ОПТИКА И СПЕКТРОСКОПИЯ, 2014, том 117, № 4, с. 616-622
НЕЛИНЕЙНАЯ И КВАНТОВАЯ ОПТИКА
УДК 535.14
EFFECT OF THERMAL ENVIRONMENT ON QUANTUM NONLOCALITY OF SUPERPOSITION OF TWO COHERENT STATES (я/2 OUT OF PHASE)
© 2014 г. D. K. Mishra
Physics Department, V. S. Mehta college of Science, Kaushambi-212201, U. P., India E-mail: kndmishra@rediffmail.com Received December 30, 2013; in final from, April 24, 2014
Recently we studied the test of Bell-type inequalities with superposition of two coherent states (я/2 out of phase) [D. K. Mishra, Acta Phys. Pol. A. 123, 21 (2013)]. Here, we study the effect of thermal environment on the Bell-CHSH inequality. As the state interacts with the thermal environment, the time evolution of the linear entropy to study the purity loss in the state has also been examined.
DOI: 10.7868/S0030403414100171
1. INTRODUCTION
Quantum nonlocality [1] confirms the interpretation and validity of quantum mechanics against the local-realistic theories by violations of the constraints on the correlation between local measurement outcomes. Mathematical expression of such a constraint is known as Bell's inequality [2], of which many variants have been formulated [3, 4]. For example, well known Clauser, Horne, Shimony, and Holt's (CHSH) inequality [3] and Clauser and Horne (CH) [4] are used for the verification of nonlocal correlations in a two-dimensional Hilbert space. Nonlocal correlations play crucial role for the device-independent versions of quantum information protocols, such as cryptography, random number generation, state estimation, and entangled measurement certification.
Quantum continuous variables (CV) [5] of light have been applied to realize some of the standard informational tasks traditionally based on the discrete variables, the qubits. Bell-inequality tests were performed by using qubits which statisfy the spacelike separation between two local parties. But, the difficulty of the detection loophole demanded to look at other approaches for Bell-inequality tests. CV states are of recent interest in order to suggest proposals for loophole-free Bell-inequality tests [6] and have advantage over discrete variable states due to the highly efficient and well experimentally developed method of detection for CV states. Bell's inequality tests in the phase space have been studied by Banaszek and Wodkiweicz (BM) [7] in terms of the Wigner (Q) function based upon the photon number parity (on/off) measurements and the displacement operation. The Wigner function approach supports the Bell's inequality version of CHSH [3], while the Q function supports the version of CH [4].
Schrodinger cat states, superposition of two coherent states in free-traveling optical fields, are non-
Gaussian CV states. These states have been generated and detected [8, 9], where the size of the states was reasonably large for fundamental tests of quantum theory and quantum-information processing [10]. Nonclassical features [11] ofSchrodinger cat states are of great interest in the quantum information processing applications. R. Zeng etal. [12] discussed nonclassical features, such as the sub-Poissonian photon statistics, quadrature squeezing, and the negativity of the Wigner function, of the superposition of two coherent states which are n/2 out of phase (we will abbreviate this state as "SCSP" for simplicity throughout the paper),
№> = (N/72 )(|a) + e* |/a>), (1)
with the normalization constant
N2 = [ 1 + exp (-| a|2) cos (^ + |a|2 )]-1.
Here, |a> is the coherent state defined by the eigenvalue equation, aja) = aja). Recently, we studied the quantum nonlocality test for SCSP by using photon parity and on/off measurements [13]. We studied violations of the Bell-CHSH and Bell-CH inequalities for the SCSP state and observed strong violations establishing the quantum nonlocality.
Coherent superposition is at the heart of quantum mechanics and finds applications in a number of technological advances. When a system interacts with its environment, the coherence destroys and the superposition drives to an incoherent mixture [14]. So, it is important to know the robustness of SCSP taking into account of the effect of the thermal environment. Robustness against a thermal environment on different superposition states composed of several states usually employed in both theoretical [15] and experimental studies [16] have been studied using the linear entropy [17]. This approach has been used in the present paper
for studying the robustness of the state against the thermal environment [18].
The paper is organized as follows. In Sec. 2, we give an introduction about the Bell-CHSH inequalities using photon parity measurement scheme. In Sec. 3, we study about the robustness of the SCSP under thermal environment and its effect on the Bell-CHSH inequality. In Sec. 4, we study the robustness of the SCSP against the thermal environment using linear entropy, and in Sec. 5, we conclude the results.
2. BELL-CHSH INEQUALITY WITH PHOTON PARITY MEASUREMENT SCHEME
Wigner function [19] of a quantum state described by the density operator, p , is
where we call B = |BCHSH|, the Bell-CHSH function. The two-mode Wigner function at a given phase point
described by p and y is W(p, y) = (4/n2)Tr[pn (p, y)], where p is the density operator of the field. Then the Wigner representation of Bell-CHSH inequality is
5 = (7)
= (n2/4)|< W(0, 0) + W(p, 0) + W(0, y) - W(p, y»| < 2.
The Cirel'son bound is B = |BCHSH| < 2,J2 in the generalized BW formalism.
