entangled state for constructing a generalized phase-space representation and its statistical...

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Entangled state for constructing a generalized phase-space representation and its statistical behavior Li-yun Hu 1,2 and Hong-yi Fan 2 1 College of Physics and Communication Electronics, Jiangxi Normal University, Nanchang 330022, People’s Republic of China 2 Department of Physics, Shanghai Jiao Tong University, Shanghai 200030, People’s Republic of China Received 9 February 2009; published 19 August 2009 We construct a generalized phase-space representation GPSR based on the idea of Einstein-Podolsky- Rosen quantum entanglement, i.e., we generalize the Torres-Vega-Frederick phase-space representation to the entangled case, which is characteristic of the features when two particles’ relative coordinate, total momentum operators, and their conjugative variables, respectively, operate on the GPSR. This representation is complete and nonorthogonal. The Weyl-ordered form of the density operator of GPSR is derived, and its identification with the generalized Husimi operator is recognized, which clearly exhibit its statistical behavior. The minimum uncertainty relation obeyed by the GPSR is also demonstrated. DOI: 10.1103/PhysRevA.80.022115 PACS numbers: 03.65.Ud, 42.50.p I. INTRODUCTION Phase-space formalism of quantum mechanics began with Wigner’s celebrated paper 1 in 1932, since then the gen- eralized phase-space techniques have found useful applica- tions in various branches of physics 25. The main idea of this formalism is to represent the density operator as a qua- sidistribution function over the classical phase-space q , p. Phase-space formalism implies that there is no a unique way to represent a quantum state as a wave function in the stan- dard approach one usually uses the coordinate q or the momentum p representations. By observing this, Torres- Vega and Frederick TF constructed a quantum mechanical phase-space representation in 1993 6, = , where denotes a point in the q , p space characteristic of param- eters , , , , using it the standard Schrödinger equation i t q, t = - 1 2m 2 q 2 + Vq q, t, =1 1 changes to the following quantum Liouville equation in phase-space i t , t = - 1 2m 2 q 2 + V q + i p , t , 2 where obtained from q via = - dq qq, i.e., Torres-Vega and Frederick employed Dirac symbol to associate Eq. 2 with the state vector . The bra exhib- its the remarkable features when coordinate operator Q and momentum operator P, respectively, operates on it, i.e., Q = q + i p , 3 P = p + i q , 4 so is a state of phase-space variables q , p; , , , and are all real number parameters specifying a whole family of states, satisfying - = 1. 5 It has been demonstrated that this representation best fits the correspondence between the classical and quantum Liouville equations and have wide applications 612. To make the TF theory complete, in Ref. 13, we have found the explicit form of in Fock space, 2 - 1/2 exp q 2 2 - p 2 2 + 2q + ipa 1 + + 2 a 1 †2 0 1 , 6 where 0 1 is the single-mode vacuum state, a 1 and a 1 are the Bose annihilation and creation operators, respectively, obey- ing a 1 , a 1 = 1. By using the technique of integration within an ordered product IWOP of operators 14,15, we have shown that is a special coherent squeezed state which makes up a quantum mechanical representation. contains elements of both the momentum eigenstate and the coordi- nate eigenstate. Remarkably, reduces to the Husimi operator for some special values of , , , . When we move to tackling quantum entanglement in two degree of freedom DOF case, enlightened by the paper of Einstein-Podolsky-Rosen EPR in 1935 16, who noticed that two particles’ relative coordinate Q 1 - Q 2 and total mo- mentum P 1 + P 2 are commutative and can be simultaneously measured, we are naturally led to consider first the common eigenvector of the relative coordinate operator Q 1 - Q 2 and the momentum sum operator P 1 + P 2 , as well as the com- mon eigenvector of their conjugative variables P 1 - P 2 and Q 1 + Q 2 , since Q 1 + Q 2 , P 1 + P 2 =2i and Q 1 - Q 2 , P 1 - P 2 =2i. Correspondingly, because in the entangled case only those states simultaneously describing two entangled particles can be endowed with physical meaning, phase- space should be understood with regard to and . Be- sides, = re i also obeys the eigenequations PHYSICAL REVIEW A 80, 022115 2009 1050-2947/2009/802/0221158 ©2009 The American Physical Society 022115-1

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Page 1: Entangled state for constructing a generalized phase-space representation and its statistical behavior

Entangled state for constructing a generalized phase-space representationand its statistical behavior

Li-yun Hu1,2 and Hong-yi Fan2

1College of Physics and Communication Electronics, Jiangxi Normal University, Nanchang 330022, People’s Republic of China2Department of Physics, Shanghai Jiao Tong University, Shanghai 200030, People’s Republic of China

�Received 9 February 2009; published 19 August 2009�

We construct a generalized phase-space representation �GPSR� based on the idea of Einstein-Podolsky-Rosen quantum entanglement, i.e., we generalize the Torres-Vega-Frederick phase-space representation to theentangled case, which is characteristic of the features when two particles’ relative coordinate, total momentumoperators, and their conjugative variables, respectively, operate on the GPSR. This representation is completeand nonorthogonal. The Weyl-ordered form of the density operator of GPSR is derived, and its identificationwith the generalized Husimi operator is recognized, which clearly exhibit its statistical behavior. The minimumuncertainty relation obeyed by the GPSR is also demonstrated.

