Maximally entangled states and the Bell inequality

18 March 19% PHYSICS

EUEVIER

LETTERS

A

Physics Letters A 212 (1996) 123-129

Maximally entangled states and the Bell inequality Jose L. Cereceda Departamento Gestih de Red. Telefbnica de Espafia, SA., Gran Via 30, 28013 Madrid, Spain

Received 10 November 1995; accepted for publication 9 January 19% Communicated by P.R. Holland

Abstract Kar’s recent proof showing that a maximally entangled state of two spin- 4 particles gives the largest violation of a Bell inequality is extended to N spin-t particles (N > 3). In particular, it is shown that all the states yielding a direct

contradiction with the assumption of local realism do generally consist of a superposition of maximally entangled states.

Recently, Kar (see Ref. [ 11, and references therein) has shown that a maximally entangled state of two spin- 3 particles not only gives a maximum violation of the CHSH inequality [2] but also gives the largest violation attainable for any pairs of four spin observables, these pairs being noncommuting for both systems. To prove this, Kar made use of an elegant (and powerful) technique based on the determination of the eigenvectors and eigenvalues of the associated Bell operator [3]. In this Letter we would like to extend these results to the case in which N spin- $ particles (N & 3) are considered. We will show that the most general N-particle state giving the largest violation of a Bell inequality does consist of a superposition of maximally entangled states. As expected, those states giving maxhnal departure from classical expectations correspond to the class of states introduced by Greenberger, Home, and Zeilinger in proving Bell’s theorem without using inequalities [4,5]. In order to look for a violation of local realism when dealing with N spin-3 particles it is necessary to consider correlation functions involving measurements on each of the particles. A suitable generalisation of the CHSH inequality to an N-measurement scheme was obtained by Hardy [6]. Hardy’s inequalities can be rewritten in the form -2&B&2,

(1)

where (B, > denotes the expectation value of some Hermitian operator B, (the so-called Bell operator 131) acting on the 2 ‘-dimensional tensor product space associated with N spin- 3 particles. For concreteness, and for purposes of comparison with Ref. [l], in what follows we concentrate on the case N = 3. Later on, we shall consider the case of arbitrary N. For three spin- i particles the representative Bell operator is [6] B, = o(‘+(n;)+‘s)

+ o(n’,>o(‘%>o(‘r’s>

+ (+(‘~‘I)(+(&)u(%)

- ~(‘r,>a(%)o(n,), (2)

0375-9601/%/$I 2.00 0 1996 Elsevier Science B.V. All rights reserved PII SO375-9601(96)00026-6

124

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Letters A 212 (1996) 123-129

where the operator a(n,> (o(n’$ corresponds to a spin measurement on particle i (i = 1, 2, 3) along the direction ni (@. The square of this Bell operator is given by

(3) Now, replacing each commutator [ cr(n,), atn’J Bi=41+4sin

8, sin 8, crL,crL,+4sin

by its value ’ 2i sin Bi uIi, e2 sin 6$ ~~~~~~~+4sin

we get 8, sin 8, t~~i(~~~,

(4)

where Bi is the angle included between ni and n:, and (To i is the spin operator for particle i along the direction perpendicular to both R,. and rr:.. From this expression we can see at once that the largest eigenvalue for Bi is p, = 4( 1 -t- 1sin 8, sin 8, I + I sin 8, sin 8, I + 1sin 0, sin e3 I),

