# Continuous variable entanglement distillation of Non-Gaussian Mixed States

Ruifang Dong1,2,∗, Mikael Lassen1,2, Joel Heersink1,3, Christoph Marquardt1,3,
Radim Filip4, Gerd Leuchs1,3 and Ulrik L. Andersen2 1Max Planck Institute for
the Science of Light, Günther-Scharowsky-Str. 1/Bau 24, 91058 Erlangen,
Germany
2Department of Physics, Technical University of Denmark, Building 309, 2800
Lyngby, Denmark
3Institute of Optics, Information and Photonics, Friedrich-Alexander-
University Erlangen-Nuremberg, Str. 7/B2, 91058 Erlangen, Germany
4Department of Optics, Palacký University, 17. listopadu 50, 77200 Olomouc,
Czech Republic
<EMAIL_ADDRESS>
###### Abstract

Many different quantum information communication protocols such as
teleportation, dense coding and entanglement based quantum key distribution
are based on the faithful transmission of entanglement between distant
location in an optical network. The distribution of entanglement in such a
network is however hampered by loss and noise that is inherent in all
practical quantum channels. Thus, to enable faithful transmission one must
resort to the protocol of entanglement distillation. In this paper we present
a detailed theoretical analysis and an experimental realization of continuous
variable entanglement distillation in a channel that is inflicted by different
kinds of non-Gaussian noise. The continuous variable entangled states are
generated by exploiting the third order non-linearity in optical fibers, and
the states are sent through a free-space laboratory channel in which the
losses are altered to simulate a free-space atmospheric channel with varying
losses. We use linear optical components, homodyne measurements and classical
communication to distill the entanglement, and we find that by using this
method the entanglement can be probabilistically increased for some specific
non-Gaussian noise channels.

###### pacs:

42.50.Lc,42.50.Dv,42.81.Dp,42.65.Dr

##  I Introduction

Quantum communication is a promising platform for sending secret messages with
absolute security and developing new low energy optical communication systems
gisin02.rmp . Such quantum communication protocols require the faithful
transmission of pure quantum states over very long distances. Heretofore,
significant experimental progress has been achieved in free space and fiber
based quantum cryptography where communication over more than 100 km have been
demonstrated dixon08.oe ; Manderbach07.prl ; ursin07.natphy . The
implementation of quantum communication systems over even larger distances -
as will be the case for transatlantic or deep space communication - can be
carried out by using quantum teleportation. However, it requires that the two
communicating parties share highly entangled states. One is therefore faced
with the technologically difficult problem of distributing highly entangled
states over long distances. The most serious problem in such a transmission is
the unavoidable coupling with the environment which leads to losses and
decoherence of the entangled states.

Losses and decoherence can be overcome by the use of entanglement
distillation, which is the protocol of extracting from an ensemble of less
entangled states a subset of states with a higher degree of entanglement
bennett96.prl . Distillation is therefore a purifying protocol that selects
highly entangled pure states from a mixture that is a result of noisy
transmission. This protocol has been experimentally demonstrated for qubit
systems exploiting a posteriori generated polarization entangled states
kwiat01.nat ; zhao01.pra ; yamamoto03.nat ; pan03.nat ; zhao03.prl . Common
for these implementations of entanglement distillation is their relative
experimental simplicity; only simple linear optical components such as beam
splitters and phase shifters are used to recover the entanglement. This
inherent simplicity of the distillation setups arises from the non-Gaussian
nature of the Wigner function of the entangled states. It has however been
proved that in case the Wigner functions of the entangled states are Gaussian,
entanglement distillation cannot be performed by linear optical components,
homodyne detection and classical communication eisert02.prl ; fiurasek02.prl ;
giedke02.pra . This is a very important result since it tells us that standard
continuous variable entanglement generated by e.g. a second-order or a small
third-order non-linearity cannot be distilled by simple means as these kinds
of entangled states are described by Gaussian Wigner functions.

Several avenues around the no-go distillation theorem have been proposed. The
first idea to increase the amount of CV entanglement was put forward by
Opatrný et al. opatrny00.pra . They suggested to subtract a single photon from
each of the modes of a two-mode squeezed state using weakly reflecting beam
splitters and single photon counters, and thereby conditionally prepare a non-
Gaussian state which eventually could increase the fidelity of CV quantum
teleportation. This protocol was further elaborated upon by Cochrane et al.
Cochrane2002.pra  and Olivares et al. Olivares2003.pra . Other approaches
relying on strong cross Kerr nonlinearities were suggested by Duan et al.
duan00b.prl ; duan2000.pra  and Fiurášek et al. fiurasek2003.pra . The usage
of such non-Gaussian operations results in non-Gaussian entangled states. To
get back to the Gaussian regime, it has been suggested to use a Gaussification
protocol based on simple linear optical components and vacuum projective
measurements (which can be implemented by either avalanche photodiodes or
homodyne detection) browne03.pra . Distillation including the Gaussification
protocol was first considered for pure states by Browne et al. browne03.pra
but later extended to the more relevant case of mixed states by Eisert et al.
eisert03.pra .

Due to the experimental complexity of the above mentioned proposals, the
experimental demonstration of Gaussian entanglement distillation has remained
a challenge. A first step towards the demonstration of Gaussian entanglement
distillation was presented in ref. ourjoumtsev07.prl  where a modified version
of the scheme by Opatrný et al. opatrny00.pra  was implemented: single photons
were subtracted from one of the two modes of a Gaussian entangled state using
a nonlocal joint measurement and as a result, an increase of entanglement was
observed. Recently, the full scheme of Opartny et al. was demonstrated by
Takahashi et al. takahashi09.quant . They observed a gain of entanglement by
means of conditional local subtraction of a single photon or two photons from
a two-mode Gaussian state. Furthermore they confirmed that two-photon
subtraction also improves Gaussian-like entanglement.

In the work mentioned above, only Gaussian noise has been considered as for
example associated with constant attenuation. Gaussian noise is however not
the only kind of noise occurring in information channels. E.g. if the
magnitude or phase of the transmission coefficient of a channel is
fluctuating, the resulting transmitted state is a non-Gaussian mixed state.
Because of the non-Gaussianity of the transmitted state, the aforementioned
no-go theorem does not apply and thus Gaussian transformations suffice to
distill the state. Actually, such a non-Gaussian mixture of Gaussian states
can be distilled and Gaussified using an approach fiurasek07.pra  related to
the one suggested in ref. browne03.pra . Alternatively, it is also possible to
distill and Gaussify non-Gaussian states using a simpler single-copy scheme
which is not relying on interference but is based on a weak measurement of the
corrupted states and heralding of the remaining state dong08.nat . Such an
approach has been also suggested for cat state purification suzuki06.pra ,
coherent state purification wittmann08.pra  and squeezed state distillation
heersink06.prl .

The distillation of Gaussian entanglement corrupted by non-Gaussian noise was
recently experimentally demonstrated in two different laboratories. More
specifically, it was demonstrated that by employing simple linear optical
components, homodyne detection and feedforward, it is possible to extract more
entanglement out of a less entangled state that has been affected by
attenuation noise dong08.nat  or phase noise hage08.nat .

In this paper we elaborate on the work in ref. dong08.nat , largely extending
the theoretical discussion on the characterization of non-Gaussian
entanglement and, on the experimental side, testing our distillation protocol
in new attenuation channels.

The paper is organized as follows: In Section II, the entanglement
distillation protocol utilized in our experiment is fully discussed. In
Section III the experimental setup for realization of the entanglement
distillation is described, and the experimental results are shown and
discussed in Section IV.

