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Unitary equivalence of different bipartite entangled states under unitary transformations Shuai Wang a,∗ , Li-Li Hou a , Xue-Fen Xu b a b

School of Mathematics and Physics, Changzhou University, Changzhou 213164, PR China Department of Basic Course, Wuxi Institute of Technology, Wuxi 214121, China

a r t i c l e

a b s t r a c t

i n f o

Article history: Received 16 September 2013 Accepted 15 May 2014

The unitary equivalence of different bipartite entangled states with continuous variables under unitary transformations are investigated. With the help of the technique of integration within an ordered product of operators, the corresponding unitary operators are also derived. These results may deepen people’s understanding to the various bipartite entangled states, and enrich the representations and transformations theory in quantum mechanics. © 2014 Elsevier GmbH. All rights reserved.

PACS: 03.65.Ud 42.50.Dv 03.65.−w Keywords: Entangled states Unitary equivalence IWOP technique Unitary transformation

1. Introduction Entanglement plays a crucial role in quantum information processing and quantum computation [1]. The concept of quantum entanglement concept originated in the paper of Einstein, Podolsky and Rosen (EPR) [2] arguing on the incompleteness of quantum mechanics. According to the original idea of EPR that the twoparticle relative coordinate commutes with their total momentum [X1 − X2 , P1 + P2 ] = 0, Fan and Klauder [3] ﬁrst introduced the original entangled states | = 1 + i2

1 † † † † | = exp − ||2 + a1 − ∗ a2 + a1 a2 |00, 2 †

(1)

†

where a1 and a2 are two bosonic creation operators. | is the common eigenvector of X1 − X√ 2 and P1 + P2 , and obeys√the eigenvector equations, (X1 − X2 )| = 21 |, (P1 + P2 )| = 22 |. On the other hand, the common eigenvector | = 1 + i 2 of X1 + X2 and P1 − P2 in two-mode Fock space is

1 † † † † | = exp − ||2 + a1 + ∗ a2 − a1 a2 |00, 2

(2)

while | is the common eigenvector of X1 + X2 √and P1 − P2 , and obeys √ the eigenvector equations, (X1 + X2 )| = 21 |, (P1 − P2 )| = 22 |. Since then, some kinds of the bipartite entangled states [4–9], such as thermal entangled states, intermediate entangled states, parameterized entangled states and coherententangled states, have been introduced. Because of their practical applications in quantum optics and quantum information, they have brought considerable attention of physicists. For example, a new intermediate entangled state | ; [8] has been constructed to develop the squeezing transformation, Hadamard transformation, Fresnel transformation and Radon transformation. Another new parameterized entangled state |s,r [9] was also introduced to derive a two-mode entangled Fresnel operator, to formulate quantum correspondence theory of the classical circular harmonic correlator, and to develop quantum tomography of entangled systems. These bipartite entangled states possess the completeness relation and the orthogonal property. Thus, they form new complete and orthogonal quantum representations. On the other hand, the unitary equivalence of quantum states play important roles in the theory of quantum information [1]. In quantum mechanics, two quantum states and are said to be unitarily equivalent if there exists a unitary operator U such that = U † U

∗ Corresponding author. E-mail address: [email protected] (S. Wang). http://dx.doi.org/10.1016/j.ijleo.2014.06.078 0030-4026/© 2014 Elsevier GmbH. All rights reserved.

or | = U † |˚. (for pure states)

(3)

It is known that the coordinate eigenstate is unitarily equivalent to the momentum eigenstate by the Fourier transform operator

S. Wang et al. / Optik 125 (2014) 5596–5599 †

(e−i a a ). The unitarily equivalent relation between the radiation ﬁeld eigenstate and the coordinate eigenstate has been naturally established by the phase-shifting operator [10]. Nha proved that ordinary intelligent states are unitary equivalence to generalized intelligent states by a rotation operator [11]. For the local unitary equivalence of multipartite entangled pure states, one can refer to those recent results in Refs. [12,13]. Considering the various bipartite entangled states, we can put forward a interesting question: given two bipartite entangled states | and |˚, whether or not | can be transformed into |˚ by unitary operation? That is to say, those new quantum entangled states may be unitary equivalence to each other. 2. Intermediate entangled state and two-mode phase-shifting operator

d2 ||= .

F=

(4)

Using Eqs. (1) and (2), as well as the technique of integration within an ordered product of operators (IWOP) [14,15], we have

F

=

d2 † † : exp[−||2 + (a1 + a2 ) + ∗ (a1 − a2 ) † †

†

(5)

†

+a1 a2 − a1 a2 − a1 a1 − a2 a2 ] : †

Re() < 0

(6)

The parity operator can induce the following unitary transformation F (X1 − X2 )F = X1 + X2 , F † (P1 + P2 )F = P1 − P2 ,

, √ 2

Pi =

. √ 2i

d2 ||= | = F † |= ,

(9)

where the orthogonal relation | = ı2 ( − ) is used. Thus, we can say that | is unitary equivalence to | by the parity operator. Next, we prove that the intermediate entangled state | ; is unitary equivalence to the original entangled state |. Based on the commutative relation

|00

√

22 |; .

