- Email: [email protected]
The detection loophole in Hardy's nonlocality theorem and minimum detection efficiency
8 April 19%
PHYSICS LETTERS A
ELSEVIER
Physics Letters A 2 I2 (1996) 309-3 14
The detection loophole in Hardy’s nonlocality theorem and minimum detection efficiency Won Young Hwanga, In Gyu Koha, Yeong Deok Han b ’
Department
of Physics,
b Department
Korea Advanced of Physics,
Institute
of Science and Technology,
Woosuk University,
490 Hujeong-ri,
373-l
Samrye-eup,
Kusung-dong, Wanju-kun.
Yusung-ku. Cheonbuk,
Taejon, Korea
Korea
Received 4 October 1995; revised manuscript received 22 January 1996; accepted for publication 23 January 1996 Communicatedby P.R. Holland
Abstract The detection loophole is discussed in connection with Hardy’s nonlocality theorem. An inequality that must be satisfied in order that the Hardy nonlocality theorem avoids the detection loophole is derived. The minimum detector efficiency is calculated to be i. It is shown that the minimum value obtainable theoretically using a version of Bell’s inequalities is f. PACS: 03.65.B~ Keywords:
Nonlocality; Detection loophole; Minimum efficiency
1. Introduction For a long time local realism has been very successful in explaining diverse natural phenomena. Quantum mechanics, however, has begun to challenge local realism [ 1 I. Thus it has become an important problem whether local realism can coexist with quantum mechanics. In 1964, Bell [ l] discovered a mathematical method that can answer this question. Bell’s inequalities [ 1,2], which must be satisfied by all forms of local realism, are violated by quantum mechanics. However, Bell’s inequalities are not always violated by quantum mechanics. They are satisfied for experiments for which quantum mechanics gives perfect correlations. Greenberger, Horne and Zeilinger (GHZ) [ 31 showed that Bell’s theorem (quantum mechanics contradicts local realism) can also be proved for perfect correlations. GHZ’s proof was for more than three particles. Recently, Hardy showed that it is possible to prove Bell’s theorem without inequality for two particles [4] and for EPR-like experimental settings [ 51. On the other hand, the detection loophole [6] has led to controversies regarding the interpretation of the experimental results of Bell’s inequalities and there have been efforts to lower the necessary efficiency of detectors [7]. In this paper, it will be shown by an intuitive method how this detection loophole is applied to Hardy’s nonlocality theorem (without an inequality). An inequality that must be satisfied in order that the Hardy nonlocality theorem avoids the detection loophole is derived. This inequality can also be derived from a version (Eq. (8)) of Bell’s inequalities. Also, the efficiency needed in order that Bell’s theorem avoids the detection loophole is calculated for Hardy’s nonlocality theorem [5]. The necessary efficiency for the case 03759601/96/$12.00
@ 1996 Elsevier Science B.V.
PI/ SO375-9601(96)00072-2
All rights reserved
W.Y. Hwang et al./Physics Letters A 212 (1996) 309-314
310
A
a
A Fig. I. The set X is characterized by the results x, y. z and u of the correspondingmeasurements. Fig. 2. Definition of the set P
where the nonlocal effect is maximum (y of Eq. (20) of Ref. [ 51 is maximum) is about 0.84, and the minimum of the necessary efficiency is f (- 66.7%). This value was already obtained in [ 71 by computer calculation, where the experimental conditions are identical to those of Hardy. It is shown that this $ is the minimum value obtainable theoretically by Eq. (8). 2. The detection loophole in Hardy’s nonlocality theorem In the following, it will be shown by an intuitive method how local realism can coexist with Hardy’s nonlocality theorem by the detection loophole. The detection loophole also enables, in similar ways, local realism to coexist with other forms of nonlocality without an inequality, e.g. the GHZ [ 31 and Kochen-Specker [ 81 ones. In an ordinary EPR-Bell experiment, the spin is measured along the direction (~1or cy2at A which is spacelike separated from B where the spin is measured along another direction /Ii or &. In Hardy’s experiment [ 51, (~1 is set equal to fit and cy:!is set equal to fl2 (Fig. l), at = p, = a,
CY:! = p2 z b.
