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Infrared Remote Sensing and Instrumentation

Generating 2D statistical distributions from Fock to Poissonian states on variant maps using random sequences

Jeffrey Zheng, Xinming Zhang, Chris Zheng

Published Sep 9, 2019• 0 citations
Publisher

Abstract

From a quantum statistical viewpoint, four typical quantum states are Fock, Sub-Poissonian, Poissonian and SuperPoissonian states. Quantum interactions are focus among Fock and Poissonian states. Using quantum statistics, model and simulation, this paper proposes two models: matrix and variant transformations: 1. MT Matrix Transformation – eigenvalue states; 2. VT Variant Transformation – invariant states to analyze three random sequences: 1) random; 2) conditional random in a constant; 3) periodic pattern. Four procedures are proposed. Fast Fourier Transformation FFT is applied as one of MT schemes and two invariant scheme of VT schemes are applied, three random sequences are in M segments and each segment has a length m to generate a measuring sequence. Shifting operations are applied on each random sequence to create m+1 spectrum distributions. For FFT, a pair of eigenvalues are selected as the output. Two types of 1D and 2D variant maps are generated to illustrate multiple parameter selections to generate a series of results. Since sequences 1) and 3) are related simple, more cases are focus on sequences 2). Better than FFT, VT distinguishes various Fock, Sub-Poissonian, Poissonian states in random analysis to distinguish three random sequences as three levels of statistical ensembles: Micro-canonical, Canonical, and Grand-Canonical ensembles. Applying two transformations, quantum statistics, model and simulation of modern quantum theory and applications can be explored.

DOI
10.1117/12.2528254

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From a quantum statistical viewpoint, four typical quantum states are Fock, Sub-Poissonian, Poissonian and Super-Poissonian states. Quantum interactions are among Fock and Poissonian states. Using quantum statistics, model and simulation, this paper propose two models: matrix and variant transformations: 1. MT Matrix Transformation – eigenvalue states; 2. VT Variant Transformation – invariant states to analyze three random sequences: 1) random; 2) conditional random in a constant; 3) periodic pattern. Four procedures are proposed. Fast Fourier Transformation FFT is applied as one of MT schemes and two invariant scheme of VT schemes are applied, three random sequences are used in M segments, and each segment has a length m to generate a measuring sequence. Shifting operations are applied on each random sequence to create m+1 spectrum distributions. Better than FFT, VT can identify Fock, Sub-Poissonian, Poissonian states in random analysis to distinguish three random sequences as three levels of statistical ensembles: Micro-canonical, Canonical, and Grand-Canonical ensembles. Applying two transformations, quantum statistics, model and simulation of modern quantum theory and applications can be explored. Copyright © VBRI Press.
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Abstract In this chapter, a testing model is used to apply statistical probability in multiple distributions on three maps for a selected sequence to check refined stationary randomness on quantum sequences. Three random data sequences are collected from two quantum random resources: one from Australian National University (ANU) and two (initial and secure) from University of Science and Technology of China (USTC). Multiple results are created on three maps, and measurements of stationary randomness are illustrated and compared. Three samples show distinct stationary properties.
For any graph consisting of k vertices and m edges we construct an ensemble of random pure quantum states which describe a system composed of 2 m subsystems. Each edge of the graph represents a bipartite, maximally entangled state. Each vertex represents a random unitary matrix generated according to the Haar measure, which describes the coupling between subsystems. Dividing all subsystems into two parts, one may study entanglement with respect to this partition. A general technique to derive an expression for the average entanglement entropy of random pure states associated with a given graph is presented. Our technique relies on Weingarten calculus and flow problems. We analyze the statistical properties of spectra of such random density matrices and show for which cases they are described by the free Poissonian (Marchenko–Pastur) distribution. We derive a discrete family of generalized, Fuss–Catalan distributions and explicitly construct graphs which lead to ensembles of random states characterized by these novel distributions of eigenvalues.
Embedded Random Matrix Ensembles to Statistical Shell Model: Operation of qqq-normal forms
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