##  II The entanglement distillation protocol

The basic scheme of entanglement distillation is illustrated in Fig. 2. The
primary goal is to efficiently distribute bipartite entanglement between two
sites in a communication network to be used for e.g. teleportation or quantum
key distribution. Suppose the two-mode entangled state (also known as
Einstein-Podolsky-Rosen state (EPR)) is produced at site A. One of the modes
is kept at A’s site while the second mode is sent through a noisy quantum
channel. As a result of this noise, the entangled state will be corrupted and
the entanglement is degraded. The idea is then to recover the entanglement
using local operations at the two sites and classical communication between
the sites. To enable distillation, it is however required to generate and
subsequently distribute a large ensemble of highly entangled states. After
transmission, the ensemble transforms into a set of less entangled states from
which one can distill out a smaller set of higher entangled states.

A notable difference between our distillation approach and the schemes
proposed in Refs. browne03.pra ; fiurasek07.pra  is that our procedure relies
on single copies of distributed entangled states whereas the protocols in Ref.
browne03.pra ; fiurasek07.pra  are based on at least two copies. The multi
copy approach relies on very precise interference between the copies, thus
rendering this protocol rather difficult. One disadvantage of the single copy
approach is the fact that the entangled state is inevitably polluted with a
small amount of vacuum noise in the distillation machine. This pollution can,
however, be reduced if one is willing to trade it for a lower success rate.

Before describing the details of the experimental demonstration, we wish to
address the question on how to evaluate the protocol. The entanglement after
distillation must be appropriately evaluated and shown to be larger than the
entanglement before distillation to ensure a successful demonstration. One way
of verifying the success of distillation is to fully characterise the input
and output states using quantum tomography and then subsequently calculate an
entanglement monotone such as the logarithmic negativity. However, in the
experiment presented in this paper (as well as many other experiments on
continuous variable entanglement) we only measured the covariance matrix as
such measurements are easier to implement. The question that we would like to
address in the following is whether it is possible to verify the success of
distillation based on the covariance matrix of a non-Gaussian state.

###  II.1 Entanglement evaluation

In order to quantify the performance of the distillation protocol, the amount
of distillable entanglement before and after distillation ought to be
computed. It is however not known how to quantify the degree of distillable
entanglement of non-Gaussian mixed states bennett96.prl ; rains99.pra1 ;
rains99.pra2 . Therefore, as an alternative to the quantification of the
distillation protocol, one could try to estimate qualitatively whether
distillation has taken place by comparing computable bounds on distillable
entanglement before and after distillation. First we will have a closer look
at such bounds.

####  II.1.1 Upper and lower bounds on distillable entanglement

Although it is unknown how to find the amount of distillable entanglement of
non-Gaussian mixed states, we can easily find the upper and lower bounds by
computing the logarithmic negativity and the conditional entropy, respectively
vidal02.pra ; lower2 ; lower1 . These bounds can be found before and after
distillation, and the success of the distillation protocol can be
unambiguously proved by comparing these entanglement intervals: If the
entanglement interval is shifted towards higher entanglement and is not
overlapping with the interval before the distillation, the distillation has
proved successful. In other words, distillation has been performed if the
lower bound after the protocol is larger than the upper bound before. This is
illustrated in Fig. 1(a).

It has been proved that for any state, the log-negativity,

$LN(\rho)\equiv\log_{2}\left(2\mathcal{N}+1\right)=\log_{2}\left\|\rho^{T_{A}}\right\|_{1}.$
(1)

is an upper bound on the distillable entanglement; $E_{D}<LN(\rho)$
vidal02.pra . Here $\rho$ is the density matrix of the state,
$||\rho^{T_{A}}||$ is the trace norm of the partial transpose of the state
with respect to subsystem A, and the negativity is defined as

$\mathcal{N}(\rho)\equiv\frac{\left\|\rho^{T_{A}}\right\|_{1}-1}{2}.$ (2)

The negativity corresponds to the absolute value of the sum of the negative
eigenvalues of $\rho^{T_{A}}$ and it vanishes for non-entangled states.

In our experiment we were not able to measure the density matrix and thus
compute the exact value of the negativity. We therefore use another (more
strict) upper bound that is experimentally easier to estimate. As the
negativity is a convex function we have

$\mathcal{N}(\sum_{i}p_{i}\rho_{i})\leq\sum_{i}p_{i}\mathcal{N}(\rho_{i}).$
(3)

where $\rho_{i}$ denotes the $i$th hermitian component in the mixed state, and
$p_{i}$ is the weight for the $i$th component with $p_{i}\geq 0$ and
$\sum_{i}p_{i}=1$. Using this result we can find an upper bound on the log-
negativity for mixed states:

$LN(\sum_{i}p_{i}\rho_{i})\leq\log_{2}\left(1+2\sum_{i}p_{i}\mathcal{N}(\rho_{i})\right).$
(4)

This upper bound for the log-negativity will later be used to compute an upper
bound for the distillable entanglement.

Another entanglement monotone is the conditional entropy. In contrast to the
log-negativity, the conditional entropy yields a lower bound on the
distillable entanglement: $E_{D}>S(\tilde{\rho}_{A})-S(\tilde{\rho})$ lower1 ;
lower2 , where $\tilde{\rho}$ is the density matrix corresponding to Gaussian
approximation of the state and $\tilde{\rho}_{A}$ is the reduced density
matrix with respect to system A. The entropies of the states can be calculated
from the covariance matrix, $\mathbf{CM}$, using

$\displaystyle S(\tilde{\rho}_{A})$ $\displaystyle=$ $\displaystyle
f(\det\mathbf{A}),$ $\displaystyle S(\tilde{\rho})$ $\displaystyle=$
$\displaystyle\sum_{i}f(\mu_{i}),$ $\displaystyle f(x)$ $\displaystyle=$
$\displaystyle\frac{x+1}{2}\log_{2}(\frac{x+1}{2})-\frac{x-1}{2}\log_{2}(\frac{x-1}{2}),$
(5)

where

$\mu_{1,2}=\sqrt{\frac{\gamma\pm\sqrt{\gamma^{2}-4\det\mathbf{CM}}}{2}},$ (6)

are the symplectic eigenvalues of the covariance density matrix and
$\gamma=\det\mathbf{A}+\det\mathbf{B}+2\det\mathbf{C}$. Here $\mathbf{A}$,
$\mathbf{B}$, and $\mathbf{C}$ are submatrices of the covariance matrix:
$\mathbf{CM}=\\{\mathbf{A},\mathbf{C};\mathbf{C}^{T},\mathbf{B}\\}$.

It is important to note that this lower bound is very sensitive to excess
noise of the two-mode squeezed state. Even for a small amount of excess noise,
the lower bound approaches zero and thus is not very useful. This is
illustrated in Fig.1(b) which shows the distillable entanglement intervals
before and after distillation of a noisy entangled state. Although the
distillation protocol might remove the non-Gaussian noise of the state, the
Gaussian noise of the state persists, and thus the entropy (that is the lower
bound on distillable entanglement) will remain very low even after
distillation. This results in an overlap between the two entanglement
intervals and thus the comparison of computable entanglement bounds fails to
witness the action of distillation in terms of distillable entanglement.

Figure 1: Schematic demonstration of entanglement distillation of non-Gaussian
mixed states. In figure (a), the distillation with a pure state is illustrated
via the shift of the entanglement interval composed by the upper and lower
bounds on distillable entanglement before and after the distillation protocol.
Figure (b) shows the distillation with mixed states, the lower bound of which
does not manifest increase even for a small excess noise in the state.

####  II.1.2 Logarithmic negativity

In our experiment, the entangled states possess a large amount of Gaussian
excess noise and thus the prescribed method is insufficient to prove the act
of entanglement distillation using distillable entanglement as a measure.
However, in certain cases we can use the logarithmic negativity as a measure
to witness the act of entanglement distillation even though we only have
access to the covariance matrix as we will explain in the following.

