Entanglement witnesses based on second moments Author: Tatiana MIHAESCUSupervisors: Prof. Dr. Dagmar BRUß A thesis submitted in fulfilment of the… [622372]

UNIVERSITY OF DÜSSELDORF
MASTER THESIS
Entanglement witnesses based on second
moments
Author:
Tatiana MIHAESCUSupervisors:
Prof. Dr. Dagmar BRUß
A thesis submitted in fulfilment of the requirements
for the degree of master
in the
University of Düsseldorf
Institute for Theoretical Physics III
October 4, 2018

University of Düsseldorf
Abstract
Entanglement witnesses based on second moments
by Tatiana M IHAESCU
We present the separability problem for Gaussian states and introduce the concept of the entangle-
ment witness based on the covariance matrices. We also investigate the problem of entanglement
detection, given an unknown covariance matrix, by using random entanglement witnesses con-
structed with semidefinite programming technics. This idea is inspired from Ref. [ 21], where the
discrete quantum systems are analyzed.

Acknowledgements
ii

CONTENTS
Abstract i
Acknowledgements ii
Contents iii
1 Introduction 1
2 Continuous variable systems 2
2.1 Gaussian states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1.1 Symplectic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Euler decomposition of symplectic matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Separability of Gaussian states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3.1 PPT criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3.2 Uncertainty relations criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Duan criterion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Local uncertainty relations (LURs). . . . . . . . . . . . . . . . . . . . . . . . . 10
3 Entanglement witnesses based on second moments 11
3.1 Symplectic structure of the variance entanglement witnesses . . . . . . . . . . . . . . . . 11
3.2 Geometrical meaning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2.1 Relation to Logarithmic Negativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.3 The optimization problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.3.1 Two-mode EWs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.3.2 Three-mode EWs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4 Entanglement witnesses for unknown covariance matrices 20
4.1 Entanglement witnesses from variances measurements . . . . . . . . . . . . . . . . . . . 20
4.1.1 Two-mode CM reconstruction by a single homodyne detector. . . . . . . . . . . . 24
4.2 Detection scheme and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.3 Statistical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5 Conclusion 29
Bibliography 30
iv

CHAPTER 1
INTRODUCTION
Entanglement is the most valuable characteristic of quantum systems which allows for applications
in quantum information science unattainable in the framework of classical theory. A significant
problem in quantum information processing is to find out whether a quantum state is separable or
entangled. In general this is a hard task, but with some preliminary knowledge about the form of the
state it is possible to optimize the procedure of entanglement detection up to a few measurements.
In this work we analyze the possibility of applying the random entanglement witnesses
based on second moments, which could be useful in the case of completely unknown continuous
variable states. In particular, we look for the type of entanglement witness which could be efficiently
constructed using a semidefinite program.
For this purpose we first present the problem of separability for Gaussian states, which
have a very convenient theoretical description, through the covariance matrix of the system and its
symplectic eigenvalues. Next we introduce the entanglement witness based on covariance matrices,
and discuss the conditions for the efficient implementation of random entanglement witnesses by
means of semidefinite optimization procedure.
In the following M(N,R) denotes the set of N£Nreal matrices,
S(N,R)Æ{M2M(N,R) :MÆMT} (1.1)
is the set of N£Nsymmetric real matrices, and
P(N,R)Æ{M2S(N,R) :MÈ0} (1.2)
is the set of the N£Npositive-definite real symmetric matrices.
1

CHAPTER 2
CONTINUOUS VARIABLE SYSTEMS
A continuous variable (CV) system of Ncanonical bosonic modes, like the quantized electromagnetic
field with a Hamiltonian of a system of Nharmonic oscillators (modes), is defined in a Hilbert space
HÆ ­N
kÆ1Hk, each with an infinite-dimensional space HkÆL2(R) and two canonical observables
ˆxkand ˆpk, with the corresponding phase space variables xkand pk[1–3]. One can define a vector
of quadrature operators ˆRT´(ˆR1,…, ˆR2N)Æ(ˆx1,ˆp1,…, ˆxN,ˆpN) satisfying the bosonic commutation
relations [ ˆRi,ˆRj]Æi­i jˆI,i,jÆ1,…,2N, where ˆIis the identity matrix, ­i j´[­]i jare the elements
of the fundamental symplectic matrix
­ÆNM
kÆ10
@0 1
¡1 01
A, (2.1)
and we assume ß Æ 1. We denote by RTÆ(x1,p1,…,xN,pN) the vector of eigenvalues of the quadrature
operators ˆRspanning the phase space. This provides a convenient way to describe a CV system in the
phase space K(which is isomorphic to R2N) with the scalar product corresponding to the symplectic
matrix, u¢vÆuT­v, i.e.,K:Æ(R2N,­).
An alternative and complete description of any quantum state ½is provided by means
of a suitable continuous function in phase space called the characteristic function , defined as the
expectation value of the Weyl operator ˆW»Æei»T­ˆR:
½(»)ÆTr(ˆW»½), (2.2)
with»2R2N. In quantum optics, the Weyl operator coincides with the phase space displacement
operator (with a different convention for the arguments). The Wigner function W½is the inverse
2

Continuous variable systems 3
Fourier transform of the characteristic function, defined as
W½(»)Æ1
(2¼)NZ
d2N´ei»T­´Â½(´). (2.3)
It is a normalized, real-valued function in phase space and yet it can take negative values, but in the
case of Gaussian states it is always positive.
2.1 G AUSSIAN STATES
The Gaussian states play an important role in CV systems, being easy to handle mathematically and
experimentally available in many applications. By definition a Gaussian state is a state with Gaussian
characteristic function [1, 2]:
½(»)Æe¡1
4»T­°­T»¡i(­d)T», (2.4)
where dÆ hˆRi½ÆTr[ˆR½] is the displacement vector and°is the covariance matrix (matrix of second
moments of the state). Therefore, Gaussian states are fully defined by the first and second moments,
and a Gaussian state of Nmodes requires 2 N2ÅNreal parameters for its description.
The elements of a 2 N£2Ncovariance matrix (CM) °2S(2N,R) of the state are the vari-
ances of the quadrature operators defined as1
°i jÅi­i jÆ2h(ˆRi¡hˆRii)(ˆRj¡hˆRji)i½. (2.5)
The Robertson-Schrödinger uncertainty relation in terms of the CM reads:
°Åi­¸0, (2.6)
assuring that it is a true CM of a quantum state. We introduce the set of N-mode CMs defined as:
¡(2N) :Æ{°2S(2N,R) :°Åi­¸0}. (2.7)
Notice that the inequality (2.6) implies °È0 and det°¸1, while for a real symmetric matrix to
be a classical correlation matrix only the definite positivity is required. Therefore we have that:
¡(2N)½P(2N,R). Further we neglect the first moments, since entanglement is invariant under local
unitary displacements.
The vacuum state is the state with the variance of the position and momentum operators
equal to 1 and with zero thermal photon number ( ¯nÆ0), thus, its CM is just the identity matrix: °vƈI.
1Assume without loss of generality hˆRi Æ0. Consider the operator matrix ˆRˆRTwith elements decomposed as follows:
(ˆRˆRT)klƈRkˆRlÆ1
2{ˆRk,ˆRl}Å1
2[ˆRk,ˆRl]. The 2 N£2Nreal CM °for the state ½is defined by hˆRˆRTi ÆTr(½ˆRˆRT)Æ1
2°Å1
2i­,
with°klÆTr(½{ˆRk,ˆRl}).

