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\begin{document}

\title{Noise and entanglement in quantum conductors}

\classification{73.23.-b, 03.65.Nk, 03.67.Mn} \keywords
{entanglement, noise, electronic transport}

\author{G.B.\ Lesovik}{}
\author{A.V.\ Lebedev}{
  address={L.D. Landau Institute for Theoretical Physics, RAS, 119334 Moscow, Russia \\
  Theoretische Physik, Schafmattstrasse 32, ETH-Zurich, CH-8093, Zurich, Switzerland}
}


\begin{abstract}
In this article  we discuss  our two recent proposals on producing
and detecting of entangled states in quantum conductors. First we
analyze a setup where two electrons are scattered on a quantum dot
with Coulomb repulsion and became orbitally entangled. Second, for
identical noninteracting particles we suggest an operating scheme
for the deliberate generation of spin-entangled electron pairs in a
normal-metal mesoscopic structure with a fork geometry. The
spin-entangled pair is created through a post-selection in the two
branches of the fork. We also make comments on different ways of
producing and quantifying the degree of entanglement.

\end{abstract}

\maketitle

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\section{Introduction}

Quantum mechanics is a non-deterministic theory, where one can not
predict results of all measurements with certainty. A simplest
example is the tunneling of the electron through a barrier. After
the scattering electron may tunnel through the barrier with a finite
probability $T$ or may be reflected back with probability $R=1-T$.
In analogy with optics, this phenomena is known as partitioning of
the wave function. This type of phenomena requires probabilistic
description which turns out to be the only accessible way in general
situation for quantum systems, as was first pointed out by Born
\cite{born}.

The partitioning of the wave function reveals itself in the shot
noise phenomenon in coherent conductors, where Fermi statistics
suppress fluctuations in the number of particles incoming from
reservoir \cite{noise}. The shot noise can be observed by measuring
the second order current-current correlators that requires a very
elaborate experiment in practice \cite{noise_ex}. In general, noise
measurements provide more information about the system then the
measurement of conductance alone. As an example, measurement of the
shot noise allows one to probe the charge of the carriers
\cite{charge}. Going beyond the second order correlators, one could
study the full counting statistics (FCS) that is the probability to
transmit $n$ electrons through a barrier during a given time
interval \cite{levitov93}. To measure FCS, one needs even more
elaborate setup, which has been realized for electrons
\cite{ensslin}. Still the experimental setup for measuring FCS were
electrons flying ballistically remains to be challenging.

A more complicated type of uncertainty in quantum mechanics appears
in the joint state of two systems $A$ and $B$ which have been
interacted in the past. Then the outcome of the measurement on the
system $A$ will in general depend on a state of the system $B$ even
if two systems are spatially separated and no longer interact with
each other. The appearance of such correlations between two systems
is not a surprise even in the classical physics. What is amazing is
that these correlation in the quantum case, as was first shown by
Bell \cite{bell}, may be stronger than any possible classical
correlations adopting the Einstein locality principle.

The situation where the state of the system $B$ correlated with the
system $A$ is unknown can be described by the density matrix
$\rho_A$  introduced independently by Landau \cite{landau} and von
Neumann \cite{neumann}. In this case the state of the system $A$ is
mixed, i.e. can not be described by the wave function as it can be
done for pure states. To distinguish between pure and mixed states
one could calculate the purity $\Pi_A = \mbox{tr}\{\rho_A^2\}$ which
equals to $1$ for pure states and $\Pi_A<1$ for the mixed ones
\cite{lanlif}.

For a long time this type of correlation was mostly considered as
the source of the decoherence of the system $A$ by some reservoir
$B$, spoiling the quantum behaviour of the system $A$. Nowadays it
is very well understood that such mixed states with $\Pi_A<1$
describe an entangled state of two systems $A$ and $B$, a concept
first introduced by Einstein Podolsky and Rosen \cite{EPR} and
Schroedinger \cite{schroedinger}. Provided the full control on the
state of the system $B$, today these quantum correlations between
entangled systems are regarded as a resource for various sort of the
quantum information schemes like cryptography and quantum
computation. In addition, one may study fundamental aspects of the
quantum theory like locality in quantum mechanics by measuring
proper correlations between the systems $A$ and $B$, checking e.g.
Bell inequality \cite{bell,clauser}.

Various setups were proposed for producing isolated entangled
particles and detecting the presence of  the entanglement by testing
the Bell inequality via measuring cross correlators for the currents
\cite{entset,lebedev}, see Ref. \cite{benrew} for a recent review.
As we have already noted, to check the entanglement presence one may
use purity or von Neumann entropy. Nevertheless it is not always
clear what would be the experimental procedure to check all
components of the density matrix. Moreover, in general $\Pi_A<1$
does not guarantee that the system $A$ is not entangled with some
other system $C$ e.g. with an uncontrolled environment. Testing the
Bell inequality via measuring proper correlation between particles
remains to be the most reliable way to detect the entanglement.
Although such an experiment was successfully  realized in optics
\cite{aspect}, the corresponding experiment for the massive
particles remains to be challenging. To make some progress, one
could measure instead more accessible quantities of the total system
$A+B$ e.g. magnetization which may indicate the presence of
entanglement \cite{fazio}. In this case, one has to strongly rely on
a particular theoretical model, describing the system.

In this article  we discuss  our two recent proposals on producing
and detecting of entangled states in quantum conductors. The generic
way for two particles become entangled is to let them interact with
each other. In \cite{lebedev_08} we analyzed a setup where two
electrons are scattered on a quantum dot with Coulomb repulsion.
Characterizing the dot by its resonances we have derived an exact
formula for the $N$ particle scattering matrix. We make use of our
results to study the interaction-induced orbital entanglement of two
electrons incident on the dot in a spin-singlet state.

For the identical particles one can create an entangled state by a
post-selection of measured outcomes without direct interaction
\cite{bose}. In \cite{lebedev} we suggest an operating scheme for
the deliberate generation of spin-entangled electron pairs in a
normal-metal mesoscopic structure with a fork geometry. Voltage
pulses with associated Faraday flux equal to one flux unit
$\Phi=hc/e$ drive individual singlet-pairs of electrons towards the
beam splitter. The spin-entangled pair is created through a
post-selection in the two branches of the fork. We analyze the
appearance of entanglement in a Bell inequality test formulated in
terms of the number of transmitted electrons with a given spin
polarization.

\section{Entanglement due to Coulomb interaction}

In this section we follow  our recent paper~\cite{lebedev_08}, where
we derived the $N$-particle scattering matrix for electrons
propagating through quantum dot with Coulomb repulsion. The
scattering matrix, taking asymptotically free incoming states
through an interaction region and providing the free outgoing
states, is of huge basic and practical interest.  Originally
introduced by Born \cite{born} and by Wheeler and independently by
Heisenberg \cite{wheeler} in atomic and particle physics, its
application to electron transport \cite{landauer} has made it into a
central tool of mesoscopic physics. Its formulation for
non-interacting electrons provides the two-terminal conductance
\cite{landauer}, non-equilibrium noise \cite{noise} and the full
counting statistics \cite{levitov93} in terms of the transmission
probability across the scatterer. In interacting case the scattering
matrix besides the transport characteristics allows one to study
degree of entanglement of a many-particle scattered state induced by
interaction. The entanglement between electrons can be studied in a
new type of transport experiment, where specially designed electron
sources send a finite number of electrons towards the scattering
region \cite{pulses,keeling}, see Ref.\ \cite{glattli} for recent
experiments in this direction.

Within our formalism, the interaction is accounted for by the
Hamiltonian $\hat H_\mathrm{int} = e^2\hat N^2/2C$, where $\hat N$
is the dot's electron number operator and $C$ denotes its
capacitance.  We use two-particle scattering matrix to study the
wave function and degree of entanglement of two scattered electrons.
Below, we construct the two-particle scattering matrix for the
quantum dot including Coulomb interaction and generalize the result
to the $N$-particle situation. For a quantum dot with a single
resonance our model is equivalent to the Anderson impurity
model~\cite{AIM} up to the self-interaction energy $e^2\hat N/2C$,
which leads to the renormalization of the bare resonance energy
$\epsilon \rightarrow \epsilon + e^2/2C$, see below.

