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Enhancing entanglement of entangled coherent states by single-mode coherent superposition of photon subtraction and addition Li-Yun Hu a,∗ , Hao-Liang Zhang a , Yin-Quan Hu a , Zhi-Ming Zhang b a b

Department of Physics, Jiangxi Normal University, Nanchang 330022, China Laboratory of Nanophotonic Functional Materials and Devices, SIPSE & LQIT, South China Normal University, Guangzhou 510006, China

a r t i c l e

i n f o

Article history: Received 3 February 2013 Accepted 2 June 2013

Keywords: Entangled coherent states Coherent superposition Photon subtraction/addition Enhancing entanglement

a b s t r a c t We introduce a new entangled quantum state generated by applying single-mode coherent superposition of photon subtraction and addition (a† cos + a sin )m to the entangled coherent state |± (˛, 0), and then investigate the entanglement properties affected by coherent superposition operation. It is shown that this operation can be applied to enhance the entanglement of the state |+ (˛, 0). In addition, the effects of the coherent operation is better to improve the entanglement than that of the creation operation (a†m ) for |+ (˛, 0) in a small-amplitude regime and for |− (˛, 0) in any regime. © 2013 Elsevier GmbH. All rights reserved.

1. Introduction Quantum entanglement with continuous variable (CV) has been a central role in quantum information processing [1], such as CV quantum teleportation, dense coding, and quantum cloning. Generally, the higher entanglement degree of quantum states are required to realize the requirement of quantum information protocols for long-distance communication. In fact, a source of practically available is characteristics of an ﬁnite degree of entanglement, thus it is not trivial to enhance the entanglement of given entangled states. For this purpose, there have been suggestions and realizations for engineering the quantum state, which are plausible ways to conditionally manipulate a nonclassical state of an optical ﬁeld by subtracting or adding photons from/to quantum states. For example, photon subtraction or addition has been applied to improve entanglement between Gaussian states [2,3], loophole-free tests of Bell’s inequality [4,5], and quantum computing [6]. Recently, coherent superposition of photon subtraction and addition, i.e., the ta† + ra operator is proposed for quantum state engineering, which can transform a classical state to a nonclassical one [7]. Then the local coherent superposition is performed on two-mode squeezed vacuum for enhancing quantum entanglement or non-Gaussian entanglement distillation [8–10]. As another example, local photon subtraction and addition are also used to enhance quantum linear

∗ Corresponding author. E-mail address: [email protected] (L.-Y. Hu). 0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2013.06.039

ampliﬁer [11,12] in terms of intensity gain, quantum ﬁdelity and the Holevo variance of phase. Thus these local coherent operations (superposition) at the level of quantum operation can provide a useful tool in quantum information processing. In this work, we apply the single-mode coherent superposim tion operation (a† cos + a sin ) to two-mode entangled coherent states (ECSs) |± (˛, 0) (see Eq. (1) below), and then we investigate the entanglement properties of coherent superposition ECSs (CSECSs) |± (˛, m). The photon-subtraction (am , = /2) ECS and the photon-addition (a†m , = 0) ECS can be considered as two special cases of the CS-ECSs. Using the concurrence, we calculate the degree of entanglement. It is found that (i) single-mode coherent superposition operation is useful to remarkably improve the entanglement of ECSs | + (˛, 0) and that (ii) the degree of entanglement of the CS-ECSs (|+ (˛, m)) is larger than that of single-mode photonadded ECSs (a†m |+ (˛, 0)) in a certain amplitude regime, while the degree of entanglement of |− (˛, m) is larger than that of the single-mode photon-added ESCs (a†m |− (˛, 0)) in any regime. 2. Entangled coherent states by a coherent superposition of photon substraction and addition First, let us brieﬂy review the ECSs, deﬁned by [13] |± (˛, 0) = N±,˛ (|˛, ˛ ± | − ˛, −˛) ,

(1)

where |˛, ˛ ≡ |˛a ⊗ |˛b with |˛a and |˛b being the usual coherent states in a and b modes, respectively, and (N± (˛, 0))−2 = 2[1 ± 2 e−4|˛| ] is the normalization constants. Theoretically, one can introduce a new kind of continuous-variable entangled states on the

