Advances in Mathematics· 2026Q1
Discrete quantum Gaussians and central limit theorem
- 4citations
- Q1SCImago
- 2026year
Short summary
Stabilizer states in discrete-variable quantum systems are analogous to Gaussian states in continuous-variable systems, termed 'discrete quantum Gaussians'. Their convolution yields another stabilizer state, and they extremize quantum entropy and Fisher information.
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Key points
- Stabilizer states are proposed as 'discrete quantum Gaussians' due to their analogous role to Gaussian states in continuous-variable systems.
- The convolution of two stabilizer states is proven to be another stabilizer state.
- Stabilizer states extremize both quantum entropy and Fisher information.
- A quantum central limit theorem demonstrates that iterating the convolution of a zero-mean quantum state converges to a stabilizer state.
- The convergence rate of the quantum central limit theorem is bounded by the 'magic gap'.
AI-generated from the title and abstract; the full text is not read.
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.
The authors' abstract, as published at the source. Advances in Mathematics, 2026 · DOI ↗
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Field: Statistical and Nonlinear Physics
Statistical and Nonlinear PhysicsPhysics and Astronomy