DOI: 10.1103/PhysRevA.80.022115 PACS number�s�: 03.65.Ud, 42.50.�p

I. INTRODUCTION

Phase-space formalism of quantum mechanics began withWigner’s celebrated paper �1� in 1932, since then the �gen-eralized� phase-space techniques have found useful applica-tions in various branches of physics �2–5�. The main idea ofthis formalism is to represent the density operator as a qua-sidistribution function over the classical phase-space �q , p�.Phase-space formalism implies that there is no a unique wayto represent a quantum state as a wave function �in the stan-dard approach one usually uses the coordinate ��q� or themomentum ��p� representations�. By observing this, Torres-Vega and Frederick �TF� constructed a quantum mechanicalphase-space representation in 1993 �6�, �=����, where �denotes a point in the �q , p� space characteristic of param-eters �� ,� ,� ,��, using it the standard Schrödinger equation

i�

�t��q,t� = �−

1

2m

�2

�q2 + V�q����q,t�,� = 1 �1�

changes to the following quantum Liouville equation inphase-space

i�

�t���,t� = �−

1

2m

�2

�q2 + V�q + i�

�p�����,t� , �2�

where ���� obtained from ��q� via ����=− dq� �q���q�,

i.e., Torres-Vega and Frederick employed Dirac symbol toassociate Eq. �2� with the state vector ���. The bra �� exhib-its the remarkable features when coordinate operator Q andmomentum operator P, respectively, operates on it, i.e.,

��Q = ��q + i��

�p��� , �3�

��P = ��p + i��

�q��� , �4�

so �� is a state of phase-space variables �q , p�; �, �, �, and� are all real number parameters specifying a whole familyof �� states, satisfying

�� − �� = 1. �5�

It has been demonstrated that this representation best fits thecorrespondence between the classical and quantum Liouvilleequations and have wide applications �6–12�.

To make the TF theory complete, in Ref. �13�, we havefound the explicit form of ��� in Fock space,

��� �2�− �����1/2

exp��q2

2�−

�p2

2�+ �2��q + i�p�a1

† +�� + ��

2a1

†2��0�1,

�6�

where �0�1 is the single-mode vacuum state, a1 and a1† are the

Bose annihilation and creation operators, respectively, obey-ing �a1 ,a1

†�=1. By using the technique of integration withinan ordered product �IWOP� of operators �14,15�, we haveshown that ��� is a special coherent squeezed state whichmakes up a quantum mechanical representation. ��� containselements of both the momentum eigenstate and the coordi-nate eigenstate. Remarkably, ����� reduces to the Husimioperator for some special values of �� ,� ,� ,��.

When we move to tackling quantum entanglement in twodegree of freedom �DOF� case, enlightened by the paper ofEinstein-Podolsky-Rosen �EPR� in 1935 �16�, who noticedthat two particles’ relative coordinate Q1−Q2 and total mo-mentum P1+ P2 are commutative and can be simultaneouslymeasured, we are naturally led to consider first the commoneigenvector �� of the relative coordinate operator Q1−Q2and the momentum sum operator P1+ P2, as well as the com-mon eigenvector �� of their conjugative variables P1− P2and Q1+Q2, since �Q1+Q2 , P1+ P2�=2i and �Q1−Q2 , P1− P2�=2i. Correspondingly, because in the entangled caseonly those states simultaneously describing two entangledparticles can be endowed with physical meaning, phase-space should be understood with regard to �� and ��. Be-sides, ��=rei � also obeys the eigenequations

PHYSICAL REVIEW A 80, 022115 �2009�

1050-2947/2009/80�2�/022115�8� ©2009 The American Physical Society022115-1

Page 2: Entangled state for constructing a generalized phase-space representation and its statistical behavior

Q1 − Q2

��Q1 − Q2�2 + �P1 + P2�2�� = rei � = cos �� = rei � ,

and

P1 + P2

��Q1 − Q2�2 + �P1 + P2�2�� = rei � = sin �� = rei � ,

by noticing ��Q1−Q2�2+ �P1+ P2�2���=rei �=2r2��=rei �,we can define a phase operator. Therefore we should con-struct generalized phase-space representation based on twomutually conjugative EPR variables �sum and differencevariables�. As one can see shortly later, this generalizationwould result in the development of both the Wigner functionand the Husimi function theory for entangled states. It isexpected that states that are entangled or partly entangledwould form a compact “blob” in the sum and differencephase-space, as opposed to the strongly oscillating Wignerfunction that they would have in the “old” �Q , P� represen-tation. This would have two useful consequences: �1� itwould become much easier to “eyeball” what is going onphysically when, for example, some sort of complicated dy-namics is going on; and �2� compact distributions are muchbetter for Monte Carlo sampling if one were to make calcu-lations this way.