(1)
A set X of hidden variables A, which give the value (x) when (a) is measured and (y) when (b) is measured at A, and give the value (z) when (a) is measured and (u) when (b) is measured at B, where x, y, z, u = f respectively, is represented as in Fig. 1, Hardy’s nonlocality is shown by the following equations (Eq. ( 14) of Ref. [51),
n++(b,b) =O,
(2)
n+_(a, b) = 0,
(3)
n_+(b,a)
=O,
(4)
n++(a, a) > 0,
(5)
where nij ( (Y,p) is the number of events with result (i) when the spin is measured at A along the direction (a) and with result (j) when the spin is measured at B along the direction (p) (i, j = &) . With the representation of Fig. 1, Hardy nonlocality theorem is expressed as follows. By Eq. (5), there is a nonempty set P of A with x = + and z = +. In order not to contradict Eqs. (3) and (4), y and u must be (+) for P (Fig. 2). This fact contradicts Eq. (2). Thus Eqs. (2)-(5), predicted by quantum mechanics, are not consistent with local realism. However, a loophole can be found in this contradiction, by assuming the existence of the sets Q, R and S (Fig. 3) which are consistent with Eqs. (2)-( 5) if only the coincident detections are taken into account. In
W.Y. Hwnng er al./Physics Letters A 212 11996) 309-314
311
0
0
L!4
L
Q/B
+
0
L4
R/B
L
+
A
A
Fig. 3. Definitions of the sets Q, R and S. 0 denotes an undetected event
Fig. 3, 0 means an undetected event. The element of P must be an element of Q or R or S in order not to contradict Eqs. (2)-(4). Therefore, n(P) = n(Q) + n(R) + n(S),
(6)
where n(Z) is the number of elements of set Z. By Eq. (6), the following inequality is derived, fi++(a,a)
< n+o(a,b) +no+(b,a).
(7)
(The elements of R and S give the same counts to both sides of this inequality. The elements of Q give more counts on the right-hand side than on the left-hand side by a factor of two. Thus, in total, the right-hand side has more counts than the Ieft-hand side.) This is an inequality that must be satisfied along with Eqs. (2)-(4) in order that Hardy’s nonlocality avoids the detection loophole. Here the contradiction is avoided by inserting 0 (an undetected event) where the contradiction occurs in the loop of deduction by quantum mechanical predictions. Using this method, local realism might also coexist with other forms of nonlocality without an inequality, such as the GHZ [ 31 and Kochen-Specker [8] ones. Eq. (7) can also be derived in another way using a version (Eq. (12) of Ref. [7] ) of Bell’s inequalities, which is for the case of spin-; particles
In Hardy’s case, with Eq. ( l), Eq. (8) becomes n++(a,a)
-n+-(a,b)
From Eqs. (2)-(4) 3. Calculation
-n-+(6,a)
-n+o(a,b)
-no+(ha)
< n++(b,b).
(9)
and Eq. (9), Eq. (7) is derived.
of the minimum
detection efficiency
For imperfect detectors of efficiency 77 (0 6 r) < 1), the actual detection counts are related to quantum mechanical probabilities by the following equations,
312
W.Y. Hwang et al./ Physics Letters A 212 (19%) 309-314 n++ (a, a> = Np++ (a, a> v2,
(10)
n+da,b)
= N[p++(a.b)
+p+-(a,b)lrl(l
-T,J),
(11)
no+(ha)
= Np++(ha)
+P-+(ha)lrl(l
- 3).
( 12)
In the above equations, N is the total number of measurements and pij denotes quantum mechanical probabilities. Substituting Eqs. (lo)-( 12) into Eq. (7), 7.. <
p++(aJ) +p+-(a,b) p++(a,a)
+p++(a,b)
+p++(b,a)
+p+-(a,b)
+p_+(b,a)
+p++(b,a)
+p_+(b,a)
(13)
is obtained. Also, by Eqs. (3) and (4), p+_(a,b)
=p_+(b,a)
=O.