Usually, the scattering matrix connects states at given energies;
here, we start with the propagator describing the scattering of wave
packets in coordinate space. We start with a (properly symmetrized,
spin indices are suppressed) incident two-electron wave function at
time $t_1$, $\Psi^\mathrm{in}(\vec y,t_1)$, with $\vec
y=\{y_1,y_2\}$. The scattered wave function at later times $t_2 >
t_1$ can be obtained with the help of the two-particle propagator
$K^{\scriptscriptstyle (2)}(\vec x,t_2;\vec y,t_1)$ describing the
evolution of two particles from the initial positions $\vec y$ at
time $t_1$ to the final positions $\vec x$ at $t_2$,
%
\begin{equation}
      \Psi^\mathrm{out}(\vec x,t_2)
      =\int d^2y \,  K^{\scriptscriptstyle (2)}
      (\vec x,t_2; \vec y,t_1)\,
      \Psi^\mathrm{in}(\vec y,t_1).
      \label{out}
\end{equation}
%
The two-particle propagator $K^{\scriptscriptstyle (2)}$ can be
defined through a Feynman path integral over trajectories $\vec
x(t)$,
%
\begin{equation}
      K^{\scriptscriptstyle(2)}(\vec{x},t_2;\vec{y},t_1) \!=\!\!
      \int\! {\cal D}[\vec{x}\,]\,
      \exp \Bigl( \frac{i}{\hbar} \int_{t_1}^{t_2}\!\!dt\,
      L^{\scriptscriptstyle (2)}(\vec{x};\dot{\vec{x}}\,) \Bigr),
\end{equation}
%
with the boundary conditions $\vec x(t_1) = \vec y$. Here,
$L^{\scriptscriptstyle (2)}(\vec{x}, \dot{\vec{x}}\,)$ is the
system's Lagrangian including kinetic ($\propto m$), dot potential
($U$), and interaction ($\propto U_c = 2e^2/C$) energies,
%
\begin{equation}
      L^{\scriptscriptstyle (2)} = \!
      \sum_{i=1}^2 \Bigl[\frac{m \dot{x}_i^2}{2} - U(x_i) \Bigr]
      \!-\!  \frac{U_c}{4}
      \bigl[\chi_\mathrm{d}(x_1)\!+\!\chi_\mathrm{d}(x_2) \bigr]^2,
      \label{lag}
\end{equation}
%
with the characteristic function $\chi_\mathrm{d}(x)$ of the dot
equal to unity within the dot and zero outside.

Without interaction, the two-particle propagator factorizes,
$K^{\scriptscriptstyle (2)}(\vec x,t_2;\vec y,t_1) = \Pi_i
K^{\scriptscriptstyle (1)}(x_i,t_2;y_i,t_1)$ with
$K^{\scriptscriptstyle (1)}(x,t_2;y,t_1)$ the one-particle
propagator, while the interaction mixes the particle trajectories. A
Hubbard-Stratonovich transformation with the real auxiliary field
$z(t)$ allows us to decouple the quadratic interaction
%
\begin{eqnarray}
      &&K^{\scriptscriptstyle (2)}(\vec{x},t_2;\vec{y},t_1)
      = \int {\cal D}[z] \exp \Bigl[i\frac{U_c}{\hbar}
      \int dt\, z^2(t) \Bigr] \label{zav}\\
      &&\qquad\times\,
      K_{[z]}^{\scriptscriptstyle (1)}(x_1,t_2;y_1,t_1) \,
      K_{[z]}^{\scriptscriptstyle (1)}(x_2,t_2;y_2,t_1),
      \nonumber
\end{eqnarray}
%
where $K_{[z]}^{\scriptscriptstyle (1)}(x,t_2;y,t_1)$ is the
one-particle propagator in the presence of a fluctuating potential
$U_c(t) = U_c z(t)$,
%
\[
      K^{\scriptscriptstyle (1)}_{[z]}
      =\!\int\!\!\! {\cal
      D}[x] \exp\Bigl[ \frac{i}{\hbar} \int_{t_1}^{t_2} \!\!\! dt \,
      \Bigl(\frac{m\dot{x}^2}{2}\!-U(x)\!-U_c z(t)\chi_\mathrm{d}(x)\Bigr)
      \Bigr].
\]
%

Next, we introduce the scattering matrix
$S_{\alpha\beta}^{\scriptscriptstyle (1)}(\varepsilon)\equiv {\bf
S}^{\scriptscriptstyle (1)}(\varepsilon)$ of the dot in the absence
of the fluctuating potential $U_c(t)$; the indices $\alpha,\beta
\in\{{\scriptstyle\rm L,R}\}$ specify the lead indices for the
outgoing ($\alpha$) and incoming ($\beta$) scattering channels. We
describe the dot through the resonance positions ($\epsilon_j$) and
(identical) widths ($\Gamma$); the scattering matrix
$S_{\alpha\beta}^{\scriptscriptstyle (1)} (\varepsilon)$ then takes
the form
%
\begin{equation}
      S_{\alpha\beta}^{\scriptscriptstyle (1)}(\varepsilon) =
      r_{\alpha\beta} +
      \sum_{j} \frac{i\Gamma/2}{\varepsilon-\epsilon_j +
      i\Gamma/2}\, s_{\alpha\beta}^{\scriptscriptstyle (j)},
      \label{1sc_en}
\end{equation}
%
where the constant $2\times 2$ matrices ${\bf r}$ and ${\bf
s}^{\scriptscriptstyle (j)}$ can be found from the unitarity
conditions. The Fourier transform provides the real time ($\tau$)
representation
%
\begin{equation}
      S_{\alpha\beta}^{\scriptscriptstyle (1)}(\tau)
      = \delta(\tau) r_{\alpha\beta}
      + \theta(\tau) \sum_j\frac{\eta}{2} e^{-i\omega_j\tau}
      e^{-\eta\tau/2} \, s_{\alpha\beta}^{\scriptscriptstyle (j)},
      \label{1sc_time}
\end{equation}
%
where $\eta = \Gamma/\hbar$ is the inverse dwell time, $\omega_j=
\epsilon_j/\hbar$ is the resonance frequency, and $\delta(\tau)$,
$\theta(\tau)$ are the usual $\delta$- and Heaviside functions.  The
first term in Eq.~(\ref{1sc_time}) describes the reflection of a
particle that has not penetrated into the dot, while the subsequent
terms correspond to processes where the particle has spent a time
$\tau$ inside the dot; the factor $e^{-i\omega_j\tau}$ describes the
accumulated phase. The presence of the fluctuating potential
$U_c(t)$ contributes an additional phase to the one-particle
scattering matrix (\ref{1sc_time}),
%
\begin{equation}
      S_{\alpha\beta,[z]}^{\scriptscriptstyle (1)}(t_2,t_1) =
      S_{\alpha\beta}^{\scriptscriptstyle (1)}(t_2-t_1)
      \exp \Bigl( - \frac{i}{\hbar} \int_{t_1}^{t_2} U_c(t) dt
      \Bigr),
      \label{1sc_time_z}
\end{equation}
%
where $t_1$ and $t_2$ denote the arrival and escape times of the
particle (we assume escape amplitudes that depend weakly on energy).

