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arXiv:0901.3391 · AIP conference proceedings

Many-Body Entanglement: a New Application of the Full Counting Statistics

Israel Klich, Leonid S Levitov, Vladimir Lebedev, M. V. Feigel’man

Published Jan 1, 2009• 26 citations
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Abstract

Entanglement entropy is a measure of quantum correlations between separate parts of a many‐body system, which plays an important role in many areas of physics. Here we review recent work in which a relation between this quantity and the Full Counting Statistics description of electron transport was established for noninteracting fermion systems. Using this relation, which is of a completely general character, we discuss how the entanglement entropy can be directly measured by detecting current fluctuations in a driven quantum system such as quantum point contact.

DOI
10.1063/1.3149497

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Entanglement entropy is a measure of quantum correlations between separate parts of a many‐body system, which plays an important role in many areas of physics. Here we review recent work in which a relation between this quantity and the Full Counting Statistics description of electron transport was established for noninteracting fermion systems. Using this relation, which is of a completely general character, we discuss how the entanglement entropy can be directly measured by detecting current fluctuations in a driven quantum system such as quantum point contact.
We present an exact expression for the entanglement entropy generated at a quantum point contact between noninteracting electronic leads in terms of the full counting statistics of charge fluctuations, which we illustrate with examples from both equilibrium and nonequilibrium transport. The formula is also applicable to ground-state entanglement entropy in systems described by noninteracting fermions in any dimension, which in one dimension include the critical spin-1/2 XX\mathit{XX}XX and Ising models where conformal field theory predictions for the entanglement entropy are reproduced from the full counting statistics. These results may play an important role in experimental measurements of entanglement entropy in mesoscopic structures and cold atoms in optical lattices.
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.
Transport in a quantum point contact (QPC) can be used to generate many-body entanglement of Fermi seas in the leads. A universal relation is found between the generated entanglement entropy and the fluctuations of electric current, which is valid for any protocol of driving the QPC. This relation offers a basis for direct electric measurement of entanglement entropy. In particular, by utilizing space-time duality of 1D systems, we relate electric noise generated by opening and closing the QPC periodically in time with the seminal S=1/3logL prediction of conformal field theory.
Entanglement entropy has become an important theoretical concept in condensed matter physics because it provides a unique tool for characterizing quantum mechanical many-body phases and new kinds of quantum order. However, the experimental measurement of entanglement entropy in a many-body system is widely believed to be unfeasible, owing to the nonlocal character of this quantity. Here, we propose a general method to measure the entanglement entropy. The method is based on a quantum switch (a two-level system) coupled to a composite system consisting of several copies of the original many-body system. The state of the switch controls how different parts of the composite system connect to each other. We show that, by studying the dynamics of the quantum switch only, the Rényi entanglement entropy of the many-body system can be extracted. We propose a possible design of the quantum switch, which can be realized in cold atomic systems. Our work provides a route towards testing the scaling of entanglement in critical systems as well as a method for a direct experimental detection of topological order.
The entanglement entropy generated by quantum transport, similar to any physical observable quantity, is a stochastic variable that has its distribution and can fluctuate. The fundamental question is how to define the entanglement entropy operator which allows one to discuss entanglement entropy fluctuation. By introducing the entanglement entropy operator, we develop a theoretical framework to calculate the entanglement entropy fluctuation as well as its higher order cumulants generated by electronic transport in open systems. The distribution of entanglement entropy generated by opening or closing a quantum point contact (QPC) is solved exactly. When the transmission coefficient of QPC is one-half, the entanglement entropy is maximized and fluctuationless. We also establish a general relation between the generated entanglement entropy fluctuation and charge fluctuation. We apply our theory to electronic transport through a quantum dot and study the generated entanglement entropy in the transient regime. Universal behavior is found for the cumulants of entanglement entropy at short times.
Entanglement entropy serves as a fundamental diagnostic for quantum many-body correlations, distinguishing quantum entanglement from classical stochastic fluctuations. This review systematically explores the scaling behaviors of entanglement entropy, beginning with the foundational area law in Hermitian systems and progressing to the exotic regimes of non-Hermitian physics. We focus on the mathematical apparatus of the Fisher-Hartwig theorem in fermionic chains, the biorthogonal formalism in non-unitary systems, and the emergence of negative entanglement entropy driven by exceptional boundary states and non-unitary conformal field theories.