Continuous variable systems 4
The thermal states are fundamental Gaussian states with block diagonal CMs °iÆ(2¯niÅ1)ˆI2£2, where
°thÆ ©N
iÆ1°iis the CM of the single mode labeled iwith thermal photon number ¯ni[3].
2.1.1 S YMPLECTIC ANALYSIS
The unitary operators acting on the quantum states in Hilbert space are equivalent to the symplectic
transformations which preserve the commutation relations of canonical coordinates: for any linear
unitary transformation ˆUthere exists a symplectic transformation Swith detSÆ1, so that ˆUˆRˆU†Æ
SˆR.
Definition 1.[22] The real symplectic group is defined by:
Sp(2N,R)Æ{S2M(2N,R) :S­STÆ­}, (2.8)
where S is a symplectic transformation on the phase space variables2
Symplectic transformations act by congruence on CMs: °0ÆST°S. Every matrix M2
P(2N,R) can be brought to a diagonal form through symplectic transformations according to
Williamson theorem [ 4], in the same way as symmetric matrices can be diagonalized with orthogonal
matrices.
Theorem 1. (Williamson Normal Form [4]):Let°be a 2N£2Npositive-definite real
symmetric matrix. Then there exists a symplectic transformation S 2Sp(2N,R)such that
°wÆS°STÆ ©N
kÆ10
@șk 0
0șk1
A, (2.9)
with symplectic eigenvalues șkÈ0for all k Æ1,… N .
The symplectic eigenvalues form the symplectic spectrum of°, and we define the symplectic
trace of °asstr°ÆPN
kÆ1șk, i.e., the sum of symplectic eigenvalues counted once. A Gaussian state
with CM °is completely characterized by its symplectic spectrum. The uncertainty relation (2.6)
implies the following restrictions on the symplectic eigenvalues:
șk¸1, for all kÆ1,…, N. (2.10)
The determinant of °is invariant under symplectic transformations: det°ÆQ
kș2
k, where we have
used the proprieties of the symplectic matrices: if S2Sp(2N,R), then detSÆ1 and ST2Sp(2N,R). In
order to find the symplectic spectrum of a CM °, we can proceed to finding the suitable symplectic
2The phase space variables of canonical operators ˆxand ˆprepresent ….? (some reference) : R!R0ÆSR,R2R2N.

Continuous variable systems 5
transformation which brings the CM to Williamson form, but a simpler way is to calculate the eigen-
values of ji­°jwhich coincide with symplectic eigenvalues of °3[1, 3].
One can observe that the procedure in Eq. (2.9) is the normal mode decomposition which results
from decoupling a coupled system of harmonic oscillators in thermal equilibrium [ 3]. Then the
transformed CM is the CM of Nuncoupled modes in thermal state, where șkÆ2¯nkÅ1, with ¯nkthe
thermal photon number of the mode k. Therefore, there arises an immediate conclusion that every
Gaussian state can be obtained by applying suitable symplectic operations on a thermal state.
The relevant part of physics for the study of quantum correlations in Gaussian states, in
special quantum entanglement, is contained in the study of the local symplectic spectrum of the
composite system’s CM. The two-mode Gaussian states are the simplest case to investigate, from
which the representation of the multimode states are derived.
Due to the fact that the tensor product in the Hilbert space is translated into direct sum in phase
space we can write the CM of the bipartite system as follows:
°Æ0
@A C
CTB1
A, (2.11)
where Aand Bare the CMs of the subsystems and Cis the matrix describing the correlations between
the modes. Note that if ½ABƽA­½Bis a product state, then its block CM reduces to diagonal form
°ÆA©B, as Ci jbecome zero 8i,j. Further we can apply a generic local transformation ( SA©SB)
with SA,SB2Sp(2,R) which locally acts on °:
A!SAAST
A,B!SBBST
B,C!SAC ST
B, (2.12)
bringing the submatrices to the so called standard form :
°Æ0
BBBBB@a 0c0
0a0d
c0b0
0d 0b1
CCCCCA, (2.13)
where aand bare known as symplectic main diagonal elements , orlocal symplectic eigenvalues , which
are in fact the symplectic eigenvalues of the 2 £2 main diagonal blocks. All locally equivalent states
have the same standard form of the CM. Now we can distinguish the invariants at the local symplectic
transformations [ 5,6]:aÆp
detA,bÆp
detB,cdÆdetC. In addition, the determinant and the sum
3HerejMjstands for the absolute value of the diagonalizable matrix Min the usual operatorial sense: if MÆO¡1DO,
where D is diagonal and O is a diagonalizing operator, then jMj ÆO¡1jDjO, where jDjis just the diagonal matrix with the
absolute values of the eigenvalues of Mas entries.