Next, we express the propagator $K^{\scriptscriptstyle (1)}_{[z]}$
through the scattering matrix~(\ref{1sc_time_z}). To simplify
matters, we linearize the spectrum, $\varepsilon(k) = \hbar v k$; a
particle escaped out of the dot then never returns.  In terms of
trajectories, the scattering process involves three stages: {\it i)}
the ballistic motion with velocity $v$ towards the dot, {\it ii)}
the dwell time in the dot, and, {\it iii)} the ballistic propagation
away from the dot.  We define the coordinates in the left ($x<0$)
and right ($x>0$) leads with respect to the left ($x=0^-$) and right
($x=0^+$) dot boundaries and express the propagator
$K_{[z]}^{\scriptscriptstyle (1)}$ through the scattering
matrix~(\ref{1sc_time_z}), $K^{\scriptscriptstyle
(1)}_{\alpha\beta,[z]} (x,t_2;y,t_1) =
S_{\alpha\beta,[z]}^{\scriptscriptstyle (1)}(\tau,s)/v$, where $s =
t_1+|y|/v$ and $\tau = t_2-|x|/ v$ are the arrival and escape times
of the particle to and from the dot. Similar definitions
($s_i=t_1+|y_i|/v$ and $\tau_i = t_2-|x_i|/v$) apply to the
two-particle scattering matrix: ${\bf
S}^{\scriptscriptstyle(2)}_{[z]} (\tau_1,\tau_2;s_1,s_2)=
S^{\scriptscriptstyle
(1)}_{\alpha_1\beta_1,[z]}(\tau_1,s_1)S^{\scriptscriptstyle
(1)}_{\alpha_2\beta_2,[z]}(\tau_2,s_2)$. Taking the average with
respect to the fluctuating Gaussian field $z(t)$, $\langle z(t_2)
z(t_1) \rangle = (i/2\omega_c) \delta(t_2-t_1)$, $\omega_c =
U_c/\hbar$ one could arrive to the result
%
\begin{equation} \label{2sc_time}
      {\bf S}^{\scriptscriptstyle (2)}(\vec\tau;\vec{s})
      = {\bf \tilde S}^{\scriptscriptstyle(1)}
      (\tau_1\!-\!s_1)\otimes
      {\bf \tilde S}^{\scriptscriptstyle(1)}(\tau_2\!-\!s_2)
      \exp (-i\frac{\omega_c\tau_{12}}2),
\end{equation}
%
where ${\bf \tilde S}^{(1)}$ is the scattering
matrix~(\ref{1sc_time}) with renormalized resonance energies
$\tilde\epsilon_j =\epsilon_j + U_c/4$ and $\tau_{12}$ is the time
the two particles spend together in the dot,
%
\[
      \tau_{12} = {\scriptstyle \frac12} (|\tau_1-s_2|+|\tau_2-s_1| -
      |\tau_1-\tau_2| - |s_1-s_2|).
\]
%
This two-particle scattering matrix (\ref{2sc_time}) is the key
result of this Section. All effects of Coulomb interaction are
accounted for by renormalized resonance energies due to
self-interaction of individual electrons in the dot and an
additional phase accumulated by the electrons during their
simultaneous presence in the quantum dot. The self-interaction of
electrons may arise due to the screening environment in realistic
quantum dots, while in the absence of the screening (e.g. in
Anderson impurity model) one has to use the bare resonance energies
$\epsilon_j$ in Eq.~(\ref{2sc_time}).

Below we concentrate on a single level quantum dot with no
self-interaction present. An inverse Fourier transformation provides
us with the energy representation
%
\begin{eqnarray}
      &&{\bf S}^{\scriptscriptstyle (2)}
      (\vec{\varepsilon}^\prime;\vec\varepsilon)
      =\delta_{\varepsilon_1\varepsilon_1^\prime}
      \delta_{\varepsilon_2\varepsilon_2^\prime}\,
      {\bf S}^{\scriptscriptstyle (1)}(\varepsilon_1)
      \otimes
      {\bf S}^{\scriptscriptstyle (1)}(\varepsilon_2)
      \\
      \nonumber
      &&\quad+
      2\pi\delta(\varepsilon_1\! +\! \varepsilon_2\! -\!
      \varepsilon_1^\prime\! -\! \varepsilon_2^\prime)
      {\bf S}_{U_c}(\vec\varepsilon^\prime;\vec\varepsilon),
\end{eqnarray}
%
where the second term accounts for inelastic processes where only
the total energy $E =\varepsilon_1+\varepsilon_2$ is conserved:
%
\begin{eqnarray}\label{2sc_energy}
      \nonumber
      && {\bf S}_{U_c}(\vec\varepsilon^\prime;\vec\varepsilon)=
      \frac{(iU_c/2)\,
      {\bf s} \otimes {\bf s}}{
      \varepsilon_1\! +\! \varepsilon_2\! -\!
      2\epsilon_0\!-\! \frac{U_c}2\!+\!i\Gamma}
      \frac{i\frac{\Gamma}{2}}{\varepsilon_1\! -\!
      \epsilon_0 \!+\!i\frac{\Gamma}{2}}
      \\
      &&\times
      \frac{i\frac{\Gamma}{2}}{\varepsilon_2\! -\! \epsilon_0
      \!+\!i\frac{\Gamma}{2}}
      \biggl( \frac{1}{\varepsilon_1^\prime\! -\! \epsilon_0
      \!+\!i\frac{\Gamma}{2}}
      + \frac{1}{\varepsilon_2^\prime\! -\!\epsilon_0
      \!+\!i\frac{\Gamma}{2}}
      \biggr).
\end{eqnarray}
%
The Coulomb interaction generates additional pole at $E=2\epsilon_0
+U_c/2-i\Gamma$ of the total energy $E$. This interaction-induced
singularity cannot be obtained via a perturbative expansion for
large $U_c\gg\Gamma$.

The above derivation for the two-particle scattering matrix can be
generalized to $N$ particles, where appear an additional phase
factors accounting for the pairwise interaction of particles
residing simultaneously (for a time $\tau_{jk}$) on the dot,
%
\[
      S_{\{\alpha_j\beta_j\}}^{\scriptscriptstyle(N)}
      \bigl(\{\tau_j;s_j\}\bigr) =
      \prod_{j> k}^N
      e^{-i\omega_c \tau_{jk}/2} \prod_{j=1}^N
      \tilde S^{\scriptscriptstyle (1)}_{\alpha_j
      \beta_j}(\tau_j\!-\!s_j).
\]
%
The above result also holds true for a multichannel setup, with
$\alpha_j,~\beta_j$, $j=1,\dots,N$ turning into multichannel
indices. In particular, the results can be straightforwardly applied
to the experimental setup \cite{marcus07} with two parallel leads
feeding/emptying two capacitively coupled dots that has been
recently used to measure interaction-induced cross correlations, see
also Ref.~\cite{goorden07}.

In applying our results to realistic mesoscopic problems, we have to
avoid mixing between the scattered particles and the electrons in
the Fermi sea. Hence, we do not consider situations with levels
within the distance $\Gamma$ around the Fermi energy
$\varepsilon_{\rm \scriptscriptstyle F}$ and assume that $U_c$ does
not shift a level across $\varepsilon_{\rm \scriptscriptstyle F}$;
the latter allows us to ignore complications due to the Kondo effect
\cite{GM}.  In the following, we study the scattering problem of two
single-electron excitations created above the Fermi sea and a
quantum dot with only one resonance at $\epsilon_0$ above the Fermi
energy $\varepsilon_{\rm \scriptscriptstyle F}$,
$\epsilon_0-\varepsilon_{\rm \scriptscriptstyle F} \gg \Gamma$. The
scattering matrix (\ref{2sc_energy}) then tells, that (the
non-trivial component of) the scattered wave function involves
energies near $\epsilon_0$ and $\epsilon_+ = \epsilon_0 + U_c/2$.

We start from a two-electron state $\Psi^\mathrm{in} (x_1,x_2)$
disentangled with the Fermi sea created at time $t=0$ in the left
lead and moving towards to the dot. This could be achieved either by
applying a unit-flux voltage pulse of the Lorenzian
form~\cite{pulses} or by using a specially designed single electron
injector~\cite{keeling,glattli}. The scattered wave is given by
Eq.~(\ref{out}) and can be expressed in terms of retarded variables
$\xi_{1,2} = |x_{1,2}| - v_{\rm \scriptscriptstyle F} t$, with
$v_{\rm \scriptscriptstyle F}$ the Fermi velocity.  The scattered
wave to the right of the dot involving tunneling of both electrons
assumes the form $\Psi(\xi_1,\xi_2)=\Psi_{sq}(\xi_1,\xi_2)
+\Psi_{U_c}(\xi_1,\xi_2)$, with ($Y\equiv y_1+y_2$)
%
\begin{eqnarray}
      &&\Psi_{sq} =\frac{s_{\scriptscriptstyle\rm RL}^2}{\ell^2}
      \!\int_{\xi_<}^{\xi_>}\!\!\!\! dy_1
      \!\int_{\xi_>}^0 \!\!\! dy_2\,
      e^{ik_0 (\xi_1+\xi_2-Y)} e^{(\xi_1+\xi_2-Y)/\ell}
      \nonumber\\
      &&\times \bigl[
      \theta(\xi_2\!-\!\xi_1)\Psi^\mathrm{in}(y_1,y_2)
      \!+\!\theta(\xi_1\!-\!\xi_2) \Psi^\mathrm{in}(y_2,y_1)
      \bigr],\nonumber
\end{eqnarray}
%
\begin{eqnarray}
      &&\Psi_{U_c}=
      \frac{s_{\scriptscriptstyle\rm RL}^2}{\ell^2}
      e^{ik_+\xi_>} e^{ik_0 \xi_<}
      \int_{\xi_>}^0 \!\!\! dy_1 dy_2\,
      \Psi^\mathrm{in}(y_1,y_2)
      \nonumber\\
      &&\times \,
      e^{-ik_c(Y-|y_1-y_2|)/2} e^{-ik_0 Y}
      e^{(\xi_1+\xi_2-Y)/\ell}
\end{eqnarray}
%
where $\ell=2\hbar/\Gamma v_{\rm\scriptscriptstyle F}$ is the
real-space width of the scattered wave, $\xi_>=\max\{\xi_1,\xi_2\}$,
$\xi_<=\min\{\xi_1, \xi_2\}$, $k_0 =
\omega_0/v_{\rm\scriptscriptstyle F}$, $k_c = \omega_c/ 2v_{\rm
\scriptscriptstyle F}$, and $k_+ = k_0+k_c$. The first term
describes the process where the electrons do not overlap in the dot,
while the term $\propto e^{ik_+\xi_>} e^{ik_0\xi_<}$ deals with the
case where both electrons occupy the dot simultaneously during
scattering.  For electrons in a spin-triplet state with
anti-symmetric orbital wave function $\Psi^\mathrm{in} (y_1,y_2)$,
this term vanishes and no interaction effects survive, a consequence
of the Pauli principle.

