\documentclass[ ,final % use final for the camera ready runs %% ,draft % use draft while you are working on the paper %% ,numberedheadings % uncomment this option for numbered sections %% , % add further options here if necessary ] {aipproc} \usepackage{amsmath,amssymb} \layoutstyle{8x11double} %\documentclass[amsmath,prb,aps,twocolumn]{revtex4} % \documentclass[amsmath,prl,aps,preprint]{revtex4} \usepackage{amsthm,amsfonts,graphicx,verbatim} %\usepackage[active]{srcltx} \newcommand{\ppder}[1]{\frac{\partial^2}{{\partial #1}^2}} \newcommand{\der}[1]{\frac{d}{d #1 }} \newcommand{\pder}[1]{\frac{\partial}{\partial #1 }} \newcommand{\Tr}{{\rm Tr}} \newcommand{\tr}{{\rm tr}\/} \newcommand{\curl}{{\bf \nabla}\times} \def\req#1{(\ref{#1})} \newcommand{\D}{{\rm d}} \newcommand{\tc}{{\cal J}_1} \renewcommand{\theequation}{\arabic{equation}} \renewcommand{\cite}[1]{[\onlinecite{#1}]} %%% added by LL: \newcommand{\mpar}[1]{\marginpar{\small \it #1}} \newcommand{\be}{\begin{equation}} \newcommand{\ee}{\end{equation}} \newcommand{\bea}{\begin{eqnarray}} \newcommand{\eea}{\end{eqnarray}} \newcommand{\HH}{{\cal H}} \newcommand{\LL}{{\cal L}} \newcommand{\KK}{{\cal K}} \newcommand{\GG}{{\sf G}} %% \newcommand{\tr}{{\rm tr\/}} \newcommand{\p}{\partial} \newcommand{\s}{\sigma} \newcommand{\la}{\langle} \newcommand{\ra}{\rangle} \newcommand{\lb}{\left[} \newcommand{\rb}{\right]} \newcommand{\lp}{\left(} \newcommand{\rp}{\right)} \renewcommand{\vec}[1]{{\bf #1}} \renewcommand{\epsilon}{\varepsilon} \def\nn{\nonumber\\} \begin{document} \title{Many-Body Entanglement: a New Application of the Full Counting Statistics} %% Quantum Noise as an Entanglement Meter} \classification{03.65.Ud} \keywords{entanglement, full counting statistics} \author{Israel Klich}{address={Department of Physics, University of Virginia, Charlottesville VA 22904}} \author{Leonid Levitov}{address={Department of Physics, Massachusetts Institute of Technology, Cambridge MA 02139}} \begin{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. % {\bf % Entanglement entropy, which is a measure of quantum correlations between separate parts of a many-body system, has emerged recently as a fundamental and useful quantity in broad areas of theoretical physics, from cosmology and field theory~\cite{Bombelli86,Holzhey94} to condensed matter theory \cite{VidalLatorreetal,Refael Moore,Cardy,CalabreseCardyTimeDependentEnt,Bravyi06,EisertOsborne,Cramer08} and quantum information~\cite{Bennett96,Verstraete04,Kitaev Preskill,Levin Wen,Fradkin}. The universal appeal of the entanglement entropy concept in such different physical systems is related, in part, to the fact that it is defined solely in terms of the many-body density matrix of the system, with no relation to any particular observables. However, for the same reason, it has not been clear how to access this quantity experimentally, as measuring the entire density matrix of a many-body systems is in most cases impractical. In this paper we unveil a connection between entanglement entropy and the fluctuations of current flowing through a quantum point contact (QPC) which opens a way to perform a measurement of entanglement entropy. This connection is universal and robust with respect to the details of driving the QPC. In particular, we propose measuring the entropy generated by opening and closing the QPC periodically in time, and show that by utilizing space-time duality of 1d systems the entropy generated in this process can be used to test the seminal ${\cal S}=\frac13 \log L$ prediction of conformal field theory~\cite{Holzhey94}. Our results provide a method for theoretical and experimental investigation into the nature of many-body entanglement, and in particular, of its build up in non-equilibrium processes in quantum systems. % Recent years have witnessed a burst of interest in the phenomena of quantum entanglement, and in particular, in the entanglement entropy, a fundamental characteristic describing quantum many-body % correlations between two parts of a quantum system. % This quantity first emerged in field theory and cosmology~\cite{Bombelli86,Holzhey94} under the name of ``geometric entropy,'' and subsequently was adopted by quantum information theory. The notion of entanglement entropy has provided a framework for analyzing quantum critical phenomena \cite{VidalLatorreetal,Refael