Continuous variable systems 6
of the invariants are also invariants:
det°Æa2b2Åc2d2¡ab(c2Åd2),
4(°)ÆdetAÅdetBÅ2det C.
One can express the symplectic eigenvalues through symplectic invariants as follows:
ș2
1,2Æ1
2³
4§q
42¡4det °´
. (2.14)
The uncertainty relation constraint on the CM in terms of these invariants reads as:
4 · det°Å1. (2.15)
2.2 E ULER DECOMPOSITION OF SYMPLECTIC MATRICES
Every symplectic transformation can be decomposed using Euler decomposition, which represents
the singular value decomposition for real symplectic matrices. Any real symplectic matrix Scan be
written as [1, 22]:
SÆKh
©N
iÆ1S(ri)i
L, (2.16)
where K,Lare symplectic and orthogonal, while S(ri) :Æ0
@e¡r0
0 er1
Aare one mode squeezing matrices
(symplectic and nonorthogonal).
The symplectic and orthogonal matrices form the maximal compact subgroup K(n) within
the noncompact Sp(2R) [22]. The group K(n) is isomorphic to the group of n£ncomplex unitary
matrices U(n) through the relation:
K(n)Æ{S(X,Y)jX¡i Y2U(n)}, (2.17)
where the corresponding symplectic matrices are of the following form:
S(X,Y)Æ0
@X Y
¡Y X1
A2Sp(2n,R). (2.18)
Such transformations describe multiport interferometers and are called passive canonical unitaries,
which preserve the photon number. The active canonical unitaries correspond to nonorthogonal
symplectic transformations, such as one mode squeezing.

Continuous variable systems 7
2.3 S EPARABILITY OF GAUSSIAN STATES
Historically the 1935 paper of Einstein, Podolsky and Rosen [ 28] argued the incompleteness of
quantum mechanics theory by showing the existence of peculiar quantum phenomena characterized
by strong quantum correlations among the involved systems, giving rise to nonlocal properties of the
state. For instance, there exist states of composite systems which cannot be separ ated or factorized
to pure local states of the subsystems, named entangled states [7].
Definition 2. (Separable state) [8]:A state ½of a bipartite quantum system on HÆ
HA­HBis separable if and only if it can be written as a convex combination of product states :
½ÆX
kpk½(A)
k­½(B)
k, (2.19)
where p kare the probabilities withP
kpkÆ1and½(A)
k,½(B)
kbelong to HAandHB, respectively.
According to this definition a separable state can be prepared by means of operations acting locally
on the subsystems, separated from each other. Yet, any correlations which can arise are attributed to
possible classical communication between subsystems, and hence are of classical origin.
In the case of CV states where the tensor product in Hilbert space of the subsystems A and
B,HÆHA­HBis translated into direct sum in phase space KÆKA©KB, an equivalent criterion
was defined for CMs [10], known as covariance matrix criterion (CMC) .
Theorem 3. (CMC) [10]:Let°be a CM of the state ½which is separable with respect to parties
A and B. Then there exist proper CMs °A,°Bcorresponding to two parties, such that
°¸°A©°B. (2.20)
Conversely, if this condition is satisfied, then the Gaussian state with CM °is separable.
This relation states that for a separable Gaussian state ½, there exists a product Gaussian
state ¾and a matrix °0Æ°¡°A©°B¸0, which is the covariance of a classical Gaussian probability
distribution P. Hence, any separable Gaussian state can be prepared from a product Gaussian state by
addition of noise, i.e., averaged phase space translations. The CMC in Eq. (2.20) is a special instance
of a solvable semidefinite program in the primal form [ 13,19]. However, this is an optimization
problem for a specific guess only of the form of CM.
We define the set of separable CMs with respect to the split AjBas:
¡AjB(2N)Æ{°2¡(2N) :°ÆPÅ°A©°B,P¸0},
for some °A2¡(2NA) and °B2¡(2NB) with NÆNAÅNB.
Lemma 1. Let°be a CM, element of ¡(2N) (¡AjB(2N)), P a positive matrix and ®¸1, then [14]

Continuous variable systems 8
1.°0Æ°ÅP and °"Æ®°are still elements of ¡(2N) (¡AjB(2N)), and
2. all principal submatrices of °containing N rmodes are elements of ¡(2Nr) (¡AjB(2Nr)).
The multimode quantum state is defined on the set of Northonormal modes which can
be decomposed into many partitions, which means that there are different types of entanglement
with respect to different partitions. The term of full inseparability was coined for the presence of
entanglement in systems with 1-mode per partition.
The set of fully separable CMs is closed and convex [13], with the boundary given by the product
states with CMs ©N
iÆ1°isatisfying the condition ©N
iÆ1°i¸i­. Such matrices define a closed convex
cone which is a subset of the space of matrices ©N
iÆ1°i. The convex combination of fully separable
CMs preserves the semi-definite constraint imposed by the uncertainty condition: if ©N
iÆ1°i¸i­and
©N
iÆ1´i¸i­, then
®©N
iÆ1°iÅ(1¡®)©N
iÆ1´i¸i­, (2.21)
where ®2[0,1].
The separability problem for Gaussian states has been solved in Ref. [ 11], by defining a
nonlinear map which preserves entanglement properties of the CM. By iteration, one obtains a CM
for which it is easy to check separability.
2.3.1 PPT CRITERION
A simple and widely used criterion to decide whether the states are separable or not is the positive
partial transpose criterion (PPT criterion) [ 9], which states that if the state of a composite system ½is
separable then its partial transpose is a non-negative density operator with trace one. Consequently,
we may claim that
if½TB6¸0, then ½is entangled.
The PPT criterion is necessary and sufficient for 2 ­2 and 2 ­3 systems of qubits.
The action of partial transposition in phase space corresponds to a mirror reflection performed on
only one of the two subsystems, since the complex conjugate has the physical meaning of the time
reversal in the Schrödinger equation, and corresponds to a change in sign of the momentum variables.
This results in the following transformation for the Wigner function [1, 2]:
PT:W½(R)! ˜W½(R) :ÆW½(¤R),
where
¤:Æ0
@1 0
0 11
A©0
@1 0
0¡11
A. (2.22)

