Correlation matrices of two-mode bosonic systems
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Pirandola, Stefano , Alessio Serafini, and Seth Lloyd.
“Correlation matrices of two-mode bosonic systems.” Physical
Review A 79.5 (2009): 052327. (C) 2010 The American Physical
Society.
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http://dx.doi.org/10.1103/PhysRevA.79.052327
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PHYSICAL REVIEW A 79, 052327 共2009兲
Correlation matrices of two-mode bosonic systems
1
Stefano Pirandola,1 Alessio Serafini,2 and Seth Lloyd1,3
Research Laboratory of Electronics, MIT, Cambridge, Massachusetts 02139, USA
Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom
3
Department of Mechanical Engineering, MIT, Cambridge, Massachusetts 02139, USA
共Received 17 February 2009; published 20 May 2009兲
2
We present a detailed analysis of all the algebraic conditions an arbitrary 4 ⫻ 4 symmetric matrix must
satisfy in order to represent the correlation matrix of a two-mode bosonic system. Then, we completely clarify
when this arbitrary matrix can represent the correlation matrix of a separable or entangled Gaussian state. In
this analysis, we introduce alternative sets of conditions, which are expressed in terms of local symplectic
invariants.
DOI: 10.1103/PhysRevA.79.052327
PACS number共s兲: 03.67.⫺a, 03.65.Ud, 42.50.⫺p, 02.10.Ud
I. INTRODUCTION
Statistical moments of second order represent a key element of the quantum mechanical paradigm. Besides providing the “language” in which the uncertainty principles are
expressed, they serve as indicators in a number of applications of the theory, with both applied and fundamental interest. In particular, second moments are central to the description of bosonic fields in second quantization 共as is the case,
for instance, in quantum optics, where second order coherence is characterized in terms of second moments兲 and of
nonrelativistic particles in first quantization. Such systems
do, in fact, share the same formal description 关1兴, which
hence extends its domain to a variety of fields ranging from
atomic physics to quantum optics, from superconductors’
physics to nanomechanical systems.
During the last decade, the rise of quantum information
science has renewed the focus on these areas because of their
potential for coherent quantum manipulations, and has concomitantly brought new problems and questions to the attention of theorists, which resulted in the birth of the field of
“continuous variable” 共CV兲 quantum information 关2兴 共see
Refs. 关3–5兴 for some literature on CV quantum computation,
CV quantum teleportation, and CV quantum key distribution,
respectively兲. A systematic analysis of the properties of continuous variable quantum states inferred from the structure of
their second moments has been thereby carried out, which
lead to a well-established, extensive theoretical picture 共see,
for instance 关6–9兴兲. Such an analysis proved to be most relevant also in view of the experimental prominence of the
class of Gaussian states 关10,11兴, which are completely determined by their first and second moments. In first, seminal
endeavors 关6,7兴, the qualitative characterization of the quantum correlations 共“entanglement”兲 of Gaussian states of two
degrees of freedom has been successfully achieved. Now,
while very well established and relatively simple, this result
is still often expressed in a nonrigourous or incomplete manner, which is prone to confound the unacquainted reader.
Because it constitutes one of the basic building blocks on
which the theoretical characterization of Gaussian states has
been constructed, it seems to us extremely important for it to
be re-derived and re-expressed in a rigourous manner and
full detail: this is one of the motifs and central aims of the
present paper.
1050-2947/2009/79共5兲/052327共10兲
In general, the main question we will address and rigourously answer is the following: 共i兲 What are the algebraic
conditions that a 4 ⫻ 4 real symmetric matrix V must satisfy
in order to represent the correlation matrix of a two-mode
bosonic system 关12兴? Then, we shall move on to answer a
closely connected question: 共ii兲 What are the algebraic conditions to be satisfied by V in order to represent the correlation matrix of a separable 共or entangled兲 Gaussian state of
two bosonic modes?
Both these questions are thoroughly answered providing
complete sets of conditions, which are expressed in terms of
global or local symplectic invariants. In particular, the set of
local conditions, i.e., given in terms of local invariants, is
completely new in literature. Then, by specifying some of
these algebraic results in the case of positive-definite matrices, we can make a direct comparison with the previous
work of Ref. 关6兴 and provide a rigorous and correct interpretation of its seminal results.
The paper is organized as follows. In Sec. II we review
basic notions about bosonic states, correlation matrices, and
symplectic transformations. In Sec. III we present the mathematical tools to be used in the derivations of Secs. IV and V.
In Sec. IV we provide two sets of algebraic conditions for the
physical genuinity of the correlation matrix of two bosonic
modes. These conditions are expressed in terms of global or
local symplectic invariants. In Sec. V we provide similar
conditions for separability. Next, in Sec. VI, we specify some
of our results for making a direct comparison with the previous achievements of Ref. 关6兴. Finally, Sec. VII is for conclusions. Notice that in the paper we will denote by M共n , R兲
the set of n ⫻ n real matrices. Then, we use the compact
notation
S共n,R兲 = 兵M 苸 M共n,R兲:M = MT其,
共1兲
for the set of the n ⫻ n symmetric real matrices, and
P共n,R兲 = 兵M 苸 S共n,R兲:M ⬎ 0其,
共2兲
for the set of the n ⫻ n positive-definite real matrices. We
will also consider the set 共group兲 of proper rotations
052327-1
SO共n兲 = 兵M 苸 M共n,R兲:MTM = I, det M = 1其.
共3兲
©2009 The American Physical Society
PHYSICAL REVIEW A 79, 052327 共2009兲
PIRANDOLA, SERAFINI, AND LLOYD
II. BOSONIC SYSTEMS AND SYMPLECTIC
TRANSFORMATIONS
one can always construct a set of complex vectors z = u0
+ iau1, for a 苸 R, such that
A. Correlation matrix of a bosonic system
0 ⱕ 共zⴱ兲T共V + i⍀兲z = 2auT1 ⍀u0 + a2uT1 Vu1 .
Let us consider a bosonic system composed by n modes,
labeled by an index k. Such a system can be described by an
n
Hk and a vector
infinite-dimensional Hilbert space H = 丢 k=1
T
of quadrature operators x̂ ª 共q̂1 , p̂1 , ¯ , q̂n , p̂n兲. In particular,
these operators satisfy the commutation relations
关x̂l,x̂m兴 = 2i⍀lm ,
共4兲
where l , m = 1 , ¯ , 2n and ⍀lm are the entries of the simplectic form
n
⍀=
丣 ␻,
k=1
␻ª
冉 冊
0
1
−1 0
.
