Bipartite Fluctuations as a Probe of Many-Body Entanglement

TL;DR

This paper explores how bipartite fluctuations of particle number and spin can serve as practical probes of many-body entanglement, providing analytical, numerical, and experimental insights into their relationship with entanglement entropy.

Contribution

It introduces bipartite fluctuations as a new, effective tool for studying many-body entanglement, with explicit formulas relating fluctuations to entanglement spectra and entropy.

Findings

Fluctuations fully encode entanglement entropy in non-interacting fermion systems.

Analytical and numerical methods demonstrate the tractability of fluctuations in various models.

Measurement of charge cumulants can indicate many-body entanglement in experiments.

Abstract

We investigate in detail the behavior of the bipartite fluctuations of particle number $\hat{N}$ and spin $\hat{S}^z$ 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 non-interacting 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\'{e}nyi entropies. In this connection we derive a simple formula that explicitly relates the eigenvalues of the reduced density matrix to the R\'{e}nyi 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.