Continuous variable systems 9
Therefore, when we replace ½in Eq. (7) by its partial transpose the order of factors of ˆRkˆRlin the
CM is reversed when both factors belong to subsystem B and the momenta of the subsystem B are
reversed. We find that the CM results in ˜°Æ¤°¤, and if the partial transpose of ½is positive then we
must have ˜°Åi­¸0, which is equivalent with °Åi˜­¸0, where ˜­Æ¤­¤ .
For two-mode Gaussian states, i.e., 1 £1 case, one can define the separability criterion
in terms of invariants of the partially transposed CM ˜°, since entanglement is invariant to local
symplectic transformations SÆS1©S2,S1,S22Sp(2,R) [3,6]. As a result, the PPT criterion is reduced
to a simpler condition in terms of symplectic eigenvalues: ˜ș1,2¸1, or in terms of local symplectic
invariants:
˜4 · det°Å1, (2.23)
where ˜4 Æ 4¡ 4cd.
2.3.2 U NCERTAINTY RELATIONS CRITERION
DUAN CRITERION . In Ref. [ 12] Duan proposed a sufficient inseparability criterion for bipartite
CV systems, based on the total variance of the Einstein-Podolsky-Rosen (EPR) type operators [ 12],
defined as follows:
ˆuÆ jajˆx1Å1
aˆx2,
ˆvÆ jajˆp1¡1
aˆp2, (2.24)
for which the uncertainty relation reads as: h(4ˆu)2ih(4ˆv)2i ¸1
4ja2¡1
a2j2, where a2R,a6Æ0. The
criterion states that for any separable quantum state ½the total variance of a pair of EPR-like operators
2.24, satisfies the inequality
h(4ˆu)2i½Åh(4ˆv)2i½¸a2Å1
a2. (2.25)
This criterion has been proven to be necessary and sufficient for Gaussian states with CMs in standard
form (2.13). The reason behind this choice of operators is that the maximally entangled state4is the
null eigenvector of x1Åx2and p1¡p2operators, i.e., the EPR operators represent the nulifiers of the
maximally entangled state.
Denoting by {Kk}the set of EPR-like operators and by {Mk}the set of canonical observables
of two bosonic modes, they are connected through a basis transformation KiÆPN
jÆ1Oi jMj, with
4Squeezed vacuum state (known also as EPR states) with infinite squeezing [1].

Continuous variable systems 10
some matrix O. Then the CM °({Kk}) for the set of EPR-like operators is given by:
°({Kk})ÆO°({Mk})OT, (2.26)
where °({Mk}) is the CM in terms of canonical operators. The relation (24) represents simply the
trace of the CM °({Kk}), and therefore, we can choose the basis in which we want to express our CM,
allowing to define simpler inseparability criteria.
LOCAL UNCERTAINTY RELATIONS (LUR S). The LURs can be interpreted as uncertainty relations for
separable states coming from uncertainty in the reduced states [ 20]: Let {ˆAk}and {ˆBk}be observables
in the two different systems Aand Brespectively, which have bounded variances on single systems:
X
k±2(ˆAk)¸UAandX
k±2(ˆBk)¸UB, (2.27)
where ±2(ˆM) :Æ h(M¡hMi)2i Æ h M2i¡h Mi2is the variance of an observable M. Then, for separable
states the following relation
X
k±2(ˆAk­ˆIňI­ˆBk)¸UAÅUB (2.28)
holds, and its violation implies entanglement.
Note the similarity between LURs criterion and Duan criterion in Eq. (24). In fact it has
been shown that a state violates LURs criterion if and only if it violates the CM criterion [20].

CHAPTER 3
ENTANGLEMENT WITNESSES BASED ON
SECOND MOMENTS
The set of CMs and the set of separable CMs form closed convex subsets of the space of real 2 N£2N
symmetric matrices, and due to the corollary of the Hahn-Banach theorem [ 15], we have for a CM
¹62¡AjB(2N) that there exists a symmetric matrix M2S(2N,R) and a number c2Rsuch that
Tr(M¹)Çc, while Tr(M°)Ècfor all °2¡AjB(2N). The operators Mare hyperplanes separating
entangled CMs from the set of separable CMs, hence they are entanglement witnesses (EWs) based on
variances of canonical variables [13, 14].
As we have seen, the information about entanglement of a CM is comprised in its symplectic
spectrum rather than in its eigenvalues, which impose certain conditions on the symplectic trace of a
potential EW.
3.1 S YMPLECTIC STRUCTURE OF THE VARIANCE ENTANGLEMENT
WITNESSES
In the following we will revisit the notion of EW and its particularity for the CV systems. Although the
results presented are for general multimode Gaussian states, some comments are made for the case
of two-mode CMs for simplicity.
Definition 3. A Hermitian operator ˆW is an EW if [16]
hˆWi ÆTr(½ˆW)¸0, for all ½separable,
hˆWi ÆTr(¾ˆW)Ç0, for at least one ¾. (3.1)
11

Entanglement witnesses based on second moments 12
Following Refs. [16], any witness operator can be written as:
ˆWƈL¡¸ˆI, (3.2)
where ˆLis a positive semi-definite operator, and ¸is the minimum of the test operator ˆLover all
separable quantum states:
¸:Æinf{hÂjˆLjÂi:Â2X}, (3.3)
whereXis the set of separable states.
Now let us consider a general class of two-mode positive semi-definite Hermitian operators
of the form
ˆLƈRTZˆRÅaTˆRÅc, (3.4)
where aTÆ(®1,¯1,®2,¯2)2R4is an arbitrary vector, and Zis a real Hermitian matrix:
ZÆ0
@Z1Zc
ZT
cZ21
A. (3.5)
For a state ½with CM °and displacement ~dthe expectation value of the operator ˆLis related to the
trace of matrix Zas follows:
hˆLi½ÆTr[ˆL½]Æ4X
k,lÆ1ZklTr[ˆRkˆRl½]
Æ1
24X
k,lÆ1Zkl(°klÅ2dkdl)Æ1
2Tr[Z°]Å~dTZ~d, (3.6)
where we have ignored the linear part and the constant in the operator ˆLwhich induce displacements
in phase space. Then the test matrix Zin Eq. (3.5) is referred to as the EW based on second moments
[13, 14].
In the following we try to evaluate the parameter ¸in Eq. 3.3 for two-mode separable
Gaussian states. For a separable state with CM °2¡1j2(2) the following inequality is fullfiled:
Tr[Z°]¸Tr[Z°1©°2]ÆTr[Z1°1]ÅTr[Z2°2], (3.7)
where we have used °Æ°1©°2ÅP, with P¸0, and Tr[Z P]¸0 [14]. In our case °1and°2are single-
mode CMs, but the same procedure can be applied for multimode and multipartite states. Now we
use the Williamson decomposition for a single mode entanglement witness ZiÆST
iZw
iSi,iÆ1,2,