First we note that in general, the Gaussian logarithmic negativity is an
insufficient measure of entanglement distillation of non-Gaussian states as
this measure only yields an upper bound, and with upper bounds of LN both
before and after distillation a conclusion cannot be drawn. However, if the
state after distillation is perfectly Gaussified its Gaussian LN becomes the
exact LN, and if this exact value of LN is larger than the upper bound of LN
before distillation (computed from (eqn. 4)), one may successfully prove the
action of entanglement distillation entirely from the covariances matrices.
This condition will be used for some of the experiments presented in this
paper. More specifically, we will use this approach for testing entanglement
distillation in a binary transmission channel. For other transmission channels
investigated in this paper, the state will not be perfectly Gaussified in the
distillation process and the approach cannot be applied. For such cases,
however, we will resort to evaluations of the Gaussian part of the state in
terms of Gaussian entanglement.

####  II.1.3 Gaussian entanglement

In addition to an increase in distillable entanglement and logarithmic
negativity, the protocol can also be evaluated in terms of its Gaussian
entanglement. Although the Gaussian entanglement is not accounting for the
entanglement of the entire state (but only considers the second moments), it
is quite useful as it directly yields the amount of entanglement useful for
Gaussian protocols, a prominent example being teleportation of Gaussian
states.

In a Gaussian approximation, the state can be described by the covariance
matrix $\mathbf{CM}$ walls94.book . The logarithmic negativity ($LN$) can then
be found as

$LN=-\log_{2}\nu_{min}.$ (7)

where $\nu_{min}$ is the smallest symplectic eigenvalue of the partial
transposed covariance matrix. The symplectic eigenvalues can be calculated
from the covariance matrix using

$\nu_{1,2}=\sqrt{\frac{\delta\pm\sqrt{\delta^{2}-4\det\mathbf{CM}}}{2}}$ (8)

where $\delta=\det\mathbf{A}+\det\mathbf{B}-2\det\mathbf{C}$, $\mathbf{A}$,
$\mathbf{B}$ and $\mathbf{C}$ represent the submatrices in the correlation
matrix vidal02.pra . Then by finding the smallest eigenvalue of the covariance
matrix and inserting it in eqn. (7), a measure of the Gaussian entanglement of
the state can be found.

###  II.2 Theory of our protocol

Figure 2: Schematics of the entanglement distillation protocol. A weak
measurement on beam B is diagnosing the state and subsequently used to herald
the highly entangled components of the state.

We now undertake our experimental setup a theoretical treatment in light of
the results of the previous section.

The schematic of our protocol is shown in Fig. 2. The two-mode squeezed or
entangled state is produced by mixing two squeezed Gaussian states at a beam
splitter. The squeezed states are assumed to be identical with variances
$V_{S}$ and $V_{A}$ along the squeezed and anti-squeezed quadratures,
respectively. The beam splitter has a transmittivity of $T_{S}$ and a
reflectivity of $R_{S}=1-T_{S}$. One mode (beam A) from the entangled pair is
given to Alice and the other part (beam B) is transmitted through a fading
channel. The loss in the fading channel is characterized by the transmission
factor $0\leq\eta(t)\leq 1$ which fluctuates randomly. The probability
distribution of the fluctuating attenuation can be divided into $N$ different
slots each associated with a sub-channel with a constant attenuation. The
transmission of sub-channel $i$ is $\eta_{i}$ and it occurs with the
probability $p_{i}$ so that $\sum^{N}_{i=1}p_{i}=1$. For a particular $i$th
sub-channel with transmission of $\eta_{i}$, the transmitted state is Gaussian
and can be fully characterised by the covariance matrix $\mathbf{CM}_{i}$:

$\displaystyle\mathbf{CM}_{i}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\mathbf{A}_{i}&\mathbf{C}_{i}\\\
\mathbf{C}^{T}_{i}&\mathbf{B}_{i}\end{array}\right),\quad\mathbf{A}_{i}=\left(\begin{array}[]{cc}V_{AX,i}&0\\\
0&V_{BX,i}\end{array}\right),$ (13) $\displaystyle\mathbf{B}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}V_{AP,i}&0\\\
0&V_{BP,i}\end{array}\right),\quad\mathbf{C}=\left(\begin{array}[]{cc}C_{X,i}&0\\\
0&C_{P,i}\end{array}\right).$ (18)

where the elements are given by:

$\displaystyle V_{AX,i}$ $\displaystyle=$ $\displaystyle
T_{S}V_{S}+R_{S}V_{A},$ $\displaystyle V_{BX,i}$ $\displaystyle=$
$\displaystyle\eta_{i}(T_{S}V_{A}+R_{S}V_{S})+(1-\eta_{i}),$ $\displaystyle
V_{AP,i}$ $\displaystyle=$ $\displaystyle T_{S}V_{A}+R_{S}V_{S},$
$\displaystyle V_{BP,i}$ $\displaystyle=$
$\displaystyle\eta_{i}(T_{S}V_{S}+R_{S}V_{A})+(1-\eta_{i}),$ $\displaystyle
C_{X,i}$ $\displaystyle=$ $\displaystyle-
C_{P,i}=\sqrt{\eta_{i}}\sqrt{R_{S}T_{S}}(V_{A}-V_{S}).$ (19)

Then according to eqn. (4) we can find an upper bound for the log-negativity
of the state after transmission in the fluctuating channel
(using$||\rho_{i}^{T}||=1/\nu_{min,i}$):

$\displaystyle
LN(\sum_{i}p_{i}\rho_{i})\leq\log_{2}\sum_{i}(p_{i}/\nu_{min,i}).$ (20)

where $\nu_{min,i}$ corresponds to the smallest symplectic eigenvalue of the
$i$th partial transposed covariance matrix. This means that the right hand
side of this expression is also an upper bound on the distillable entanglement
of the non-Gaussian noisy state. Therefore, to truly prove that the
entanglement has increased, this bound must in principle be surpassed.

We now consider the Gaussian entanglement of our states using the Wigner
function formalism. The Wigner function of the total state and the $i$th state
can be described as

$\displaystyle W(\mathbf{X},\mathbf{P})$ $\displaystyle=$
$\displaystyle\sum_{i}^{N}p_{i}W_{i}(\mathbf{X},\mathbf{P}),$ $\displaystyle
W_{i}(\mathbf{X},\mathbf{P})$ $\displaystyle=$
$\displaystyle\frac{\exp\left(-\mathbf{X}\mathbf{V}_{X,i}^{-1}\mathbf{X}^{T}-\mathbf{P}\mathbf{V}_{P,i}^{-1}\mathbf{P}^{T}\right)}{4\pi^{2}\sqrt{\mbox{det}\mathbf{V}_{X,i}\;\mbox{det}\mathbf{V}_{P,i}}},$
(21)

where $\mathbf{X}=(x_{A},x_{B})$ and $\mathbf{P}=(p_{A},p_{B})$.
$\mathbf{V}_{X,i}$ and $\mathbf{V}_{P,i}$ are given by

$\displaystyle\mathbf{V}_{X,i}=\left(\begin{array}[]{cc}V_{AX,i}&C_{X,i}\\\
C_{X,i}&V_{BX,i}\end{array}\right),\;\mathbf{V}_{P,i}=\left(\begin{array}[]{cc}V_{AP,i}&C_{P,i}\\\
C_{P,i}&V_{BP,i}\end{array}\right).$ (26)

From the Wigner function the second moments of the quadratures can be
calculated through integration:

$\displaystyle\left\langle\hat{Z}\hat{Y}\right\rangle=\int
dx_{A}dx_{B}dp_{A}dp_{B}zyW(x_{A},x_{B},p_{A},p_{B})$
$\displaystyle=\sum_{i}p_{i}\int
dx_{A}dx_{B}dp_{A}dp_{B}zyW_{i}(x_{A},x_{B},p_{A},p_{B}),$ (27)

where $\hat{Z},\hat{Y}=\hat{X}_{A},\hat{X}_{B},\hat{P}_{A},\hat{P}_{B}$. As
the first moments of the vacuum squeezed states in both the quadratures are
zero, the variances directly correspond to the second moments. Therefore, the
elements of the total covariance matrix are simply the convex sum of the
symmetrical moments (II.2):

$\langle\hat{Z}\hat{Y}\rangle=\sum_{i}p_{i}\langle\hat{Z}\hat{Y}\rangle_{i}.$
(28)

Since all the moments (II.2) are just linear combinations of the transmission
factors $\eta_{i}$ and $\sqrt{\eta_{i}}$, the covariance matrix of the mixed
state has the following elements:

$\displaystyle V_{AX}$ $\displaystyle=$ $\displaystyle T_{S}V_{S}+R_{S}V_{A},$
$\displaystyle V_{BX}$ $\displaystyle=$
$\displaystyle\left<\eta\right>(T_{S}V_{A}+R_{S}V_{S})+(1-\left<\eta\right>),$
$\displaystyle V_{AP}$ $\displaystyle=$ $\displaystyle T_{S}V_{A}+R_{S}V_{S},$
$\displaystyle V_{BP}$ $\displaystyle=$
$\displaystyle\left<\eta\right>(T_{S}V_{S}+R_{S}V_{A})+(1-\left<\eta\right>),$
$\displaystyle C_{X}$ $\displaystyle=$ $\displaystyle-
C_{P}=\left<\sqrt{\eta}\right>\sqrt{R_{S}T_{S}}(V_{A}-V_{S}).$ (29)

where the symbol $\left<\cdot\right>$ denotes averaging over the fluctuating
attenuations. Comparing this set of equations with the set in (II.2)
associated with the second moments for the single sub-channels, we see that
the attenuation coefficient $\eta$ is replaced by the averaged attenuation
$\left<\eta\right>$, and $\sqrt{\eta}$ is replaced by
$\left<\sqrt{\eta}\right>$. It is interesting to note that if the attenuation
factor is constant (which means that the transmitted state will remain
Gaussian) there will always be some, although small, amount of Gaussian
entanglement left in the state. On the other hand, if the attenuation factor
is statistically fluctuating as in our case, the Gaussian entanglement of the
non-Gaussian state will rapidly degrade and eventually completely disappear.

To implement entanglement distillation, a part of the beam B is extracted by a
tap beam splitter with transmittivity $T$. A single quadrature is measured
(for example the amplitude quadrature, $\hat{X}_{t}$) and based on the
measurement outcome the remaining state is probabilistically heralded; it is
either kept or discarded depending on whether the measurement outcome is above
or below the threshold value $x_{th}$. The conditioned Wigner function of the
output signal state after the distillation is

$\displaystyle W_{p}(x_{A},p_{A},x^{\prime}_{B},p^{\prime}_{B})=$
$\displaystyle\int_{x_{th}}^{\infty}dx_{t}\int_{-\infty}^{\infty}dp_{t}\sum_{i=1}^{N}p_{i}W_{i}(x_{A},p_{A},x_{B},p_{B})W_{0}(x_{v},p_{v}).$

where $x_{B}=\sqrt{T}x^{\prime}_{B}-\sqrt{1-T}x_{t}$,
$p_{B}=\sqrt{T}p^{\prime}_{B}+\sqrt{1-T}p_{t}$,
$x_{v}=\sqrt{T}x_{t}-\sqrt{1-T}x^{\prime}_{B}$ and
$p_{v}=\sqrt{1-T}p_{t}+\sqrt{T}p^{\prime}_{B}$, the Wigner function
$W_{0}(x_{v},p_{v})$ represents the vacuum mode entering the asymmetric tap
beam splitter. After integration, the Wigner function can be written as

$\displaystyle W_{p}(x_{A},p_{A},x^{\prime}_{B},p^{\prime}_{B})=$ (31)
$\displaystyle\frac{1}{P_{S}}\sum_{i}p_{i}W^{\prime}_{X,i}(x_{A},x^{\prime}_{B};x_{th})\times
W^{\prime}_{P,i}(p_{A},p^{\prime}_{B}).$

This is a product mixture of two non-Gaussian states which should be compared
to the state before distillation which was a mixture of Gaussian states.
$P_{S}$ is the total probability of success.

The $\hat{X}$ related elements of the covariance matrix can be calculated from
this Wigner function directly by computing the symmetrically ordered moments:

$\displaystyle\langle X_{A}\rangle^{P}$ $\displaystyle=$
$\displaystyle\frac{\sum_{i}p_{i}\langle
X^{\prime}_{A}\rangle_{i}^{P}}{P_{S}},$ $\displaystyle\langle
X^{\prime}_{B}\rangle^{P}$ $\displaystyle=$
$\displaystyle\frac{\sum_{i}p_{i}\langle
X^{\prime}_{B}\rangle_{i}^{P}}{P_{S}},$ $\displaystyle\langle
X^{2}_{A}\rangle^{P}$ $\displaystyle=$
$\displaystyle\frac{\sum_{i}p_{i}\langle
X_{A}^{{}^{\prime}2}\rangle_{i}^{P}}{P_{S}},$ $\displaystyle\langle
X^{{}^{\prime}2}_{B}\rangle^{P}$ $\displaystyle=$
$\displaystyle\frac{\sum_{i}p_{i}\langle
X_{B}^{{}^{\prime}2}\rangle_{i}^{P}}{P_{S}},$ $\displaystyle\langle
X_{A}X^{\prime}_{B}\rangle^{P}$ $\displaystyle=$
$\displaystyle\frac{\sum_{i}p_{i}\langle
X_{A}X^{\prime}_{B}\rangle_{i}^{P}}{P_{S}}.$ (32)

with

$\displaystyle\langle X_{A}\rangle_{i}^{P}$ $\displaystyle=$
$\displaystyle\frac{C_{X,i}\sqrt{R}}{\sqrt{2\pi
V^{\prime}_{DX,i}}}\exp\left(-\frac{x_{th}^{2}}{2V^{\prime}_{DX,i}}\right),$
$\displaystyle\langle X^{\prime}_{B}\rangle_{i}^{P}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{TR}(V_{BXi}-1)}{\sqrt{2\pi
V^{\prime}_{DX,i}}}\exp\left(-\frac{x_{th}^{2}}{2V^{\prime}_{DX,i}}\right),$
$\displaystyle\langle X^{2}_{A}\rangle_{i}^{P}$ $\displaystyle=$
$\displaystyle\frac{RC_{X,i}^{2}x_{th}}{\sqrt{2\pi
V^{{}^{\prime}3}_{DX,i}}}\exp\left(-\frac{x_{th}^{2}}{2V^{\prime}_{DX,i}}\right)+$
$\displaystyle\frac{V_{AX,i}}{2}\mbox{Erfc}\left[\frac{x_{th}}{\sqrt{2V^{\prime}_{DX,i}}}\right],$
$\displaystyle\langle X^{{}^{\prime}2}_{B}\rangle_{i}^{P}$ $\displaystyle=$
$\displaystyle\frac{RT(V^{\prime}_{DX,i}-1)^{2}x_{th}}{\sqrt{2\pi
V^{{}^{\prime}3}_{DX,i}}}\exp\left(-\frac{x_{th}^{2}}{2V^{\prime}_{DX,i}}\right)+$
$\displaystyle\frac{RT(V_{BX,i}-1)^{2}+V_{BX,i}}{2V^{\prime}_{DXi}}\mbox{Erfc}\left[\frac{x_{th}}{\sqrt{2V^{\prime}_{DX,i}}}\right],$
$\displaystyle\langle X_{A}X^{\prime}_{B}\rangle_{i}^{P}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{T}R(V^{\prime}_{DX,i}-1)C_{Xi}}{\sqrt{2\pi
V^{{}^{\prime}3}_{DXi}}}\exp\left(-\frac{x_{th}^{2}}{2V^{\prime}_{DX,i}}\right)+$
(33)
$\displaystyle\frac{\sqrt{T}C_{X,i}V^{\prime}_{DX,i}}{2}\mbox{Erfc}\left[\frac{x_{th}}{\sqrt{2V^{\prime}_{DX,i}}}\right].$

where $V^{\prime}_{DX,i}=RV_{BX,i}+T$ is the output variance of the detected
mode and