(14)
And also, by symmetry, p++(a.b)
=p++(ka).
(15)
Using Eqs. (14) and (15), Eq. (13) is simplified to rl<
ZP++ (a, b) p++(a,a)
(16)
+2p++(a,b)’
Thus, if the efficiency q is higher than the right-hand side of Eq. (16), then the Hardy nonlocality theorem can be proven by the detection loophole. The value of the right-hand side of Eq. ( 16) depends on the particular experimental situation: spin measurement directions and quantum mechanical state. In the following, the necessary efficiency is calculated for some specific cases. In Ref. [ 51, the nonlocality order parameter is y (3 p++ (a, a) = [(a - p)aj?/( 1 - a/3)12) and it has its maximum value 0.09 for a = 0.9070 and /3 = 0.4211. In this case, p++(a, b) is calculated using Eq. ( 13.b) of Ref. [5], p++(a,b)
= jNA2B1* = - a2p2
1 -cup
= 0.24.
(17)
Inserting these values into Eq. (16) we obtain 776 0.84.
(18)
(The case cr. /3 2 0, is considered here without loss of generality.) Thus detectors of efficiency higher than 0.84 are needed in this case. Next, the minimum of the necessary efficiency is calculated. Using the expressions of p++(a, a) and p++(a, b), Eq. (16) becomes 2 V G (a-kW/(l
--a$)
+2’
(19)
The minimum of the right-hand side is obtained when ( CY- p)2/( 1 - crp) has its maximum value 1, which is obtained in the following limit
(20)
W.Y. Hwang et al./Physics
Letters A 212 (1996) 309-314
313
In this limit, 2 rl
G (cw-/?)2/(1
2 -o/3)
+2
(21)
+ 3’
The minimum efficiency needed to violate locality is thus f. The limits of the state and the directions of measurement are given by the equations that connect cy and p with the directions of spin measurements (Eqs. (18) and (19) of Ref. [5]), I$) = aI+)I+)
- PI-)I-)
-+
(22)
l+N+L
b -+ -z.
a --+ --I,
(23)
4. Discussion It is interesting that the minimum of the necessary efficiency is obtained in the limit that 1s) becomes a direct product state (Eq. (22)) which does not show any nonlocality. The minimum 3 (N 66.7%) is just the value found in Table 2 of Ref. [7]. The state and directions of measurement obtained in Bqs. (22) and (23) are just Eqs. (32)-( 34) of Ref. [ 71 if the basis of polarization directions of the second particle is rotated by 90”. In fact, i is just the minimum value obtainable theoretically by Eq. (8). This can be shown using the following relations, n+-(w,P2)
n++(al,Pl)
=~r12P++(WPi),
n-+(a2,/31)
= ~r12P-+(~2,Pd,
= m2P+-hP2L
n+o(4,/32)
= NT41 - 7?)[P++(“19/32)
+P+-(al,P2)1,
no+(~2vPl)
=m4l-7))[P++(~:!,P1)
+p-+(~2,/31)19
?~++(a29/32) = Nq2p++b2,P2L
By Eqs. (24)
(24)
and (8),
(25) = {[P++(~t,P2)
+P+-(m,P:!)l+
{P++(al*Pl)
+ [P++(m,P;!)
-[P++(~29P2)
+P+-(al,Pz)
[p++(a2,P1)
+P+-(al,p2)1
+p-+(a2,p,)l} + [p++(a2,P1)
+p-+(w,pl)l
+p-+hpdl}-‘.
(26)
In Eq. (26), P,j
> 0
for all
i, j.
