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arXiv:2302.08423 · Advances in Mathematics

Discrete quantum Gaussians and central limit theorem

Kaifeng Bu, Weichen Gu, Arthur Jaffe

Published Sep 4, 2026• 4 citations
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Abstract

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.

DOI
10.1016/j.aim.2026.111204

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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.
We introduce a framework to study discrete-variable (DV) quantum systems based on qudits. It relies on notions of a mean state (MS), a minimal stabilizer-projection state (MSPS), and a new convolution. Some interesting consequences are: The MS is the closest MSPS to a given state with respect to the relative entropy; the MS is extremal with respect to the von Neumann entropy, demonstrating a "maximal entropy principle in DV systems." We obtain a series of inequalities for quantum entropies and for Fisher information based on convolution, giving a "second law of thermodynamics for quantum convolutions." We show that the convolution of two stabilizer states is a stabilizer state. We establish a central limit theorem, based on iterating the convolution of a zero-mean quantum state, and show this converges to its MS. The rate of convergence is characterized by the "magic gap," which we define in terms of the support of the characteristic function of the state. We elaborate on two examples: the DV beam splitter and the DV amplifier.
In this work, we investigate the behavior of quantum entropy under quantum convolution and its application in quantifying magic. We first establish an entropic, quantum central limit theorem (q-CLT), where the rate of convergence is bounded by the magic gap. We also introduce a new quantum divergence based on quantum convolution, called the quantum Ruzsa divergence, to study the stabilizer structure of quantum states. We conjecture a “convolutional strong subadditivity” inequality, which leads to the triangle inequality for the quantum Ruzsa divergence. In addition, we propose two new magic measures, the quantum Ruzsa divergence of magic and quantum-doubling constant, to quantify the amount of magic in quantum states. Finally, by using the quantum convolution, we extend the classical, inverse sumset theory to the quantum case. These results shed new insight into the study of the stabilizer and magic states in quantum information theory.
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We investigate the extremality of stabilizer states to reveal their exceptional role in the space of all nnn-qubit/qudit states. We establish uncertainty principles for the characteristic function and the Wigner function of states, respectively. We find that only stabilizer states achieve saturation in these principles. Furthermore, we prove a general theorem that stabilizer states are extremal for convex information measures invariant under local unitaries. We explore this extremality in the context of various quantum information and correlation measures, including entanglement entropy, conditional entropy and other entanglement measures. Additionally, leveraging the recent discovery that stabilizer states are the limit states under quantum convolution, we establish the monotonicity of the entanglement entropy and conditional entropy under quantum convolution. These results highlight the remarkable information-theoretic properties of stabilizer states. Their extremality provides valuable insights into their ability to capture information content and correlations, paving the way for further exploration of their potential in quantum information processing.
Gaussian states play a central role in continuous quantum mechanics, and an appropriate generalization might play a similar role in discrete quantum mechanics. A common-sense discretization of these turns out to be trivial. In this work, we describe two new discretizations based on recent developments in discrete phase space methods. We also extend existing notions of convergence between continuous and discrete systems and apply some of them to our constructions. Accompanying this work is a numerical software package "qWeyl" which we use to give examples and provide visualizations.
In this paper we introduce the concept of stabilizer convolution and demonstrate that the only convolutions that can be used for stability testing are stabilizer convolutions. Moreover, using any stabilizer convolution yields similarly good results. Through the Choi-Jamio\l{}kowski isomorphism, this method can also be applied to Clifford testing for quantum gates. Leveraging the invariance of circuit magic classes of mean states under stabilizer self-convolution, we formulate a central limit theorem for circuit magic classes of quantum states. Moreover, employing the Choi-Jamio\l{}kowski correspondence, we generalize this finding to establish a central limit theorem for circuit magic classes of completely positive and trace-preserving channels.