共5兲
Let us denote by D共H兲 the space of density operators acting
on H. It is known that an arbitrary density operator ␳
苸 D共H兲 has an equivalent representation in a real symplectic
space K = K共R2n , ⍀兲 called the phase space. This is a real
vector space which is spanned by the singular eigenvalues
xT = 共q1 , p1 , ¯ , qn , pn兲 of x̂T 共representing the “continuous
variables” of the system兲 and associated to a symplectic
product u · v = uT⍀v. In this space, ␳ corresponds to a quasiprobability distribution known as the Wigner function W
= W共x兲. In general, such a function is fully characterized by
the entire set of its statistical moments 关13兴. However, in the
particular case of Gaussian states, the Wigner function is
Gaussian and, therefore, fully characterized by the first and
second moments only. These two moments are also known as
the displacement vector d ª 具x̂典 and the correlation matrix
共CM兲 V, whose generic entry is defined by
1
Vlm ª 具⌬x̂l⌬x̂m + ⌬x̂m⌬x̂l典,
2
V + i⍀ ⱖ 0.
共7兲
In other words, an arbitrary V 苸 S共2n , R兲 is a bona fide quantum CM if and only if Eq. 共7兲 holds. Equivalently, we can
introduce the set of n-mode quantum CM’s to be defined as
qCM共n兲 ª 兵V 苸 S共2n,R兲:V + i⍀ ⱖ 0其.
共8兲
Notice that the condition of Eq. 共7兲 implies a first relevant
constraint on the matrix V:
Lemma 1 (Definite positivity of V兲. For every V
苸 S共2n , R兲 satisfying V + i⍀ ⱖ 0, one has
V ⬎ 0.
Values of a such that the inequality above is violated can
always be found regardless of the values of uT1 ⍀u0 and
uT1 Vu1. This implies uTVu ⫽ 0 for every u 苸 R2n and, therefore, V ⬎ 0.
䊏
According to lemma 1, we then have
qCM共n兲 債 P共2n,R兲.
共9兲
Proof. Let u 苸 R2n, then 0 ⱕ uTVu + iuT⍀u = uTVu because
⍀ is antisymmetric. Hence V ⱖ 0. To prove definite positivity suppose, ad absurdum, that a nontrivial real vector u0
exists such that uT0 Vu0 = 0. Another vector u1 苸 R2n such that
uT1 ⍀u0 ⫽ 0 always exists as nul共⍀兲 = 0. As a consequence,
共11兲
Furthermore, it is trivial to show positive-definite matrices
which violate Eq. 共7兲, so that we actually have
qCM共n兲 傺 P共2n,R兲.
共12兲
Notice that definite positivity is the only requirement for a
real symmetric matrix to be a classical correlation matrix.
B. Symplectic transformations
The most general real linear transformation of the quadratures
S:x̂ → x̂⬘ ª Sx̂,
共13兲
must preserve Eq. 共4兲 in order to be a physical operation.
This happens when the matrix S 苸 M共2n , R兲 preserves the
symplectic form of Eq. 共5兲, i.e.,
S⍀ST = ⍀.
共14兲
The set of all the matrices S 苸 M共2n , R兲 satisfying Eq. 共14兲
forms the so-called real symplectic group
共6兲
where ⌬x̂l ª x̂l − 具x̂l典. According to the definition of Eq. 共6兲,
the CM of n bosonic modes is a real and symmetric matrix in
2n dimension, i.e., V 苸 S共2n , R兲. As a direct consequence of
Eq. 共4兲, such a matrix must also satisfy the uncertainty principle 关9,14兴
共10兲
S p共2n,R兲 ª 兵S 苸 M共2n,R兲:S⍀ST = ⍀其,
共15兲
whose elements are called symplectic or canonical transformations. As a consequence, the most general real linear
transformation in phase space S : x → x⬘ ª Sx must be symplectic. Its action on the Wigner function is simply given by
W共x兲 → W共S−1x兲, so that the displacement is linearly modified while the CM is transformed according to the congruence
V → SVST .
共16兲
Symplectic transformations are very important since every S
acting in the phase space K corresponds to a Gaussian unitary Û共S兲 acting on the Hilbert space H, i.e., a unitary operator preserving the Gaussian statistics of the quantum
states. These unitaries are the ones generated by bilinear
Hamiltonians and can always be decomposed into singlemode squeezers and multimode interferometers 关10,15兴. In
particular, local symplectic transformations
n
S=
丣 Sk 苸 Sp共2,R兲 丣
k=1
¯ 丣 S p共2,R兲
correspond to local Gaussian unitaries
052327-2
共17兲
PHYSICAL REVIEW A 79, 052327 共2009兲
CORRELATION MATRICES OF TWO-MODE BOSONIC SYSTEMS
n
Û共S兲 =
丢 Ûk .
共18兲
k=1
Local symplectic transformations can always be decomposed
as products of local rotations and local squeezings. In fact,
thanks to the following characterization
S p共2,R兲 = 兵S 苸 M共2,R兲:det S = 1其,
共19兲
we have that every S 苸 S p共2 , R兲 can be expressed as a product of proper rotations
R共␸兲 ª
冉
sin ␸ − cos ␸
cos ␸
and squeezing matrices
S共␰兲 ª
冉
sin ␸
冊
␰1/2 0
,
0 ␰−1/2
冊
,
␰ ⬎ 0.
共20兲
共21兲
Besides, it is also important to identify which quantities of
a CM are preserved under the application of symplectic
transformations. In general, for a given CM V, we say that a
functional
f:V → f共V兲 苸 R
f共V兲 = f共SVS 兲,
A C
,
CT B
共25兲
where A, B 苸 S共2 , R兲, and C 苸 M共2 , R兲. In this case the
symplectic spectrum 兵␯1 , ␯2其 ª 兵␯− , ␯+其 can be computed via
the simple formula 关22兴
␯⫾ =
共23兲
for every S 苸 S p共2n , R兲. Then, we say that f共V兲 is a local
symplectic invariant if Eq. 共23兲 holds for every S
苸 S p共2 , R兲 丣 ¯ 丣 S p共2 , R兲 关16兴. Notice that we can extend
the notion of symplectic invariance also to a property of a
matrix. For instance, the definite positivity of V is a global
symplectic invariant since V ⬎ 0 ⇒ SVST ⬎ 0.