(27)
From the impossibility of superluminal communication or directly from quantum mechanics, P++(m,P1>
+P+-(W,Pl)
=p++(LyI,p2)
P++(m,P1)
+P-+(w*Pl)
=p++((Y2,P1) +p-+(az,pI)
must be satisfied. By Eqs. (27)
and (28))
+p+-(crl,p2),
(28)
314
W.Y. Hwang et d/Physics
P++(w*Pl)
Letters A 212 (1996) 309-314
+P-+(cy:!,PI>
(29)
hold obviously. By Eqs. (27) and (29), the minimum of the right-hand side of Eq. (26) is obtained to be 3 when the following equations are satisfied, P++(~l*Pl)
=P++(cfI7P2)
+P+-(w,a)
=P++(a2rPl>
P++(a2vP2)
=p+-(Q19P2)
=P-+(~29Pl)
=o.
+P-+(cu:!,PI),
(30) (31)
Eq. (31) is just the conditions (Eqs. (2)-(4)) of Hardy’s nonlocality theorem. Eq. (30) is satisfied in the case (Eq. (21) ) when the minimun 3 is obtained. Acknowledgement
The authors express their gratitude to Dr. J.W. Lee for good suggestions. This work is supported in part by the KOSEF. References 1 II J.S. Bell, Physics I (1964) 195; E Selleri, ed., Quantum mechanics versus local realism, The Einstein-Podolsky-Rosen paradox (Plenum, New York, 1988). 12 1J.F. Clauser and M. Home, Phys. Rev. D 10 ( 1974) 526. 13 I D.M. Greenberger, M. Home, A. Shimony and A. Zeilinger, Am. J. Phys. 58 ( 1990) 1131. 141 L. Hardy, Phys. Rev. Lett. 68 (1992) 2981. 15I L. Hardy, Phys. Rev. Lett. 71 (1993) 1665. 16 I PG. Kwiat, PH. Eberhard, A.M. Steinberger and R.Y. Chiao, Phys. Rev. A 49 ( 1994) 3209. 17 ] P.H. Eberhard, Phys. Rev. A 47 ( 1993) R747. 18 I S. Kochen and E.P Specker, J. Math. Mech. 17 ( 1967) 59.
PHYSICS LETTERS A
ELSEVIER
Physics Letters A 2 I2 (1996) 309-3 14
The detection loophole in Hardy’s nonlocality theorem and minimum detection efficiency Won Young Hwanga, In Gyu Koha, Yeong Deok Han b ’
Department
of Physics,
b Department
Korea Advanced of Physics,
Institute
of Science and Technology,
Woosuk University,
490 Hujeong-ri,
373-l
Samrye-eup,
Kusung-dong, Wanju-kun.
Yusung-ku. Cheonbuk,
Taejon, Korea
Korea
Received 4 October 1995; revised manuscript received 22 January 1996; accepted for publication 23 January 1996 Communicatedby P.R. Holland
Abstract The detection loophole is discussed in connection with Hardy’s nonlocality theorem. An inequality that must be satisfied in order that the Hardy nonlocality theorem avoids the detection loophole is derived. The minimum detector efficiency is calculated to be i. It is shown that the minimum value obtainable theoretically using a version of Bell’s inequalities is f. PACS: 03.65.B~ Keywords:
Nonlocality; Detection loophole; Minimum efficiency
1. Introduction For a long time local realism has been very successful in explaining diverse natural phenomena. Quantum mechanics, however, has begun to challenge local realism [ 1 I. Thus it has become an important problem whether local realism can coexist with quantum mechanics. In 1964, Bell [ l] discovered a mathematical method that can answer this question. Bell’s inequalities [ 1,2], which must be satisfied by all forms of local realism, are violated by quantum mechanics. However, Bell’s inequalities are not always violated by quantum mechanics. They are satisfied for experiments for which quantum mechanics gives perfect correlations. Greenberger, Horne and Zeilinger (GHZ) [ 31 showed that Bell’s theorem (quantum mechanics contradicts local realism) can also be proved for perfect correlations. GHZ’s proof was for more than three particles. Recently, Hardy showed that it is possible to prove Bell’s theorem without inequality for two particles [4] and for EPR-like experimental settings [ 51. On the other hand, the detection loophole [6] has led to controversies regarding the interpretation of the experimental results of Bell’s inequalities and there have been efforts to lower the necessary efficiency of detectors [7]. In this paper, it will be shown by an intuitive method how this detection loophole is applied to Hardy’s nonlocality theorem (without an inequality). An inequality that must be satisfied in order that the Hardy nonlocality theorem avoids the detection loophole is derived. This inequality can also be derived from a version (Eq. (8)) of Bell’s inequalities. Also, the efficiency needed in order that Bell’s theorem avoids the detection loophole is calculated for Hardy’s nonlocality theorem [5]. The necessary efficiency for the case 03759601/96/$12.00
@ 1996 Elsevier Science B.V.