Let us show that the Coulomb interaction in the dot leads to an
orbital entanglement of the two particles (for interaction-induced
spin entanglement in a quantum dot, see Ref.\ \cite{oliver}). Here,
we concentrate on the component of the wave function where two
electrons are transmitted to the right and estimate its degree of
entanglement, which is entirely due to the interaction in the dot.
We analyze the situation where the length of the incoming wave
packet is small with respect to $\ell$. Then the normalized wave
function on the right has the universal form:
%
\begin{equation}
      \Psi_{\scriptscriptstyle\rm RR}
      (\xi_1,\xi_2) =(2/\ell)\,
      e^{ik_+ \xi_>} e^{ik_0 \xi_<} \,
      e^{(\xi_1+\xi_2)/\ell},
      \label{2wave_sg}
\end{equation}
%
where $\xi_{1,2} <0$. Eq.~(\ref{2wave_sg}) describes a two-electron
state with different momenta $k_+$ and $k_0 < k_+$, as has to be
expected since the first electron escaping carries an energy shifted
up by the Coulomb interaction. The state~(\ref{2wave_sg}) can be
rewritten in a form
%
\begin{equation}
      \Psi_{\scriptscriptstyle\rm RR}
      =(2/\ell)
      e^{i(k_0+\frac{k_c}{2})(\xi_1+\xi_2)}\, e^{i\frac{k_c}{2}|\xi_1-\xi_2|}
      e^{(\xi_1+\xi_2)/\ell},
      \label{2wave_epr}
\end{equation}
%
reminding about the original Einstein-Podolsky-Rosen state
\cite{EPR}: $\Psi_{EPR}(x_1,x_2) = \delta(x_1-x_2+x_0)$ describing
the orbitally entangled state of two particles with the fixed
relative position $x_0$ but unknown center of mass coordinate (note
that this EPR state yet has to be properly normalized). To quantify
its entanglement, one may calculate the von Neumann entropy $E$ of
the reduced density matrix $\rho(x,x^\prime) = \int dx_2\,
\Psi_{\scriptscriptstyle\rm RR}(x,x_2) \Psi_{\scriptscriptstyle\rm
RR}^*(x^\prime,x_2)$. Instead, we determine the purity $\Pi(\rho) =
\mbox{tr}\, \rho^2$, which is unity only for separable states and
provides the lower limit $E>1-\Pi$. With $A \equiv ik_c\ell/
(2-ik_c\ell)$, we find the density matrix
%
\begin{eqnarray} \label{density}
      &&\rho(\xi,\xi^\prime) = (2/\ell)\,\theta(-\xi)
      \theta(-\xi^\prime) e^{(\xi+\xi^\prime)/\ell}
      e^{ik_0(\xi-\xi^\prime)}\nonumber\\
      &&\>\times \bigl[ 1
      + A\,\theta(\xi\!-\!\xi^\prime) (e^{2\xi/\ell}-
      e^{ik_c(\xi-\xi^\prime)}  e^{2\xi^\prime/\ell})
      \nonumber\\
      &&\quad
      + A^*\theta(\xi^\prime\!-\!\xi) (e^{2\xi^\prime/\ell}-
      e^{ik_c(\xi-\xi^\prime)} e^{2\xi/\ell}) \bigr],
\end{eqnarray}
%
that results into a purity $\Pi = [1+2/(1+(k_c\ell/4)^2)]/3$.  We
conclude that at finite $U_c$ the state~(\ref{2wave_epr}) is
entangled and the degree of entanglement saturates as the Coulomb
interaction becomes larger than the resonance width, $k_c\ell =
U_c/\Gamma \gg 1$, i.e., when the energies of the escaped particles
become distinguishable. In this case one could find $\Pi=\frac13$
and $\mbox{tr}\, \rho^3=\frac2{15}$ and the degree of entanglement
may be estimated more precisely using an expansion $E=
-\mbox{tr}\{\rho\log_2\rho\} = \sum_{n=1} \frac1n\,
\mbox{tr}\{\rho(1-\rho)^n\}/\ln2$ which gives $E>1.3$.

\section{Entanglement of noninteracting identical particles}

In the present section we closely follow our previous
work~\cite{lebedev}. We discuss a scheme generating pulsed
spin-entangled electron pairs in a normal-metal mesoscopic structure
arranged in a fork geometry, see Fig.\ \ref{fig:fork}. In this
device, spin-entangled electron pairs are generated via the
injection of spin-singlet pairs into the source lead from the
reservoir. This entanglement is made accessible by splitting the
pair into the two leads `u' and `d' and subsequent projection
(through the Bell measurement) to that part of the wave function
describing separated electrons travelling in different leads
\cite{lebedev_04}. Rather then quantum pumping with a cyclic
potential as in Refs.\ \cite{samuelsson_04b,beenakker_05}, our
proposal makes use of definite voltage pulses generating
spin-entangled electron pairs. Below we discuss a scheme where
voltage pulses of specific form accumulating one unit of flux
$\Phi_0 = -c \int dt \, V(t)$ and applied to the source lead `s'
generate pairs of spin-entangled electrons which then are
distributed between the two outgoing leads of the fork, the upper
and lower arms denoted as `u' and `d'. These spin-entangled electron
states are subsequently analyzed in a Bell experiment \cite{bell}
involving the measurement of cross-correlations \cite{nike_02}
between the number of electrons transmitted through the
corresponding spin filters in the two arms of the fork, see Fig.\
\ref{fig:fork}. Using time resolved correlators, we are in a
position to analyze arbitrary forms of voltage pulses and determine
the resulting degree of violation in the Bell setup. We find that
Lorentzian shaped pulses generate spin-entangled pairs with 50 \%
probability, corresponding in efficiency to the optimal performance
of one entangled pair per two cycles as found by Beenakker {\it et
al.} \cite{beenakker_05}. The reduction in efficiency to 50 \% is
due to the competing processes where the spin-entangled pair
generated by the voltage pulse propagates into only one of the two
arms. In order to make use of this structure as a deterministic
entangler, the Bell measurement setup has to be replaced through a
corresponding projection device (post-) selecting that part of the
wave function with the two electrons distributed between the two
arms; alternatively, this post-selection may be part of the
application device itself, as is the case in the Bell inequality
measurement.
\begin{figure}
  \includegraphics[height=.3\textheight, width=.45\textwidth]{fig_Lesovik}
  \caption{}
   \label{fig:fork}
\end{figure}