Entanglement witnesses based on second moments 13
with Si2Sp(2,R) and Zw
iÆdiag( zi,zi). Then the above relation can be written in a simpler form:
2X
iÆ1Tr[Zi°i]Æ2X
iÆ1Tr[ST
iZw
iSi°i]Æ2X
iÆ1Tr[Zw
iSi°iST
i]
Æ22X
iÆ1ziTr[Si°iST
i]Æ22X
iÆ1ziTr[°0
i]
Æ22X
iÆ1zi(2h¯niiÅ1)¸22X
iÆ1zi (3.8)
where °0
iÆSi°iST
iis also a CM andP2
iÆ1str[Zi]ÆP2
iÆ1zi. We used the fact that the trace of a single-
mode CM equals the energy of a single harmonic oscillator Tr°iÆ hx2
iiÅh p2
ii Æ2h¯niiÅ1, where ¯niis
the thermal photon number of the mode i.
The test operator usually considered for the entanglement test of a CV system is the total
energy of this system ˆLƈH[14,16,17]. In this case the minimum of the Hamiltonian for separable
states, ¸Æ1, is reached when the system is in the ground state. In addition, note that in this case the
following inequality holds for the case of bipartite CM: str[Z1]Åstr[Z2]¸1
2[14]. The results have a
straightforward generalization for N-mode CM, as is stated in the following definition and theorem
presented in Ref. [13].
Definition 4. An EW based on second moments can be characterized by a real symmetric
matrix Z ¸0such that:
Tr[Z°s]¸1 for all (fully) separable CMs °s,
Tr[Z°]Ç1 for some entangled CM °. (3.9)
Theorem 4. The 2N£2N matrix Z 2S(2N,R)is an EW on CMs if and only if
Z¸0, (3.10)
NX
kÆ1str[Zk]¸1
2, (3.11)
str[Z]Ç1
2(3.12)
hold, where Z kis the block on the diagonal of Z acting on system labeled k.
In conclusion, the first two conditions in Theorem 4 assure that such an EW gives a value
larger than 1 for all separable CMs °2¡AjB(2N). The condition which characterizes a proper EW on
CMs is the inequality 3.12, since one can see that if str[Z]¸1
2, then Tr[Z°]¸18°2¡(2N), i.e., such
matrix does not detect entanglement [ 14].Therefore, one can always define separating hyperplanes
of the set of fully separable CMs for a general N-mode system with the following the theorem above.

Entanglement witnesses based on second moments 14
3.2 G EOMETRICAL MEANING
The geometrical meaning of EWs based on CMs is very much the same as for EWs based on expecta-
tion values [ 16]: it is a separating hyperplane for which all CMs that correspond to separable states
are on one side of the hyperplane with Tr[Z°]¸1, while on the other side are the second moments of
the entangled states giving Tr[Z°]Ç1. As a result, all entangled CMs can be detected by EWs. The
optimal EW Zoptdetects more entangled states that any other witness, and therefore it is closest to
the set of separable CM (see Fig.1), i.e., there exists a separable CM °with Tr[ Zopt°]Æ1.
The minimal EW for a CM °comes from the insight that for the set of separable CMs
there can be defined closed, convex sets ¡p
AjB(2N) parametrized by a parameter p2(0,1], and which
include each other: for example ¡1
AjB(2N)½¡0.5
AjB½¡0.2
AjB(2N)½¡(2N). The CMs °2¡p
AjBare called
p-separable [14].
Definition 4. For any °2¡(2N)CM of a bipartite system AB, and any p 2(0,1] we define
°is p – separable ()°
pis separable (3.13)
with respect to the split A jB.
This results from Lemma 1, where ®Æ1
p¸1. Note that by definition 1-separability is the
separability with the usual meaning. Therefore, when measuring an EW Zthe relation Tr[Z°]¸p
assures the p-separability of °, and orders the entangled CMs in sets of p-separable CMs. In addition,
the smaller is the number p, the more entangled the CM can be [ 14], and the best estimate of the
degree of entanglement of a CM °is given by the minimal EW, Tr[ Zmi n°]Æpmi n (see Fig.1).
Fig. 1. Schematic representation of the entanglement witness Zon CMs as a separating hyperplane from the convex set
¡AjB(2N) of CMs consistent with separable Gaussian states . The optimal entanglement witness Zopt detects the
entangled CMs from regions I,I Iand I I I, while the minimal entanglement witness Zmi n detects entanglement only for
CMs from region I. The minimal witness gives the minimal value pmi n for the CM °[14].
From an EW detecting entanglement of a CM there can be defined LUR observables detecting

Entanglement witnesses based on second moments 15
entanglement of the state as well. In order to show this one can use the spectral theorem ZÆP
l¸lal
ial
j
with¸¸0. Define ˆNlÆp
¸lP
ial
iˆMi, with ˆMi:ƈAi­ˆIňI­ˆBi. Then Tr[Z°]ÆP
l±2(ˆNl). This is very
useful since in practice it is difficult to find a LUR detecting entanglement of a given state. Therefore,
Duan criterion is an example of EW based on second moments.
The EW based on second moments are powerful criteria for inseparability even for bound
entangled states, while the PPT criterion and LURs are just special examples of EWs on CMs. But, for
two-mode and 1 £NGaussian states the PPT is a necessary and sufficient criterion.
3.2.1 R ELATION TO LOGARITHMIC NEGATIVITY
The negativity is an entanglement monotone which derives from the PPT criterion of separability. It
quantifies the amount of the partial transposition which fails to be positive and is defined as:
N(½)Æk½Tik1¡1
2, (3.14)
where kˆAk1ÆTrp
ˆA†ˆAis the trace norm of the operator ˆA.
In continuous variable systems the logarithmic negativity EN(½) is used very often, which
is also very convenient and easy to compute. It is defined as:
EN(½)Ælnk½Tik1Æln(1Å2N(½)), (3.15)
such that if all the eigenvalues pkof½Tiare positive, then the logarithmic negativity is zero since
Trq
½Ti½TiÆTr½TiÆ1. According to [ 31] the logarithmic negativity of a Gaussian state with CM °is
given by
EN(½)Æ ¡X
kln˜șk, for ˜șkÇ1, (3.16)
where {˜șk}are the symplectic eigenvalues of the partially transposed CM ˜°. The condition ˜șkÇ1
corresponds to entangled state, and otherwise the logarithmic negativity is zero. From the last relation
it is obvious that the smaller the symplectic eigenvalues (ranging between zero and one) are, the more
entangled this Gaussian state is.
As it is argued in ref [ 14], the entanglement witness Zon CMs does also provide a good
estimate for the degree of entanglement of an unknown CM. Furthermore, if the CM is not completely
unknown to the experimenter then this knowledge allows to determine the minimal entanlgment
witness for the CM, which provides the best estimate of the degree of entanglement. For instance, the
following relation for logarithmic negativity in the case that PPT-criterion is necessary and sufficient,
proved in ref. [14] holds
EN(°)¸ln1
p, (3.17)