Moore,Cardy} and quantum quenches~\cite{CalabreseCardyTimeDependentEnt,Bravyi06,EisertOsborne,Cramer08}. Recently it was used as a probe of complexity of topologically ordered states~\cite{Kitaev Preskill,Levin Wen,Fradkin}. % In addition, this quantity is of fundamental interest for quantum information theory as a measure of the resources available for quantum computation~\cite{Bennett96} as well % as for numerical approaches to strongly correlated systems~\cite{Verstraete04}. % {\it Can the entanglement entropy be measured? } % Here we identify a system where such a measurement is possible, thereby offering an affirmative answer to this question. % %% by linking the entanglement entropy with electron transport. % In particular, we establish a relation between the entropy and quantum noise in a quantum point contact (QPC) \cite{van Wees}, an electron beam-splitter with tunable transmission and reflection, that serves as a door between electron reservoirs, which can be opened and closed on demand. % % Here we are interested in the situation when % % %% In that, the QPC transmission coefficient varies in time so that % % the reservoirs, initially disconnected, are connected at a certain time, and then are disconnected again at a later time, as illustrated in Fig.\ref{fig1}. % % We show that the fluctuations of electric current flowing through the QPC can be used to quantify the entanglement generated by the connection, and thereby measure the entanglement entropy. % %% % We show that the entanglement entropy generated by connecting fermionic reservoirs through a QPC is related in a universal way to the fluctuations of current flowing between the reservoirs. We demonstrate that the entropy can be inferred from the full counting statistics of current fluctuations, and propose noise measurement as a tool to test our understanding of many-body entanglement in a realistic setting. In particular, current fluctuations in a QPC driven by periodic pulses can be used to probe the seminal $\frac13\log L$ prediction for the entanglement entropy. % } \end{abstract} \maketitle \section{Introduction} Density matrices as a tool for describing quantum-mechanical systems when only partial information about the quantum state is available were introduced in quantum mechanics by Landau in 1927 \cite{Landau}. In this work, called ``The problem of damping in wave mechanics,'' he was interested in irreversibility of certain quantum mechanical processes, such as the spontaneous decay of excited atomic states. Such irreversibility is not inherent to quantum mechanics, it arises from a fully reversible quantum evolution of a larger system, including the variables describing radiation. More generally, Landau was concerned with %% Specifically, he was interested in the situation when some variables needed to completely describe the system cannot be measured. In such cases the imprecision of our knowledge renders the quantum state vector $\psi$ a useless quantity. Instead, a density matrix must be used, which is defined in the subspace of the system's Hilbert space spanned by those states which can be measured, $\rho=\sum_{ij} \rho_{ij}|i\rangle\langle j|$. Another generalization of quantum mechanics in which density matrices feature prominently is quantum statistical mechanics. It was developed, also in 1927, by von Neumann \cite{vonNeumann}, as a way to introduce statistical description in quantum theory. In this approach, a quantum system can occupy states $|i\rangle$, forming an orthonormal set, with statistical probabilities $0\le p_i\le 1$. Such a system is described by $\rho=\sum_{i}p_i|i\rangle\langle i|$, which is nothing but a diagonal representation of the density matrix. In this work, von Neumann first wrote his famous formula for the entropy, %% expressed in quantum-mechanical terms as % \be\label{eq:vN_entropy} {\cal S} = -\Tr\lp \rho\log\rho\rp = -\sum_{i}p_i\log p_i , \ee which is a proper extension of the Gibbs entropy (and the Shannon entropy) to the quantum case. For a set of $N$ states, the entropy (\ref{eq:vN_entropy}) can take values varying between $0$ (for a pure state) and $\log N$, realized when all states are occupied with equal probabilities. %% , $\rho_{ij}=N^{-1}\delta_{ij}$. \begin{figure}[hb] \includegraphics*[width=3.0in]{Levitov_Landau100_fig1.eps} \caption{ Entanglement between two regions, $A$ and $B$, probed by an observer who can perform measurements only in region $A$ but not in region $B$. The quantum state of the entire system $A+B$, described by the state vector $|0\rangle$, must be ``projected'' on $A$, giving $\rho_A=\Tr_B\rho_0$, where $\rho_0=|0\rangle\langle0|$. Due to entanglement between variables in $A$ and in $B$, the projected state is typically a mixed state, $\rho_A\ne\rho_A^2$, even when the whole system is in a pure state. The von Neumann entropy of the projected state, Eq.