\subsection{Bell Inequality with Number Correlators}\label{sec:BI}

The Bell inequality~\cite{clauser} is based on the Lemma saying
that, given a set of real numbers $x$, $\bar x$, $y$, $\bar y$, $X$,
$Y$ with $|x/X|$, $|\bar x/X|$, $|y/Y|$, and $|\bar y/Y|$ restricted
to the interval $[0,1]$, the inequality $|xy - x\bar y +\bar x y +
\bar x \bar y|\leq 2|XY|$ holds true. We define the operator of
electric charge $\hat N_i(t_\mathrm{ac})$ transmitted through the
$i$-th spin detector during the time interval $[0,t_\mathrm{ac}]$,
where $t_\mathrm{ac}>0$ is the accumulation time. The charge
operator $\hat N_i(t_\mathrm{ac})$ can be expressed via the electric
current $\hat I_i(t)$ flowing through the $i$-th detector, $\hat
N_i(t_\mathrm{ac})= \int_0^{t_\mathrm{ac}} dt^\prime\, \hat
I_i(t^\prime)$. In the Bell test experiment, see
Fig.~\ref{fig:fork}, one measures the number of transmitted
electrons with a given spin polarization, $N_i$, $i=1,\dots,4$, and
defines the quantities $x=N_1-N_3$, $y=N_2-N_4$, $X=N_1+N_3$, and
$Y=N_2+N_4$ for fixed orientations ${\bf a}$ and ${\bf b}$ of the
polarizers (and similar for $\bar x$ and $\bar y$ for the
orientations $\bar {\bf a}$ and $\bar{\bf b}$), see
Ref.~\cite{nike_02}. Our Bell setup measures the correlations ${\cal
K}_{ij}({\bf a},{\bf b})= \langle \hat N_i(t_\mathrm{ac}) \hat
N_j(t_\mathrm{ac}) \rangle$ between the number of transmitted
electrons $N_{i}$, $i=1,3$, in the lead `u' with spin polarization
along $\pm{\bf a}$ and their partners $N_j$, $j=2,4$, in lead `d'
with spin polarization along $\pm{\bf b}$. Using the above
definitions for $x$, $y$, $X$, and $Y$, we obtain the normalized
particle-number difference correlator,
%
\begin{eqnarray}
      E({\bf a},{\bf b}) &=&
      \frac{\langle [\hat N_1 -\hat N_3][\hat N_2 -\hat N_4]\rangle}
      {\langle [\hat N_1 +\hat N_3][\hat N_2 +\hat N_4]\rangle},
\end{eqnarray}
%
and evaluating the correlators for the four different combinations
of directions ${\bf a},~\bar{\bf a}$ and ${\bf b},~\bar{\bf b}$, we
arrive at the Bell inequality
%
\begin{equation}
      E_{\scriptscriptstyle\rm BI} =
      | E({\bf a},{\bf b}) - E({\bf a},\bar {\bf b})
      + E(\bar {\bf a},{\bf b}) + E(\bar {\bf a},\bar {\bf b}) |
      \leq 2.
      \label{BI1}
\end{equation}
%

We proceed further by extracting in ${\cal K}_{ij}$ the irreducible
particle number correlators $K_{ij}(t_\mathrm{ac})=\langle
\delta\hat N_i(t_\mathrm{ac}) \delta\hat N_j(t_\mathrm{ac})\rangle$
and rewrite $E({\bf a},{\bf b})$ in the form
%
\begin{equation}
      E({\bf a},{\bf b}) =
      \frac{K_{12} - K_{14} - K_{32} + K_{34} + \Lambda_-}
      {K_{12} + K_{14} + K_{32} + K_{34}+\Lambda_+},
\end{equation}
%
where we have defined $\Lambda_\pm = [\langle \hat{N_1} \rangle \pm
\langle \hat{N_3} \rangle][\langle \hat{N_2}\rangle \pm \langle
\hat{N_4} \rangle]$ and $K_{ij}$ may be rewritten in terms of
irreducible current correlators $C_{ij}({\bf a},{\bf b};
t_1,t_2)=\langle \delta \hat I_i(t_1) \delta \hat I_j(t_2) \rangle$
with $\delta\hat I_i(t)= \hat I_i(t)- \langle \hat I_i(t)\rangle$,
%
\begin{eqnarray}
   \label{K}
   K_{ij}(t_\mathrm{ac})
   &=& \int_0^{t_\mathrm{ac}} dt_1 dt_2 \,
   C_{ij}({\bf a},{\bf b};t_1,t_2).
   \nonumber
\end{eqnarray}
%
The average currents are related via $\langle \hat I_1(t)
\rangle=\langle\hat I_3(t)\rangle=\langle\hat I_\mathrm{u}
(t)\rangle/2$ and $\langle \hat I_2(t)\rangle =\langle \hat
I_4(t)\rangle =$ $\langle \hat I_\mathrm{d}(t)\rangle/2$ and thus
$\Lambda_-=0$, $\Lambda_+=\langle \hat N_\mathrm{u}\rangle \langle
\hat N_\mathrm{d}\rangle$. The irreducible current-current
correlator factorizes into a product of spin and orbital parts,
$C_{ij}({\bf a},{\bf b};t_1,t_2) =|\langle {\bf a}_i|{\bf b}_j
\rangle|^2 C_\mathrm{ud}(t_1,t_2)$ with ${\bf a}_{1,3}= \pm{\bf a}$
and ${\bf b}_{2,4}=\pm{\bf b}$. The spin projections involve the
angle $\theta_{{\bf a}{\bf b}}$ between the directions $\bf a$ and
$\bf b$ of the polarizers, $\langle \pm {\bf a}|\pm {\bf b}\rangle =
\cos^2 (\theta_{{\bf a}{\bf b}}/2)$ and $\langle \pm {\bf a}|\mp
{\bf b}\rangle = \sin^2 (\theta_{{\bf a}{\bf b}}/2)$, and the Bell
inequality assumes the form
%
\begin{equation}
      \left|\frac{K_\mathrm{ud}
      [\cos\theta_{{\bf a}{\bf b}}-\cos\theta_{{\bf a}\bar{\bf b}}
      +\cos\theta_{\bar{\bf a}{\bf b}}+\cos\theta_{\bar{\bf a}\bar{\bf b}}]}
      {2K_\mathrm{ud}+\langle \hat N_\mathrm{u}\rangle \langle
      \hat N_\mathrm{d}\rangle}
      \right|\leq1,
      \label{BI2}
\end{equation}
%
where $K_\mathrm{ud}(t_\mathrm{ac})= \int_0^{t_\mathrm{ac}} dt_1
dt_2\, C_\mathrm{ud}(t_1,t_2)$ is the (irreducible) number
cross-correlator between the upper and lower leads of the fork. The
maximal violation of the Bell inequality is attained for the
standard orientations of the detector polarizations $\theta_{{\bf
a}{\bf b}}=\theta_{\bar{\bf a}{\bf b}} =\theta_{\bar{\bf a}\bar{\bf
b}}=\pi/4$, $\theta_{{\bf a}\bar{\bf b}} =3\pi/4$; the Bell
inequality (\ref{BI2}) then reduces to
%
\begin{equation}
      E_{\scriptscriptstyle\rm BI} = \left|
      \frac{2K_\mathrm{ud}}{2K_\mathrm{ud}+\langle \hat N_\mathrm{u}
      \rangle \langle \hat N_\mathrm{d}\rangle}\right| \leq
      \frac1{\sqrt2}.
      \label{BI3}
\end{equation}
%

\subsection{Number Correlators for a Single
Pulse}\label{sec:onepulse}

The orbital part $C_\mathrm{ud}(t_1,t_2)$ of the current
cross-corre\-lator between the upper and lower leads can be
calculated within the standard scattering theory of noise
\cite{noise}. We assume that the time dependent voltage drop $V(t)$
at the splitter can be treated adiabatically (i.e., the voltage
changes slowly during the electron scattering time). Such approach
have first been used in the calculation of the spectral noise power
in an $ac$-driven system \cite{ll_94} and its validity has been
confirmed in several experiments \cite{ksp_00}.

