# Full counting statistics of information content and particle number

## TL;DR

This paper develops a theoretical framework to analyze the joint probability distribution of particle number and information content in a bipartite quantum conductor, extending Green function techniques to derive entropies and efficiency distributions.

## Contribution

It introduces a method to calculate the joint probability distribution of particle number and information content using extended Keldysh Green functions, including energy-dependent transmission.

## Key findings

- Derived the time dependence of accessible entanglement entropy for energy-independent transmission.
- Analyzed joint probability distributions under resonant and sequential tunneling conditions.
- Discussed the probability distribution of information transfer efficiency per electron.

## Abstract

We consider a bipartite quantum conductor and discuss the joint probability distribution of particle number in a subsystem and the self-information associated with the reduced density matrix of the subsystem. By extending the multi-contour 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 transfered by a single electron.