$P_{S,i}=\frac{1}{2}\mbox{Erfc}\left[\frac{x_{th}}{\sqrt{2V^{\prime}_{DX,i}}}\right]$
(34)

is the success probability of distilling the $\i$-th constituent of the mixed
state. The total probability of success $P_{S}$ is then given by
$P_{S}=\sum_{i}p_{i}P_{S,i}$.

Since the first moments of the $\hat{P}$ quadrature are vanishingly small, the
$\hat{P}$ related elements of the covariances matrix are directly given by

$\displaystyle\langle P^{2}_{A}\rangle^{P}$ $\displaystyle=$
$\displaystyle\frac{1}{P_{S}}\sum_{i}p_{i}P_{S,i}V_{AP,i},$
$\displaystyle\langle P^{{}^{\prime}2}_{B}\rangle^{P}$ $\displaystyle=$
$\displaystyle\frac{1}{P_{S}}\sum_{i}p_{i}P_{S,i}(TV_{BP,i}+R),$
$\displaystyle\langle P_{A}P^{\prime}_{B}\rangle^{P}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{T}}{P_{S}}\sum_{i}p_{i}P_{S,i}C_{P,i}.$

The covariance matrix $\mathbf{CM}^{P}$ can then be constructed from these
elements. This covariance matrix fully characterizes the Gaussian part of the
state and thus yields the Gaussian log-negativity by using eqn. (7).

As we discussed in Section II.1, to successfully demonstrate entanglement
distillation of non-Gaussian states, the upper bound on distillable
entanglement before distillation must be surpassed by the lower bound on
distillable entanglement after the distillation (see also Fig. 1). Due to the
fragility of the lower bound, this can be only achieved for almost pure states
as mentioned above. A theoretical demonstration is given in Fig. 3. Here we
consider the transmission of entanglement in a channel which is randomly
blocked: The entangled state is perfectly transmitted with the probability
$p_{1}=0.2$ and completely erased with the probability $p_{2}=0.8$. We assume
the two squeezed states which produce entanglement have variances along the
squeezed quadrature as $V_{S}=0.1$, the entangling beam splitter is symmetric
($T_{S}=50\%$) and the tap beam splitter has a transmission of $T=0.7$. The
distillation with a pure entangled state ($V_{A}=1/V_{S}$) as well as a mixed
state ($V_{A}=1/V_{S}+10$) are investigated as a function of the success
probability, and shown in Fig. 3 (a) and (b) respectively. Following the
theory of section II.1.1, we calculate the upper bound on distillable
entanglement of the non-Gaussian state before distillation as shown by the
bold straight lines in Fig. 3. The lower bounds on distillable entanglement
after distillation are computed and shown in Fig. 3 by the dashed lines. We
see that the proof of entanglement distillation of non-Gaussian states already
fails for a mixed state with a small amount of excess noise. The upper bounds
on Gaussian entanglement after distillation are also computed and shown in
Fig. 3 by solid lines. As the success probability reduces, the Gaussian
entanglement increases. Furthermore, when it surpasses the upper bound on
distillable entanglement before distillation (bold solid lines) at a certain
low success probability, the distilled state is Gaussified as well and thus we
can justify a Gaussian state in the entanglement measure.

Figure 3: (color online) Theoretical simulations of distillable entanglement
of non-Gaussian mixed states as a function of success probability. The two
plots are corresponding to two different purities of the entangled input
states. In figure (a), the distillation with initially pure state is plotted
($V_{A}=1/V_{S}$), while figure (b) shows the distillation with initially
mixed states ($V_{A}=1/V_{S}+10$). The other parameters are taken as:
$V_{S}=0.1$, $V_{N}=1$, $T_{S}=0.5$, $T=0.7$, $\eta_{1}=1$, $\eta_{2}=0$,
$p_{1}=0.2$, $p_{2}=1-p_{1}$. In both plots, the lower dashed line shows the
lower bound on distillable entanglement, and the upper solid line is the upper
bound on Gaussian entanglement. The bold straight line is the upper bound of
the non-Gaussian distillable entanglement before distillation.

##  III Experimental realization

Figure 4: Schematics of the experimental setup for the preparation,
distillation and verification of the distillation of entanglement from a non-
Gaussian mixture of polarization entangled states.

The experimental realization of the distillation of corrupted entangled states
consists of three parts as schematically illustrated in Fig. 4: the
preparation, distillation and verification. In the following we describe each
part.

###  III.1 Generation of polarisation squeezing and entanglement

The generation of polarization squeezed beams serves as the first step for the
demonstration of entanglement distillation. Here we exploit the Kerr
nonlinearity of silica fibers experienced by ultrashort laser pulses for the
generation of quadrature squeezed states. Fig. 5 depicts the setup for the
generation of a polarization squeezed beam. A pulsed (140 fs) Cr4+:YAG laser
at a wavelength of 1500 nm and a repetition rate of 163 MHz is used to pump a
polarization-maintaining fiber. Two linearly polarized light pulses with
identical intensities are traveling in single pass along the orthogonal
polarization axes ($x$ and $y$) of the fiber. Two quadrature squeezed states,
the squeezed quadrature of which are skewed by $\theta_{\mathrm{sq}}$ from the
amplitude direction, are thereby independently generated. After the fiber the
emerging pulses are overlapped with a $\pi/2$ relative phase difference. The
relative phase difference is achieved using a birefringence pre-compensation,
an unbalanced Michelson-like interferometer heersink03.pra ; heersink05.ol ;
dong08.ol ; fiorentino01.pra . This is controlled by a feedback locking loop
based on a $S_{2}$ measurement of a small portion ($\leq$0.1%) of the fiber
output. The measured error signal is fed back to the piezo-electric element of
the pre-compensation via a PI controller, so that the $S_{2}$ parameter of the
output mode vanishes. This results in a circularly polarized beam at the fiber
output ($\langle\hat{S}_{1}\rangle=\langle\hat{S}_{2}\rangle=0$,
$\langle\hat{S}_{3}\rangle=\langle\hat{S}_{0}\rangle=\alpha^{2}$). The
corresponding Stokes operator uncertainty relations are reduced to a single
nontrivial one in the so-called $\hat{S_{1}}-\hat{S_{2}}$ dark plane:
$\Delta^{2}\hat{S}_{\theta}\,\Delta^{2}\hat{S}_{\theta+\pi/2}\geq|\langle\hat{S}_{3}\rangle|^{2}$,
where $\hat{S}(\theta)=\cos(\theta)\hat{S}_{1}+\sin(\theta)\hat{S}_{2}$
denotes a general Stokes parameter rotated by $\theta$ in the dark
$\hat{S_{1}}-\hat{S_{2}}$ plane with $\langle\hat{S}_{\theta}\rangle=0$.
Therefore, polarization squeezing occurs if
$\Delta^{2}\hat{S}_{\theta}<|\langle\hat{S}_{3}\rangle|=\alpha^{2}$, in which
$\Delta^{2}\hat{S}_{\theta}$ can be directly measured in a Stokes measurement
heersink05.ol . As the noise of Stokes parameters $\hat{S}_{\theta}$ is linked
to the quadrature noise of the Kerr squeezed modes in the same angle
($\Delta^{2}\hat{S}_{\theta}\approx\alpha(\delta\hat{X}_{x,\theta}-\delta\hat{X}_{y,\theta})/\sqrt{2}\approx\alpha^{2}\Delta^{2}\hat{X}_{\theta}$
heersink05.ol ), the squeezed Stokes operator is
$\hat{S}(\theta_{\mathrm{sq}})$ and the orthogonal, anti-squeezed Stokes
operator is $\hat{S}(\theta_{\mathrm{sq}}+\pi/2)$. Due to the equivalence
between the polarization squeezing and vacuum squeezing josse03.prl , we
utilize the conjugate quadratures $\hat{X}$ and $\hat{P}$ to denote the
polarization squeezed and anti-squeezed Stokes operators.

