7 October 1996 PHYSICS LETTERS A ELSEVIER
Physics Letters A 221 (1996) 283-286
Explicit solutions for negative-probability measures for all entangled states Yeong Deok Han a*‘, Won Young Hwang b,2, In Gyu Koh b a Depurtment of Physics, Woosuk Universi~, 490 Hujeong, Samrye, Wanju. Cheonbuk, South Korea h Department of Physics, Korea Advanced Institute of Science and Technology, 373-l Kusung, Yusung, Taejon, South Korea received 16 July 1996; accepted for publication 30 July 1996 Communicated by P.R. Holland
Received 31 May 1996; revised manuscript
Abstract We obtain explicit solutions for probability measures that reproduce quantum mechanical predictions for some spinmeasurement directions for all entangled states. The necessity of negative probability in this case is shown. This constitutes another proof of Gisin’s theorem that all entangled states are incompatible with any local hidden-variable models (all entangled states can violate Bell’s inequality). A degree of freedom that remains in the solution is noted. PACS: 03.65.B~ Keywords: Bell inequality;
1. Introduction Since Bell proved in 1964 [ 11 that any hiddenvariable model without nonlocal interactions [ 2,3] cannot reproduce all the quantum mechanical predictions for EPR-like experiments [ 451, much theoretical [ 5,6] and experimental [ 7,8] work has been done on this subject. In local hidden-variable (LHV) models [ 1,561, one makes the assumption: the outcome of a measurement is determined by the hidden variable A and the conditions of the measurement, and the outcome of the measurement is independent of the conditions of other measurements which are space-like separated from the measurement. Now, we consider the set of all A. As we have assumed, whatever measurement is ’ E-mail: [email protected]
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done, the outcome is determined for a given A before the measurements. Therefore, the set of all A can be divided into several disjoint subsets At, AZ,. . ., according to the outcomes of the measurements. If the involved measurements are MI, M2,. . ., M, and the number of distinct outcomes is assumed Ti for Mi (i = n) then the maximal number of subsets is $; ,: ’ In order to reproduce the same predictions (probabilities, correlations, etc.) as those of quantum mechanics, suitable probability measures have to be assigned to each subset. However, the positivity of the probability measure of each subset acts as an obstacle. In fact, under some conditions the quantum mechanical predictions cannot be reproduced without violating the positivity of probability measures. Bell’s inequality and later generalizations of it [ 5,6] can be regarded as constraints on some combinations of correlation functions by the positivity of the probability measures.
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Y.D. Han et al./ Physics Letters A 221 (1996) 283-284
On the other hand, some papers [ 9,10 ] have considered extended probability measures (including negative ones), though there are controversies concerning their physical meaning. In this Letter, we obtain explicit solutions for extended probability measures that reproduce quantum mechanical predictions for some measurement directions for all entangled states. The necessity of negative probability in this case is shown. This constitutes another proof of Gisin’s theorem [ 121 that all entangled states are incompatible with any local hidden-variable models (all entangled states can violate Bell’s inequality). A degree of freedom that remains in the solution is observed.
2. Negativity of solutions for probability measures for entangled states and a degree of freedom in solutions
IS-) = c(u+) + d/u-), with /aI2 + lb12 = [cl2 + ldl2 = 1,
UC* + bd* = 0.
With this new basis, I+) can be expressed as I$) = c~al~+)l~+)
= c~a~u+)Is+) + ablu-)ls+) - pclu+)\s-)
Al 122 A3 A4 A5
A6 A7 A8
(aab - /3cd) Iu+)Iu-)
+ (ab* - pd2)Iu-)lu-).
where (s&)i denote the orthonormal basis of the ith particle (i = 1,2), and LYand p are real numbers satisfying cy2 + p2 = 1. For this state, the outcome of each spin-measurement for particle 1 along the s direction and that for particle 2 along the s direction are 100% correlated. We introduce another orthonormal basis luzk) defined by the unitary transformation, IS+) =++j
Subset of A
+ (aab - /3cd)lu-)lu+)
We consider general entangled states of two spin-i particles. By choosing an appropriate basis for each particle, the state IJI) can be written as (by Schmidt decomposition) [ 111, I$) = aIs+)
Now, we consider a LHV model with its hidden variable A for the above system. The set of all A can be divided into 16 (= 24) disjoint subsets according to the outcomes of the four measurements, spinmeasurements along s and u for particles 1 and 2. However, since their outcomes are 100% correlated for st and s2 measurements, the number of subsets is reduced to 8 (Table 1) . Each set Ai is characterized by the signs in its row. For example, A:! is the subset having the following properties; if A of a given particle pair belongs to this subset, then when the spin of particle 1 of the pair is measured along the s direction the result will be -, when the spin of particle 1 of the pair is measured along the u direction the result will be +, when the spin of particle 2 of the pair is measured along the s direction the result will be -, and when the spin of particle 2 of the pair is measured along the u direction the result will be +. mi denotes the probability measure of the ith subset Ai. Under the condition that this model must give the same results as quantum mechanical ones (from Eqs. (1) and (3)-(5)), we obtain several equations for ml,. . . , m3, P( s+; s+) = ff2 = ml + m3 + m5 + m7,
u+> = cu21al” = ml + ms.