Recalling the form of �� in two-mode Fock space �17�

��� = exp�−1

2���2 + �a1

† − ��a2† + a1

†a2†��00�, � = �1 + i�2,

�7�

where ai, ai† �i=1,2� are the two-mode Bose annihilation and

creation operators obeying �ai ,aj†�=�ij, we see that ��� satis-

fies the eigenequations:

�Q1 − Q2���� = �2�1���, �P1 + P2���� = �2�2��� , �8�

and possesses the orthogonal-complete relation

� d2�

������ = 1, ������ = ����� − ������� − ��� . �9�

EPR entanglement involved in ��� can be clearly seen fromits Schmidt decompositions, i.e., ���=e−i�1�2−

dq�q�1 � �q−�2�1�2ei�2�2q, where �q�i �i=1,2� is the eigenvector of co-ordinate Qi. Based on �� and �� we generalize Eq. �6� to anenlarged phase-space representation e��, where the subscript�e� implies entanglement. Then how to construct this repre-sentation? Similar in spirit to Eqs. �3� and �4� we start withconsidering what are the four self-consistent and reasonableequations �involving �, �, �, and �� in two DOF case. Afterdoing tries, we find that ���e is characteristic of the featureswhen Q1−Q2 and P1+ P2, respectively, operates on it,

e��Q1 − Q2

�2= ���1 + i�

��2�e�� ,

e��P1 + P2

�2= ���2 − i�

��1�e�� , �10�

where �1+ i�2=� and �1+ i�2=� are complex variables indi-cating the phase-space representation e��. Simultaneously,under the action of the center-of-mass operator Q1+Q2 andthe relative momentum operator P1− P2, the state e�� shouldexhibit

e��Q1 + Q2

�2= ���1 − i�

��2�e�� ,

e��P1 − P2

�2= ���2 + i�

��1�e�� . �11�

Equations �10� and �11� are the nontrivial extension of Eqs.�3� and �4�, so e�� corresponds to a generalized phase-spacerepresentation �GPSR�.

The work of extending �� to the entangled case e�� issomehow like the extension from the single-mode squeezedstate to the two-mode squeezed state �an entangled state too��18�. In the following we shall derive the explicit form of���e and further analyze its properties, in so doing, theTorres-Vega-Frederick theory can be developed and en-riched.

Our paper is arranged as follows. In Sec. II using the ���representation we shall derive the explicit form of ���e intwo-mode Fock space and then analyze its properties. In Sec.III, the completeness relation and nonorthogonality of ���eare demonstrated. In Sec. IV the minimum uncertainty rela-tion of two pairs of quadrature operators in ���e is shown. InSecs. V and VI we derive the Weyl-ordered form of ���ee��,which yields its classical correspondence, and then examineits marginal distributions in “��� direction” and “��� direc-tion.” Sections VII and VIII are devoted to the identificationof the density operator ���ee�� with the generalized Husimioperator, and to the derivation of Wigner function of ���e,respectively.

II. STATE ��‹e IN TWO-MODE FOCK SPACE

We find that the explicit form of the state ���e in two-mode Fock space is �see the Appendix�,

���e 2�− ���� exp�����2

2�−

����2

2�+ ��� + ���a1

+ ���� − ����a2† − ��� + ���a1

†a2†��00� , �12�

where �, �, �, and � satisfy the relation Eq. �5�; to satisfy thesquare integrable condition for wave function in the phase-space of ���e,

�� �0, and �

� �0 are demanded. From Eq. �12�we see that the representation ���e involves elements of both��� and ��� representations, which are both entangled states:i.e., the set of values ��=−1, �=0 gives the standard EPRentangled state ���e→ ��=��� �comparing with Eq. �7��;while the set ��=1, �=0 yields ���e→ ��=��� �comparingwith Eq. �39� below�. To certify that Eq. �12� really obeysEqs. �10� and �11� we operate ai on ���e,

