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arXiv:2401.12105 · Physical Review Letters

Magic Resource Can Enhance the Quantum Capacity of Channels

Kaifeng Bu, Arthur M. Jaffe

Published Feb 6, 2025• 18 citations
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Abstract

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.

DOI
10.1103/physrevlett.134.050202

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Physical Review A•May 13, 2025•5 citations
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 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
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=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.
In the stabilizer formalism of quantum error correction and fault-tolerant quantum computation, stabilizer states, as common eigenstates (with the eigenvalue 1) of operators in maximal Abelian subgroups of the Pauli group, play the role of classical states, while magic (non-stabilizer) states are quantum resources enabling universal quantum computation when injected into stabilizer circuits.The natural issue arises as how to characterize and quantify the ability of quantum channels (including quantum gates) in generating magic resource (non-stabilzerness) from stabilizer states.In this work, we introduce two intuitive quantities characterizing magic-resource generating power of quantum channels and reveal their basic properties.We evaluate these quantifies for several prototypical quantum channels and quantum gates.In particular, we demonstrate that the qubit T -gate (i.e., π/8-gate) and its qutrit generalization are optimal in generating magic resource in a class of diagonal unitaries.This highlights the significance of the T -gate and its extensions from a resource-theoretic perspective and provides a theoretical support of their wide applications in stabilizer quantum computation.
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 .
Abstract Beam‐splitters are basic and instrumental tools not only in the study of foundational issues in quantum mechanics, but also in various applications of quantum information processing. In the continuous‐variable scenario involving quantum optical states and Gaussian quantum information, interference phenomena are widely investigated and characterized via beam‐splitters. In this work, within the framework of stabilizer quantum computation in finite quantum systems, entanglement and magic correlations generated by discrete beam‐splitters are explored. Two types of magic correlations are distinguished: mutual magic (analogous to quantum mutual information) and non‐local magic. Explicit formulas for these correlations are derived in the qubit scenario where the beam‐splitter is implemented by the controlled‐NOT gate, with the ‐type magic states playing a particularly prominent role as input states. Extending to prime‐dimensional qudit systems, it is shown that fiducial states of mutually unbiased bases also play a remarkable role. Basic properties of these magic correlations are revealed and illustrated through some typical examples.
According to the celebrated Gottesman-Knill theorem, magic resource (nonstabilizerness) is necessary to drive universal quantum computation. Thus characterization and quantification of magic resource for quantum states and quantum channels are significant issues. In this work, we propose a method to construct quantifiers of magic resource for quantum channels from those for quantum states by employing the channel-state duality to convert quantum channels to the corresponding Choi states. We focus on a specific quantifier of magic resource based on characteristic functions (Fourier transforms) of quantum states. This quantifier is well defined for all dimensions and easy to calculate, as opposed to those defined via discrete Wigner functions. It satisfies some desirable properties and provides an upper bound to the mana (a kind of logarithmic negativity of discrete Wigner functions) of quantum channels. Using this quantity, we evaluate the amount of magic resource for some typical quantum channels and compare them with some existing quantifiers of generating-power of magic resource. For the dephasing channels in prime dimensional systems, our quantifier of magic resource essentially characterizes the average generating power of magic resource, while, for general channels, they are quite different and some challenging issues arise.