PI/ SO375-9601(96)00072-2
All rights reserved
W.Y. Hwang et al./Physics Letters A 212 (1996) 309-314
310
A
a
A Fig. I. The set X is characterized by the results x, y. z and u of the correspondingmeasurements. Fig. 2. Definition of the set P
where the nonlocal effect is maximum (y of Eq. (20) of Ref. [ 51 is maximum) is about 0.84, and the minimum of the necessary efficiency is f (- 66.7%). This value was already obtained in [ 71 by computer calculation, where the experimental conditions are identical to those of Hardy. It is shown that this $ is the minimum value obtainable theoretically by Eq. (8). 2. The detection loophole in Hardy’s nonlocality theorem In the following, it will be shown by an intuitive method how local realism can coexist with Hardy’s nonlocality theorem by the detection loophole. The detection loophole also enables, in similar ways, local realism to coexist with other forms of nonlocality without an inequality, e.g. the GHZ [ 31 and Kochen-Specker [ 81 ones. In an ordinary EPR-Bell experiment, the spin is measured along the direction (~1or cy2at A which is spacelike separated from B where the spin is measured along another direction /Ii or &. In Hardy’s experiment [ 51, (~1 is set equal to fit and cy:!is set equal to fl2 (Fig. l), at = p, = a,
CY:! = p2 z b.
(1)
A set X of hidden variables A, which give the value (x) when (a) is measured and (y) when (b) is measured at A, and give the value (z) when (a) is measured and (u) when (b) is measured at B, where x, y, z, u = f respectively, is represented as in Fig. 1, Hardy’s nonlocality is shown by the following equations (Eq. ( 14) of Ref. [51),
n++(b,b) =O,
(2)
n+_(a, b) = 0,
(3)
n_+(b,a)
=O,
(4)
n++(a, a) > 0,
(5)
where nij ( (Y,p) is the number of events with result (i) when the spin is measured at A along the direction (a) and with result (j) when the spin is measured at B along the direction (p) (i, j = &) . With the representation of Fig. 1, Hardy nonlocality theorem is expressed as follows. By Eq. (5), there is a nonempty set P of A with x = + and z = +. In order not to contradict Eqs. (3) and (4), y and u must be (+) for P (Fig. 2). This fact contradicts Eq. (2). Thus Eqs. (2)-(5), predicted by quantum mechanics, are not consistent with local realism. However, a loophole can be found in this contradiction, by assuming the existence of the sets Q, R and S (Fig. 3) which are consistent with Eqs. (2)-( 5) if only the coincident detections are taken into account. In
W.Y. Hwnng er al./Physics Letters A 212 11996) 309-314
311
0
0
L!4
L
Q/B
+
0
L4
R/B
L
+
A
A
Fig. 3. Definitions of the sets Q, R and S. 0 denotes an undetected event
Fig. 3, 0 means an undetected event. The element of P must be an element of Q or R or S in order not to contradict Eqs. (2)-(4). Therefore, n(P) = n(Q) + n(R) + n(S),
(6)
where n(Z) is the number of elements of set Z. By Eq. (6), the following inequality is derived, fi++(a,a)
< n+o(a,b) +no+(b,a).