In the limit of linear dispersion the irreducible current
cross-correlator $C_\mathrm{ud}(t_1,t_2) = \langle \delta \hat
I_\mathrm{u} (x,t_1) \delta \hat I_\mathrm{d} (y,t_2) \rangle$
measured at the positions $x$ and $y$ in the leads `u' and `d' can
be splitted into two terms, one due to equilibrium fluctuations,
$C_\mathrm{ud}^\mathrm{eq} (t_1-t_2)= \int
(d\omega/2\pi)\,S^\mathrm{eq}(\omega)e^{i\omega(t_1-t_2)}$ with
%
\begin{equation}
      S^\mathrm{eq}(\omega) = -\frac{2e^2}{h}\, T_\mathrm{ud}
      \cos(\omega\tau^+)\,\frac{\hbar\omega}{1-e^{\hbar\omega/\theta}},
\end{equation}
%
and a second term describing the excess correlations at finite
voltage,
%
\begin{equation}
      C_\mathrm{ud}^\mathrm{ex}(t_1,t_2) = -\frac{4e^2}{h^2}
      T_\mathrm{u} T_\mathrm{d}
      \sin^2 \frac{\phi(\xi_1)\!-\!\phi(\xi_2)}2\,
      \alpha(\tau\!-\!\tau^-,\theta),
      \label{cor_eq_ex}
\end{equation}
%
with $\alpha(\tau,\theta)= \pi^2\theta^2 / \sinh^2 [ \pi\theta\tau /
\hbar]$ ($\theta$ is the temperature of electronic reservoirs),
$\tau=t_1-t_2$, $\tau^\pm =(x\pm y)/v_{\scriptscriptstyle\rm F}$,
$\xi_1 = t_1-x/v_{\scriptscriptstyle\rm F}$, and
$\xi_2=t_2-y/v_{\scriptscriptstyle\rm F}$. The coefficients
$T_\mathrm{u}$, $T_\mathrm{d}$, and $T_\mathrm{ud}$ denote the
transmission probabilities from the source to the `up', `down'
leads, and between the `down' and the `up' leads.

The equilibrium correlator $C_\mathrm{ud}^\mathrm{eq}$ describes the
correlations of the electrons in the Fermi sea propagating
ballistically from lead `u' to lead `d' (or vice versa) with the
retardation $\tau^+=(x_1+x_2)/v_{\scriptscriptstyle\rm F}$. The
corresponding equilibrium part of the particle-number
cross-correlator, $K_\mathrm{ud}^\mathrm{eq} =
\int_0^{t_\mathrm{ac}} dt_1 dt_2 \,
C_\mathrm{ud}^\mathrm{eq}(t_1-t_2)$ then takes the form
%
\begin{equation}
      K_\mathrm{ud}^\mathrm{eq} \approx \frac{e^2}{\pi^2}
      T_\mathrm{ud} \ln \frac{t_\mathrm{ac}}{\tau},\quad
      \tau = \mbox{max}\{\hbar/\epsilon_{\scriptscriptstyle\rm
      F}, \tau^+\}, \label{ncceq}
\end{equation}
%
where we have assumed the zero temperature limit and an accumulation
time $t_\mathrm{ac}\gg\tau$. The logarithmic divergence in
$t_\mathrm{ac}$ reduces the violation of the Bell inequality
Eq.~(\ref{BI3}) at large accumulation times and one has to suppress
the equilibrium correlations between the upper and the lower leads
in the setup. This can be achieved via a reduction in the
transmission probability $T_\mathrm{ud}$, however, in the fork
geometry of Fig.~\ref{fig:fork}(a) the probability $T_\mathrm{ud}$
cannot be made to vanish. Alternatively, one may chose a setup with
a reflectionless four-terminal beam splitter as sketched in
Fig.~\ref{fig:fork}(b) with no exchange amplitude between the upper
and lower outgoing leads and thus $K_\mathrm{ud}^\mathrm{eq}=0$.

Next, we concentrate on the excess part $K_\mathrm{ud}^\mathrm{ex}$
of the particle-number cross-correlator $\langle \hat N_\mathrm{u}
(t_\mathrm{ac}) \hat N_\mathrm{d}(t_\mathrm{ac}) \rangle $. Note
that the excess fluctuations are the same for both setups
Fig.~\ref{fig:fork}(a) and (b) and we can carry out all the
calculations for the fork geometry. We consider a sharp voltage
pulse applied at time $t_0$, $0<t_0<t_\mathrm{ac}$, with short
duration $\delta t$. The total accumulated phase $\phi(t)$ then
exhibits a step-like time dependence with the step height
$\Delta\phi= \phi(t_0 +\delta t/2)-\phi(t_0-\delta t/2)= -2\pi
\Phi/\Phi_0$, where we have introduced the Faraday flux $\Phi=-c\int
V(t) dt$ and $\Phi_0 = hc/e$ is the flux quantum. The excess part of
the particle-number cross-correlator $K_\mathrm{ud}$ then takes the
form (we consider again the zero temperature limit)
%
\begin{equation}
      K_\mathrm{ud}^\mathrm{ex} = -
      \frac{e^2}{\pi^2}T_\mathrm{u} T_\mathrm{d}\!\!
      \int\limits_0^{t_\mathrm{ac}} \!\! dt_1 dt_2\,
      \frac{\sin^2[(\phi(t_1)-\phi(t_2))/2]}{(t_1-t_2)^2}.
      \label{ncord}
\end{equation}
%
For a sharp pulse with $\delta t \ll t_0, t_\mathrm{ac}$ we can
identify two distinct contributions arising from the integration
domains $|t_1-t_2|\ll \delta t$ and $|t_1-t_2|\gg \delta t$, cf.\
Refs.\ \cite{levitov93,ll_93}; we denote them with $K^<$ and $K^>$.
Introducing the average and relative time coordinates
$t=(t_1+t_2)/2$ and $\tau=t_1-t_2$ and expanding the phase
difference $\phi(t_1)-\phi(t_2)= \phi(t+\tau/2)-\phi(t-\tau/2)
\approx \dot \phi(t)\tau$, the first contribution $K^<$ reads
%
\begin{equation}
      K^< \approx -\frac{e^2}{2\pi} T_\mathrm{u} T_\mathrm{d}
      \int\limits_0^{t_\mathrm{ac}} dt\, |\dot \phi(t)|.
      \label{K<}
\end{equation}
%
Assuming $\phi(t)$ is a monotonic function of $t$ Eq.~(\ref{K<}) can
be rewritten in terms of the Faraday flux $\Phi$,
%
\begin{equation}
      K^< = -e^2
      T_\mathrm{u} T_\mathrm{d}\,\frac{|\Phi|}{\Phi_0}.
\end{equation}
%
Assuming the Lorenzian form of the voltage pulse currying for
integer $n=|\Phi|/\Phi_0$ exactly $n$ (spinless) electrons
\cite{pulses}, the particle-number cross-correlator
$K_\mathrm{ud}^\mathrm{ex}$ describes the correlations arising from
the $n$ additional particles pushed through the fork by the voltage
pulse $V(t)$.

The second contribution $K^>$ to $K_\mathrm{ud}^\mathrm{ex}$
originates from the time domains $0<t_{1(2)}<t_0-\delta t/2$ and
$t_0+\delta t/2<t_{2(1)}<t_\mathrm{ac}$, where $|\phi(t_1)
-\phi(t_2)|=2\pi \Phi/\Phi_0$, hence
%
\begin{equation}
      K^> \approx
      -\frac{2 e^2}{\pi^2} T_\mathrm{u} T_\mathrm{d} \sin^2
      \frac{\pi\Phi}{\Phi_0}\ln \frac{t_\mathrm{m}}{\delta t};
      \label{response}
\end{equation}
%
here, we have kept the most divergent term in the measurement time
$t_\mathrm{m}=t_\mathrm{ac}-t_0$, the time during which the pulse
manifests itself in the detector. The above expression describes the
response of the electron gas to the sudden perturbation $V(t)$; the
logarithmic divergence in the measurement time $t_\mathrm{m}$ can be
interpreted \cite{ll_93} along the lines of the orthogonality
catastrophe \cite{anderson}, with the isolated perturbation in
space, the impurity, replaced by the sudden perturbation in time.
The periodicity of the response in the Faraday flux $\Phi$ is due to
the discrete nature of electron transport as expressed through the
binomial character of the distribution function of transmitted
particles \cite{levitov93,ll_93}. Remarkably, the above
logarithmically divergent contribution to
$K_\mathrm{ud}^\mathrm{ex}$ vanishes for voltage pulses carrying an
integer number of electrons.