(\ref{eq:EE}), is a measure of the entanglement between $A$ and $B$. % Schematic of a quantum point contact (QPC) with transmission changing in time. % The left and right leads are initially % disconnected, then connected at $t_00$) wavepacket states in the leads. % Here $v_F$ % is the Fermi velocity and $A$, $B$ are the reflection and transmission amplitudes. Crucially, the FCS generating function (\ref{eq:chi_Pn}) can be expressed through the evolved many-body density matrix of the initial state, projected on one of the leads after the evolution is completed, as in Eq.\eqref{eq:evolution+projection}. This relation, derived in Refs.\cite{Abanov Ivanov,KlichLevitov08a}, is outlined below, and then used to find the entanglement entropy. % The state evolves as a pure state, and is made impure only after projection is performed. For example, for the evolution illustrated in Fig.\ref{fig2}b, the many-body state evolves as a pure state while the QPC is open, after which the reduced density matrix of the lead $L$ is given by % % % \be\label{eq:evolution+projection} % \rho_L(t_1)={\bf Tr}_R ({\bf U}(t_1,t_0)\rho_0{\bf U}^{\dag}(t_0,t_1)) % . % \ee % % % Here $\rho_0$ is the initial density matrix of the system, % ${\bf U}$ is the many-body evolution between $t_0$ and $t_1$, and ${\bf Tr}_{R}$ is a partial trace over degrees of freedom in the lead $R$. There are certain general properties of the evolved density matrix in a noninteracting fermion system which are best understood by considering evolution of a gaussian state % \be\label{eq:gaussian} \rho=\frac1{Z}\exp\lp -\sum_{ij}H_{ij}a_i^\dagger a_j\rp ,\quad Z=\det(e^{-H}+1) , \ee % where $H$ is a general hermitian operator in the single-particle Hilbert space, and $Z$ is the normalization factor. The state of interest, describing the QPC at zero temperature, represents a particular case of Eq.(\ref{eq:gaussian}). One property which greatly simplifies the analysis, is that the state (\ref{eq:gaussian}) remains gaussian under the evolution (\ref{eq:evolution+projection}). This follows from the observation that the Schr\"odinger evolution of $a_i$'s is equivalent to the single-particle Heisenberg evolution of $H_{ij}$. Gaussian form of the state is also preserved under projection \cite{KlichLevitov08a}. This property can be used to reduce the many-body quantities of interest to certain one-particle quantities. Indeed, any gaussian state (\ref{eq:gaussian}) can be described by a matrix in the single-particle Hilbert space defined as % \be n_{ij}=\Tr\lp \rho a_i^\dagger a_j\rp=\lb \lp e^H+1\rp^{-1}\rb_{ij} . \ee % In particular, for a fully filled Fermi sea in both reservoirs the matrix $n$ is a projector on the subspace of all states with negative energy, $\epsilon_L<0$, $\epsilon_R<0$. As a projector, the matrix $n$ satisfies the relation $n^2=n$. In what follows, we will need to consider evolution of the projector $n$, %%describing a Fermi sea in one of the reservoirs. For the left reservoir, it projects on the states with $\epsilon_L<0$. We will denote this matrix $\tilde n$. Evolution of $n$, followed by projection on the left reservoir $L$, % Its evolution followed by projection, Eq.(\ref{eq:evolution+projection}), which is described by % \be\label{eq:M_definition} n_U=UnU^\dagger ,\quad M=P_Ln_UP_L , \ee % where $P_L$ is a projection on the modes in $L$. The matrix $n_U$ is a projector describing evolved Fermi sea, whereas the matrix $M$, which is of main interest for us, is given by a product of three projectors. Thus generally $M$ is not a projector. %% {\bf OK? L or L+R?