(7)
(The elements of R and S give the same counts to both sides of this inequality. The elements of Q give more counts on the right-hand side than on the left-hand side by a factor of two. Thus, in total, the right-hand side has more counts than the Ieft-hand side.) This is an inequality that must be satisfied along with Eqs. (2)-(4) in order that Hardy’s nonlocality avoids the detection loophole. Here the contradiction is avoided by inserting 0 (an undetected event) where the contradiction occurs in the loop of deduction by quantum mechanical predictions. Using this method, local realism might also coexist with other forms of nonlocality without an inequality, such as the GHZ [ 31 and Kochen-Specker [8] ones. Eq. (7) can also be derived in another way using a version (Eq. (12) of Ref. [7] ) of Bell’s inequalities, which is for the case of spin-; particles
In Hardy’s case, with Eq. ( l), Eq. (8) becomes n++(a,a)
-n+-(a,b)
From Eqs. (2)-(4) 3. Calculation
-n-+(6,a)
-n+o(a,b)
-no+(ha)
< n++(b,b).
(9)
and Eq. (9), Eq. (7) is derived.
of the minimum
detection efficiency
For imperfect detectors of efficiency 77 (0 6 r) < 1), the actual detection counts are related to quantum mechanical probabilities by the following equations,
312
W.Y. Hwang et al./ Physics Letters A 212 (19%) 309-314 n++ (a, a> = Np++ (a, a> v2,
(10)
n+da,b)
= N[p++(a.b)
+p+-(a,b)lrl(l
-T,J),
(11)
no+(ha)
= Np++(ha)
+P-+(ha)lrl(l
- 3).
( 12)
In the above equations, N is the total number of measurements and pij denotes quantum mechanical probabilities. Substituting Eqs. (lo)-( 12) into Eq. (7), 7.. <
p++(aJ) +p+-(a,b) p++(a,a)
+p++(a,b)
+p++(b,a)
+p+-(a,b)
+p_+(b,a)
+p++(b,a)
+p_+(b,a)
(13)
is obtained. Also, by Eqs. (3) and (4), p+_(a,b)
=p_+(b,a)
=O.
(14)
And also, by symmetry, p++(a.b)
=p++(ka).
(15)
Using Eqs. (14) and (15), Eq. (13) is simplified to rl<
ZP++ (a, b) p++(a,a)
(16)
+2p++(a,b)’
Thus, if the efficiency q is higher than the right-hand side of Eq. (16), then the Hardy nonlocality theorem can be proven by the detection loophole. The value of the right-hand side of Eq. ( 16) depends on the particular experimental situation: spin measurement directions and quantum mechanical state. In the following, the necessary efficiency is calculated for some specific cases. In Ref. [ 51, the nonlocality order parameter is y (3 p++ (a, a) = [(a - p)aj?/( 1 - a/3)12) and it has its maximum value 0.09 for a = 0.9070 and /3 = 0.4211. In this case, p++(a, b) is calculated using Eq. ( 13.b) of Ref. [5], p++(a,b)
= jNA2B1* = - a2p2
1 -cup
= 0.24.
(17)
Inserting these values into Eq. (16) we obtain 776 0.84.
(18)
(The case cr. /3 2 0, is considered here without loss of generality.) Thus detectors of efficiency higher than 0.84 are needed in this case. Next, the minimum of the necessary efficiency is calculated. Using the expressions of p++(a, a) and p++(a, b), Eq. (16) becomes 2 V G (a-kW/(l
--a$)
+2’
(19)
The minimum of the right-hand side is obtained when ( CY- p)2/( 1 - crp) has its maximum value 1, which is obtained in the following limit
(20)
W.Y. Hwang et al./Physics
Letters A 212 (1996) 309-314
313
In this limit, 2 rl
G (cw-/?)2/(1
2 -o/3)
+2
(21)
+ 3’
The minimum efficiency needed to violate locality is thus f. The limits of the state and the directions of measurement are given by the equations that connect cy and p with the directions of spin measurements (Eqs. (18) and (19) of Ref. [5]), I$) = aI+)I+)
- PI-)I-)
-+
(22)
l+N+L
b -+ -z.
