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arXiv:1704.02679 · Physical review. B./Physical review. B

Full counting statistics of information content and particle number

Yasuhiro Utsumi

Published Aug 9, 2017• 6 citations
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Abstract

We consider a bipartite quantum conductor and discuss the joint probability distribution of a particle number in a subsystem and the self-information associated with the reduced density matrix of the subsystem. By extending the multicontour Keldysh Green-function technique, we calculate the R\'enyi entropy of a positive integer order $M$ subjected to the particle number constraint, from which we derive the joint probability distribution. For energy-independent transmission, we derive the time dependence of the accessible entanglement entropy, or the conditional entropy. We analyze the joint probability distribution for energy-dependent transmission probability at the steady state under the coherent resonant tunneling and the incoherent sequential tunneling conditions. We also discuss the probability distribution of the efficiency, which measures the information content transferred by a single electron.

DOI
10.1103/physrevb.96.085304

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We consider a bipartite quantum conductor and discuss the joint probability distribution of a particle number in a subsystem and the self-information associated with the reduced density matrix of the subsystem. By extending the multicontour Keldysh Green-function technique, we calculate the R'enyi entropy of a positive integer order MMM subjected to the particle number constraint, from which we derive the joint probability distribution. For energy-independent transmission, we derive the time dependence of the accessible entanglement entropy, or the conditional entropy. We analyze the joint probability distribution for energy-dependent transmission probability at the steady state under the coherent resonant tunneling and the incoherent sequential tunneling conditions. We also discuss the probability distribution of the efficiency, which measures the information content transferred by a single electron.
The performance of communication through a quantum conductor is limited by the laws of physics. The optimum channel capacity, the maximum rate at which information can be transmitted under a given signal power, i.e., the heat current, connects the theory of communication, quantum physics, and thermodynamics. Here, the author calculates the distribution of fluctuations of self-information and finds a Jarzynski equality like universal relation connecting the fluctuations of information, the R\'enyi entropy of order zero, and the optimum capacity.
The Renyi entropies for an interacting quantum dot are computed under a multi-contour Keldysh formalism. With this in hand, the self-information generating function is determined and the probability distribution of self-information is computed, including the effects of Coulomb interactions in the Hartree approximation. The author argues through these computations that fluctuations in the self-information and the Renyi entropies are tied together intimately and that these fluctuations thus offer a way to give the Renyi entropies physical meaning.
We investigate in detail the behavior of the bipartite fluctuations of particle number $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{N}$ and spin ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{S}}^{z}$ in many-body quantum systems, focusing on systems where such U(1) charges are both conserved and fluctuate within subsystems due to exchange of charges between subsystems. We propose that the bipartite fluctuations are an effective tool for studying many-body physics, particularly its entanglement properties, in the same way that noise and full counting statistics have been used in mesoscopic transport and cold-atomic gases. For systems that can be mapped to a problem of noninteracting fermions, we show that the fluctuations and higher-order cumulants fully encode the information needed to determine the entanglement entropy as well as the full entanglement spectrum through the R\'enyi entropies. In this connection, we derive a simple formula that explicitly relates the eigenvalues of the reduced density matrix to the R\'enyi entropies of integer order for any finite density matrix. In other systems, particularly in one dimension, the fluctuations are in many ways similar but not equivalent to the entanglement entropy. Fluctuations are tractable analytically, computable numerically in both density matrix renormalization-group and quantum Monte Carlo calculations, and in principle accessible in condensed-matter and cold-atom experiments. In the context of quantum point contacts, measurement of the second charge cumulant showing a logarithmic dependence on time would constitute a strong indication of many-body entanglement.
We calculate analytically the entanglement and Renyi entropies, the negativity and the mutual information together with all the density and many-particle correlation functions for free bosons on a lattice in the ground state. We show that those quantities can be derived from a multinomial form of the reduced density matrix in the configuration space whose diagonal elements dictate the statistics of the particle distribution, while the off-diagonal coherence terms control the quantum fluctuations. We clarify by this analysis how to reconcile the logarithmic behavior of the entanglement entropy with the volume law of the particle number fluctuations.
We construct the entanglement entropy of a general scattering process between one particle species (denoted as AAA) and the rest of the final state. Assuming a pure initial momentum eigenstate, we show that the reduced density matrix in particle AAA's Fock space is block-diagonal in zero-, one- and multi-particle states, and is diagonal in the momentum basis for the one-particle state. As a result, we could write down the Von Neumann entropy of particle AAA, SAS_ASA​, in terms of scattering probabilities and differential cross sections, assuming no more than one particle AAA is produced in the final state. SAS_ASA​ contains the information of both the total scattering probabilities (or cross sections) and the differential distributions --- the latter depends on a phase-space resolution factor as a result of the infinite-dimensional momentum Hilbert space. Our result is general and model-independent since we consider the full final state from time evolution (S∣i⟩S|i\rangleS∣i⟩) rather than a specific outcome, and is applicable as long as the probability for producing multiple particle AAA is subleading. We also obtain the n≥2n\geq2n≥2 Tsallis and Rényi entropies in our framework and find out that, at the leading order, they are proportional to the total scattering probability and carry no differential information.
We calculate analytically the entanglement and R\'enyi entropies, the negativity, and the mutual information together with all the density and many-particle correlation functions for free bosons on a lattice in the ground state, for both homogeneous and inhomogeneous systems. We show that all those quantities can be derived from a multinomial form of the reduced density matrix in the configuration space whose diagonal elements dictate the statistics of the particle distribution, while the off-diagonal coherence terms control the quantum fluctuations. We provide by this analysis a unified approach based on a reduced density matrix technique useful to calculate both the entanglement properties and an infinite number of correlation functions.