III. SYMPLECTIC ANALYSIS
Here, we review some basic tools that can be used for the
symplectic manipulation of the CM’s. In particular, the central tool in this trade is Williamson’s theorem 关17兴, which
ensures the possibility of carrying out the symplectic diagonalization of real matrices in even dimension under the definite positivity constraint 共as in the case of the CM’s兲.
Lemma 2 (Williamson’s theorem). For every V
苸 P共2n , R兲, there exists a symplectic matrix S 苸 S p共2n , R兲
such that
冢
冉 冊
冑
⌬共V兲 ⫾ 冑⌬共V兲2 − 4 det V
,
2
共26兲
where
T
SVS =
V=
共22兲
is a 共global兲 symplectic invariant if
T
The corresponding Williamson form W is unique up to a
permutation of the symplectic spectrum 共i.e., of the bosonic
modes兲. Fixing this permutation, the diagonalizing symplectic matrix S of Eq. 共24兲 is defined up to local rotations
n
丣 k=1Rk with Rk 苸 SO共2兲. For the sake of completeness, we
report in Appendix A a simple proof of Williamson’s theorem, originally presented in Ref. 关19兴. Using this proof, we
show in Appendix B an algorithm which finds the diagonalizing symplectic matrix S of Eq. 共24兲. This algorithm is not
the fastest but it can be helpful in studying problems like the
optimal discrimination of Gaussian states 关20兴 and the quantum illumination 关21兴.
Let us consider the case of a 4 ⫻ 4 positive-definite matrix
V 苸 P共4 , R兲, as in the case of CM’s describing two bosonic
modes. This matrix can be expressed in the block form
␯1
␯1
␯n
␯n
冣
ª W ⬎ 0,
⌬共V兲 ª det A + det B + 2 det C.
Here, the quantities det A, det B, and det C are local symplectic invariants, while det V and ⌬共V兲 are global symplectic invariants, which can also be written as
det V = ␯−2␯+2,
⌬共V兲 = ␯−2 + ␯+2 .
共28兲
Another important tool in the symplectic analysis is the reduction to standard form by local symplectic transformations
关6,7兴. In general, such a reduction holds for symmetric matrices V 苸 S共4 , R兲 with positive diagonal blocks, as we easily
show in the following. In particular, it can be applied to
positive-definite matrices V 苸 P共4 , R兲 and, therefore, to CMs
V 苸 qCM共2兲.
Lemma 3 (Standard form). For every
V=
冉 冊
A C
苸 S共4,R兲,
CT B
共29兲
with A , B ⬎ 0, there exists some S 苸 S p共2 , R兲 丣 S p共2 , R兲 such
that
共24兲
where the n positive quantities 兵␯1 , ¯ , ␯n其 are the “symplectic eigenvalues” of V, and the diagonal matrix W is its “Williamson form” 共or “normal form”兲.
The symplectic spectrum 兵␯1 , ¯ , ␯n其 can be computed as
the standard eigenspectrum of the matrix 兩i⍀V兩 where the
modulus must be understood in the operatorial sense 关18兴.
共27兲
SVST =
冢
a
c+
a
c+
c−
b
c−
b
冣
ª VI ,
共30兲
where the real parameters a , b , c+ , c− satisfy
det A = a2,
det B = b2,
det C = c+c− ,
共31兲
and
052327-3
det V = det VI = 共ab − c+2兲共ab − c−2兲.
共32兲
PHYSICAL REVIEW A 79, 052327 共2009兲
PIRANDOLA, SERAFINI, AND LLOYD
Proof. Let us consider a pair of single-mode symplectic
transformations SA, SB 苸 S p共2 , R兲 and a pair of single-mode
proper rotations R共␪A兲, R共␪B兲 苸 SO共2兲. By applying the local
symplectic transformation
S = R共␪A兲SA 丣 R共␪B兲SB
to the matrix V, we get
SVST =
冉
冊
A⬘ C⬘
,
C ⬘T B ⬘
共33兲
B⬘ ª
␯− ⱖ 1 ⇔ ⌬共V兲 − 2 ⱖ 冑⌬共V兲2 − 4 det V ⇔ max兵2,2冑det V其
ⱕ ⌬共V兲 ⱕ 1 + det V ⇔
共34兲
where
A⬘ ª R共␪A兲共SAASAT兲RT共␪A兲,
Eq. 共40兲 for a generic V 苸 S共4 , R兲. Under the definite positivity assumption V ⬎ 0, one can also use Eq. 共26兲 to prove
the equivalences
共35兲
R共␪B兲共SBBSBT兲RT共␪B兲,
共36兲
C⬘ ª R共␪A兲共SACSBT兲RT共␪B兲.
共37兲
Since A , B 苸 P共2 , R兲, we can apply Williamson’s theorem.
This means that we can choose SA and SB such that
A⬘ = R共␪A兲aI RT共␪A兲 = aI,
共38兲
B⬘ = R共␪B兲bI RT共␪B兲 = bI,
共39兲
where a共b兲 is the symplectic eigenvalue of A共B兲 while the
angle ␪A共␪B兲 is arbitrary. Since the pair 兵␪A , ␪B其 can be chosen freely, we can always choose a pair 兵¯␪A , ¯␪B其 in Eq. 共37兲
such that C⬘ = diag共c+ , c−兲 关23兴. As a consequence, Eq. 共34兲 is
globally equal to Eq. 共30兲. Finally, since the transformation
of Eq. 共33兲 is local and symplectic, all the determinants relative to the blocks and the global matrix are preserved, so that
Eqs. 共31兲 and 共32兲 are trivially implied.
䊏
V ⬎ 0,
␯− ⱖ 1,
␯− ⱖ 1 ⇔
V ⬎ 0,
det V ⱖ 1,
⌬共V兲 ⱕ 1 + det V.