Finally, the average number of transmitted (spinless) particles are
given by $\langle \hat N_\mathrm{u(d)} (t_\mathrm{ac}) \rangle =
\int_0^{t_\mathrm{ac}} dt\, \langle \hat I_\mathrm{u(d)}(x,t)
\rangle$, where  within the scattering matrix approach,
%
\begin{equation}
      \langle \hat I_\mathrm{u(d)}(x,t) \rangle = \frac{e}{h}
      T_\mathrm{u(d)}\, eV(t-x/v_{\scriptscriptstyle\rm F}),
      \label{Iud}
\end{equation}
%
and hence,
%
\begin{equation}
      \langle \hat N_\mathrm{u(d)}(t_\mathrm{ac})\rangle
      = e T_\mathrm{u(d)}\,\frac{\Phi}{\Phi_0}.
      \label{N_Phi}
\end{equation}
%

\subsection{Pulse with integer flux}\label{sec:if}

Substituting the above expressions for the particle-number
cross-correlators and for the average number of transmitted
particles into (\ref{BI3}) we arrive at the following general result
for the Bell inequality
%
\begin{equation}
      E_{\scriptscriptstyle\rm BI} =
      \left|
      \frac{n+(2/\pi^2)\sin^2(\pi n) \ln(t_\mathrm{m}/\delta t)}
      {2n^2-n-(2/\pi^2)\sin^2(\pi n) \ln(t_\mathrm{m}/\delta t)}
      \right|.
      \label{BI4}
\end{equation}
%

For Lorenzian voltage pulse with integer $n$ all logarithmic terms
vanish, leaving us with the Bell inequality
%
\begin{equation}
      E_{\scriptscriptstyle\rm BI} =
      \left|
      \frac{1}{2n-1}
      \right|\leq\frac1{\sqrt{2}},
      \label{BI5}
\end{equation}
%
which we find maximally violated for $n=1$ and never violated for
larger integers $n>1$ --- any additional particle accumulated in the
detector spoils the violation of the Bell inequality. Furthermore,
this violation is independent of the transparencies $T_\mathrm{u}$,
$T_\mathrm{d}$ and hence universal; moreover, the Bell
inequality~(\ref{BI5}) does not depend on duration of the voltage
pulse but involves only the number of electrons $n$ carried by it.

The Lorenzian voltage pulse with $n=1$ pushes two electrons with
opposite spin polarization towards the beam splitter. Such a pair
appears in a singlet state and can be described by the wave function
$\Psi_\mathrm{in}^{\scriptscriptstyle 12} =
\phi_\mathrm{s}^{\scriptscriptstyle 1}
\phi_\mathrm{s}^{\scriptscriptstyle 2}
\chi_\mathrm{sg}^{\scriptscriptstyle 12}$ with the spin-singlet
state $\chi_\mathrm{sg}^{\scriptscriptstyle 12} =
[\chi_{\uparrow}^{\scriptscriptstyle 1}
\chi_{\downarrow}^{\scriptscriptstyle 2} -
\chi_{\downarrow}^{\scriptscriptstyle 1}
\chi_{\uparrow}^{\scriptscriptstyle 2}]/\sqrt{2}$; $\phi_\mathrm{s}$
is the orbital part of the wave function describing a particle in
the source lead `s' and the upper indices 1 and 2 denote the
particle number. This local spin-singlet pair is scattered at the
splitter and the wave function $\Psi_\mathrm{in}^{\scriptscriptstyle
12}$ transforms to $\Psi_\mathrm{scat}^{\scriptscriptstyle 12} =
t_\mathrm{su}^2 \phi_\mathrm{u}^{\scriptscriptstyle 1}
\phi_\mathrm{u}^{\scriptscriptstyle 2}
\chi_\mathrm{sg}^{\scriptscriptstyle 12}+ t_\mathrm{sd}^2
\phi_\mathrm{d}^{\scriptscriptstyle 1}
\phi_\mathrm{d}^{\scriptscriptstyle 2}
\chi_\mathrm{sg}^{\scriptscriptstyle 12}+ t_\mathrm{su}
t_\mathrm{sd} [\phi_\mathrm{u}^{\scriptscriptstyle 1}
\phi_\mathrm{d}^{\scriptscriptstyle 2} +
\phi_\mathrm{d}^{\scriptscriptstyle 1}
\phi_\mathrm{u}^{\scriptscriptstyle
2}]\chi_\mathrm{sg}^{\scriptscriptstyle 12}$, where the last term
describes two particles in a singlet state shared between the upper
and lower leads of the fork. The Bell inequality test is only
sensitive to pairs of particles propagating in different arms,
implying a projection of the scattered wave function
$\Psi_\mathrm{scat}^{\scriptscriptstyle 12}$ onto the spin-entangled
component. Thus the origin of the entanglement is found in the
post-selection during the cross-correlation measurement effectuated
in the Bell inequality test~\cite{lebedev_04}. From an experimental
point of view it may be difficult to produce voltage pulses driving
exactly one (spinless) particle $n=1$. However, as follows from the
full expression Eq.\ (\ref{BI4}), for a sufficiently small deviation
$|\delta n| =|n-1| \ll 1$ the logarithmic terms are small in the
parameter $(\delta n)^2$ and thus can be neglected, provided the
measurement time $t_m$ satisfies the condition $(\delta n)^2\ln
(t_\mathrm{m}/\delta t)\ll 1$. The same argument applies to the case
of pumping with an alternating signal \cite{samuelsson_04b,
beenakker_05}. Even if the two-particle Bell inequality may be
violated when the average injected current vanishes still it does
not guarantee the creation of entangled electron pairs, see Ref.
\cite{lebedev} for details.


\section{Conclusion}\label{sec:conc}

We have studied two different setups where entanglement of an
electron pair is produced either by involving interaction between
electrons or by projective measurement or post-selection. In
contrast to the first method the entanglement due to post-selection
appears only for the indistinguishable particles. In the interacting
case a singlet electron pair is scattered on a single level quantum
dot with Coulomb interaction. The resulting two-particle scattering
state becomes orbitally entangled for $U_c\gg \Gamma$ ($U_c$ -
Coulomb repulsion energy, $\Gamma$ is the resonance width). In order
to quantify the produced entanglement we have estimated the von
Neumann entropy of the single-electron density matrix $\rho$.
Although the density matrix in principle can be measured directly we
have not described the realistic scheme for such measurement.
Calculating the purity $\Pi=\mbox{tr}\{\rho^2\}$ and
$\mbox{tr}\{\rho^3\}$ the degree of entanglement of the scattered
wave function component where both electrons are transmitted through
the dot can be estimated as $E> 1.3$ of the spin-singlet entangled
state.

In the setup involving a post-selection the following steps lead to
the appearance of the spin-entangled state \cite{lebedev_04}: i) the
Fermi statistics allows one to extract a electron spin-singlet state
with the same orbit out of the Fermi sea; ii) the beam splitter
directs the mixed product state into the two leads thus organizing
their spatial separation, iii) a coincidence measurement projects
the mixed product state onto its (spin-)entangled component
describing the electron pair split between the two leads. In this
situation we have suggested a realistic scheme for the Bell test
involving the measurement of particle number cross-correlators. It
is shown that the corresponding Bell inequality is maximally
violated indicating the maximally entangled spin state of the
electron pair.

Both these schemes rely on the existence of the electron source
which allows one to sent a finite number of electrons in a pure
quantum state, see Ref.~\cite{keeling} for theoretical proposal and
Ref.~\cite{glattli} for recent experimental advances in this
direction. Alternatively~\cite{pulses} one may apply to a electronic
reservoir a voltage pulse of the Lorenzian form with integer Faraday
flux $\Phi = -c \int dt V(t) = n \Phi_0$ extracting exactly $n$
electrons (per spin component) not entangled with the Fermi sea.





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\begin{theacknowledgments}
We acknowledge the financial support from the Swiss National
Foundation (through the program MaNEP and the CTS-ETHZ), the Russian
Foundation for Basic Research (08-02-00767a), and the program
'Quantum Macrophysics' of the RAS.
\end{theacknowledgments}

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%% if you don't like the warning.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\IfFileExists{\jobname.bbl}{}
% {\typeout{}
%  \typeout{******************************************}
%  \typeout{** Please run "bibtex \jobname" to optain}
%  \typeout{** the bibliography and then re-run LaTeX}
%  \typeout{** twice to fix the references!}
%  \typeout{******************************************}
%  \typeout{}
% }



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% The following lines show an example how to produce a bibliography
%% without the help of the BibTeX program. This could be used instead
%% of the above.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{thebibliography}{99}

\bibitem{born} M.\ Born, Z.\ Physik {\bf 37}, 863 (1926).