(11)

(12)

d2 † † |; | = exp[i(a1 a1 + a2 a2 )],

F1 =

(13)

which is the two-mode phase-shifting operator [17]. We note that F1 rotates the quadratures †

F1 Xi F1 = Xi cos + Pi sin ,

(14)

† F1 Pi F1

(14)

= Pi cos − Xi sin ,

which are the rotated transformations of X and P in phase space. Naturally, if both sides of Eq. (13) are multiplied by |, we have d2 |; || = F † |

(15)

Therefore, | ; is unitary equivalence to | by the unitary operator † † exp[i(a1 a1 + a2 a2 )], and they have the same set of eigenvalues. As the application of unitarily equivalent relation between | ; and |, squeezing operator with complex parameter in the intermediate entangled representation | ; can be rewritten as

S(r, ) =

d2 | ; ; | = F † S(r)F

(10)

= =

(16)

d2

d2 d2 ∗ − ∗ ∗ exp |; ; | 2 2 2

F † U()F

where U() =

d2 2

d2

exp

∗ −∗ ∗ 2

(17)

|| is the Hadamard

operator in the entangled state representation | [19]. It can be seen that the rotated single-mode squeezed operator and the rotated Hadamard operator in | ; representation can be obtained by a rotation transformation in the | representation. 3. Parameterized entangled state and two-mode Fresnel operator †

Recently, considering the commutative relation [a1 + ia2 , a1 −

† ia2 ]

= 0, Meng et al. [9] constructed the following unity of the Gaussian operator integration within normal ordering

[(X1 − X2 ) cos − (P1 − P2 ) sin , (X1 + X2 ) cos + (P1 + P2 ) sin ] = 0,

2

For our purpose, we introduce the following non-symmetric integration projective operator

(8)

From Eq. (4), we can immediately obtain that there exists a unitarily equivalent relation between | and |. If both sides of Eq. (4) are multiplied by |, we have | =

which is the common eigenvector of [(X1 − X2 ) cos − (P1 − P2 ) sin ] and [(X1 + X2 ) cos + (P1 + P2 ) sin ], i.e., √ [(X1 − X2 ) cos − (P1 − P2 ) sin ]|; = 21 |; ,

(7)

where † ai − ai

† †

1 2 † −i † † † − || + a e − ∗ a2 e−i + a1 a2 e−2i 1 =e 2 |00,

U(, )

† ai + ai

†

where S(r) = | | squeezing operator with real squeezing parameter [18]. While the two-mode continuous rotated Hadamard operator in | ; can be rewritten as

†

Xi =

†

−i(a1 − ∗ a2 ) sin + a1 a2 cos − i sin

|; =

which is just the parity operator in quantum mechanics [16], and the sign : : denotes the normal ordering of operator. In deriving Eq. † † (5), we have used the operator identity e a a =: exp[(e − 1)a2 a2 ] :, as well as the integral formula d2 z |z|2 +z+z∗ 1 e = − exp − ,

1 † † = exp − ||2 + (a1 − ∗ a2 ) cos 2

|;

†

=: exp[−2a2 a2 ] := exp[ a2 a2 ],

Xu et al. [8] constructed a intermediate entangled state. The explicit expression of | ; in two-mode Fock space is

[(X1 + X2 ) cos + (P1 + P2 ) sin ]|; =

In order to answer the above question, we ﬁrst make a brief review about the original entangled states [3] and prove that | is unitary equivalence to |. We introduce the following nonsymmetric integration projective operator

5597

d2 † † : exp[−( ∗ − a1 − ia2 )( − a1 + ia2 )] := 1.

(18)

5598

S. Wang et al. / Optik 125 (2014) 5596–5599

After decomposing the exponential function in Eq. (18) into a pair of mutually conjugate vector, they obtained

1

†

†

† †

| = exp − ||2 + a1 − i ∗ a2 + ia1 a2 |00, 2

(19)

where = 1 + i 2 . Actually, | is also the common eigenstate of X1 − P2 and P1 − X2 , √ √ (X1 − P2 )| = 21 |, (P1 − X2 )| = 22 | (20)

Because |s,r posses the orthogonal property s,r |s,r = ı2 ( − )

U(s, r)

d2 † ||= = exp −i a2 a2 2

F2 =

†

†

(22)

Therefore, | is unitary equivalence to the original entangled state | by the Fourier transformation operator

d2 † ||= | = F2 |= .

| =

(23)

