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1、精品论文 型纠缠相干态的产生及贝尔型不等式的研究宋思谕1,2 , 王书浩1,2, 龙桂鲁1,21 清华大学物理学院低维量子物理国家重点实验室，北京1000842 清华大学清华信息科学技术国家实验室，北京100084 摘要：我们提出了一种新的纠缠相干态 - 型纠缠相干态的产生方案。这个过程是通过克尔介 质和线性光学器件来实现的。基于光子对称性测量和光子阈值测量，我们研究了 型纠缠相 干态的量子非局域性。对于贝尔型不等式，它显示出强的违背。 关键词：纠缠相干态，态产生，贝尔型不等式中图分类号： O43Generation of the -type entangled coherent state
2、and violations of Bell-type inequalitySi Yu Song1,2, Shuhao Wang1,2, Gui Lu Long1,21 State Key Laboratory of Low-Dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing, 1000842 Tsinghua National Laboratory For Information Science and Technology, TsinghuaUniversity, Beiji
3、ng, 100084Abstract: We propose a protocol to generate a new type of state, called the -type entangled coherent state. The process is realized by using one -cross-Kerr medium and linear optical components. The quantum non-locality of the -type entangled coherent state is investigated based on photon-
4、parity and photon-threshold measurements. We show that stronger violations of Bell-type inequality appear in the -type entangled state.Key words: entangled coherent states, state generation, Bell-type inequality0 INTRODUCTIONEntanglement is a key feature in quantum mechanics, which contributes to th
5、e non-local and non-classical behavior of quantum systems. Entanglement has great potential in quantum information processing, including quantum cryptography 1, 2, 3, 4, superdose coding 5, 6, 7,基金项目： National Natural Science Foundation (11175094)，National Basic Research Program (2009CB929402), Na-
6、tional Basic Research Program (2011CB9216002), the SRFDP under Grant (20110002110007)作者简介： Song Si Yu(1986-)，female，PHD，major research direction：quantum information. Correspondence author：Long Gui Lu (1962-)，male，professor，major research direction：quantum information.- 14 -quantum teleportation 8, 9
7、, 10, 11, 12, 18, 19, 20, 17, 22, 13, 14, 21, 16, 15, and quantum computation 28, 23, 24, 25, 26, 27. The Bell inequality was rst proposed in 1964 29, and violations of this inequality can be regard as evidence of entanglement in two-qubit systems 30. The Bell-Merman inequality 31 is the multi-qubit
8、 version of the Bell inequality in Ref. 29. This inequality is based on the assumption that every party is allowed to choose between two measurement settings. The Bell-type inequality for continuous variable entangled states of two-mode electromagnetic elds are given by Banaszek and Wdkiewicz, who s
9、how that such inequality can be obtained through a Wigner distribution function 32. The multi-mode continuous variable states have been thoroughly investigated thereafter 33, 26.Entangled photons are important information carriers. The discrete degree of freedoms of photons, for instance, the polari
10、zation, are typically used for information processing. Such entangled photons can be generated through parametric down conversion. However, another realization of information carriers is facilitated by coherent superposed states (CSSs)|CSSEven/Odd = N (| | ),(1)where the subscript Even/Odd correspon
11、ds to the even and odd CSSs, and N is a normalization21constant equal to N = 2(1 + e2| ) 2 . The encoding method of entangled coherent states(ECSs) 23 using CSSs as qubits is similar to that of entangled discrete photon system. Themethod has the advantage of being robust against decoherence 34, thus
12、 the state has been widely used 18, 19, 20, 22, 21, 23, 24, 25, 26.The coherent states are the eigenstates of the annihilation operator a, i.e., a| = |, where is the eigenvalue corresponding to |, which is complex. The coherent states can beexpressed as |2 n| = e 2 |n = D()|0 (2)n=0 n! with the numb
13、er state is |n = (a)n/n!|0, and the displacement operator isD () =exp(a a). The coherent states | and | are encoded as 0 and 1, respectively, where we assume that is real. Given that |0|1|2 = | |2 = exp(4|2 ), the coherent statesare orthogonal when is suciently large.If the multiples of annihilation
14、 operators are involved in the coherent state, such state is called a multi-mode coherent state. Multi-mode ECSs have several types: the GHZ-type ECS 19, 20, 26, 21, the W-type ECS 22, 26, 35, 36, the cluster-type ECS 37, etc. These ECSs can be constructed by using beam splitters (BSs), phase shifte