Proof. For every V 苸 P共4 , R兲, the application of Williamson’s theorem to Eq. 共7兲 gives V + i⍀ ⱖ 0 ⇔ ␯− ⱖ 1 共recalling
that ␯− is the smallest symplectic eigenvalue兲. Since V + i⍀
ⱖ 0 ⇒ V ⬎ 0 共see lemma 1兲, we can write V + i⍀ ⱖ 0
⇔ 共V ⬎ 0 ∧ ␯− ⱖ 1兲 which proves the bona fide condition of
冎
共42兲
再
det V ⱖ 1
⌬共V兲 ⱕ 1 + det V,
冎
共43兲
冉 冊
A C
苸 S共4,R兲
CT B
共44兲
is a quantum CM if and only if it satisfies
A,B ⬎ 0,
共45兲
⌬共V兲 ⱕ 1 + det V,
共46兲
2冑det A det B + 共det C兲2 ⱕ det V + det A det B. 共47兲
Proof. Under the assumption A , B ⬎ 0, the matrix V can be
reduced to the standard form VI of Eq. 共30兲 via a local symplectic transformation S. Since S⍀ST = ⍀, the Heisenberg
principle can be written in the equivalent form 关25兴
共40兲
共41兲
2冑det V ⱕ ⌬共V兲 ⱕ 1 + det V.
which states the equivalence between to Eqs. 共40兲 and 共41兲,
where the underlying assumption V ⬎ 0 is also shown. 䊏
Notice that, crucially, Williamson’s theorem could be applied to V because of its definite positivity, which thus implies the existence of well-defined symplectic eigenvalues
关24兴. The condition ␯− ⱖ 1 alone is therefore not, by itself,
fully equivalent to the uncertainty principle V + i⍀ ⱖ 0, unless definite positivity is also assumed. The essential role of
the prescription V ⬎ 0, often neglected in the literature, is
especially clear in the formulation of Eq. 共41兲: in fact, the
other two inequalities in Eq. 共41兲 only depend on the squared
symplectic eigenvalues and cannot thus distinguish between
positive and negative eigenvalues.
Besides the algebraic requirements of the previous theorem, we can derive an alternative set of conditions by applying the reduction to standard form. These conditions are expressed in terms of local symplectic invariants.
Theorem 5. An arbitrary
V=
or, equivalently,
det V ⱖ 1
According to Eq. 共28兲 the condition 2冑det V ⱕ ⌬共V兲 in Eq.
共42兲 corresponds to 2␯−␯+ ⱕ ␯−2 + ␯+2, which is trivially satisfied. Then, we have
IV. GENUINENESS OF A TWO-MODE CORRELATION
MATRIX
By applying the symplectic tools of Sec. III, we can
now derive very simple algebraic conditions for characterizing the genuineness of a two-mode CM. In other words,
starting from a generic 4 ⫻ 4 real and symmetric matrix
关V 苸 S共4 , R兲兴, we give the algebraic conditions that such a
matrix must satisfy in order to represent the CM of two
bosonic modes, i.e., a bona fide two-mode quantum CM
关V 苸 qCM共2兲兴. As a consequence of Williamson’s theorem,
we have the following algebraic conditions in terms of global symplectic invariants 关9兴.
Theorem 4. An arbitrary V 苸 S共4 , R兲 is a quantum CM if
and only if it satisfies
再
V + i⍀ ⱖ 0 ⇔ VI + i⍀ ⱖ 0.
共48兲
Since the matrix VI + i⍀ is Hermitian, its four eigenvalues
␭++ , ␭−+ , ␭+− , ␭−− are real. It is then easy to show that
where
052327-4
+
2␭⫾
= a + b + 冑␮ ⫾ 2冑␯ ,
共49兲
−
= a + b − 冑␮ ⫾ 2冑␯ ,
2␭⫾
共50兲
PHYSICAL REVIEW A 79, 052327 共2009兲
CORRELATION MATRICES OF TWO-MODE BOSONIC SYSTEMS
␮ ª 4 + 共a − b兲2 + 2共c+2 + c−2兲 ⱖ 4,
共51兲
and
where
␯ ª 4共a − b兲 + 共c+ + c−兲 关4 + 共c+ − c−兲 兴 ⱖ 0.
2
Since
PT:W共x兲 → W̃共x兲 ª W共⌳x兲,
␭+−
2
2
共52兲
⌳ª
is the minimum eigenvalue, we have that
VI + i⍀ ⱖ 0 ⇔ ␭+− ⱖ 0 ⇔ a + b − 冑␮ + 2冑␯ ⱖ 0
⇔
再
冦
共a + b兲2 ⱖ ␮ + 2冑␯
a+bⱖ0
⇔ 共a + b兲2 − ␮ ⱖ 0
a + b ⱖ 0.
1
1
1
丣
−1
共62兲
.
For the corresponding CM V 苸 qCM共2兲, the PT transformation is given by
冎
关共a + b兲2 − ␮兴2 ⱖ 4␯
冉 冊 冉 冊
共61兲
PT:V → Ṽ ª ⌳V⌳,
冧
共53兲
共63兲
where the partially transposed matrix Ṽ belongs to P共4 , R兲
关25兴 but not necessarily to qCM共2兲. By writing V in the
block form of Eq. 共25兲, one easily checks that the action of
the PT transformation ⌳ reduces to the following sign flip
det C → − det C,
共64兲
Last condition a + b ⱖ 0 in Eq. 共53兲 is trivially included in
A ⬎ 0 and B ⬎ 0 which gives a ⬎ 0 and b ⬎ 0 共for congruence
with the diagonal matrices aI and bI兲. The other two conditions
at the level of the local symplectic invariants. As a consequence, the positive-definite matrix Ṽ has
关共a + b兲2 − ␮兴2 ⱖ 4␯ ,
˜ 共V兲,
⌬共Ṽ兲 = det A + det B − 2 det C ª ⌬
共54兲
共65兲
and symplectic eigenvalues
and
共a + b兲2 − ␮ ⱖ 0,
共55兲
can be recast in terms of the local symplectic invariants. In
fact, by inserting Eqs. 共51兲 and 共52兲 in Eq. 共54兲, we get
a2 + b2 + 2c+c− ⱕ 共ab − c+2兲共ab − c−2兲 + 1,
共56兲
which is equivalent to Eq. 共46兲 by using Eq. 共32兲 and
⌬共V兲 = ⌬共VI兲 = a2 + b2 + 2c+c−. Finally, by inserting Eq. 共51兲
in Eq. 共55兲, we get 2ab − c+2 − c−2 ⱖ 2, which is equivalent to
2a2b2 − ab共c+2 + c−2兲 ⱖ 2ab,
共57兲
since ab ⬎ 0. In terms of local symplectic invariants, last
inequality is equal to
2 det A det B − I4 ⱖ 2冑det A det B,
˜␯⫾ =
冑
˜ 共V兲 ⫾ ⌬
˜ 共V兲2 − 4 det V
⌬
.