Figure 5: Setup for the generation of polarisation squeezing. The fiber is a
13.2 meters long polarization-maintaining 3M FS-PM-7811 fiber with a mode
field diameter of 5.7 $\mu$m and a beat length of 1.67 mm. The interferometer
in front of the fiber introduces a phase shift $\delta\phi$ between the two
orthogonally polarised pulses to pre-compensate for the birefringence.
$\lambda/4$, $\lambda/2$: quarter–, half–wave plates, PBS: polarising beam
splitter. PZT: Piezo-electric element.

To generate polarization entanglement two identical polarization-maintaining
fibers are used. Two polarization squeezed beams, labeled A and B, are then
generated. By balancing the transmitted optical power of the two fibers, the
two resultant polarization squeezed beams have identical squeezing angles,
squeezing and anti-squeezing properties. The two polarization squeezed beams
are then interfered on a 50/50 beam splitter (Fig. 4) with the interference
visibility aligned to be $>98\%$. The relative phase between the two input
beams is locked to $\pi/2$ so that the two output beams after the beam
splitter have equal intensity and are maximally entangled. The two entangled
outputs remain circularly polarised, thus the quantum correlations between
them are lying in the dark $\hat{S}_{1}-\hat{S}_{2}$ plane with the signatures
$\hat{S}_{\mathrm{A}}(\theta_{sq})+\hat{S}_{\mathrm{B}}(\theta_{sq})\rightarrow
0$ and
$\hat{S}_{\mathrm{A}}(\theta_{sq}+\pi/2)-\hat{S}_{\mathrm{B}}(\theta_{sq}+\pi/2)\rightarrow
0$ (or $\hat{X}_{\mathrm{A}}+\hat{X}_{\mathrm{B}}\rightarrow 0$ and
$\hat{P}_{\mathrm{A}}-\hat{P}_{\mathrm{B}}\rightarrow 0$).

###  III.2 Preparation of a non-Gaussian mixed state

The preparation of a non-Gaussian mixture of polarization entangled states is
implemented by transmitting one of the entangled beams, e.g. beam B, through a
controllable neutral density filter (ND). The filter is implemented to produce
a lossy channel with $N=45$ different transmittance levels, ranging from 0.1
to 1 in steps of 0.9/44. The entangled beam is then transmitted through the
lossy channel with 45 realizations. Combining all these realizations a non-
Gaussian mixed state, such as the one described by eqn. (II.2), is achieved,
with the probabilities $p_{i}$ all being identical. However, after the
measurement we can select a certain probability envelope function to give the
different channels pre-specified probability weights. With this technique we
can easily implement different transmission scenarios (see e.g. Fig. 9-1, Fig.
10-1, and Fig. 11-1). As a result of the lossy transmission, the Gaussian
entanglement between the two beams A and B are degraded or completely lost.

###  III.3 Entanglement distillation

The distillation operation consists of a measurement of $\hat{X}$ on a small
portion of the mixed entangled beam. This is implemented by tapping 7% of beam
B after the ND filter using a beam splitter. The measurement is followed by a
probabilistic heralding process where the remaining state is kept or
discarded, conditioned on the measurement outcomes: e.g. if the outcome of the
weak measurement is larger than the threshold value, $X_{th}$, then the state
is kept. Note that the signal heralding process could in principle be
implemented electro-optically to generate a freely propagating distilled
signal state. However, to avoid such complications, our conditioning is
instead based on digital data post-selection using a verification measurement
on the conjugate quadratures $\hat{X}$ and $\hat{P}$ of the beams A and B.

###  III.4 The tap and verification measurement

The tap and verification measurement are accomplished simultaneously by three
independent Stokes measurement apparatuses. Each measurement apparatus
consists of a half-wave plate and a polarizing beam splitter (PBS). Since the
light beam is circularly ($S_{3}$) polarized, a rotation of the half-wave
plate enables the measurement of different Stokes parameters lying in the
’dark’ polarization plane. For the tap measurement the half-wave plate is
always set at the angle corresponding to $\hat{X}$ in the ’dark’ polarization
plane. Via the verification measurement setup, the Gaussian properties of the
entangled states are characterized by measuring the entries of the covariance
matrix. By generating near symmetric states and choosing a proper reference
frame, we assume that the intra-correlations (such as
$\langle\hat{X}_{A}\hat{P}_{A}\rangle$) are zero. The measurements of these
entries are accomplished by applying polarization measurements of beam A and B
with both the half-wave plates set to the angle corresponding to either
$\hat{X}$ or $\hat{P}$ in the ’dark’ polarization plane. The outputs of the
PBS are detected by identical pairs of balanced photo-detectors based on 98%
quantum efficiency InGaAs PIN-photodiodes and with an incorporated low-pass
filter in order to avoid ac saturation due to the laser repetition
oscillation. The detected AC photocurrents are passively pairwise subtracted
and subsequently down-mixed at 17 MHz, low-pass filtered (1.9 MHz), and
amplified (FEMTO DHPVA-100) before being oversampled by a 16-bit A/D card
(Gage CompuScope 1610) at the rate of $10^{7}$ samples per second. The time
series data are then low-passed with a digital top-hat filter with a bandwidth
of 1 MHz. After these data processing steps, the noise statistics of the
Stokes parameters are characterized at 17 MHz relative to the optical field
carrier frequency ($\approx$200 THz) with a bandwidth of 1 MHz. The signal is
sampled around this sideband to avoid classical noise present in the frequency
band around the carrier schiller96.prl . For each polarization measurement,
the detected photocurrent noise of beam A and B and the tap beam were
simultaneously sampled for $2.4\times 10^{8}$ times, thus the self and cross
correlations between the data set of A and B could thereby easily be
characterized. The covariance matrix was subsequently determined and the log-
negativity was calculated according to eqn. (7).

##  IV Experimental results

For perfect transmission (corresponding to no loss in beam B), the marginal
distributions of the entangled beams, A and B, along the quadrature $X$ and
$P$ are plotted in Fig. 6. In Fig. 6(a) the procedure of realizing the
different noisy channels is shown. The sampled data of different attenuation
channels is concatenated according to the different weights of the
transmission probabilities. These samples then provide the measurement data
for the distillation procedure. From Fig. 6(a) and Fig. 6 (b) we can see that,
each individual mode exhibits a large excess noise (measured fluctuation $>17$
dB). However the joint measurements on the entangled beams A and B exhibit
less noise fluctuation than the shot noise reference, as shown in Fig. 6(c).
The observed two-mode squeezing between beam A and B is $-2.6\pm 0.3$ dB and
$-2.4\pm 0.3$ dB for $\hat{X}$ and $\hat{P}$, respectively. From the
determined covariance matrix we compute the log-negativity to be $0.76\pm
0.08$.