LI-YUN HU AND HONG-YI FAN PHYSICAL REVIEW A 80, 022115 �2009�

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Page 3: Entangled state for constructing a generalized phase-space representation and its statistical behavior

a1���e = ���� + ��� − ��� + ���a2†����e,

a2���e = ����� − ���� − ��� + ���a1†����e. �13�

Then noting the relation Qi= �ai+ai†� /�2, Pi= �ai−ai

†� / ��2i�,and Eq. �5� as well as

�� e�� = e������

2�− �a2�,

��� e�� = e�����

2�+ �a1� ,

�� e�� = e���−���

2�+ �a2�,

��� e�� = e���−��

2�+ �a1� ,

�14�

we see, for example,

e��Q1 + Q2

�2= e���− ���a1 + �a2� − i��2 + ��1�

= ���1 + �� �

��−

�����e��

= ���1 − i��

��2�e�� , �15�

which is the first equation in Eq. �11�. In a similar way, we

can check that e�� satisfies the other relations in Eqs. �10�and �11�. Using Eqs. �10� and �11� and noticing the commu-tator �

Q1�Q2�2

,P1�P2

�2�= i, we have

e���Q1 � Q2

�2,P1 � P2

�2� = i��� − ���e�� , �16�

which results in the condition ��−��=1 as shown in Eq.�5�.

III. PROPERTIES OF ��‹e

A. Completeness relation of ��‹e

Next we prove the completeness relation of ���e in Eq.�12�. Using the normally ordered form of the vacuum pro-jector

�00�00� ¬ exp�− a1†a1 − a2

†a2�: , �17�

where the symbol : : denotes the normal product, whichmeans all the bosonic creation operators are standing on theleft of annihilation operators in a monomial of a† and a �19�,and remembering that a normally ordered product of opera-tors can be integrated with respect to c numbers provided theintegration is convergent, we can use Eq. �12� and the IWOPtechnique to perform the following integration

1

�2�2� d2�d2�

4�2 ���ee��

= −��

��� d2�d2�

�2 :exp�����2

�+ ���a1

† − a2� + ����a1 − a2†� − a1

†a1

−����2

�+ ���a1

† + a2� + ����a2† + a1� − ��� + ����a1

†a2† + a1a2� − a2

†a2�ª −��

��� d2�d2�

�2 :exp��

��� + ��a1 − a2

†�����

+ ��a1† − a2�� −

��� − ��a2

† + a1����� − ��a1† + a2���ª :exp�− �a1

†a1 + a2†a2���� − �� + 1�� ª 1, �18�

where we have used the integral formula �20�

� d2�

�e����2+��+���

= −1

�e− ��

� , Re � � 0. �19�

Thus ���e is capable of making up a quantum mechanicalrepresentation.

B. Nonorthogonality of ��‹e

Noticing the overlap

e��z1,z2� = 2�− ���� exp�−�z1�2

2+

����2

2�−

����2

2�

+ ���� + ����z1�exp�−�z2�2

2+ ��� − ���z2

− ��� + ���z1z2� , �20�

where �z�=exp�−�z�2 /2+za†��0� is the coherent state �21,22�and using the overcompleteness relation of coherent states

d2z1d2z2

�2 �z1 ,z2�z1 ,z2�=1, we can derive the inner product

e� ����e, �����e has the same �, �, �, and � as in ���e�,

ENTANGLED STATE FOR CONSTRUCTING A… PHYSICAL REVIEW A 80, 022115 �2009�

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Page 4: Entangled state for constructing a generalized phase-space representation and its statistical behavior

e�����e =� d2z1d2z2

�2 e��z1,z2�z1,z2����e

= − 4����� d2z1d2z2

�2 exp�− �z1�2 + ���� + ����z1

+ ���� + ����z1�� − �z2�2 + ��� − ���z2 + �����

− �����z2� − ��� + ���z1z2 + � �

2�����2 + ����2�

−�

2�����2 + ����2� − ��� + ���z1

�z2�� . �21�

With the aid of the integral formula in Eq. �19�, we performthe integral over d2z1d2z2 in Eq. �21� and finally obtain

e�����e = exp� �

4����� − ���2

−1

4������� − ���� + ���� − ����� +

4�����

− ���2 −�� + ��

4�������� − ����� + ��� − ����� .

�22�

From Eq. �22� one can see that e� ����e is nonorthogonal,only when �=�� and �=�� Eq. �22� reduces to e� ���e=1.

IV. MINIMUM UNCERTAINTY RELATION FOR ��‹e

Due to Heisenberg’s uncertainty principle, it is impossiblethat ���e is the simultaneous eigenvectors of both �Q1−Q2 , P1+ P2� and �Q1+Q2 , P1− P2�. Note that ��� and ��� arerelated to each other by

���� =1

2exp� ��� − ���

2� , �23�

we see

����e =� d2�

���������e =� d2�

2�e����−����/2����e,

����e =� d2�

���������e =� d2�

2�e����−����/2����e,

�24�

which are conjugated each other. It then follows from Eq.�24� that once the value of �Q1−Q2 , P1+ P2� has been mea-sured, one can find the system with any value for �Q1+Q2 , P1− P2�, and vice versa.