2
共66兲
Once the PT transformation has been extended, also the
Peres criterion can be consequently extended via the logical
implication
␳AB separable ⇒ ˜␳AB 苸 D共H兲 ⇒ Ṽ 苸 qCM共2兲, 共67兲
which becomes an equivalence for Gaussian states under 1
⫻ n mode bipartitions 关6,8兴.
Theorem 6 (Separability). Let us consider a Gaussian state
␳AB with CM V 苸 qCM共2兲. Then, ␳AB is separable if and
only if
共58兲
where
冑
Ṽ 苸 qCM共2兲,
共68兲
˜␯− ⱖ 1,
共69兲
˜ 共V兲 ⱕ 1 + det V.
⌬
共70兲
or, equivalently,
I4 ª Tr共A␻C␻B␻C ␻兲 =
T
ab共c+2
+
c−2兲
共59兲
is another local symplectic invariant. In fact, the quantity I4
is connected to the other local symplectic invariants by
det V = det A det B + 共det C兲2 − I4 ,
共60兲
which holds for every V 苸 S共4 , R兲. Using Eq. 共60兲 in Eq.
共58兲, we then get Eq. 共47兲.
䊏
V. SEPARABILITY OF A TWO-MODE CORRELATION
MATRIX
Few years ago, Ref. 关6兴 showed how to extend the partial
transposition and the Peres entanglement criterion 关26兴 to
bipartite bosonic systems. In fact, partial transposition
PT: ␳AB → ˜␳AB corresponds in phase space to a “local time
reversal” which inverts the momentum of only one of two
subsystems. This means that we have the following transformation for the Wigner function:
or, equivalently
Proof. The proof of Eq. 共68兲 follows exactly the same
steps of the one in Ref. 关6兴, where the P representation is
exploited 共see Ref. 关27兴 for recent connections between P
representation and separability.兲 In order to prove Eqs. 共69兲
and 共70兲, let us apply theorem 4 to the positive-definite matrix Ṽ 苸 P共4 , R兲. Then, we get
Ṽ 苸 qCM共2兲 ⇔ ˜␯− ⱖ 1 ⇔
再
det Ṽ ⱖ 1
˜ 共V兲 ⱕ 1 + det Ṽ
⌬
冎
共71兲
where Eq. 共69兲 is trivially proven. Now, since det Ṽ
= det共⌳V⌳兲 = det V ⱖ 1, the first condition in Eq. 共71兲 is al-
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PHYSICAL REVIEW A 79, 052327 共2009兲
PIRANDOLA, SERAFINI, AND LLOYD
ways satisfied and, therefore, the separability condition is
reduced to Eq. 共70兲.
䊏
Let us now derive the algebraic conditions that a generic
symmetric matrix must satisfy to represent the CM of a separable or entangled Gaussian state. The following corollary
gives an easy recipe to check if a symmetric matrix is a good
or bad candidate for this aim.
Corollary 7. An arbitrary
V=
冉 冊
A C
苸 S共4,R兲
CT B
共72兲
represents the CM of a separable Gaussian state if and only if
it satisfies
V ⬎ 0,
␯− ⱖ 1,
˜␯− ⱖ 1,
共73兲
˜ 共V兲 ⬎ 1 + det V for the entanglethe separability and with ⌬
ment.
䊏
VI. RELATION WITH THE PREVIOUS WORK
BY SIMON
In order to make a direct comparison with the previous
work by Simon 关6兴, we have to specify some of our results,
given for arbitrary symmetric matrices V 苸 S共4 , R兲, to the
case of positive-definite matrices, i.e., V 苸 P共4 , R兲. As an
immediate consequence of theorem 4, we have the following
result.
Corollary 8. An arbitrary V 苸 P共4 , R兲 is a two-mode
quantum CM V 苸 qCM共2兲, i.e., V + i⍀ ⱖ 0, if and only if
␯− ⱖ 1,
or, equivalently,
V ⬎ 0,
det V ⱖ 1,
⌫共V兲 ⱕ 1 + det V,
共74兲
共82兲
or, equivalently,
or, equivalently,
A,B ⬎ 0,
⌫共V兲 ⱕ 1 + det V,
det V ⱖ 1,
共75兲
2冑det A det B + 共det C兲2 ⱕ det V + det A det B, 共76兲
␯− ⱖ 1,
˜␯− ⬍ 1,
共77兲
or, equivalently,
V ⬎ 0,
det V ⱖ 1,
˜ 共V兲,
⌬共V兲 ⱕ 1 + det V ⬍ ⌬
共78兲
or, equivalently,
A,B ⬎ 0,
˜ 共V兲,
⌬共V兲 ⱕ 1 + det V ⬍ ⌬
Ṽ 苸 qCM共2兲.
det A det B + 共1 − det C兲2 − I4 ⱖ det A + det B,
共85兲
+ 共ab − c+2兲共ab − c−2兲.
共86兲
Note that Eq. 共86兲 can be equivalently written as
a2b2 + 共1 − c+c−兲2 − ab共c+2 + c−2兲 ⱖ a2 + b2 .
共81兲
The bona fide condition V 苸 qCM共2兲 is equivalently expressed by the conditions of Eqs. 共40兲 and 共41兲 in theorem 4.
Then, for every V 苸 qCM共2兲, the separability condition Ṽ
苸 qCM共2兲 is equivalent to Eqs. 共69兲 and 共70兲 in theorem 6.
By combining Eq. 共40兲 with Eq. 共69兲 and Eq. 共41兲 with Eq.
共70兲, one easily gets Eqs. 共73兲 and 共74兲, where
˜ 共V兲其 ⱕ 1 + det V ⇔ ⌫共V兲 ⱕ 1 + det V. According
max兵⌬共V兲 , ⌬
to theorem 6, for every V 苸 qCM共2兲 the entanglement con˜ 共V兲 ⬎ 1
dition is expressed by ˜␯− ⬍ 1 or, equivalently, by ⌬
+ det V. Then, it is trivial to derive the corresponding Eqs.