Entanglement witnesses based on second moments 16
where p2(0,1) is the outcome of the measured EW Zon CM °:Tr[Z°]Æp. The equality holds in the
case of two-modes, where only one of the symplectic eigenvalues ˜șkcan be smaller than one, and
the measured Zmi n giving the smallest possible value pmi n. Note that the EW detect also the bound
entangled states, for which the logarithmic negativity is zero.
3.3 T HE OPTIMIZATION PROBLEM
The problem of finding a witness operator Z, given the repeated measurements Pi, is resumed to
finding the coefficients cisuch that:
Tr(Z°s)ÆX
iciTr(Pi°s)¸1, (3.18)
for any separable CM °s, and
Tr(Z°e)ÆX
iciTr(Pi°e)Ç1, (3.19)
for some entangled CM °e. From Theorem 4 this is equivalent to the following optimization problem:
minimize: c¢m
subject to: Z¸0
NX
kÆ1str[Zk]¸1
2
where Zkis the block on the diagonal of Zacting on system labeled k, and mÆTr(P°), with PÆ
Pn
iÆ1Pi. The second constraint is a nonlinear one since it implies the calculation of the symplectic
eigenvalues of Zkthat can be obtained from the eigenvalues of ji­Zkj, which coincide with symplectic
eigenvalues of Zk. In the following we propose an analog of linear inequality for the constraint in our
optimization problem for two-mode and multimode CMs.
3.3.1 T WO-MODE EW S
The entanglement witness based on variances of a bipartite Gaussian state state is a real, positive
semi-definite, symmetric 4 £4 matrix Z, with the following block form:
ZÆ0
@Z1Zc
ZT
cZ21
A, (3.20)

Entanglement witnesses based on second moments 17
where Z1and Z2are 2£2 matrices. The necessary condition on Zin order to be an entanglement
witness as stated in Theorem 4 is given by:
str[Z1]Åstr[Z2]¸1
2, (3.21)
where str[Z] is the symplectic trace of Z, which is the sum of symplectic eigenvalues of Zcounted
once. As Z1and Z2are 2£2 matrices, they have only one symplectic eigenvalue. We denote by z1and
z2the symplectic eigenvalue of Z1and Z2respectively. Then, the condition in Eq. (3.21) is:
z1Åz2¸1
2. (3.22)
Now, we analyze the following inequality:
Z1Åi x­¸0, where ­Æ0
@0 1
¡1 01
A,x2R. (3.23)
By symplectic transformations Sthe positive matrix above can be brought to Williamson normal
form as follows:
S(Z1Åi x­)STÆZw
1Åi x­Æ0
@z1 xi
¡xi z 11
A, (3.24)
where Zw
1Ædi ag (z1,z1). The positive eigenvalues ®of this matrix are determined from the equation:
(z1¡®)2¡x2Æ(z1¡®¡x)(z1¡®Åx)Æ0, (3.25)
and hence
z1§xÆ®¸0, (3.26)
i.e., the the polynomial above has two roots. However, the single-mode symplectic eigenvalue has
to be a positive value since they represent the variance in canonical operators of the single-mode
thermal state. Therefore, we will consider also the variable x¸0 to be positive, and z1¸x.
A similar inequality can be written for the block matrix Z2:
Z2Åi(1
2¡x)­¸0, (3.27)
and consequently, we obtain the following inequality for the symplectic eigenvalue z2and the
parameter x:
z2¸1
2¡x, for x·1
2. (3.28)
The above inequalities assures that the condition in Eq. (3.21) is always fulfilled for x2[0,1
2]. In
conclusion, we obtain that the optimization problem with nonlinear condition (3.21) is equivalent to

Entanglement witnesses based on second moments 18
the following optimization problem:
minimize: c¢m
subject to: ZÆX
iciPi
Z¸0
Z1Åi x­¸0
Z2Åi(1
2¡x)­¸0
0·x·1
2
where Z1and Z2are defined in Eq. (3.20).
3.3.2 T HREE -MODE EW S
For multimode CMs there are classes of separability arising from different possible splittings of the
modes. By example, for three-mode Gaussian states one can check the entanglement with respect to
bipartite splittings but also the genuine tripartite entanglement [ 24]. The EW then is a 6 £6 matrix
fulfilling different constraints corresponding to each type of separability.
The entanglement with respect to 1 £2 splitting is verified by the following EW :
ZÆ0
BBB@Z1 Z12 Z13
ZT
12Z2 Z23
ZT
13ZT
23Z31
CCCA, (3.29)
where all the elements are 2 £2 block matrices, and the block diagonal matrices under interest are Z1
and Z0Æ0
@Z2 Z23
ZT
23Z31
A. By following the same reasoning as for the bipartite case the constraints in the
optimization program in Section 3.3 in this case read as follows:
Z1Åi x­¸0
Z0Åi1
2(1
2¡x)­¸0
0·x·1
2.
In a similar way one can write the constraints for the EW detecting entanglement in other bipartite
splitting. For the detection of genuine tripartite entanglement the EW has to satisfy the following

Entanglement witnesses based on second moments 19
conditions, which involve each 2 £2 block diagonal matrices of Z:
Z1Åi x­¸0
Z2Åi1
2(1
2¡x)­¸0
Z3Åi1
2(1
2¡x)­¸0
0·x·1
2.