} To illustrate the time evolution of single-particle quantities, such as $n_U$, we recall that in the FCS approach it is convenient to work in a time representation~\cite{LevitovLesovik04}, labeling states by times of arrival at the scatterer $t_x$. In this representation the initial Fermi projection is given by $n(t,t')={1\over 2\pi i(t-t'+i0)}I$ with $I$ a $2\times 2$ identity matrix in the $L$, $R$ basis. The evolved state $n_U$ is given by % \begin{eqnarray}\nonumber n_U(t,t')=U(t)n(t,t')U^{\dag}(t') ,\quad U(t)= \begin{bmatrix} B(t) & A(t) \\ -A^*(t) & B^*(t) \\ \end{bmatrix} . \end{eqnarray} % % These relations are inessential for the derivation of our main result, they serve here illustration purpose only. The evolution operator is diagonal with respect to the arrival time label $t$, which is precisely why this representation is so convenient. The FCS generating function \eqref{eq:chi_Pn} can be expressed in terms of the single-particle quantities, such as $n$, $n_U$, $P_L$ and $M$, in several different ways. The first representation of this kind, found in Ref.\cite{LevitovLesovik}, involves a functional determinant % Our next step is to relate the quantity $M$, Eq.(\ref{eq:M_definition}), % and the FCS generating function \eqref{Def of chi} which can be expressed as a functional determinant~\cite{LevitovLesovik}: % \be\label{FCS} \chi(\lambda)= \det(1-n +n U^{\dag}e^{i\lambda P_L}U e^{-i\lambda P_L}) . \ee % %% In the case of interest, for infinitely deep Fermi sea, t This determinant, which must be properly regularized for infinitely deep Fermi sea \cite{MuzukantskiiAdamov,Avron Graf}, can be explicitly evaluated, yielding the FCS for various driving protocols of the QPC~\cite{Ivanov93,Levitov96,Ivanov97}. Another useful representation of the functional determinant giving $\chi(\lambda)$ was obtained recently in Ref.\cite{Abanov Ivanov} (a similar relation was derived in a related problem of particle number fluctuations \cite{Klich06,KlichRefaelSilva}), % \be\label{eq:chi-M} \chi (\lambda)=\det\Big((1-M +M e^{i\lambda })e^{-i\lambda (n P_L)_U}\Big) , \ee % where $(n P_L)_U=Un P_LU^\dagger$ (see discussion in \cite{KlichLevitov08a}). Here $M$ is the Fermi sea in $L$, evolved by $U$ and projected back to $L$ by $P_L$ (see Eq.\eqref{eq:M_definition}). The unitary operator $e^{-i\lambda (n P_L)_U}$ contributes a multiplicative factor of the form $e^{ix\lambda}$ to the determinant, which may only affect the first cumulant $C_1$ of the FCS generating function, Eq.(\ref{eq:chi_Cn}). The representation (\ref{eq:chi-M}) is of interest because it reveals certain general features of $\chi(\lambda)$. It is convenient to introduce the spectral density of $M$, defined by $\mu(z)=\Tr\, \delta(z-M)$. Since $M$ is a product of three projectors, $M=P_Ln_UP_L$, all its eigenvalues lie in the interval $0\le z\le 1$ (see Refs.\cite{Vanevic,Abanov Ivanov,Sherkunov}). Using the spectral density, we can rewrite (\ref{eq:chi-M}) as % \be \log\chi(\lambda)=ix\lambda+\int_0^1 dz\mu(z)\log\lp 1-z+ze^{i\lambda }\rp . \ee % % Here we used the fact that all eigenvalues of $M$ lie in the interval $0\le z\le 1$, which was proven in several different ways in Refs.\cite{Vanevic,Abanov Ivanov,Sherkunov}. This expression indicates that the FCS always assumes a generalized binomial form, $\chi(\lambda)\propto \prod_z \lp 1-z+ze^{i\lambda }\rp^{\mu(z)}$, with the product taken over the entire spectrum of $M$. \section{The relation between entanglement and Full Counting Statistics} The entanglement generated as a result of evolution of two Fermi seas coupled through a QPC is characterized by the von Neumann entropy of the density matrix \eqref{eq:evolution+projection}, ${\cal S}=-\Tr\, \rho_L(t_1)\log\rho_L(t_1)$. As we discussed above, this matrix is of a gaussian form, Eq.\eqref{eq:gaussian}. This property can be used to express entropy through single-particle quantities (see Refs.\cite{Peschel,KlichLevitov08a}), giving % \be\label{EntropyM} {\cal S}=-\Tr\, \lp M\log M+(1-M)\log(1-M)\rp \ee % where $M$ is defined in \eqref{eq:M_definition}, and the trace is taken in the space of single-particle modes in $L$. If the spectral density of $M$ is known, the entropy \eqref{EntropyM} can be written as % \be\label{eq:S_mu(z)} {\cal S}=-\int_0^1dz\mu(z)\lp z\log z+(1-z)\log(1-z)\rp \ee % At the same time, as we discussed above, the FCS generating function is also expressed through the spectral density of $M$. Furthermore, the spectral density is encoded in the generating function. This can be seen most easily by rewriting Eq.