a --+ --I,
(23)
4. Discussion It is interesting that the minimum of the necessary efficiency is obtained in the limit that 1s) becomes a direct product state (Eq. (22)) which does not show any nonlocality. The minimum 3 (N 66.7%) is just the value found in Table 2 of Ref. [7]. The state and directions of measurement obtained in Bqs. (22) and (23) are just Eqs. (32)-( 34) of Ref. [ 71 if the basis of polarization directions of the second particle is rotated by 90”. In fact, i is just the minimum value obtainable theoretically by Eq. (8). This can be shown using the following relations, n+-(w,P2)
n++(al,Pl)
=~r12P++(WPi),
n-+(a2,/31)
= ~r12P-+(~2,Pd,
= m2P+-hP2L
n+o(4,/32)
= NT41 - 7?)[P++(“19/32)
+P+-(al,P2)1,
no+(~2vPl)
=m4l-7))[P++(~:!,P1)
+p-+(~2,/31)19
?~++(a29/32) = Nq2p++b2,P2L
By Eqs. (24)
(24)
and (8),
(25) = {[P++(~t,P2)
+P+-(m,P:!)l+
{P++(al*Pl)
+ [P++(m,P;!)
-[P++(~29P2)
+P+-(al,Pz)
[p++(a2,P1)
+P+-(al,p2)1
+p-+(a2,p,)l} + [p++(a2,P1)
+p-+(w,pl)l
+p-+hpdl}-‘.
(26)
In Eq. (26), P,j
> 0
for all
i, j.
(27)
From the impossibility of superluminal communication or directly from quantum mechanics, P++(m,P1>
+P+-(W,Pl)
=p++(LyI,p2)
P++(m,P1)
+P-+(w*Pl)
=p++((Y2,P1) +p-+(az,pI)
must be satisfied. By Eqs. (27)
and (28))
+p+-(crl,p2),
(28)
314
W.Y. Hwang et d/Physics
P++(w*Pl)
Letters A 212 (1996) 309-314
+P-+(cy:!,PI>
(29)
hold obviously. By Eqs. (27) and (29), the minimum of the right-hand side of Eq. (26) is obtained to be 3 when the following equations are satisfied, P++(~l*Pl)
=P++(cfI7P2)
+P+-(w,a)
=P++(a2rPl>
P++(a2vP2)
=p+-(Q19P2)
=P-+(~29Pl)
=o.
+P-+(cu:!,PI),
(30) (31)
Eq. (31) is just the conditions (Eqs. (2)-(4)) of Hardy’s nonlocality theorem. Eq. (30) is satisfied in the case (Eq. (21) ) when the minimun 3 is obtained. Acknowledgement
The authors express their gratitude to Dr. J.W. Lee for good suggestions. This work is supported in part by the KOSEF. References 1 II J.S. Bell, Physics I (1964) 195; E Selleri, ed., Quantum mechanics versus local realism, The Einstein-Podolsky-Rosen paradox (Plenum, New York, 1988). 12 1J.F. Clauser and M. Home, Phys. Rev. D 10 ( 1974) 526. 13 I D.M. Greenberger, M. Home, A. Shimony and A. Zeilinger, Am. J. Phys. 58 ( 1990) 1131. 141 L. Hardy, Phys. Rev. Lett. 68 (1992) 2981. 15I L. Hardy, Phys. Rev. Lett. 71 (1993) 1665. 16 I PG. Kwiat, PH. Eberhard, A.M. Steinberger and R.Y. Chiao, Phys. Rev. A 49 ( 1994) 3209. 17 ] P.H. Eberhard, Phys. Rev. A 47 ( 1993) R747. 18 I S. Kochen and E.P Specker, J. Math. Mech. 17 ( 1967) 59.