Figure 6: (color online). Experimentally measured marginal distributions
associated with the (a) $X$ and $P$ of beam A, (b) $X$ and $P$ of beam B and
(c) the joint measurements $X_{A}+X_{B}$ and $P_{A}-P_{B}$. The black and red
curves are the distributions for shot noise and the quadrature on measurement,
respectively.

To experimentally demonstrate the distillation of entanglement out of non-
Gaussian noise, three different lossy channels are considered: the discrete
erasure channel, where the transmission randomly alternates between two
different levels, and two semi-continuous channels, where the transmittance
alternates between 45 different levels with specified probability amplitudes.
The probability distributions of the transmittance for the discrete channel
and the continuous channels are shown in Fig. 9-1, Fig. 10-1, and Fig. 11-1,
respectively.

###  IV.1 The discrete lossy channel

The discrete erasure channel alternates between full (100%) transmission and
25% transmission at a probability of $0.5$. Each realisation is concatenated
to each other with identical weights. The concatenation procedure yields the
same statistical values as true randomly varying data. After transmission the
resulting state is a mixture of a highly entangled state and a weakly
entangled state. In the inset of Fig. 7, we show marginal distributions
illustrating the single beam statistics of the individual components of the
mixture. The statistics of beam B is seen to be contaminated with the
attenuated entangled state thus producing non-Gaussian statistics. For this
state we measure the correlations in $\hat{X}$ and $\hat{P}$ to be above the
shot noise level by $5.5\pm 0.3$ dB and $5.6\pm 0.3$ dB, respectively, and the
Gaussian LN to be $-1.63\pm 0.02$. The Gaussian entanglement is completely
lost as a result of the introduction of such time-dependent loss. This is in
stark contrast to the scenario where only stationary loss (corresponding to
Gaussian loss) is inflicting the entangled states. In that case, a certain
degree of Gaussian entanglement will always survive, although it will be small
for high loss levels.

Figure 7: (color online). Experimentally measured marginal distributions
illustrating the effect of distillation. (a) Example of concatenated sampled
data and the resulting marginal distribution for the amplitude quadrature in
the tap measurement. The vertical line indicates the threshold value chosen
for this realization. (b) Marginal distributions associated with the
measurements of $X$ and $P$ of beam B (two left figures) and (c) the joint
measurements $X_{A}+X_{B}$ and $P_{A}-P_{B}$ (two right figures). The black,
blue and red curves are the distributions for shot noise, the mixed state
before distillation and after distillation, respectively. Inset: phase-space
representation of the non-Gaussian mixed state and the post-selection
procedure used in the measurements. The black vertical line indicates the
threshold value.

The state is then fed into the distiller and we perform homodyne measurements
on beam A, beam B and the tap beam simultaneously. By measuring $\hat{X}$ in
the tap we construct the distribution shown by the red curve in the left hand
side of Fig. 7(a). The data trace of the mixed tap signal is plotted
accordingly on the right hand side. The measurements of $\hat{X}$ and
$\hat{P}$ of the signal entangled states were recorded as well. For
simplicity, we only show the distribution for the beam B (in Fig.
7(b-1),(b-2)) and the joint distribution of beam A and B (in Fig.
7(c-1),(c-2)). The blue (dashed) and red curves denote the distributions
before and after the post-selection process, respectively. From the blue
curves shown in Fig. 7, we can see that the entanglement between A and B is
lost due to the non-Gaussian noise. Performing postselection on this data by
conditioning it on the tap measurement outcome (denoted by $X_{th}=9.0$), we
observe a recovery of the entanglement. That is, the correlated distribution
of the signal turns out to be narrower than that of the shot noise (as shown
by red curves in Fig. 7(c).

Using the data shown in Fig. 7, a tomographic reconstruction of the covariance
matrices of the distilled entangled state was carried out. From these data we
determined the most significant eight of the ten independent parameters of the
covariance matrix, namely the variances of four quadratures
$\hat{X}_{A}$,$\hat{X}_{B}$, $\hat{P}_{A}$, $\hat{P}_{B}$ and covariances
between all pairs of quadratures of the entangled beams A and B. As mentioned
before, the intra-correlations were ignored. The resulting covariance matrices
are plotted in Fig. 8 for ten different postseletion threshold values from
$X_{th}=0.0$ to $X_{th}=9.0$ with a step of 1.0. With increasing postselection
threshold the distillation becomes stronger, as shown by the reduction of the
quadrature variances of $\hat{X}_{A}$, $\hat{X}_{B}$ and the increase of the
quadrature variances of $\hat{P}_{A}$, $\hat{P}_{B}$. Moreover, the reduction
(or increase) of the covariances $C(\hat{X}_{A},\hat{X}_{B})$
($C(\hat{P}_{A},\hat{P}_{B})$) was shown slightly slower. Consequently, the
entanglement of the two modes A and B was enhanced by the distillation.

Figure 8: (color online). Reconstructed covariance matrices of distilled
entangled states. The brown segmented plane shows the region for the
individual elements in the covariance matrix. The sub-bars represent the
results of our distillation protocol for 10 different threshold values
postseletion threshold values from $X_{th}=0.0$ to $X_{th}=9.0$ with a step of
1.0.

Furthermore, the distilled entanglement, or log-negativity, was investigated
as a function of the success probability, as shown in Fig. 9 by black open
circles. The error bars of the distilled log-negativity depend on two
contributions: First the measurement error, which is mainly associated with
the finite resolution of the A/D converter and noise of the electronic
amplifiers. This is considered by estimating the experimental error for all
the elements of the covariance matrix as ’0.03’. The measurement error for the
LN can be simulated by a Monte-Carlo model. Second, the statistical error is
due to the finite measurement time and the postselection process. It is
considered by adding a scaled term $\sqrt{2/(N-1)}$, where $N$ denotes the
number of postselected data frieden83.book . The probability distributions of
the two superimposed states in the mixture after distillation are shown for
different postselection thresholds, corresponding to
$X_{th}=0.0,2.0,4.0,6.0,9.0$, labeled by 1-5 in order. The plots explicitly
show the effect of the distillation protocol, when the postselection threshold
increases, the Gaussian LN increases, ultimately approaching the LN of the
input entanglement without losses. The probability distribution tends to a
single valued distribution, therefore the mixture of the two Gaussian
entangled states reduces to a single highly entangled Gaussian state, thus
demonstrating the act of Gaussification. However, the amount of distilled
data, or success probability, decreases, causing an increase in the
statistical error on the distillable entanglement. Based on the experimental
parameters, a theoretical simulation is plotted by the red curve and shows a
very good agreement with the experimental results.

Figure 9: (color online). Experimental and theoretical results outlining the
distillation of an entangled state from a discrete lossy channel. The
experimental results are marked by circles and the theoretical prediction is
plotted by the red solid line. The bound for Gaussian entanglement is given by
the blue line, and the upper bound for total entanglement before distillation
is given by the black dashed line. Both bounds are surpassed by the
experimental data. The weight of the two constituents in the mixed state after
distillation for various threshold values is also experimentally investigated
and shown in the plots labeled by 1-5. The error bars of the log-negativity
represent the standard deviations.