\eqref{eq:chi-M} as % \be\label{eq:chi(z)} \chi (z)=\det\lp (z-M) e^{-i(n P_L)_U\lambda(z)}(1-e^{i\lambda(z)})\rp , \ee % where the parameter $\lambda$ was changed to $z=(1-e^{i\lambda})^{-1}$. Thus the resolvent of $M$ is given by the derivative $\partial_z\log \chi(z-i0)$ up to a sum of two terms $\frac{a_0}{z}+\frac{a_1}{z-1}$ arising from the last two factors in \eqref{eq:chi(z)}. Using this observation the entanglement entropy can be expressed through the FCS cumulants $C_m$ % defined in Eq.\eqref{eq:chi_Cn} \cite{KlichLevitov08a}. Writing $\mu(z)$ in Eq.\eqref{eq:S_mu(z)} as $\frac1{\pi}{\rm Im}\,\partial_z\log\chi(z-i0)$ and plugging into it the relation \eqref{eq:chi_Cn}, we integrate over $z$ to obtain %% (see Ref.\cite{KlichLevitov08a}) % \begin{eqnarray}\label{Entropy from cumulants} %% {\cal S}=\sum_{m\,({\rm even})\,>0} {(2\pi)^{m}|B_{m}|\over m !}C_{m} {\cal S}=\sum_{m>0} {\alpha_m\over m !}C_{m} ,\quad \alpha_m= % 2^{m-1}\int_{-\infty}^{\infty}\D u {u^m\over % {\sinh}^2{u} }= \Big\{ \begin{array}{cc} (2\pi)^m |B_{m}|, & $m$\,\,{\rm even} \\ 0, & $m$\,\,{\rm odd} \end{array} , \end{eqnarray} % where $B_m$ are Bernoulli numbers, $B_2=\frac16$, $B_4=-\frac1{30}$, $B_6=\frac1{42}$... These numbers are defined by the generating function $\frac{x}{e^x-1}=\sum_{n=0}^{\infty}B_n {x^n\over n!}$. Interestingly, only even cumulants contribute to the entropy. The first few terms in the series \eqref{Entropy from cumulants} are: \begin{eqnarray}\label{Entropy C2 C4 C6} {\cal S}={\pi^2\over 3} C_2+{\pi^4\over 15}C_4+{2\pi^6\over 945}C_6+... \end{eqnarray} % Asymptotically, $|B_{n}|\approx{2\, n!\over (2\pi)^{n}}$ for large even $n$, which means that the coefficients in (\ref{Entropy from cumulants}) stay bounded, $\alpha_m/m!\approx 2$ for large $m$. % high order cumulants. It is clear from this derivation that the relations \eqref{Entropy from cumulants} and \eqref{eq:S_mu(z)} are completely general and valid for arbitrary driving. The formula \eqref{Entropy from cumulants} can be used to determine the entanglement entropy from the values of FCS moments, whereas the formula \eqref{eq:S_mu(z)} can be used when the spectral density $\mu(z)$ is known. %% Similar relation can be derived for other quantities of interest, such as Renyi entropies, or single copy entropy \cite{Eisert Cramer}. As an example illustrating these relations, it is instructive to consider a QPC biased with a DC voltage. The FCS for a DC-biased QPC with constant transmission $0\le D\le 1$ is known to have binomial form \cite{LevitovLesovik}, described by the generating function \eqref{eq:binomial}, % \be \chi(\lambda)=\lp 1-D+De^{i\lambda}\rp^N ,\quad N=eV\Delta t/h , \ee % where $N$, given by the product of the bias voltage $V$ and the measurement time $\Delta t=t_1-t_0$, is interpreted as the ``number of attempts.'' Comparing this expression with \eqref{eq:chi-M}, we infer that the spectral density of $M$ in this case is a delta function, $\mu(z)=N\delta(z-D)$. Plugging this result in the expression \eqref{eq:S_mu(z)}, we find that entanglement entropy is generated at a constant rate given by % \be\label{eq:dS/dt_DC} d{\cal S}/dt=-\lp D\log D+(1-D)\log(1-D)\rp eV/h \ee % Production of entanglement in a DC-biased QPC was considered in Ref.\cite{Beenakker_dS/dt}, were the result \eqref{eq:dS/dt_DC} was obtained. The rate of entanglement production is zero for $D=0,1$ and maximal for $D=1/2$. In this case, using the series \eqref{Entropy from cumulants} turns out to be not too convenient because of a large number of high-order cumulants that contribute to the result. This can be seen most directly in the limit of small $D\ll1$, when the entanglement production rate scales as $D\log(1/D)$, while the cumulants $C_m\sim D$. This means that there are about $\log(1/D)$ terms in the series \eqref{Entropy from cumulants} giving contributions of the same order of magnitude. \section{Connecting and disconnecting Fermi seas} % Here we discuss several examples of the relation between entanglement entropy and FCS. It is instructive to start with a QPC biased with a DC voltage. Production of entanglement in such a system was considered in Ref.\cite{Beenakker_dS/dt}, where it was found that it is generated % where % we recover the result \eqref{eq:dS/dt_DC}. Here we shall discuss the protocol of driving the QPC in which it is opened at $t_0