Wigner function of the state SCSP, |y), is [12]
W(p) = (N2/2)[Wla}(p) + Wlia}(p) + Wintm, (8) where
W(a) = Tr [p Jexp (a^* - a*^)D(^)n-1 . (2) Wja>(ß) = (2/n) exp [-2 (|a|2 + |ß|2 - a * ß - aß* )],
Here the displacement operator [20], D (£,), is defined
as D (£,) = exp(£,at — £,* a). The Wigner function W(a) is real-valued, uniformly continuous, and square-integrable function of a for all density operators pa . In a coherent state representation, the Wigner
function can be written as W(a) = (2/n)(y|H (a)|y), where n (a) = D (a) nD (a) is the parity operator n = (—1)a a, shifted in the phase space by a, with the
help of displacement operator, Da (a). The displacement operator can be experimentally realizable by a beam splitter with the transmission coefficient close to one and a strong coherent state being injected into the other input port [7]. Correlated parity measurement [7] can be described by the following POVM operators [21]:
n+(ß) = D(ß) £ |2k><2k|D(ß),
(3)
k = 0
n-(ß) = D(ß) £ |2k + 1>(2k + 1|Dt(ß). (4)
k = 0
Corresponding operator for the correlated measurement of the parity on modes "a" and "b" of two parties, say Alice and Bob, may be defined as
n (ß,Y) =
= [n l+) (ß) - n (ß)] ® [n i+)(Y) - n <f) (Y)].
(5)
The outcome of the measurements is either +1 or —1 (i. e., dichotomic). Then the Bell-CHSH inequality is
B = B
CHSHl
= |<n(ß, Y) + n(ß, Y') + n(ß', Y) - n(ß', Y')> ^ 2,
(6)
(ß) =
= ( 2/n) exp [-2 (|a|2 + \ß\2 + i a* ß - /aß* )], Wint(ß) = (4/n)cos(^ - |a|2 + A) x x exp(- |a|2 - 21ß2 + A),
(9) (10)
(11)
and
A = a*p + ap* - ia*p + iap*. (12)
We can visualize the nonclassical nature of the state |y) in terms of the negative regions of its Wigner function [13]. In order to make the Bell-CHSH inequality test, the single-mode SCSP state, |y), with the Wigner function given by Eq. (8) is divided by a beam splitter to generate a two-mode state shared by distant parties, say Alice and Bob. The beam splitter operator acting
on modes a and a is represented as
B(0) = exp [0(a1 b - ab1 )/2],
(13)
where the beam splitter reflectivity and transmittivity are defined as R = sin2(0/2) and T = 1 — R, respectively. When the state |y) passes through a 50 : 50 beam splitter, the Wigner function of the resulting state is
Wout(p, y) = w,}w>}(p^+i-), (14) where W|0>(y) is the Wigner function of the vacuum:
Wo> (Y) = ( 2/n) exp (-2| Y2 ).
(15)
The two-mode state ^out(p, y), given by Eq. (14), can be used to calculate the Bell-CHSH function given by Eq. (7). We can find [16] several situations when the Bell-CHSH inequality, Eq. (7), is violated by the state
and the Cirel'son bound, B = |BCHSH| < 2^/2 , is also followed.
go
yj
0.4
^ 0-2
cd
0 frntm.
~ 'tfliitt
-0.2 - 'kwttj-
-0.4 - H
6a -
Fig. 1. Evolution of the Wigner function of SCSP, W(P, t), under thermal environment with s(t) and (9a — 9p) for n = 0.4, ^ = n/2, J = 1.0.
3. EVOLUTION OF THE SCSP UNDER THERMAL ENVIRONMENT AND ITS EFFECT ON BELL-CHSH INEQUALITY
When the SCSP evolves in the thermal environment, the evolution of the density matrix can be described [22] by the master equation
dp/dt = y(n + 1 )(2apat - atap - pata) + + yn(2atpa - a atp - pa at),
(16)
where y represents the dissipative coefficient and n denotes the average thermal photon number of the environment. When n = 0, Eq. (16) reduces to the master equation describing photon-loss channel. The evolution of the Wigner function described by Eq. (8) under thermal environment is governed by the following integration equation [23]:
W(p,T) = -1- \d\Wh(Z)Wp ~s(T)Z,T = o),
t(T)2j V t(T) ; (17)
whose convergent condition is Re(Z ± f ± g) < 0 and Re(Z2 - 4fg)/(Z ± f ± g) < 0, the integration in Eq. (17) can be performed easily and, hence, the Wigner function for the initial single-mode state, described by Eq. (8), evolving in thermal environment is obtained as
wth(Z) =
n( 1 + 2n) Using the integration formula [24]:
\d2z exp (Z| z\2 + £z + n z * + fz2 + gz* ) =
Jz'
: exp
- 4fg
- Z+ £ g + n z2 - 4fg
W(P, t) =
= (N2/2)[Wla}(P, t) + Wlia}(P, t) + Wnt(P, T)], where
Wa) (P,T) = = ( 2/nA ) exp [ a| a|2 + p|2 + c (ap* + a *p)],
(20)
(21a)
WVa) (P,T) = = (2/nA)exp[a\a|2 + P\2 + c(aP* - a*P)], 4 cos F
Wnt(P,T) =
n[ t (t)2 + s(T)2( 1 + 2 n )]
(21b) exp ( U), (21c)
A = t (t)2 + s (t)2 ( 1 + 2 n ), a = [ - 4A + s(T)2t(T)2 ( 1 + 2 n )]/2A, b = [- 2/1(t)2 + 2s(t)2( 1 + 2n)/A], c = [ - 1 / t(T) + s(T)2t(T)( 1 + 2 n ) /A ], u = 2
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