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L.-Y. Hu et al. / Optik 125 (2014) 61–63

basis of coherent entangled states, by operating repeatedly coherent superposition operation (a† cos + a sin ) on the a mode of ECSs (CS-ECSs), expressed as |± (˛, m) = N±,˛,m (a† cos + a sin )

m

× (|˛, ˛ ± | − ˛, −˛), (2)

where N±,˛,m represents the normalization factor. In particular, when = 0, Eq. (2) becomes |± (˛, m) = N± (˛, m) a†m (|˛, ˛ ± | − ˛, −˛), whose properties of entanglement is discussed in detail in Ref. [14]; while for = /2, Eq. (2) reduces to |± (˛, m)=N± (˛, m) am (|˛, ˛ ± | − ˛, −˛), corresponding to the two-mode “Schröinger cat” states. Next, we shall derive the normalization factor N±,˛,m , which is important for further discussing the properties of statistics and entanglement for quantum states. Using the formula eA+B = eA eB e−[A,B]/2 = eB eA e[A,B]/2 , which is valid for [A, [A, B]] = [B, [A, B]] = 0, one can ﬁnd

a† cos + a sin 2

= e−1/2|˛|

m

|˛

m

∂ 1/4s2 sin 2+˛s sin (s cos +˛)a† e e |0|s=0 , ∂ sm

(3)

thus the normalization factor N±,˛,m can be calculated as [N±,˛,m ]−2 = 2 (A1 ± A2 ) ,

(4)

where |a , | a and |b , |ıb are normalized states of system a and b, respectively, and the normalization constant is given by

A1 =

C=

22m−l l![(m − l)!]

In particular, when = ± , the concurrence C reduces to

2

A2 = e−4|˛|

m m (−1)m−l (m!)2 sin 2 2

l=0

22m−l l![(m − l)!] tanl

|Hm−l ˇ |2 ,

and Hm (x) is single-variable√ Hermite polynomials, as well as ˛ =√ (˛∗ sin + ˛ cos )/(i sin 2) and ˇ = (˛∗ sin − ˛ cos )/(i sin 2). It is noted that when m = 0, Eq. (4) becomes −2 the usual ECSs in Eq. (1) with N±,˛,0 = 2[1 ± exp(−4|˛|2 )], and for the case of = 0, corresponding to the single-mode excited ECSs (SE-ECSs), Eq. (4) reduces to [N±,˛,m ]−2 = 2m![Lm (−|˛|2 ) ± exp(−4|˛|2 )Lm (|˛|2 )], as expected [14], where Lm (x) is the m-order Laguerre polynomial. For convenience, we introduce the following normalized states: −1/2

|˛m = A1

a sin + a† cos

m

|˛,

(6)

then the CS-ECSs |± (˛, m) can be rewritten as 1/2

|± (˛, m) = N±,˛,m A1

(|˛m ⊗ |˛ ± | − ˛m ⊗ | − ˛) .

(7)

3. The degree of entanglement for the CS-ECSs

| = N |a ⊗ |b + | a ⊗ |ıb ,

1 ± Re p1 p∗2

,

(11)

which only depends on the overlaps | and |ı. Note that the systems in Eq. (7) are two-state system essentially, thus the entanglement can be measured by using the concurrence. Accordingly, the overlaps are obtained as A2 2|˛|2 e , A1

p1

= m ˛| − ˛m =

p2

= − ˛|˛ = e−2|˛| .

(12)

2

Substituting Eq. (12) into Eq. (11) yields 2

A21 − A22 e4|˛| A1 ± A2

2

1 − e−4|˛|

1/2

,

(13)

where A1 and A2 are deﬁned in Eq. (5). Eq. (13) is the analytical expression of concurrence for the CS-ECSs. 2 In particular, when m = 0, leading to A1 = 1 and A2 = e−4|˛| , then 2 we have C− = 1, and C+ = tanh(2|˛| ), which indicates that |− (˛, 0) is a maximum entangled state, and that |+ (˛, 0) is a partly entangled state whose concurrence increases with the increasing amplitude |˛| [14]. On the other hand, when = 0, corresponding to the single-mode excited ECSs (SE-ECSs), then A1 = Lm (− |˛|2 ), and 2 A2 = e−4|˛| Lm (|˛|2 ), thus Eq. (13) becomes 2

C±,=0 =

2 (−|˛|2 ) − e−4|˛| L2 (|˛|2 )] [Lm m

1/2

Lm (−|˛|2 ) ± e−4|˛| Lm (|˛|2 ) 2

,

(14)

which is agreement with the result in Ref. [14]. While for = /2, corresponding to the case of photon-subtraction (am ), A1 = 2 2 |˛|2m , A2 = (−1)m |˛|2m e−4|˛| , then C±,=/2 = (1 − e−4|˛| )/[1 ± (−1)m e−4|˛| ]. This implies that the concurrence C±,=/2 keep unchanged for the even number photon-subtraction case (i.e., the state | ± is not changed); while for the odd number photonsubtraction case, the states shall present a transformation from the maximum entangled state to the partially entangled state, and vice verse. 2

4. Results and discussions

In this section, we calculate the amount of entanglement of the CS-ECSs and investigate the inﬂuence of the single-mode coherent superposition operation on the entanglement of the CS-ECSs. For continuous-variables-type entangled states like (7), the degree of quantum entanglement of the bipartite entangled states can be measured in terms of the concurrence [15–17]. The concurrence equals one for a maximally entangled state. For two systems involving nonorthogonal states deﬁned as [15–17]