(5)

which attains a maximum value of pmax = 16. From (4) it is also apparent that to every eigenvalue of Bg there corresponds a pair of (degenerate) eigenvectors, namely, 1uL1, crL2, a13> and I -rII, -vL2, -(rL3), where 1c-r1i> (I - vI i)) is the eigenvector of oL i with eigenvalue uI i = f 1 (- uI i = T 1). In particular, the eigenvectors corresponding to the largest eigenvalue (5) are (in an obvious notation): 1T , t, 7 ) and I&, 4, .l.> for sgnkin ~,)=s~(sin e,)=sgn(sin 0,>; it, t, J> and 14, 1, 7) for sgn(sin @,>= sgn(sin f?,) # sgdsin 8,); I t , .J, 1) and 15 , r, T > for sgnkin 8,) f sgn(sin 8,) = sgn(sin 8,); and, finally, I t, _1, t > and I 4, f , J> for sgn(sin 0,) = sgn(sin 8,) + sgn(sin 0,). On the other hand, it can be easily seen that the minimum possible eigenvalue for Bi is zero. So, for example, I f , 7, 1) is an eigenvector of B,!, with zero eigenvalue whenever 8, = 8, = 8, = r/2. Of course, the operator Bi cannot have negative eigenvalues because this would imply the (Hermitian) operator B, has a complex spectrum. As every eigenvalue for B$ must lie in the interval f0, 161 it follows that the eigenvalues for B, are necessarily restricted to lie in the interval f - 4, 41. Consequently, inequality (1) for the Bell operator (2) will be violated for those eigenvectors of B, with eigenvalues A fulfilling 2 < I A I *E 4. Note that the maximum amount of violation of Hardy’s inequality predicted by quantum mechanics is by a factor of 2, instead of the factor fi achieved in the CHSH inequality. It is worth pointing out that the results mentioned in the present paragraph remain valid for an arbitrary number of particles. This is so because Hardy’s inequalities (1) involve only four correlation unctions regardless of the number N of me~uremen~ considered 161.This fact will be used later in considering the N-particle case. In view of Eq. (5), the largest eigenvalue for B, will be Ih,I =2(1 +/sin

8, sin e2 I + I sin 8, sin 8, I + I sin 0, sin 8, I)I”.

(6)

Now, as a product state cannot give rise to violations of local realism (see below), it follows that an eigenvector of B, with eigenvalue A such that 2 < I h 1 G 4 will necessarily consist of an entanglement of the two degenerate eigenvectors of Bf, with eigenvalue A2 [3],

Iq> = o I q, z2, z,> +@+I

-z,,

-z2,

-z,>,

(7)

where, for simplicity, we restrict ourselves to directions ni and n$ lying in the x-y plane (these directions being specified by the ~imu~al angles & and &, respectively). so that 1zi> ( 1-z,)) denotes the eigenvector

’ This expression for the commutator actually differs from that of Ref. f l] by a factor - 1. This is because, following convention, we take here the angle 8, as the negative of that used in Ref. [I].

the usual

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Letters A 212 (1996) 123-129

125

of the spin operator along the z-axis for particle i, with eigenvalue z, = f 1 (- zi = T 1). It is easy to show that, whenever 2 < I A I < 4, the relative signs of z,, z2, and z3 in (7) are uniquely determined by A (for fixed values of el, e2, and f?,.) Likewise, the real coefficients a and p (which are assumed to satisfy the normalisation condition a2 + p ’ = 1) as well as the phase factor 4 will depend on the eigenvalue h. We will now show that for the case in which’ the eigenvector (7) is associated with the largest eigenvalue (6), these coefficients must fulfill the condition I a I = I p I = I/ fi. In other words, the largest possible violation occurs for maximally entangled states. This can be seen by directly evaluating the expectation value (%PI B, IP) for the state vector (7). This is given by

+cos(+-zp#J’,

-z2(6;

-z~~3~-~~~((P-z1~,-z2~2-z~~p3)1.