In order to see clearly how the state ���e obeys uncertaintyrelation, we introduce two pairs of quadrature phase ampli-tudes for two-mode field

Q� =Q1 � Q2

�2, P� =

P1 � P2

�2, �Q�,P�� = i . �25�

In similar to deriving Eq. �22�, using Eqs. �7�, �12�, and �39��see below�, we calculate the overlap between �� and ���e,

����e =�−��

��exp� ��

2����

�+ ��2

+1

2������ − ���� − ���� − ����� , �26�

and the overlap between �� and ���e

����e =�−��

��exp� ��

2��� �

�− ��2

−1

2������ − ���� − ���� − ����� . �27�

Then employing the completeness relation of ��� and Eq.�26�, we have

Q−� =� d2�

��1�����e�2 = −

�1

�,

Q−2� =� d2�

��1

2�����e�2 =�1

2

�2 −��

2��, �28�

which leads to

P−� =�2

�, P−

2� =�2

2

�2 −��

2��. �29�

It then follows

�Q−2� = Q−

2� − Q−�2 = −��

2��,

�P−2� = P−

2� − P−�2 = −��

2��, �30�

which yields

��Q−2��P−

2� =1

2. �31�

Similarly, using Eq. �27� we can derive

Q+� =�2

�, Q+

2� =�2

2

�2 −��

2��,

P+� =�1

�, P+

2� =�1

2

�2 −��

2��, �32�

and

��Q+2��P+

2� =1

2. �33�

Equations �30�–�33� show that ���e is a minimum uncertaintystate for the two pairs of quadrature operators.

V. WEYL-ORDERED FORM OF ��‹eeŠ��

For a bipartite operator A, we can convert it into its Weylordering form by using the formula �23–25�

LI-YUN HU AND HONG-YI FAN PHYSICAL REVIEW A 80, 022115 �2009�

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Page 5: Entangled state for constructing a generalized phase-space representation and its statistical behavior

A = 4� d2z1d2z2

�2

:

:− z1,− z2�A�z1,z2�exp�2�a1

†a1 + a1z1�

− z1a1†� + 2�a2

†a2 + a2z2� − z2a2

†��:

:, �34�

where the symbol ::

:: denotes the Weyl ordering, �zi� is the

coherent state, −zi �zi�=exp�−2�zi�2�. Note that the order ofBose operators ai and ai

† within a Weyl-ordered product canbe permuted. That is to say, even though �a ,a†�=1, we canhave :

:aa† :: = :

:a†a :

: . Substituting Eq. �12� into Eq. �34� andperforming the integration by virtue of the technique of in-tegration within a Weyl ordered product �IWWOP� of opera-tors �26�, we finally obtain

���ee�� = − 16����� d2z1d2z2

�2

:

:exp�− �z1�2 + ���� + ��� − 2a1

†�z1 + �2a1 − �� − ���z1� − �z2�2 + ��� − �� − 2a2

†�z2

+ �2a2 − ��� + ����z2� − ��� + ����z1

�z2� + z1z2� +

����2

�−

����2

�+ 2a1

†a1 + 2a2†a2� :

:

= 4:

:exp���

����

�+ �a1 − a2

†�����

�+ �a1

† − a2�� +��

��� �

�− �a1 + a2

†��� ��

�− �a2 + a1

†��� :

:, �35�

or

���ee�� = 4:

:exp���

�����1

�+

Q1 − Q2

�2�2

+ ��2

�+

P1 + P2

�2�2� +

��

���� �1

�−

Q1 + Q2

�2�2

+ � �2

�−

P1 − P2

�2�2�� :

:, �36�

which is the Weyl ordering form of ���ee��. We should no-tice the difference between Eq. �35� and Eq. �18�, since theyare in different operator ordering.

VI. MARGINAL DISTRIBUTIONS OF ��‹eeŠ��

The merit of Weyl ordering lies in the Weyl ordering in-variance under similar transformations �27�, which bringsconvenience for us to obtain the marginal distributions of���ee��. From the Weyl-ordered form Eq. �36� we obtain themarginal distributions of ���ee��,

� d2�

����ee�� = −

4���

:

:exp���

���� �1

�−

Q1 + Q2

�2�2

+ � �2

�−

P1 − P2

�2�2�� :

:. �37�

Noting �Q1+Q2 , P1− P2�=0, there is no operator orderingproblem involved in Eq. �37�, so the symbol :

::: in Eq. �37�

can be neglected, i.e.,

� d2�

����ee�� = −

4���

�exp���

���� �1

�−

Q1 + Q2

�2�2

+ � �2

�−

P1 − P2

�2�2�� . �38�

By using the simultaneous eigenstate ��� of the commutativeoperators �Q1+Q2 , P1− P2� in two-mode Fock space �17�

��� = exp�−1

2���2 + �a1

† + ��a2† − a1

†a2†��00�, � = �1 + i�2,

�39�

which satisfies the eigenequations �Q1+Q2����=�2�1���,�P1− P2����=�2�2���, and the complete-orthogonal relation,

� d2�

������ = 1, d2� = d�1d�2, �40�

����� = ����� − ������� − ��� , �41�

we see that the marginal distribution of function �e� ����2 in“� direction” is given by

��� d2�

����ee���� = ��� d2�

����

��� d2�

����ee��� d2��

����������

= −4���

�� d2�

�������2exp���

��� �

�− ��2� , �42�

which is a GAUSSIAN-broadened version of quantal distribu-tion ������2 �measuring two particles’ relative momentumand center-of-mass coordinate�. Similarly, we can obtain an-other marginal distribution by performing the integration d2�over ���ee��,

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� d2�

����ee�� = −

4���

�exp���

�����1

�+

Q1 − Q2

�2�2

+ ��2

�+

P1 + P2

�2�2�� . �43�

By using Eqs. �7�, �8�, and �43� we see that the other mar-ginal distribution of �e� ����2 in “� direction” is

��� d2�

����ee����

= −4���

�� d2�

�������2exp���

����

�+ ��2� , �44�

a GAUSSIAN-broadened version of the distribution ������2�measuring two particles’ relative coordinate and total mo-mentum�. Equations �42� and �44� describe the relationshipbetween wave functions in the e�� representation and thosein EPR entangled state ��� ����� representation, respectively.

VII. ��‹eeŠ�� AS A GENERALIZED HUSIMIOPERATOR

In Ref. �28�, we have derived the Weyl-ordered form ofthe two-mode Wigner operator �w�� ,�� �in its entangledform�

�w��,�� =:

:��a1 − a2

† − ����a1† − a2 − �����a1 + a2

† − ��

��a1† + a2 − ���

:

:, �45�

where �� . . . � denotes delta-function, �w�� ,��=�1�q1 , p1�� �2�q2 , p2�, �= �− ��, �= �+ ��, �= �q1+ ip1� /�2, and �= �q2+ ip2� /�2. Equation �45� indicates that the Weyl quanti-zation scheme, for bipartite entangled system, is to take thefollowing correspondence,

� → �a1 − a2†�, � → �a1 + a2

†� , �46�

then from the form of Eq. �35� we see that the classical Weylfunction corresponding to ���ee�� is

4 exp���

����

�+ ��2

+��

��� �

�− ��2� h��,�� . �47�

Thus the Weyl quantization in this case is expressed as

���ee�� = 4� d2�d2�:

:��a1 − a2

† − ����a1† − a2 − ���

��a1 + a2† − ����a1

† + a2 − ���

:

:exp���

����

�+ ��2

+��

��� �

�− ��2�

= 4� d2�d2��w��,��exp���

����

�+ ��2

+��

��� �

�− ��2� . �48�

In particular, when �=−�=1, and �= �1+� , �= 1

1+� , Eq. �48�becomes

���ee�� → 4� d2�d2��w��,��exp�−1

��� − ��2 − ��� − ��2� ,

�49�

which is the generalization of single-mode Husimi operator�29,30�, so Eq. �48� is a generalized two-mode Husimi op-erator, the result after averaging over a “coarse graining”function exp� ��

�� � �� +��2+ ��

�� � �� −��2� for the Wigner operator

�w�� ,��. On the other hand, Eq. �48� can be considered as aWeyl correspondence formula, in this sense the course grain-ing function can be considered as the Weyl classical corre-spondence of the density operator ���ee��.

We can further check the validity of Eq. �48�, recalling thenormally ordered form of �w�� ,�� �31�,

�w��,�� =1

�2 :exp�− �a1 − a2† − ���a1

† − a2 − ���

− �a1 + a2† − ���a1

† + a2 − ����: , �50�

then substituting Eq. �50� into Eq. �48� to perform the inte-gration yields the normal ordered form of ���ee��,

���ee�� = 4� d2�d2�

�2 :exp�− �a1 − a2† − ���a1

† − a2 − ���

+��

����

�+ ��2

− �a1 + a2† − ���a1

† + a2 − ���

+��

��� �

�− ��2� :

= − 4����:exp��

��� + ��a1 − a2

†����� + ��a1† − a2��

−�

��� − ��a2

† + a1����� − ��a1† + a2���: , �51�

which confirms Eq. �18�.