15、rs (PSs), and nonlinear Kerr media 26, 37, which have been used in numerous proposed schemes.! # % & $(图 1: Schematic diagram of ECS generation, where BS denotes beamsplitter. (a) Input-outputthrough the -cross-Kerr media; (b) Generation of the GHZ state (| , , , + | 1/2 22 2 2222 , , )/22; (c) Gene
16、ration of the -type ECS, its input state is the outputstate in process (b).|1,2,3,4=N(|, , , |, , , |, , , + |, , , +| , , , + | , , , +| , , , + | , , , )1,2,3,4 ,In this paper, another type of multi-mode ECS, i.e., the -type ECS(3)21where N = 8(1 + e4| ) 2 , is generated by using one -cross-Kerr m
17、edium and linearoptical components. The entanglement performance is investigated based on photon-parityand photon-threshold measurements. The Bell-Mermin inequality 31 is applied to |, suchthat violations appear in the range of small amplitude of .The remainder of this paper is organized as follows:
18、 In Sec. 1, the generation scheme for| is given, and the eciency and the delity are discussed. In Sec. 2, the non-locality of| is discussed by means of the Bell Mermin inequality based upon two measurements. InSec. 3, we give a short summary.1 ENTANGLED STATES GENERATIONTo demonstrate the generation
19、 process of -type ECSs, we rst give a brief review of the interaction between the two-mode elds in a -cross-Kerr medium as well as of the process of generating a GHZ-type four-mode ECS in free-traveling elds from a CSS. When two-modecoherent states |(1)sa and |(1)tn are sent to the -cross-Kerr mediu
20、m (see Fig. 1(a),they become entangled as a result of the interaction between the two modes 40, 41K an|(1)sa|(1)tn = |(1)sa(| + | )n + (1)t|(1)s+1 a(| | )n/2 (4)with s, t 0, 1. The four-mode GHZ-type ECS |GHZ =1 (| , , , 42 2 22 2+ | 2, 2, 2, 2 )1,2,3,4 (5)can be generated by inputting the coherent
21、superposed state N (|2 + | 2) throughthree BSs with dierent reectivities. The generation process is shown in Fig. 1(b) 26, where21N = 2(1 + e4| ) 2 .The eect of the BS acting on two arbitrary modes a1 and a2 isB (, ) = exp (eia a eia a ),(6)21 2 1 222where the reectivity r and the transitivity t are
22、 determined by as r = sin and t = cos ,and indicates the phase angle between the reected and transmitted elds. According to the principle of BSs, we can obtain the four-mode GHZ-type ECS when considering the case wherer1 = 1 , r2 = 1 , and r3 = 1 .24 3 Taking the GHZ-type four-mode ECS as the input
23、state, we can generate -type ECSsby using PSs, balanced BSs, and -cross-Kerr media. The implementation process is shown in Fig. 1(c). The balanced BS is realized by taking = /2 and = 0. Thus, in this case,B = exp(a1 a a a2 )/4. When two coherent states |1 and |2 are input into a balanced2 1BS with e
24、qual reectivity and transitivity, the output state isBS|, 1,2 |( )2 ,( + )2 1,2 .(7)4The input state |GHZ passes through two PSs with phase shifting for modes 2 and4. The displacement operation with the displacement of 2is then performed on four modes.The displacement operation can be eectively perf
25、ormed by using a strong local coherent eld|LO injected into the other port of a BS with the transmission coecient T , which is extremelylarge 42, 43, 44. When we take 2= LO 1 T with T 1, the four modes obtain thedisplacement . Thereafter, phase shifting is performed on the rst mode to obtain then2we
26、 can get the output state at point b, which is|b = N (| 2, 0,2, 0 + |0, 2, 0, 2)1,2,3,4 ,(8)21where N = 2(1 + e4| ) 2 .We send the state |b of the four modes through the -cross-Kerr medium. However, onlytwo modes are actually input into the nonlinear medium simultaneously. It shows that modes 1and 3
27、 pass through the nonlinear medium simultaneously while modes 2 and 4 are the vacuum states (or modes 2 and 4 pass through the -cross-Kerr medium simultaneously while modes 1 and 3 are the vacuum states) in Eq. (7). Considering the fact that only two modes are inputto the -cross-Kerr medium simultan
28、eously, we can assume that mode 1 (or mode 2) and mode3 (or mode 4) interact with each other and become entangled.For input state |1,3/2,4 =|1, 2, the state becomes|11,3/2,4 =2 | 21/2 (| 2 + | 2)3/4 + | 21/2 (| 2 | 2)3/4 . (9)For input state |1,3/2,4 =| 1, 2, the state becomes|21,3/2,4 =2 | 21/2 (|
