteristic function of the state. We elaborate on our general results with a discussion of some examples, as well as extending many of them to quantum channels.
Abstract Stabilizer convolution channels in odd-prime dimensions have a natural description in finite-field phase space. We analyze these channels using Clifford symmetry and show that, for a fixed environmental state, the maximal single-letter coherent information is unchanged under Clifford-equivalent choices of the stabilizer convolution matrix. This reduces the apparently quadratic family of admissible stabilizer convolutions, at the one-use level, to sign-orbit classes of a single finite-field parameter. We also isolate quadratic cases in which the induced Clifford symmetry becomes either the identity or phase-space inversion; these cases give symmetry-protected families with zero maximal single-letter coherent information. To complement this symmetry analysis, we introduce a minimal weighted magic entropy and prove that it sharpens the magic-relative-entropy upper bound for positive convolution channels at the single-letter level. Finally, we define the single-letter coherent-information power of a convolution and show that the optimizing environmental state may be taken to be pure. Small-dimensional numerical examples illustrate the sign-orbit classification and show that coherent-information enhancement is governed by the finite-field convolution parameter rather than by dimension alone.
To achieve universal quantum computation via general fault-tolerant schemes, stabilizer operations must be supplemented with other non-stabilizer quantum resources. Motivated by this necessity, we develop a resource theory for magic quantum channels to characterize and quantify the quantum 'magic' or non-stabilizerness of noisy quantum circuits. For qudit quantum computing with odd dimension d, it is known that quantum states with non-negative Wigner function can be efficiently simulated classically. First, inspired by this observation, we introduce a resource theory based on completely positive-Wigner-preserving quantum operations as free operations, and we show that they can be efficiently simulated via a classical algorithm. Second, we introduce two efficiently computable magic measures for quantum channels, called the mana and thauma of a quantum channel. As applications, we show that these measures not only provide fundamental limits on the distillable magic of quantum channels, but they also lead to lower bounds for the task of synthesizing non-Clifford gates. Third, we propose a classical algorithm for simulating noisy quantum circuits, whose sample complexity can be quantified by the mana of a quantum channel. We further show that this algorithm can outperform another approach for simulating noisy quantum circuits, based on channel robustness. Finally, we explore the threshold of non-stabilizerness for basic quantum circuits under depolarizing noise.