VIII. WIGNER FUNCTION OF ��‹e

The Wigner function �32–35� plays an important role instudying quantum optics �18,36,37� and quantum statistics�38�. It gives the most analogous description of quantummechanics in the phase-space to classical statistical mechan-ics of Hamilton systems and is also a useful measure forstudying the nonclassical features of quantum states. For abipartite system, the two-mode Wigner operator in the en-tangled state ��� representation is expressed as �31�

�w��,�� =� d2�

�3 �� − ��� + ��e���−���. �52�

Then the Wigner function W�� ,�� of ���e is

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W��,�� = Tr����ee���w��,���

=� d2�

�3 e���� − ��� + ����ee���−���

. �53�

Substituting Eq. �26� into Eq. �53� and using the formula inEq. �19�, we obtain

W��,�� =� d2�

�3 e��� − ��� + ����ee���−���

= −��

��� d2�

�3 exp���

�����2 + ��� −

��

���

+ � �

�− ���� +

2����2���2

+ 2�2���2 + 2����� + ������=

1

�2exp���

����

�+ ��2

+��

��� �

�− ��2� . �54�

Comparing Eq. �54� with Eq. �48�, we see that Wigner func-tion of ���e is just the coarse graining function �up to a con-stant �2��2�, this is another understanding of ���ee��. Thusthe state ���e can be such introduced as its Wigner function isjust 1

�2 exp� ���� � �

� +��2+ ���� � �

� −��2�. From Eq. �54� we see thatthe Wigner function’s marginal distribution in “� direction”is a general GAUSSIAN form exp� ��

�� � �� +��2�, while its mar-

ginal distribution in “�-direction” is exp� ���� � �

� −��2�. When��+��=0 and �=−�=1, Eq. �54� reduces to W�� ,��= 1

�2 e−�� + ��2−�� − ��2, which is just the Wigner function of two-mode canonical coherent state.

In summary, based on the concept of quantum entangle-ment of Einstein-Podolsky-Rosen, we have introduced theentangled state ���e for constructing generalized phase-spacerepresentation, which possesses well-behaved properties.The set of ���e make up a complete and nonorthogonal rep-resentation, so it may have some applications, for examples:�1� It can be chosen as a good representation for solvingdynamic problems for some Hamiltonians which include ex-plicitly the function of quadrature operators Q�, and/or P�;�2� ���ee�� may be considered as a generalized Husimi op-erator, since from Eq. �48� we see that it is expressed assmoothing out the usual Wigner operator by averaging over acoarse graining function exp� ��

�� � �� +��2+ ��

�� � �� −��2�, and the

corresponding generalized distribution function is positivedefinite. �3� The representation ���e may be used to analyzeentanglement degree for some entangled states. �4� The ���estate may be taken as a quantum channel for quantum tele-portation, such channel may make the teleportation fidelityflexible, since it involves adjustable parameters �, �, �, and�. We expect these applications would work in the near fu-ture.

ACKNOWLEDGMENTS

We sincerely thank the anonymous referees for helpfulsuggestions. We would like to acknowledge support from the

National Natural Science Foundation of China under GrantsNo. 10775097 and No. 10874174, and the Research Founad-tion of the Education Department of Jiangxi Province.

APPENDIX: DERIVATION OF EQ. (12)

In this Appendix, we show how to derive Eq. �12�. In ���representation we have

Q1 + Q2

�2��� = − i

��2��� = � �

��−

������� , �A1�

P1 − P2

�2��� = i

��1��� = i� �

��+

������� . �A2�

Therefore, as required by Eqs. �10� and �11�, we see

��

2�� + ��� + �� �

��−

�����e���� = � �

��−

����e���� ,

�A3�

� �

2i�� − ��� − i�� �

��+

�����e���� =� − ��

2i e���� ,

�A4�

and

��

2�� + ��� − �� �

��−

�����e���� =� + ��

2 e���� ,

�A5�

� �

2i�� − ��� + i�� �

��+

�����e���� = i� �

��+

����e���� ,

�A6�

Combining Eqs. �A3�–�A6� yields

��� + 2��

����e���� = �e���� , �A7�

���� − 2��

���e���� = ��

e���� , �A8�

���� + 2��

���e���� = 2

�� e���� �A9�

��� − 2��

����e���� = − 2�

��� e���� . �A10�

The solution to Eqs. �A7�–�A10� is

e���� = C exp� ��

2����

�+ ��2

+1

2������ − ��� − ���� − ������ , �A11�

where C is the normalization constant determined by

e� ���e=1.

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Using the completeness relation of EPR entangled state Eq. �9� and the integral formula in Eq. �19�, we obtain

e�� =� d2�

� e������

= C00�� d2�

�exp�−

1

2���2 + ��a1 − �a2 + a1a2�

exp� �

2����� + ���2 +

1

2������ − ��� − ���� − ������

= 00�C exp�����2

2�−

����2

2�+ ���� + ����a1 + ��� − ���a2 − ��� + ���a1a2� , �A12�

which is Eq. �12� while C=2�−����.