2

| = exp − | | 2

† + a1

† + i ∗ a2

† † − ia1 a2

|00

(24)

a1 → s a1 − ra2 ,

∗

a2 → s a2 − ra1 ,

(25)

they further constructed new parameterized entangled states |s,r [9]

|s,r =

†

+

r + is † † |00 ∗a a s∗ + ir 1 2

(26) ∗

†

where ss∗ − rr∗ = 1. |s,r is the eigenstate of [(s∗ + ir )a1 − (s + ir)a2 ] †

∗

and [(s − ir)a1 − (r ∗ − is )a2 ], i.e., †

∗

[(s∗ + ir )a1 − (s + ir)a2 ]|s,r = |s,r , †

∗

[(s − ir)a1 − (r ∗ − is )a2 ]|s,r = ∗ |s,r .

(27)

Introducing the real parameters A, B, C, D through the relation s=

1 [A + D − i(B − C)], 2

r=

1 [B + C + i(A − D)], 2

s

(31)

a1 a2 ,

which is just the two-mode Fresnel operator because it corresponds to the classical optical Fresnel transform. U(s, r) satisfy U(s, r)U† (s, r) = 1, and is a unitary operator. The operator U(s, r) induces the transform in Eq. (25), and leads to †

†

∗

U † (a1 + ia2 )U = (s∗ + ir )a1 − (s + ir)a2 , †

†

∗

U † (a1 − ia2 )U = (s − ir)a1 − (r ∗ − is )a2

(32)

(33)

Based on Eqs. (33) and (32), we can see that |s,r is unitary equivalence to | by the two-mode Fresnel operator. In addition, from Eqs. (23) and (33) we further have †

|s,r = U † (s, r)F2 |= .

(34)

Therefore, |s,r is unitary equivalence to the original entangled state | by the unitary operator U(s, r)F2 , and they have the same set of eigenvalues. 4. Conclusion

†

∗ a1 ∗a 1 s∗ − ir 2 − ∗ 2∗ ∗ exp − ∗ || + ∗ ∗ s + ir s + ir 2(s + ir ) r − is ∗

r∗

|s,r = U † (s, r)|.

which is also unitary equivalence to | by the Fourier transformation operator. Based on the |, by simultaneously making the replacements ∗

Thus, we can obtain that

Similarly, the common eigenstate | of X1 + P2 and P1 + X2 ,

1

× exp

(21)

F2 (P1 + P2 )F2 = P1 − X2 .

d2 | s,r |

which is just the single-mode Fourier transformation operator. And F2 induces the unitary transform F2 (X1 − X2 )F2 = X1 − P2 ,

=

r † † 1 † † = exp − a1 a2 exp (a1 a1 + a2 a2 + 1) ln s s

We can easily prove that

2

and d |s,rs,r | = 1, |s,r forms a new quantum representation. In order to prove that |s,r is unitary equivalence to |, similarly we introduce the following non-symmetric integration projective operator

(28)

In summary, with the help of the IWOP technique, we prove that a kind of bipartite entangled states are unitary equivalence to each other. The intermediate the bipartite entangled state | ; is unitary equivalence to the original entangled state | by a twomode phase-shifting operator. While the parameterized entangled state |s,r is unitary equivalence to | by unitary operator U(s, r)F2 , where U(s, r) is the two-mode Fresnel operator, while F2 is the Fourier transformation operator. Since they are unitary equivalence to each other, they have the same set of eigenvalues. These results may deepen people’s understanding to the various continuousvariable bipartite entangled states. And these results also show that the IWOP technique is a useful tool to deal with the quantum compute and quantum representations transformation. Acknowledgements

where ss∗ − rr∗ = 1 becomes AD − BC = 1, |s,r becomes

|s,r

†

†

a1 ∗ a2 1 D − iC exp − ||2 + = − A + iB 2(A + iB) A + iB B − iA

B + iA † † a a |00, + A + iB 1 2

(29)

which is also the common eigenstate of [A(X1 − P2 ) − B(P1 + X2 )] and [A(P1 − X2 ) + B(X1 + P2 )], i.e., √ [A(X1 − P2 ) − B(P1 + X2 )]|s,r = 21 |s,r [A(P1 − X2 ) + B(X1 + P2 )]|s,r =

√

22 |s,r

(30)

Supported by the National Natural Science Foundation of China under Grant No. 11174114, the Natural Science Foundation of Jiangsu Province of China under Grant No.SBK2014041321, and the Natural Science Foundation of Wuxi Institute of Technology of China (Grant No. 401301293). References [1] M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, 2000. [2] A. Einstein, B. Podolsky, N. Rosen, Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47 (1935) 777–780. [3] H.Y. Fan, J.R. Klauder, Eigenvectors of two particles’ relative position and total momentum, Phys. Rev. A 49 (1994) 704–707.

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