To further investigate whether the total entanglement is increased after
distillation, we compute the upper bound for the LN before distillation and
verify that this bound can be surpassed by the Gaussian LN after distillation.
The upper bound of LN without the Gaussian approximation is computable from
the LN of each Gaussian state in the mixture vidal02.pra , and we find
$LN_{upper}=0.49$, which is shown in Fig. 9 by the dashed black line. We see
that for a success probability around $10^{-4}$ the Gaussian LN crosses the
upper bound for entanglement. Since the state at this point is perfectly
Gaussified we may conclude that the total entanglement of the state has indeed
increased as a result of the distillation. Fig. 9-5 gives another explicit
explanation by showing that the probability contribution from the 75%
attenuated data reaches 0 when the post selection threshold is set to
$X_{th}=9.0$, which corresponds to the distilled entanglement of
$LN^{P}_{S}=0.67\pm 0.08$ with a success probability $P_{S}=1.69\times
10^{-5}$. On the other hand, from Fig. 9-3 and Fig. 9-4, we see that even a
small contribution from the 75% attenuated data will reduce the useful
entanglement for Gaussian operations.

###  IV.2 The continuous lossy channel

We now generalize the lossy channels to have a continuously transmittance
distribution. The channel transmittance distribution is simulated by taking 45
different transmission levels as opposed to the two levels in the previous
section. In Ref. dong08.nat  we reported a channel whose transmittance is
given by an exponentially decaying function with a long tail of low
transmittances, which simulates a short-term free-space optical communications
channel where atmospheric turbulence causes scattering and beam pointing noise
book . We showed that the entanglement available for Gaussian operations can
be successfully distilled from $-0.11\pm 0.05$ to $0.39\pm 0.07$ with a
success probability of $1.66\times 10^{-5}$. However, in practical scenarios
for a transmission channel, the highest transmittance level may not have the
biggest weight in the probability distribution and therefore the distributed
peak may be displaced from the 100% transmittance level. Further, there might
be more than one peak in the probability distribution diagram. For instance,
due to some strong beam pointing noise another distributed peak will appear in
the area of low transmittance levels. In the following we will test the
performance of the distillation protocol for two different transmittance
distributions. First, when the mixed state has a peak of the transmittance
distribution which is displaced from 1 to 0.8 (Fig. 10-1). Second, when we
incorporate a second peak which is located around the transmittance level of
0.3 (Fig. 11-1).

Figure 10: (color online). Experimental and theoretical results outlining the
distillation of an entangled state from a simulated continuous lossy channel
in which the peak of the transmittance distribution is displaced from 1 to
0.8. The experimental results are marked by circles and the theoretical
prediction by the red solid curve. The evolution of the weights of the various
constituents in the mixed state as the threshold value is changed is shown in
the figures labeled 1-5. The error bars of the log-negativity represent the
standard deviations.

As shown in Fig. 10, after propagation through the one-peak displaced channel
the Gaussian LN of the mixed state is found to be $-0.50\pm 0.04$, which is
below the bound for available entanglement(shown by the solid blue line) and
substantially lower than the original value of $0.76\pm 0.08$. The state is
subsequently distilled and the change in the Gaussian LN as the threshold
value increases (and the success probability decreases) was investigated both
experimentally (black open circles) and theoretically (red curve). The
evolution of the mixture is directly visualized in the series of probability
distributions in Fig. 10-1 to 10-5 corresponding to the postselection
thresholds $X_{th}$=0.0, 3.0, 5.0, 7.0, 9.0 respectively. We see that the
distribution weights of the low transmittance levels is gradually reduced,
while the weights of the high transmittance levels is increased as the post-
selection process becomes more and more restrictive by increasing the
threshold value. E.g. for $X_{th}=9$ the probabilities associated with
transmission levels lower than 0.7 are decreased from 20% before distillation
to 1.4% and the probability for transmission levels higher than 0.7
transmission are increased to 98.6% as opposed to 80% before distillation. It
is thus clear from these figures that the highly entangled states in the
mixture have larger weight after distillation, and the corresponding Gaussian
LN after distillation rises to $0.19\pm 0.06$ with the success probability of
$5.16\times 10^{-5}$.

Figure 11: (color online). Experimental and theoretical results outlining the
distillation of an entangled state from a simulated continuous lossy channel
in which the transmittance levels are distributed as such that there are two
peaks at both high transmittance levels (0.8) and low transmittance levels
(0.3). The experimental results are marked by circles and the theoretical
prediction by the red solid curve. The evolution of the weights of the various
constituents in the mixed state as the threshold value is changed is shown in
the figures labeled 1-5. The error bars of the log-negativity represent
standard deviations.

We now turn to investigate the distillation after propagation through the two-
peak displaced channel as shown in Fig. 11-1. Before distillation the Gaussian
LN of the mixed state is found to be $-1.13\pm 0.02$. Likewise, the relation
between the distilled Gaussian LN and the success probability was investigated
both experimentally and theoretically. The results are shown in Fig. 11 by
black open circles and the red curve, respectively. Through the probability
distribution plots in Fig. 11-1 to 11-5, the evolution of the mixture
corresponding to different choices of postselection thresholds ($X_{th}$=0.0,
3.0, 5.0, 7.0, 9.0 respectively) was illustrated with the same trend that we
see on the distillation after the one-peak displaced channel. For $X_{th}=9$
the probabilities associated with transmission levels lower than 0.7 are
decreased from 48% before distillation to 1.6% and the probability for
transmission levels higher than 0.7 transmission are increased to 98.4% as
opposed to 52% before distillation, and the corresponding Gaussian LN after
distillation reaches $0.19\pm 0.06$ with the success probability of
$3.39\times 10^{-5}$.

After having shown the successful entanglement distillation on different
distributions of non-Gaussian noise, we should note that the successful
entanglement distillation depends on the transmittance distribution of the
lossy channel. For some distributions, the success probability for distilling
available entanglement for Gaussian operations will be extremely small or not
be possible. For example, after a channel with the transmittance uniformally
distributed, the Gaussian log-negativity $LN_{S}=-1.26\pm 0.02$ before
distillation will only be increased to $-0.76\pm 0.03$ with a success
probability of $1.32\times 10^{-5}$. In general more uniform transmittance
distributions turned out to be more difficult for the distillation procedure.
Distributions with high probabilities for high transmission levels and
pronounced tails and peaks at low transmission levels (as would be expected in
atmospheric channels) are more suited.

##  V Summary

In summary, we have proposed a simple method of distilling entanglement from
single copies of quantum states that have undergone attenuation in a lossy
channel with varying transmission. Simply by implementing a weak measurement
based on a beam splitter and a homodyne detector, it is possible to distill a
set of highly entangled states from a larger set of unentangled states if the
mixed state is non-Gaussian. The protocol was successfully demonstrated for a
discrete erasure channel where the transmittance alternates between 2 levels
and two semi-continuous transmission channels where the transmission levels
span 45 levels with specified distributions, respectively. We show that the
degree of Gaussian entanglement (which is relevant for Gaussian information
processing) was substantially increased by the action of distillation.
Moreover, we proved experimentally that the total entanglement was indeed
increased for the discrete channel. We found that the successful entanglement
distillation depends on the transmittance distribution of the lossy channel.
The demonstration of a distillation protocol for non-Gaussian noise provides a
crucial step towards the construction of a quantum repeater for transmitting
continuous variables quantum states over long distances in channels inflicted
by non-Gaussian noise.

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### Acknowledgments

This work was supported by the EU project COMPAS (no.212008), the Deutsche
Forschungsgesellschaft and the Danish Agency for Science Technology and
Innovation (no. 274-07-0509). ML acknowledges support from the Alexander von
Humboldt Foundation and RF acknowledges MSM 6198959213, LC 06007 of Czech
Ministry of Education and 202/07/J040 of GACR.