(1 − |p1 |2 )(1 − |p2 |2 )

C± =

2

(5)

(10)

||2 + | |2 + 2Re ∗ p1 p2

C± = |Hm−l (˛) | ,

1/2 . ∗

2||| | (1 − |p1 |2 )(1 − |p2 |2 )

2

l=0

(9)

with two complex parameters and , and p1 = a | a , p2 = b ı|b . The overlaps | and |ı are nonzero. By considering the general entangled nonorthogonal state as a state of two logical qubit, and using the Schmidt form of | , the concurrence of the entangled state (8) can be obtained as [18]

where we have set m m (m!)2 sin 2cotl

N −2 = ||2 + | |2 + 2Re ∗ p1 p∗2 ,

(8)

In order to see clearly the inﬂuence of the concurrence with parameters m and , C± for the state |± (˛, m) as a function ˛ are shown in Fig. 1. Here, for convenience, we plotted only with ˛ being positive real. From Fig. 1 with a given value of = /4, one can see that C± (˛, m) increases with the increase of ˛ for the given parameter m. Especially, the concurrence C± (˛, m) tends to unit for the larger ˛. In addition, for different m, C+ (˛, m) increases with the increase of m; while C− (˛, m) decreases with the increase of m. Thus the coherent superposition operation can be applied to enhance the entanglement of the state |+ (˛, m). The case is not true for the state |− (˛, m).

L.-Y. Hu et al. / Optik 125 (2014) 61–63

α

α

Fig. 1. Concurrence of entanglement of |± (˛, m) as a function of ˛ (considered as a real number) for different m values with = /4.

63

Furthermore, we should mention that the entanglement can be modulated by different . For instance, for the CS-ECSs |+ (˛, 1), it is found that the optimal value of C+ (˛, 1) is unit for any ˛ value when = /2 (see Fig. 3(a)). This is due to the fact that the photon-subtraction operation converts the state |+ (˛, 0) to the maximum entangled state |− (˛, 0). Thus, generally, |+ (˛, 1) can be considered a “middle” entangled state. However, for the other CS-ECSs, say |+ (˛, 2), the optimal value of C+ (˛, 2) is no longer unit for any ˛ value when equal a certain optimal value (not equal /2, see Fig. 3(b)). 5. Conclusions

Fig. 2. Concurrence of entanglement of |± (˛, m) and single-mode photon excited ECSs as a function of ˛ (considered as a real number) for different m values with = /4.

In this paper, we have introduced a new entangled quantum state |± (˛, m) (CS-ECSs), which is generated by coherent superposition of photon subtraction and addition (a† cos + a sin )m of single-mode light ﬁeld. Then we investigated the properties of entanglement how is affected by coherent superposition. By converting all components to its normalized form and using the concurrence as a measurement for the CS-ECSs, it is found that the coherent superposition operation can be used to improve the entanglement of the state |+ (˛, 0). In addition, the effects of the coherent operation is better to improve the entanglement than that of the creation operation (a†m ) for |+ (˛, 0) in a smallamplitude regime and for |− (˛, 0) in any regime. Thus |± (˛, m) can be considered as a new entanglement resource which can be applied extensively in quantum information processing with continuous-variable, such as quantum teleportation, dense coding, and quantum cloning. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant Nos. 11264018 and 60978009), the Major Research Plan of the National Natural Science Foundation of China (Grant No. 91121023), the “973” Project (No. 2011CBA00200) and the Natural Science Foundation of Jiangxi Province of China (No. 20132BAB212006), as well as the Young Talents Foundation of Jiangxi Normal University. References

Fig. 3. Concurrence of entanglement of |+ (˛, m) as a function of ˛ (considered as a real number) and for different m values: (a) m = 1 and (b) m = 2.

On the other hand, we make a comparison of the entanglement properties between the CS-ECSs and the SE-ECSs (see Fig. 2). From Fig. 2(a) and (b), one can see that the effects of the coherent operation are more prominent than those of the mere photonaddition, particularly in a small-amplitude regime (Fig. 2(a)) for |+ (˛, 0) with a given m value. The threshold value, for example m = 1, is about ˛ ≈ 0.55, which decreases with the increasing m value (Fig. 2(c)). While for the state |− (˛, 0), a maximum entangled state, although both two operations (CS and SE) reduce the entanglement, the entanglement of the CS-ECSs is always larger than that of the SE-ECSs for arbitrary ˛ (Fig. 2(c) and (d)) and a given m value. These results show that applying a coherent superposition operator of photon subtraction and addition on the ECSs may be better to enhance the quantum entanglement than using only a creation operator on the ECSs.

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