=ct21bj2= m3 + m7.
Y.D. Han et al. / Physics Letters A 221 (19%) 283-286
u+>=p21c12 = m2
s+) =(~~(a(* = ml + mg,
s+) =a2jb1* = mg + m7,
=p21d12 = m4
Now we show that at least one of the m; must be negative. If either ml or m2 is negative, the negativity is proven. If ml 2 0,
m2 b 0,
=/12/c1* = m2 + mq,
=p2)d)* = mg + mg,
ml = m2 = 0.
u+) = /aa2 - j?c212 = ml + m2,
= jcvab - pcd12 = mg + m4,
u+) = jaab - /?cd12 = rn5 + m&
= jab2 - pd*)* = m7 -t mg.
In the above equations, P( mx; ny) denotes the probability of obtaining the result that the outcome of the spin-measurement along m of particle 1 is x and the outcome of the spin-measurement along n of particle 2 is y, where m = s, u, n = s, u, x = f, y = &, respectively. Although the number of equations is greater than the number of variables, these equations can be solved since there are correlations between
m3 = cr2/u1* - ml,
/lc212 + ml,
mg =p*( IdI* - /cl*) + [au* - pc212 - ml.
P214 I4 + IPI ’ m7= a2(14- IPI) I4 + IPI ’ m4=m6 = ~
m‘J2(lal - IPI) bl+lPl .
By Eqs. (34) and (35),
N%l4 - IPII 26
I4 + IPI >
Therefore, either m7 or ms ImISt be negative, except for the case IcrI = IpI where I$) is maximally entangled. When Ial = IPI, we have
can be obtained from (28) and Eqs. (20) -
P(mx; ny), ml =ml,
then by Eqs. (29) and (21), we obtain
P( uf; u-)
On the contrary, there remains a degree of freedom corresponding to a variable ml. We consider a special basis Ju’f) defined by
For Eq. (37)) instead of Eq. (28) we choose a, b, c and d as
where i < r < 1. Then by Eqs. (21) and (26), obtain m;!+m7=-2(1-r)(r-i),
Then, era’ - PC* = 0.
which is negative for i < r < 1 and has a minimum -- A whenr=i. Thus either m2 or m7 is negative.
Y.D. Han et al./Physics
For Eq. (38)) we can show negativity way if we choose a, b, c and d as a=&,
in a similar
d = -ifi.
3. Discussion and conclusion The angle B between s and u is given by co& 38) = [(s + lu+)12.
Letters A 221 (1996) 283-286
tions for some spin-measurement directions for all entangled states. The necessity of negative probability measures in this case has been shown. This constitutes another proof of Gisin’s theorem [ 121 that all entangled states are incompatible with local realism with positive definite probabilities (all entangled states can violate Bell’s inequality). A degree of freedom in this solution was observed.
Acknowledgement (42) We thank Dr. J.D. Kim for helpful suggestions.
For Eq. (28), (43)
[ I] J.S. Bell, Physics 1 (1964)
for Eqs. (39) and (41), I(s + lu+)12 =
The s and u directions chosen by Eq. (28) are not the directions of observables that maximally violate the Bell-CHSH inequality [ 5,6]. This can be seen by the fact that the negativity is not proven for maximally entangled states when we choose the s and u directions of Eq. (28). In fact, we can find the positive solution m3 = m4 = mg = mtj = f in this case. The s and u directions chosen by Eqs. (39) and (41) also are not the directions of observables that maximally violate the Bell-CHSH inequality (S = $ when r = i). The fact that negativity is not proven for all choices of spin-measurement directions accords with the fact that Bell’s inequality is not violated for all choices of spinmeasurement directions. The fact that there remains a degree of freedom in solutions for the negative-probability measures suggests that there is a degree of freedom in negativeprobability LHV models [ 9, lo] that reproduce quantum mechanical predictions. We have obtained explicit solutions for probability measures that reproduce quantum mechanical predic-
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