CHAPTER 4
ENTANGLEMENT WITNESSES FOR UNKNOWN
COVARIANCE MATRICES
In this section we discuss the possibility to asses entanglement of a general unknown CM by imple-
menting randomly generated EWs using semidefinite programming, as has been done in Ref. [ 21] for
discrete systems.
The EW based on second moments is constructed from homodyne measurements, which is
an usual procedure in assessing entanglement of a Gaussian state in experiment [ 2]. This gives the
cross-correlations of the moments of quadratures up to the second order.
The arbitrary CMs are generated by exploiting the thermal decomposition (see Sec 2.1.1):
the thermal CM is created, which is a diagonal matrix with certain constraints on symplectic eigenval-
ues, and random symplectic transformations are applied. Symplectic matrices are generated with the
algorithm proposed in ref. [23], based on Euler decomposition discussed in Sec. 2.2.
4.1 E NTANGLEMENT WITNESSES FROM VARIANCES MEASUREMENTS
The homodyne measurements are phase sensitive measurements which allows for the detection of
the relative phase between the modes. The former is the main characteristic of the interferometric
phenomena.
A simple scheme for the homodyne measurements represented in a schematic Figure 2 is
composed of an balanced beam splitter (BBS) superposing the signal mode with the local oscillator
field (LO), and two photon detectors for the two output photon currents.
20

Entanglement witnesses for unknown covariance matrices 21
Fig. 2. Homodyne detection scheme: BSis the 50 : 50 beam splitter mixing the signal mode ˆaand the local oscillator ®LO.
The beam splitter transformation describes the linear mixing of the two generic modes ˆa
and ˆb, corresponding to the evolution operator of the form:
U(»)Æe»ˆa†ˆb¡»¤ˆaˆb†,
where »ÆÁei'2C, which acts on the modes as follows:
U†(»)0
@ˆa
ˆb1
AU(»)Æ0
@cosÁ ei'sinÁ
¡e¡i'sinÁ cosÁ1
A0
@ˆa
ˆb1
A.
In balanced homodyne detection the signal mode ˆais combined on an 50 : 50 BS ( 'Æ0,
ÁƼ
4) with an intense local oscillator ˆ®LOassumed to be in a coherent state with large photon
number. In the limit of strong local oscillator j®LOj ! 1 we can describe the local oscillator mode
classically, by the complex amplitude ®LOrather than by the annihilation operator ˆ®LO[26]. The
output modes are:
ˆa1Æ1p
2(®LOňa), ˆa2Æ1p
2(®LO¡ˆa),
and the detectors register the photocurrents:
ˆi1Æqˆa†
1ˆa1Æq
2(®¤
LOňa†)(®LOňa), (4.1)
ˆi2Æqˆa†
2ˆa2Æq
2(®¤
LO¡ˆa†)(®LO¡ˆa) (4.2)
with qa constant. The difference in the two photocurrents represents the quantity to be measured:
±ˆi´ˆi1¡ˆi2Æq(®¤
LOˆaÅ®LOˆa†).

Entanglement witnesses for unknown covariance matrices 22
Fig. 3. Quadrature fluctuations from a homodyne detection of a vacuum (top) and a coherent state (bottom). The phase of
LO was gradually shifted during the observation time. Each point represents a single measurement result [29].
With the introduction of the phase of the local oscillator ®LOÆ j®LOjexp iµ, we observe that
the actual quantity to be measured is
±ˆiÆqj®LOjˆxµ, (4.3)
where ˆx(µ)is the generalised quadrature operator of mode ˆadefined as:
ˆxµÆexp(¡iµ)ˆaÅexp( iµ)ˆa†
p
2ƈxcosµÅˆpsinµ. (4.4)
Forµ2[0,2¼] the whole space of quadratures is spanned by ˆxµ. By varying the reference phase
µthe homodyne data is collected (see Fig. 3. [ 29]), for which a probability distribution of the
output photocurrent can be assigned. The measured distribution is the quadrature distribution of
an attenuated mode and approaches the ideal detection p(µ,x)Æ hxµj½jxµiin the limit of strong LO
j®LOj ! 1 [26]:
p(µ,x)Æ1p
2¼¾2expn
¡x2
2¾2o
, (4.5)
where xis the outcome of measurement on ˆxµ, and ¾2is the variance in the quadrature histogram at
fixed angle µ. The quadrature histogram corresponds to marginal distribution of the Wigner function

Entanglement witnesses for unknown covariance matrices 23
(see Fig.1. [ 29]) in phase space. As follows from the definition of Gaussian states in Eq. (2.4), Sec.2.1.
the marginals of Gaussian Wigner function are Gaussian functions.
Fig. 4. The projection of the Wigner function under different angles in phase space represent the quadrature distributions
measured using homodyne detection [29].
For the construction of an EW on CMs only the variance ¾2in the quadrature probability
density is required. If we compute the variance of ˆxµ, we see that the quadrature variances represent
the projection of a covariance matrix, as follows from the relation:
hˆx2
µi¡h ˆxµi2ÆTr[u°uT]ÆTr[P°]ƾ2, (4.6)
where °is the 1-mode CM, and uÆ(cos(µ)sin(µ))Tis an unit vector parameterized by the phase of
local oscillator, and PÆuTu:
PÆ0
@cos2µ1
2sin(2µ)
1
2sin(2µ) sin2µ1
A. (4.7)
For the specific values of the angles µthe variance of the generalized quadrature in Eq. (4.6) coincides
with elements in the CM °, which allows for the CM reconstruction by homodyne measurements on
4 different angles. The CM of the mode ˆais reconstructed from the measurements of quadratures
xaÆxµÆ0and paÆx¼
2, which give the diagonal elements, while the off-diagonal elements °12Æ
1
2h{ˆx,ˆp}Åi¡h ˆxihˆpiwith ˆxand ˆpthe canonical operators, are obtained from the measurement of
additional quadratures: za´(xaÅpa)/p
2Æx¼
4and ta´(xa¡pa)/p
2Æx¡¼
4as°12Æ°21Æ1
2(hz2
ai¡
ht2
ai)¡hxaihpai.
Therefore, a general operator matrix Zcan be written as a linear combination of matrices
Pidenoting the repeated homodyne detection on the CM °as in Eq. (4.6) for a set of different angles
{µi}:
ZÆnX
iÆ1ciPi, (4.8)