共77兲 and 共78兲. The proof of Eqs. 共75兲 and 共76兲 and Eqs. 共79兲
and 共80兲 is the same as before except that now we have to
combine the Eqs. 共45兲–共47兲 of theorem 5 with Eq. 共70兲 for
共84兲
⌬共V兲 ⱕ 1 + det V ⇔ a2 + b2 + 2c+c− ⱕ 1
2冑det A det B + 共det C兲2 ⱕ det V + det A det B. 共80兲
V 苸 qCM共2兲,
det V ⱖ 1,
where I4 ª Tr共A␻C␻B␻CT␻兲.
Proof. By applying theorem 4 under the assumption V
⬎ 0, one trivially derives the equivalent conditions in Eqs.
共82兲 and 共83兲. In order to prove Eqs. 共84兲 and 共85兲, let us
reduce the positive-definite matrix V to its standard form of
Eq. 共30兲. Under local symplectic transformations, we then
have the equivalence
共79兲
Proof. In order to represent the CM of a separable Gaussian state, the symmetric matrix V 苸 S共4 , R兲 must simultaneously satisfy
共83兲
or, equivalently,
where ⌫共V兲 ª det A + det B + 2兩det C兩. Instead, it represents
the CM of an entangled Gaussian state if and only if it satisfies
V ⬎ 0,
⌬共V兲 ⱕ 1 + det V,
共87兲
In terms of local symplectic invariants, last relation can be
written as in Eq. 共85兲, where Eq. 共59兲 has been also used. 䊏
Notice that Eq. 共85兲 corresponds to the Eq. 共17兲 of Ref.
关6兴, up to notation factors 关28兴. In Ref. 关6兴, this condition is
incorrectly claimed to be equivalent to the Heisenberg principle V + i⍀ ⱖ 0 共this equivalence is claimed under the positivity contraint V ⬎ 0, which is a sufficient condition for the
reduction to standard form used in the corresponding proof
of Ref. 关6兴兲. In order to have a full equivalence with the
Heisenberg principle V + i⍀ ⱖ 0, the supplementary condition of Eq. 共84兲 is mandatory. It is indeed rather simple to
construct a positive-definite matrix V 苸 P共4 , R兲 which satisfies Eq. 共85兲 but violates V + i⍀ ⱖ 0. As an example, let us
consider the following real and symmetric matrix
052327-6
PHYSICAL REVIEW A 79, 052327 共2009兲
CORRELATION MATRICES OF TWO-MODE BOSONIC SYSTEMS
冢
1 + 4x
0
0
1 + 4x
1
V共x兲 =
2 − 1 + 4x
0
0
− 4x
− 1 + 4x
0
0
− 4x
1 + 4x
0
0
1 + 4x
冣
, 共88兲
which is positive-definite for every x ⬎ 0. It is easy to verify
that the Hermitian matrix V + i⍀ has the following real eigenvalues:
1
␭⫾ = 共1 + 8x ⫾ 冑17 − 16x + 64x2兲,
4
共89兲
1
␪⫾ = 共3 + 8x ⫾ 冑17 − 16x + 64x2兲.
4
共90兲
Since ␭− is the minimal eigenvalue, the Heisenberg principle
V + i⍀ ⱖ 0 is equivalent to ␭− ⱖ 0, which gives
x ⱖ 1/2.
共91兲
Now, let us explicitly compute Eqs. 共84兲 and 共85兲. It is easy
to show that Eq. 共84兲 is equivalent to
x共8x + 1兲 ⱖ 1 ⇔ x ⱖ 共冑33 − 1兲/16 ⯝ 0.3,
共92兲
while Eq. 共85兲 共Simon’s genuineness condition兲 is equivalent
to
1
1
1
+ x共8x − 5兲 ⱖ 0 ⇔ 0 ⬍ x ⱕ OR x ⱖ .
2
8
2
共93兲
From Eq. 共93兲, one can see that Simon’s condition alone
does not exclude the matrices V共x兲 for 0 ⬍ x ⱕ 1 / 8, which
are clearly unphysical since they violate the Heisenberg condition of Eq. 共91兲. A complete equivalence with Eq. 共91兲 is
retrieved by coupling Eq. 共93兲 with Eq. 共92兲, where the latter
equation excludes the nonphysical region 0 ⬍ x ⱕ 1 / 8.
This imprecision in Simon’s work leads to a common
misunderstanding of the subsequent separability condition
关Eq. 共19兲 of Ref. 关6兴兴, which in our notation corresponds to
det A det B + 共1 − 兩det C兩兲2 − I4 ⱖ det A + det B. 共94兲
In this condition, the Heisenberg principle is erroneously
claimed to be included 共in fact, it is only partially included兲.
Hence, Simon’s separability condition of Eq. 共94兲 is actually
valid only if V 苸 qCM共2兲, i.e., the positive-definite matrix
V 苸 P共4 , R兲 is already known to be a bona fide quantum
CM. In other words, the separability criterion of Eq. 共94兲
must be tested on positive-definite matrices which are already known to describe the second statistical moments of a
physical quantum state. However, under this assumption of
physicality, Simon’s separability criterion of Eq. 共94兲 displays a redundant modulus and must be simplified to
det A det B + 共1 + det C兲2 − I4 ⱖ det A + det B.
冉 冊
A C
苸 qCM共2兲.
CT B
det A det B + 共1 + det C兲2 − I4 ⱖ det A + det B.
共98兲
In particular, if ␳AB is a Gaussian state, then it is separable if
and only if Eq. 共97兲 关or Eq. 共98兲兴 holds.
Proof. The proof is straightforward. Since V is a quantum
CM, it is positive-definite and satisfies the condition det V
ⱖ 1. Now, suppose that the corresponding two-mode state
␳AB is separable. Then we have
␳AB separable ⇒ ˜␳AB 苸 D共H兲 ⇒ Ṽ 苸 qCM共2兲
⇔ Ṽ + i⍀ ⱖ 0.