29、2 + | 2)3/4 + | 21/2 (| 2 | 2)3/4 .(10)Thus, for input state |b, the output state is|c = N (| 2, 0,2, 0 + | 2, 0, 2, 0+ |2, 0,2, 0 | 2, 0, 2, 0 + |0,2, 0,2) + |0,2, 0, 2) + |0, 2, 0,2) |0, 2, 0, 2)1,2,3,4 ,(11)221|1,2,3,4=N (| , , , + | , , , +|, , , |, , , +| , , , + | , , , +|, , , |, , , )1,2,3,4
30、 ,where N = 8(1 + 2e4| e8| ) 2 . We then input modes 1 and 2 into a balanced BS, andinput modes 3 and 4 into the other balanced BS. After the transformation, the output state is(12)21where N = 8(1 + e4| ) 2 , and |1,2,3,4 is exactly the state |.2 Bell inequality violations2.1 Violation of the Bell-M
31、ermin inequality based upon the photon-parity measurementsTo investigate the non-locality of the continuous variable states, a dichotomic variable for a continuous variable mode is needed, which is similar to the single-particle spin or the single- photon polarization in qubit systems. Dene the four
32、-mode parity operator (1 , 2 , 3 , 4 ) as a product of the dichotomic variable operators (the displaced parity operators) 32:i=1 (1 , 2 , 3 , 4 ) = 4 i(i),(13)where i(i) = e(i) o(i)(14)iiwith ei (i) = oi (i) =D i(i)D i(i)in=0n=0|2n2n|D (i),i|2n + 12n + 1|D (i).(15)The projection operator e(i) or o(i
33、) corresponds to the measurement of an even or oddiinumber of photons. The eigenvalues of i(i) are 1 when an even number of photons aremeasured and 1 when an odd number of photons are measured. Therefore, the parity operator (1 , 2 , 3 , 4 ) is a dichotomic observable, which satises the denition. We
34、 can obtain theBell-Mermin inequality of | based on (1 , 2 , 3 , 4 ) as:B = |1234 1234 123 4 123 4 12 34 12 34 12 3 4 + 12 3 4 1 234 1 234 1 23 4 + 1 23 4 1 2 34 + 1 2 34+ 1 2 3 4 + 1 2 3 4 | 6 4,(16)where 1 2 3 4 is short for the expectation value ( , , , ), which is the correlation1 2 3 4function
35、of the measurements as the parity measurements are performed on the four modes.Based on the fact that the Wigner function is proportional to the expectation value of a displaced parity operator 451 N W ( , , ) = ( 2 )N(1 , , N ), (17)(1 , 2 , 3 , 4 ) can be calculated by using the Wigner function. T
36、hus, we can obtain theBell-Mermin inequality as4B =16 |W1234 W1234 W123 4 W123 4 W12 34 W12 34 W12 3 4 + W12 3 4 W1 234 W1 234 W1 23 4 + W1 23 4 W1 2 34 + W1 2 34+ W1 2 3 4 + W1 2 3 4 | 6 4,(18)where W1 2 3 4 is short for W ( , , , ). The Wigner function can be obtained by the1 2 3 4Fourier transfor
37、mation of its characteristic function as1 2 2 2 2 W (1 , 2 , 3 , 4 ) =8d 1 d 2 d 3 d 4 exp1 1 1 1+ 2 2 + 3 3 2 2 3 3+ 4 4 (1 , 2 , 3 , 4 ).4 4(19)Optimized B4.03.8Bmax3.63.43.23.02.80.0 0.2 0.4 0.6 0.81.0图 2: Optimized Bell-Mermin function Bmax for | versus the amplitude of under photonnumber parity
38、 measurements and displacement operations.The characteristic function can be calculated as(1 , 2 , 3 , 4 ) = T rD (1 )D (2 )D (3 )D (4 ),(20) where is the the density matrix of |. Substituting the characteristic function into Eq. (19) and integrating this function with variables 1 , 2 , 3 and 4 , th
39、e Wigner function of |can be obtained.The Bell-Mermin function B can be obtained by using the Wigner function. The optimized Bell-Mermin function Bmax for | based upon the parity measurements is studied, and the results are plotted in Fig. 2, where we plot the optimal Bell-Mermin function B against
40、the amplitude |. In the case where | 0, Bmax is 4, and no violation occurs. As | increases, we observe a violation of the inequality when 0.019 | 0.465, and the maximum violationis 4.1005 when | = 0.352. In conclusion, when the parity measurements are applied, aviolation of the Bell-Mermin inequalit
41、y occurs. Notably, the parity measurement diers inimplementation in the experiment because of the diculty in discriminating the exact number of photons.2.2 Violation of the Bell-Mermin inequality based upon the photon-threshold measurementsConsidering the threshold photon measurements and the displacement operations, we de- ne an observable asA() = D ()( 0 1 )D (), (21)with positive operator valued measures