�1� E. P. Wigner, Phys. Rev. 40, 749 �1932�.�2� Wolfgang P. Schleich, Quantum Optics in Phase Space �Wiley-

VCH, Berlin, 2001�.�3� N. L. Balazs and B. K. Jennings, Phys. Rep. 104, 347 �1984�.�4� D. Chruściński and K. Młodawski, Phys. Rev. A 71, 052104

�2005�.�5� Q. S. Li, G. M. Wei, and L. Q. Lü, Phys. Rev. A 70, 022105

�2004�.�6� Go. Torres-Vega and J. H. Frederick, J. Chem. Phys. 98, 3103

�1993�.�7� Go. Torres-Vega and J. H. Frederick, J. Chem. Phys. 93, 8862

�1990�.�8� F. Bopp, Werner Heisenberg und die Physik unserer Zeit

�Vieweg, Braunschweig, 1961�.�9� C. Jaffé and P. Brumer, J. Chem. Phys. 82, 2330 �1985�; C.

Jaffé, S. Kanfer, and P. Brumer, Phys. Rev. Lett. 54, 8 �1985�.�10� M. J. Davis, J. Phys. Chem. 92, 3124 �1988�.�11� S. K. Gray, J. Chem. Phys. 87, 2051 �1987�.�12� Klaus B. Møller, Thomas G. Jørgensen, and Go. Torres-Vega,

J. Phys. Chem. 106, 7228 �1997�.�13� L. Y. Hu, H. Y. Fan, and H. L. Lu, J. Chem. Phys. 128, 054101

�2008�.�14� H. Y. Fan, H. L. Lu, and Y. Fan, Ann. Phys. 321, 480 �2006�;

H. Y. Fan, J. Opt. B: Quantum Semiclassical Opt. 5, R147�2003�.

�15� A. Wünsche, J. Opt. B: Quantum Semiclassical Opt. 1, R11�1999�.

�16� A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777�1935�.

�17� Fan Hong-yi and J. R. Klauder, Phys. Rev. A 49, 704 �1994�;Fan Hongyi and Chen Bozhan, ibid. 53, 2948 �1996�; FanHong-yi and Ye Xiong, ibid. 51, 3343 �1995�.

�18� D. F. Walls and G. Milburn, Quantum Optics �Springer, Berlin,1994�.

�19� W. H. Louisell, Quantum Statistical Properties of Radiation�Wiley, New York, 1973�.

�20� R. R. Puri, Mathematical Methods of Quantum Optics�Springer-Verlag, Berlin, 2000�, Appendix A.

�21� R. J. Glauber, Phys. Rev. 130, 2529 �1963�; 131, 2766 �1963�.�22� J. R. Klauder and B. S. Skagerstam, Coherent States �World

Scientific, Singapore, 1985�.�23� H. Weyl, Z. Phys. 46, 1 �1927�.�24� H. Weyl, The Classical Groups �Princeton University Press,

New Jersey, 1953�.�25� H. Y. Fan, J. Phys. A 25, 3443 �1992�.�26� H. Y. Fan, Ann. Phys. 323, 500 �2008�.�27� H. Y. Fan, Ann. Phys. 322, 866 �2007�.�28� H. Y. Fan and Y. Fan, Int. J. Mod. Phys. A 17, 701 �2002�;

H.-y. Fan, Mod. Phys. Lett. A 15, 2297 �2000�.�29� K. Husimi, Proc. Phys. Math. Soc. Jpn. 22, 264 �1940�.�30� H. Y. Fan and Q. Guo, Phys. Lett. A 358, 203 �2006�.�31� H. Y. Fan, Phys. Rev. A 65, 064102 �2002�.�32� V. V. Dodonov and V. I. Man’ko, Theory of Nonclassical States

of Light �Taylor & Francis, New York, 2003�.�33� G. S. Agarwal and E. Wolf, Phys. Rev. D 2, 2161 �1970�; 2,

2187 �1970�; 2, 2206 �1970�.�34� V. Bužek and P. L. Knight, Prog. Opt. 34, 1 �1995�; V. Bužek,

C. H. Keitel, and P. L. Knight, Phys. Rev. A 51, 2575 �1995�.�35� R. F. O’Connell and E. P. Wigner, Phys. Lett. A 83, 145

�1981�; M. Hillery, R. F. O’Connell, M. O. Scully, and E. P.Wigner, Phys. Rep. 106, 121 �1984�.

�36� K. Vogel and H. Risken, Phys. Rev. A 40, 2847 �1989�; V. I.Tatarski�, Sov. Phys. Usp. 26, 311 �1983�.

�37� D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, Phys.Rev. Lett. 70, 1244 �1993�.

�38� R. P. Feynman, Statistical Mechanics �Benjamin, New York,1972�.

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