共99兲
By applying corollary 8 to the positive-definite matrix Ṽ
= ⌳V⌳, we have
Ṽ + i⍀ ⱖ 0 ⇔ det Ṽ ⱖ 1,
⌬共Ṽ兲 ⱕ 1 + det Ṽ. 共100兲
Since det Ṽ = det V, we have that det Ṽ ⱖ 1 is automatically
satisfied in the previous Eq. 共100兲. Then, we get
˜ 共V兲 ⱕ 1 + det V,
␳AB separable ⇒ ⌬
共101兲
˜ 共V兲 ª ⌬共Ṽ兲 is defined in Eq. 共65兲. Using Eq. 共60兲,
where ⌬
one easily proves the equivalence between Eqs. 共97兲 and
共98兲. Finally, the full equivalence which holds for Gaussian
␳AB is a direct application of theorem 6.
䊏
This criterion represents a simplification of Simon’s separability criterion. Now, it is important to notice that the separability criterion becomes a bit more involved when arbitrary
positive-definite matrices V 苸 P共4 , R兲 are considered, without any other a priori assumption. For a generic V
苸 P共4 , R兲, both the Heisenberg principle 共V + i⍀ ⱖ 0兲 and
the separability property 共Ṽ + i⍀ ⱖ 0兲 must be explicitly considered and combined together, in order to get a complete set
of algebraic conditions. Thanks to these conditions, one easily checks when a positive-definite matrix V 苸 P共4 , R兲 can
represent the quantum CM of a Gaussian state ␳AB which is
separable or entangled. By applying corollary 7, we get the
following criterion for positive-definite matrices.
Criterion 10. An arbitrary V 苸 P共4 , R兲 represents the CM
of a separable Gaussian state if and only if
det V ⱖ 1,
共102兲
⌫共V兲 ⱕ 1 + det V,
共103兲
det V ⱖ 1,
共104兲
or, equivalently,
共95兲
共96兲
共97兲
or, equivalently,
det A det B + 共1 − 兩det C兩兲2 − I4 ⱖ det A + det B.
Criterion 9 (Separability). Let us consider a two-mode quantum state ␳AB having quantum CM
V=
˜ 共V兲 ⱕ 1 + det V,
⌬
共105兲
Instead, it represents the CM of an entangled Gaussian state
if and only if
det V ⱖ 1,
The separability of ␳AB implies
052327-7
共106兲
PHYSICAL REVIEW A 79, 052327 共2009兲
PIRANDOLA, SERAFINI, AND LLOYD
˜ 共V兲,
⌬共V兲 ⱕ 1 + det V ⬍ ⌬
共A1兲
S = W1/2RV−1/2 ,
共107兲
with a suitable R 苸 SO共2n兲. In fact
or, equivalently,
det V ⱖ 1,
共A2兲
= W1/2IW1/2 = W.
共1 + det C兲 ⬍ det A + det B − det A det B + I4
2
ⱕ 共1 − det C兲2 .
SVST = 共W1/2RV−1/2兲V共V−1/2RTW1/2兲 = W1/2RIRTW1/2
共108兲
共109兲
The proof is a trivial application of corollary 7, together
with Eq. 共60兲, used to state the equivalences between Eqs.
共103兲 and 共105兲, and between Eqs. 共107兲 and 共109兲. According to Eq. 共109兲, positive-definite matrices with det C ⱖ 0
can only be associated to separable Gaussian states 关6兴. Notice that the original Simon’s separability criterion, i.e., Eq.
共105兲, must be coupled with the mandatory condition of Eq.
共104兲 in order to investigate correctly the separability properties of a generic positive-definite matrix.
Notice that Eq. 共A1兲 is well defined since V and W are
positive-definite 共therefore, nonsingular兲. However, the rotation R in Eq. 共A1兲 is not arbitrary but must be chosen in
order to make S symplectic.
Let us apply Eq. 共A1兲 to the symplectic condition S⍀ST
= ⍀. Then, we have
共W1/2RV−1/2兲⍀共V−1/2RTW1/2兲 = ⍀
⇔ R共V−1/2⍀V−1/2兲RT = W−1/2⍀W−1/2
⇔ RXRT = Y,
共A3兲
where
VII. SUMMARY
X ª V−1/2⍀V−1/2,
In theorem 4, we have rederived and explicitly stated all
the precise algebraic conditions a symmetric matrix must
satisfy to represent the CM of a two-mode bosonic 共or canonical兲 quantum system, including the 共critical and often
neglected兲 definite positivity condition. Such conditions are
expressed in terms of global symplectic invariants. In theorem 5, we have derived a new and alternative set of conditions, which are expressed in terms of local symplectic invariants. In these local conditions the positivity check is
restricted to the submatrices A and B. Finally, in theorem 6,
the necessary and sufficient condition for the separability of
two-mode Gaussian states has been reviewed and cast in a
compact form. We should stress that such a condition is valid
only under the assumption that physicality is also met 关V
苸 qCM共2兲兴. In corollary 7, both the physicality and separability have been explicitly taken into account. Then, we have
derived a complete set of 共global or local兲 conditions that a
generic symmetric matrix must satisfy in order to represent
the CM of a separable 共or entangled兲 Gaussian state of two
bosonic modes. In Sec. VI, some of our results have been
specified for positive-definite matrices and a comparison
with the previous results by Simon has been thoroughly presented. The rigourous agreement with all the conditions here
considered should constitute a constant reference in both the
theoretical practice and the analysis of experimental data involving quantum systems of two canonical degrees of freedom.
Y ª W−1/2⍀W−1/2
are antisymmetric 共because V and W are symmetric, while
⍀ is antisymmetric兲. In particular, we have
n
Y=
丣
k=1
冉
0
␯−1
k
− ␯−1
k
0
冊
共A5兲
.
Now the existence of R in Eq. 共A3兲 is assured by the following theorem on the block-diagonalization of real antisymmetric matrices 共specialized to even dimensions兲 关29兴
Theorem 11. For every A = −AT 苸 M共2n , R兲, there exists a
共unique兲 O 苸 SO共2n兲 such that
n
OAOT =
丣 ak␻ ª Ã,
共A6兲
k=1
where the 共unique兲 block diagonal form à has ak ⬎ 0.
