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.

11 pages. See also the companion work arXiv:2302.08423

References in corpus (21)

• Supplementary information for "Quantum supremacy using a programmable superconducting processor"
• Quantum computational advantage using photons
• Strong quantum computational advantage using a superconducting quantum processor
• Quantum Illumination with Gaussian States
• Restrictions on Transversal Encoded Quantum Gate Sets
• Phase-Programmable Gaussian Boson Sampling Using Stimulated Squeezed Light
• Application of a resource theory for magic states to fault-tolerant quantum computing
• Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy
• 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
• Tensor Networks in a Nutshell
• Efficient simulation scheme for a class of quantum optics experiments with non-negative Wigner representation
• Continuous-Variable Instantaneous Quantum Computing is hard to sample
• Limits on classical communication from quantum entropy power inequalities
• The classical capacity of quantum thermal noise channels to within 1.45 bits
• Continuous-variable entanglement distillation and non-commutative central limit theorems
• Central limit theorems for the large-spin asymptotics of quantum spins
• Discrete Quantum Gaussians and Central Limit Theorem
• Quantum convolution inequalities on Frobenius von Neumann algebras

Cited by in corpus (20)

• Nonstabilizerness via matrix product states in the Pauli basis
• Entanglement-magic separation in hybrid quantum circuits
• Pseudomagic Quantum States
• Magic-induced computational separation in entanglement theory
• Hybrid Oscillator-Qubit Quantum Processors: Instruction Set Architectures, Abstract Machine Models, and Applications
• Magic of quantum hypergraph states
• Pseudorandom unitaries are neither real nor sparse nor noise-robust
• Magic Resource Can Enhance the Quantum Capacity of Channels
• Bridging magic and non-Gaussian resources via Gottesman-Kitaev-Preskill encoding
• Stabilizer Testing and Magic Entropy via Quantum Fourier Analysis
• Certifiable Lower Bounds of Wigner Negativity Volume and Non-Gaussian Entanglement with Conditional Displacement Gates
• Mixed-state additivity properties of magic monotones based on quantum relative entropies for single-qubit states and beyond
• Quantum Ruzsa Divergence to Quantify Magic
• Momentum space magic for the transverse field quantum Ising model
• Quantum convolutional channels and multiparameter families of 2-unitary matrices
• Enhancement of non-Stabilizerness within Indefinite Causal Order
• Characterization of non-adaptive Clifford channels
• Towards Optimal Convergence Rates for the Quantum Central Limit Theorem
• Decoded Quantum Interferometry Under Noise
• Weighted $p$-Rényi Entropy Power Inequality: Information Theory to Quantum Shannon Theory