Entanglement witnesses for unknown covariance matrices 24
where ci2R.
4.1.1 T WO-MODE CM RECONSTRUCTION BY A SINGLE HOMODYNE DETECTOR .
The CM of the two-mode Gaussian state represented in block form as in Eq. (2.11) consists of 2 £2
block matrices Aand Bcorresponding to the modes ˆaand ˆbrespectively, and the correlation block
matrix Cdefined as:
CÆ0
@¢x1x2¢x1p2
¢p2x1¢p1p21
A, (4.9)
where ¢x2Æ hx2i¡h xi2denote the variance of the observable x,¢xpÆ1
2hxpÅpxi¡h xihpirepresent
the mutual correlations between the observables xand p, and the indices are refered to the two
modes ˆaand ˆbrespectively. For the reconstruction of the off-diagonal correlation terms in the CM
the additional modes have to be measured:
ˆcƈaňbp
2ˆdƈa¡ˆbp
2ˆeÆiˆaňbp
2ˆfÆiˆa¡ˆbp
2, (4.10)
as the elements of the block C are obtained from the quadratures:
°13Æ1
2(hx2
ci¡h x2
di)¡hxaihxbi, (4.11)
°14Æ1
2(hp2
ei¡h p2
fi)¡hxaihpbi, (4.12)
°23Æ1
2(hx2
fi¡h x2
ei)¡hpaihxbi, (4.13)
°24Æ1
2(hp2
ci¡h p2
di)¡hpaihpbi. (4.14)
One can notice that since hx2
fi Æ h x2
biÅh p2
ai¡h x2
eiandhp2
fi Æ h x2
aiÅh p2
bi¡h p2
ei(check this) the
measurement of f-quadrature is not necessary. Therefore we need 6 different measurement setting
to reconstruct the matrix C. These modes are obtained by implementing the quarter- and half-wave
plates, which induce the polarization rotation described mathematically the same as the beam splitter
acting on the polarization modes [ 25]. Thus, from the polarization point of view, ˆaand ˆbcorrespond
to vertical and horizontal polarizations, respectively, ˆcand ˆdare rotated polarization modes at §¼/4,
whereas ˆecorresponds to left-handed circular polarization [ 25]. The mode arriving at the detector
can be expressed in terms of the initial modes as follows:
ˆkÆexp( i')cosÁˆaÅsinÁˆb. (4.15)

Entanglement witnesses for unknown covariance matrices 25
The generalized quadrature operator corresponding to the mode kis:
x(k)
µÆexp(¡iµ)ˆkÅexp( iµ)ˆk†
p
2, (4.16)
and for the variance of this quadrature we get:
h(ˆx(k)
µ)2i¡h ˆx(k)
µi2ÆTr[P°], (4.17)
where °is the CM of the two-mode Gaussian state and
PÆuuT,uƳ
cosÁcos(µ¡') cos Ásin(µ¡') sin ÁcosµsinÁsinµ´
, (4.18)
is the matrix operator for the detection of the quadrature variance of the mode ˆk.
4.2 D ETECTION SCHEME AND RESULTS
The entanglement can be quantified by logarithmic negativity in case of two-mode Gaussian states,
or in general, allowing the full tomography of the CM (14 random measurements for two-modes
CMs), in which case the minimal EW can be constructed. The logarithmic negativity of any two-mode
CM with the block structure defined in Eq. (2.11) is calculated using the following formula [32]:
EÆmax{0, ¡1
2log2[4f]}, (4.19)
where
fÆ1
2(det AÅdetB)¡detC
¡µ·1
2(det AÅdetB)¡detC¸2
¡det¾¶1/2
. (4.20)
Edetermines the strength of entanglement for EÈ0, and if EÆ0, then the state is separable.
In Fig.5 is shown the detection efficiency of entanglement with the entanglement witness in the
random squeezed thermal states with the CM of the form:
°Æ0
BBBBB@a 0 c 0
0 a 0¡c
c 0 b 0
0¡c0 b1
CCCCCA,aÆ 2n1cosh2rÅ2n2sinh2rÅcosh2 r,
bÆ 2n1sinh2rÅ2n2cosh2rÅcosh2 r,
cÆ (n1Ån2Å1)sinh2 r,(4.21)

Entanglement witnesses for unknown covariance matrices 26
where n1,n2are the mean photon numbers of the two modes and ris the squeezing parameter.
These states are naturally accessible in many experimental situations…
The state is entangled if rÈrs, where
cosh2rsÆ(n1Å1)(n2Å1)
n1Ån2Å1.
For this particular type of states the amount of entanglement has a linear dependence on the squeez-
ing parameter EÆ…r. The latest experiments report the achievement in measuring 15 dB of squeezed
light [30], which corresponds to r¼1.73 according to the formula [31]:
1dBÆ10log10e2r.
0
14
13
12
11
1.2 10
9
No. of Measurements8
Log. Negativity2.7 7
6
5
4.2 4
3
2
5.6 1 Detection Fraction0.2
00.050.10.150.20.250.30.35
Fig. 5. 105runs of the algorithm on the two-mode squeezed vacuum states defined in Eq. 4.21 with n1Æn2Æ0 and
squeezing parameter r2[0,2].
For every value of the logarithmic negativity the entanglement detection fraction is calculated as the
number of detections normalized to the total number of repetitions….
Contrary to the expectations, in the continuous variable states the less entanglement states
are more easily detected than the high entanglement as it requires less measurements. These are
opposite results comparing to the discrete case [ 21]. However this result is linked to the behaviour
of the quadrature variance function of squeezing parameter. In Fig.7,8 is represented the result of

Entanglement witnesses for unknown covariance matrices 27
applying an EW constructed from a single measurement setting µÆ0, as a function of '2[¡¼,¼] and
Á2[¡¼,¼], for different values of the squeezing parameter rof the squeezed vacuum state with the
CM as in Eq.(4.19) with n1Æn2Æ0, which is entangled for rÈ0.
Fig. 7. The variance in the quadrature distribution on the squeezed vacuum CM ( n1Æn2Æ0), as a function of '2[¡¼,¼]
andÁ2[¡¼,¼], forµÆ0 and squeezing parameter rÆ0.2.
Fig. 8. The variance in the quadrature distribution on the squeezed vacuum CM n1Æn2Æ0, as a function of '2[¡¼,¼]
andÁ2[¡¼,¼], forµÆ0 and squeezing parameter rÆ1.
We observe that the maximum of variance in the quadrature is increasing with the squeezing param-
eter forming more narrow peaks. Therefore, the region of parameters 'andÁcorresponding to a

Entanglement witnesses for unknown covariance matrices 28
higher variance becomes smaller, which makes the measurement of higher variance less probable.
This explains why for higher squeezing more measurements are required for detection of entangle-
ment.
In Fig.6 is lustrated the entanglement detection efficiency with the entanglement witness of
the general random two-mode CM.
Fig. 6. 105runs of the algorithm on the general two-mode CMs.
4.3 S TATISTICAL ANALYSIS
In real experiments we do not have access to the ideal mean values of operators hPibut rather to the
data subjected to statistical fluctuations.
The homodyne data is governed by the Â2distribution as the quadrature xµis a random variable
having a normal distribution

CHAPTER 5
CONCLUSION
29

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