APPENDIX B: FINDING THE DIAGONALIZING
SYMPLECTIC MATRIX
Let us show a possible procedure for deriving the proper
rotation O that block-diagonalizes a generic antisymmetric
matrix A as in theorem 11. We can easily prove the following
connection between the block-diagonalization of A and its
unitary diagonalization
Theorem 12. The proper rotation O performing the blockdiagonalization of Eq. 共A6兲 is given by
共B1兲
O = ⌫U† ,
ACKNOWLEDGMENTS
S.P. was supported by a Marie Curie Action of the European Community. S.L. was supported by the W.M. Keck
foundation center for extreme quantum information theory
共xQIT兲.
共A4兲
where
⌫=
1
n
1
␥, ␥ ª
冑2 丣
冑2
k=1
冉 冊
i −i
1
1
,
共B2兲
and U is an arbitrary unitary performing the diagonalization
of A, i.e.,
APPENDIX A: SIMPLE PROOF OF WILLIAMSON’S
THEOREM
n
Let us construct the diagonalizing symplectic according to
the decomposition
052327-8
†
U AU =
丣 iak
k=1
冉 冊
−1
1
ª AD .
共B3兲
PHYSICAL REVIEW A 79, 052327 共2009兲
CORRELATION MATRICES OF TWO-MODE BOSONIC SYSTEMS
Proof. First, let us prove how A can be transformed into
the diagonal form AD of Eq. 共B3兲 by a unitary matrix. From
Eq. 共B2兲, we have that
␥†共ak␻兲␥ = iak
冉 冊
−1
1
U† =
As a consequence, by applying ⌫ to Eq. 共A6兲, we get
n
⌫†OAOT⌫ = ⌫†Ã⌫ =
丣 iak
k=1
冉 冊
−1
1
= AD .
Au2 = ia1u2 ,
共B6兲
Au2n = + ianu2n ,
共B7兲
n
more compactly denoted by 兵u2k−1 , u2k其k=1
with
Au2k−1 = − iaku2k−1,
Au2k = iaku2k .
共B8兲
These vectors are unique up to phase factors ␸
ª 兵␸1 , ¯ , ␸2n其, i.e., up to the replacements
u2k−1 → u2k−1ei␸2k−1,
u2k → u2kei␸2k .
共B9兲
ⴱ
u2n兲.
U = 共uⴱ2 u2 ¯ u2n
共B11兲
Let us explicitly compute the matrix product ⌫U†. By applying
冑2
冢
to the conjugate matrix
i −i
0
1
0
]
=
u2n,1 u2n,2
u2n,2n−1 u2n,2n
ⴱ
u2n,1
ⴱ
ⴱ
u2n,2n−1
u2n,2n
ⴱ
u2n,2
冣
,
共B13兲
1
冑2
冢
␣1,1 ␣1,2
␤1,1 ␤1,2
␣n,1 ␣n,2
␤n,1 ␤n,2
␣1,2n−1 ␣1,2n
␤1,2n−1 ␤1,2n
␣n,2n−1 ␣n,2n
␤n,2n−1 ␤n,2n
冣
, 共B14兲
where
␣k,j ª − 2 Im共u2k,j兲,
␤k,j ª 2 Re共u2k,j兲.
共B15兲
From Eqs. 共B14兲 and 共B15兲, we have that ⌫U is real for
every choice of U in Eq. 共B3兲, i.e., for every choice of the
phases ␸ in the corresponding eigenvectors. More strongly,
we have ⌫U† 苸 SO共2n兲 共since ⌫U† real implies ⌫U† orthogonal with det= + 1兲. Then, from Eq. 共B3兲, we easily get
†
⌫U†AU⌫† = ⌫AD⌫† = Ã.
共B16兲
In conclusion, for every diagonalizing unitary U, the proper
rotation ⌫U† corresponds to the unique proper rotation O
that performs the block-diagonalization of Eq. 共A6兲.
䊏
Both theorem 11 and theorem 12 can be applied to the Eq.
共A3兲, by setting O = R, A = X and à = Y. These theorems allow to reduce the computation of the rotation R in Eq. 共A3兲
to a unitary diagonalization. In fact, we have just to find a
unitary U that diagonalizes X, i.e.,
n
†
U XU =
丣 i␯−1
k
k=1
共B10兲
As a consequence, the most general unitary matrix that diagonalizes A has the specific form
1
ⴱ
u2,2n
冉 冊
1
−1
,
共B17兲
and then construct
ⴱ
u2k−1 = u2k
.
⌫=
u2,2n
ⴱ
u2,2n−1
⌫U† =
This means that, for every choice of ␸, we have an equivalent unitary matrix U = U共␸兲 in the diagonalization of A.
Now, by conjugating Eq. 共B8兲, one easily checks that
1
u2,2n−1
ⴱ
u2,2
共B5兲
]
Au2n−1 = − ianu2n−1,
u2,2
ⴱ
u2,1
one explicitly gets
In other words, there exists a unitary OT⌫ that diagonalizes
A according to Eq. 共B3兲.
Then, let us prove that, for every unitary U diagonalizing
A according to Eq. 共B3兲, we can write Eq. 共B1兲 where O
performs the block diagonalization of Eq. 共A6兲. For proving
this, let us consider the orthonormal eigenvectors
兵u1 , ¯ , u2n其 of A
Au1 = − ia1u1,
冢 冣冢
u2,1
u†2
T
u2n
†
u2n
共B4兲
.
uT2
i −i
1
1
冣
共B12兲
R = ⌫U† .
共B18兲
Once that we have R, we use Eq. 共A1兲 to get the symplectic
S. Here is the complete algorithm:
共1兲 Find the symplectic spectrum of V, i.e., its Williamson
form W.
共2兲 Compute the matrices W1/2 共immediate兲 and V−1/2
共needs orthogonal diagonalization兲.
共3兲 Construct the matrix X ª V−1/2⍀V−1/2.
共4兲 Find the eigenvectors of X and construct the corresponding unitary U.
共5兲 Compute R = ⌫U†.
共6兲 Compute S = W1/2RV−1/2.
By construction, this algorithm reduces the determination
of S to unitary diagonalizations. Actually, this task can be
achieved via faster methods when the symplectic spectrum is
nondegenerate. In general, the determination of S is equivalent to the construction of a symplectic basis 关30兴.
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PIRANDOLA, SERAFINI, AND LLOYD
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关23兴 For every M 苸 M共n , R兲 there always exists a pair of proper
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关28兴 Notice that Ref. 关6兴 adopts the commutation relations 关x̂l , x̂m兴
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Eq. 共17兲 of Ref. 关6兴.
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