\bibitem{noise} G.B.\ Lesovik, JETP Lett. {\bf 49}, 592 (1989);
Th.\ Martin and R.\ Landauer, Phys.\ Rev.\ B {\bf 45}, 1742 (1992);
Ya.M.\ Blanter and M.\ B\"uttiker, Phys.\ Rep.\ {\bf 336}, 1 (2000).

\bibitem{noise_ex} M.\ Reznikov {\it et al.}, Phys.\ Rev.\ Lett.\
{\bf 75}, 3340 (1995); A.\ Kumar {\it et al.}, Phys.\ Rev.\ Lett.\
{\bf 76}, 2778 (1996).

\bibitem{charge} L.\ Saminadayar {\it et al.}, Phys.\ Rev.\ Lett.\
{\bf 79}, 2526 (1997); R. de-Picciotto {\it et al.}, Nature {\bf
389}, 162 (1997).

\bibitem{levitov93} L.S.\ Levitov and G.B.\ Lesovik, JETP Lett.\
{\bf 58}, 230 (1993); L.S.\ Levitov, H.\ Lee, and G.B.\ Lesovik, J.\
Math.\ Phys.\ {\bf 37}, 4845 (1996).

\bibitem{ensslin} S. Gustavsson {\it et al.}, Phys.\ Rev.\ Lett.\
{\bf 96}, 076605 (2006).

\bibitem{bell} J.S.\ Bell, Physics {\bf 1},195 (1965).

\bibitem{landau} L.\ D.\ Landau, Z.\ Physik {\bf 45}, 430 (1927).

\bibitem{neumann} J.\ v on Neumann, Gottinger Nachr., 245 (1927).

\bibitem{lanlif} L.\ D.\ Landau and E.\ M.\ Lifshitz, {\it Qunatum
Mechanics} (Pergamon Press, London, 1958), Vol. 3 Sec. 14.

\bibitem{EPR} A.\ Einstein, B.\ Podolsky, and N.\ Rosen, Phys.\ Rev.\
{\bf 47}, 777 (1935).

\bibitem{schroedinger} E.\ Schroedinger, Naturwiss. {\bf 23}, 807
(1935).

\bibitem{clauser} J.F.\ Clauser and M.A.\ Horne,
Phys.\ Rev.\ D {\bf 10}, 526 (1974).

\bibitem{entset} G.\ Lesovik, T.\ Martin, and G.\ Blatter,
Eur.\ Phys.\ J.\ B {\bf 24}, 287 (2001); P.\ Recher, E.V.\
Sukhorukov, and D.\ Loss, Phys.\ Rev.\ B {\bf 63}, 165314 (2001);
C.W.J.\ Beenakker, C.\ Emary, M.\  Kindermann, and J.L.\ van Velsen,
Phys.\ Rev.\ Lett.\ {\bf 91}, 147901 (2003).

\bibitem{lebedev}  A.V.\ Lebedev, G.B.\ Lesovik, and G.\ Blatter,
Phys.\ Rev.\ B {\bf 72}, 245314 (2005).

\bibitem{benrew} C.\ W.\ J.\ Beenakker, in {\it Quantum Computers,
Algorithms and Chaos}, Proceedings of the International School of
Physics "Enrico Fermi" Varenna, 2005 (IOS, Amsterdam, 2006), Vol.
162.

\bibitem{aspect} A. Aspect, Nature {\bf 398}, 189 (1999) and
reference therein.

\bibitem{fazio} L.\ Amico, R.\ Fazio, A.\ Osterloh, and V.\ Vedral,
Rev.\ Mod.\ Phys.\ {\bf 80}, 517 (2008).

\bibitem{lebedev_08} A.V.\ Lebedev, G.B.\ Lesovik, and G.\ Blatter,
Phys.\ Rev.\ Lett. {\bf 100}, 226805 (2008).

\bibitem{bose} S.\ Bose and D.\ Home, Phys.\ Rev.\ Lett.\ {\bf 88},
050401 (2002).

\bibitem{wheeler} J.A.\ Wheeler, Phys.\ Rev.\ {\bf 52}, 1107 (1937);
W.\ Heisenberg, Z.\ Physik {\bf 120}, 513 and 673 (1943).

\bibitem{landauer} R.\ Landauer, IBM J.\ Res.\ Dev.\ {\bf 1}, 223
(1957); R.\ Landauer, Philosoph.\ Mag.\ {\bf 21}, 863 (1970).


\bibitem{pulses}  D.A.\ Ivanov and L.S.\ Levitov, JETP Lett.\
{\bf 58}, 461 (1993); J.\ Keeling, I.\ Klich, and L.S.\ Levitov,
Phys.\ Rev.\ Lett.\ {\bf 97}, 116403 (2006); F.\ Hassler, G.B.\
Lesovik, and G.\ Blatter, Phys.\ Rev.\ Lett.\ {\bf 99}, 076804
(2007).

\bibitem{keeling}  J.\ Keeling, A.V.\ Shytov, L.S.\ Levitov,
arXiv:0804.4281.

\bibitem{glattli} G.\ Feve {\it et al.}, Science {\bf 316}, 1169
(2007); A.\ Mah\'e, {\it et al.}, arXiv:0809.2727.
%H.\ Pothier {\it et al.}, Euro.\ Phys.\ Lett.\
%{\bf 17}, 249 (1992); L.P.\ Kouwenhoven {\it et al.}, Phys.\ Rev.\
%Lett.\ {\bf 67}, 1626 (1991); V.I.\ Talyanskii {\it et al.}, Phys.\
%Rev.\ B {\bf 56}, 15180 (1997); G.\ Feve {\it et al.}, Science {\bf
%316}, 1169 (2007).

\bibitem{AIM} P.W.\ Anderson, Phys.\ Rev.\ {\bf 124}, 41 (1961).

\bibitem{marcus07} D.T.\ McClure {\it et al.},
%L.\ DiCarlo, Y.\ Zhang, H.-A.\ Engel, C.M.\ Marcus, M.P.\ Hanson, A.C.\ Gossard,
Phys.\ Rev.\ Lett.\ {\bf 98}, 056801 (2007).

\bibitem{goorden07} M.C.\ Goorden and M.\ B\"uttiker,
Phys.\ Rev.\ Lett.\ {\bf 99}, 146801 (2007).

\bibitem{GM} L.I.\ Glazman and M.E.\ Raikh,
JETP Lett.\ {\bf 47}, 452 (1988).

\bibitem{oliver} W.D.\ Oliver, F.\ Yamaguchi, and Y.\ Yamamoto,
Phys.\ Rev.\ Lett.\ {\bf 88}, 037901 (2002).

\bibitem{lebedev_04} A.V.\ Lebedev, G.\ Blatter, C.W.J.\ Beenakker,
   and G.B.\ Lesovik,
   Phys.\ Rev.\ B {\bf 69}, 235312 (2004).

\bibitem{samuelsson_04b}  P.\ Samuelsson and M.\ B\"uttiker,
   Phys.\ Rev.\ B {\bf 71}, 245317 (2005).

\bibitem{beenakker_05} C.W.J.\ Beenakker, M.\ Titov, and B.\ Trauzettel,
   New J.\ Phys.\ {\bf 7}, 186 (2005).

\bibitem{nike_02} N.M.\ Chtchelkatchev, G.\ Blatter,
   G.B.\ Lesovik, and  T.\ Martin, Phys.\ Rev.\ B {\bf 66},
   161320(R) (2002).

\bibitem{ll_94} L.S.\ Levitov and G.B.\ Lesovik,
   Phys.\ Rev.\ Lett.\ {\bf 72}, 538 (1994);

\bibitem{ksp_00} A.A.\ Kozhevnikov, R.J.\ Schoelkopf, and
   D.E.\ Prober, Phys.\ Rev.\ Lett.\ {\bf 84}, 3398 (2000);
   L.-H.\ Reydellet, P.\ Roche, D.C.\ Glattli, B.\ Etienne,
   and Y.\ Jin,
   Phys.\ Rev.\ Lett.\ {\bf 90}, 176803 (2003).

\bibitem{ll_93} H.\ Lee and L.S.\ Levitov, cond-mat/9312013.

\bibitem{anderson} P.W.\ Anderson,
   Phys.\ Rev.\ Lett.\ {\bf 18}, 1049 (1967).

\end{thebibliography}

\end{document}

\endinput
%%
%% End of file `template-8d.tex'.
