Quantum Entropy and Central Limit Theorem

Abstract

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.

arXiv comment11 pages. See also the companion work arXiv:2302.08423

References in corpus (65)

• Supplementary information for "Quantum supremacy using a programmable superconducting processor"
• Gaussian Quantum Information
• Quantifying Coherence
• Quantum computational advantage using photons
• A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States
• Quantum Coherence as a Resource
• Continuous variable quantum cryptography using coherent states
• Quantum computation over continuous variables
• Strong quantum computational advantage using a superconducting quantum processor
• Violation of Bell inequalities by photons more than 10 km apart
• Teleportation of Quantum States
• Stabilizer Codes and Quantum Error Correction
• The second laws of quantum thermodynamics
• Quantum Illumination with Gaussian States
• Gaussian Boson Sampling
• Instantaneous non-local computation of low T-depth quantum circuits
• The Heisenberg Representation of Quantum Computers
• Restrictions on Transversal Encoded Quantum Gate Sets
• Positive Wigner functions render classical simulation of quantum computation efficient
• Phase-Programmable Gaussian Boson Sampling Using Stimulated Squeezed Light
• Efficient Classical Simulation of Continuous Variable Quantum Information Processes
• Application of a resource theory for magic states to fault-tolerant quantum computing
• Hudson's Theorem for finite-dimensional quantum systems
• Classical simulation of noninteracting-fermion quantum circuits
• Improved classical simulation of quantum circuits dominated by Clifford gates
• Extremality of Gaussian quantum states
• Boson Sampling from Gaussian States
• Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy
• Trading classical and quantum computational resources
• Stabilizer Rényi entropy
• Quantum Supremacy and the Complexity of Random Circuit Sampling
• Resource theory of quantum non-Gaussianity and Wigner negativity
• Measuring the quality of a quantum reference frame: the relative entropy of frameness
• Matchgates and classical simulation of quantum circuits
• Continuous-variable quantum cryptography is secure against non-gaussian attacks
• Optimum mixed-state discrimination for noisy entanglement-enhanced sensing
• Convex resource theory of non-Gaussianity
• Universal continuous-variable quantum computation: Requirement of optical nonlinearity for photon counting
• Resource Destroying Maps
• Tensor Networks in a Nutshell
• Stellar representation of non-Gaussian quantum states
• Efficient simulation scheme for a class of quantum optics experiments with non-negative Wigner representation
• Relative entropy is an exact measure of non-Gaussianity
• The entropy power inequality for quantum systems
• Continuous-Variable Instantaneous Quantum Computing is hard to sample
• On the statistical complexity of quantum circuits
• Quantum boolean functions
• A generalization of the Entropy Power Inequality to Bosonic Quantum Systems
• Boson sampling with Gaussian measurements
• Limits on classical communication from quantum entropy power inequalities
• Efficient classical simulation of Clifford circuits with nonstabilizer input states
• Energy-constrained discrimination of unitaries, quantum speed limits and a Gaussian Solovay-Kitaev theorem
• The classical capacity of quantum thermal noise channels to within 1.45 bits
• Gaussian states minimize the output entropy of one-mode quantum Gaussian channels
• Continuous-variable entanglement distillation and non-commutative central limit theorems
• Quons: A 3D Language for Quantum Information
• Quantum Fourier Analysis
• Convergence rates for the quantum central limit theorem
• Classical simulation of quantum circuits by half Gauss sums
• Central limit theorems for the large-spin asymptotics of quantum spins
• On a generalized Central Limit Theorem and Large Deviations for Homogeneous Open Quantum Walks
• On a Quantum Entropy Power Inequality of Audenaert, Datta and Ozols
• On the minimum output entropy of single-mode phase-insensitive Gaussian channels
• Discrete Quantum Gaussians and Central Limit Theorem
• Quantum convolution inequalities on Frobenius von Neumann algebras

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