(9

As the product a/3 factorises out in this expression, it is clear that in order for (8) to reach its largest value (6) it is necessary that the absolute value for ap be a maximum, i.e., I a I = 1 /3 I = l/ fi. Although it is apparent from (8) that this must be the case for I A, I = 4, it might appear less evident when I A, l < 4. That the above statement is indeed true can be best appreciated by means of a simple example. So, let us take the values 4, = 42 = & = 0, & = d; = ?r/2, and &s = 1r/4, so that 8, = 8, = ~/2 and 8, = ?r/4. For these values we have I A, I = 2(2 + &)I/‘. On the other hand, the absolute value of (T I L&.,I tY> is found to be, 41 ap I 12-‘/’ sin f$-(1 +2-‘j2) cos 4 I, where we have put z, = z2 = zj = + 1 in (8) since, for the above values for ei, sgn(sin 0,) = sgn(sin 0,) = sgn(sin 0,) (of course, the reasoning remains unchanged if we instead choose z, = z2 = zj = - 1; the important point is that z, = z2 = ZJ Therefore, for some a, p. and 4, we must have 2 I a/3 I 12-‘/2 sin (b - (1 + 2-‘12) cos 4 I = (2 + fi)“‘.

(9)

The only values for which this equality holds are I a I = I /3 I = l/ 1/2, and 4 = - 1r/8 + mr, where n = 0, . This follows at once from the fact that the function h(4) = 12-‘12 sin 4 - (1 + 2-‘/2) cos 4 I fl, *2,... reaches its maximum value for 4 = - ~/8 + nT, and this value is precisely (2 + fi)‘12. From the preceding example it is obvious that, if expression (8) is to be equal to the largest eigenvalue (61, the phase factor (b must be a suitable function depending only on the angles Cpiand 4: (or. alternatively, on the angles c#+and ei, as ei = 4: - A). This is so because for the largest eigenvalue we have always 2 1ap I = 1, and then Eq. (8) involves only the variables 4, & and 4:. This dependence can be easily obtained for the special (and important) case for which Bi = ?r/2 (i = 1, 2, 3). In this case the eigenvalue (6) attains its maximum value 4, and then the following equalities should be simultaneously fulfilled, cos(+~,-4;-~;)=

fl,

( 1Oa)

cos(+#+&-(p;)=

*1,

( 1Ob)

cos(4-&i-4;-&)=

fl,

( ‘Oc)

cos(+4i-&-&)=

Tl,

( 1Od)

where we have put z, = z2 = zj = + 1 in (8) since, as before, sgn(sin 0,) = sgn(sin 0,) = sgn(sin 0,). So, recalling the relationship 4: = C&+ 7r/2, it is a trivial matter to see that equalities (lOa>-(1Od) can be matched providedthat
IV)= i,l+, fi

+, +)*&4i++2+93)1-_

-,

-))

(11)

will be an eigenvector of the Bell operator B,=(Y(f(#*)~((b2+rr/2)0.((P3+7C/2)+P~f#,+~/2)a(cP*)o(Q13+n;/2) + W(#,

+ 7V2)+#%

+ 7V2kW

+Wcb,)+#*)o(W?

(12)

with eigenvalue T 4, whenever (Y= p = y = - S = + 1. More generally, it can be shown that any state of the form

IP)=& z,,

zr, 2,) fei~z~~~+~~42+L~~~~I-~I,-z2,

-z3))

(13)

is an eigenvector of the operator (12) with eigenvalue + 4 or - 4 for a suitable choice of the sign factors (Y, /3, y, and S (provided, in any case, that crpy6 = - 1). In this way, it is clear that the product of the quantum expectation values for the operators T, = d#+d& + 7r/2M+, + 42). T2 = dqb, -t 7r/2b(42bb#+ + v/2), T3 = a(+, I- 7r/2)(~(+~ + n/2)a(&), and T, = ~(c/.J,)(T(~~)c~(&), must be equal to - 1 when evaluated for any of tbe states (13). Indeed, the set of vectors (13) forms a basis set (with a total of eight linearly independent vectors) which simultaneously diagonal& the four (commuting) operators T,, T2, T3, and T, 171. As a result, the value - 1 for the above product of expectation values actually arises from the fact that T,T2T,T., = -I. That these quantum predictions for the expectation values indeed lead to a direct contradiction with the assumption of local realism constitutes the theorem of Greenberger, Home, and Zeilinger [4,5] (see also Refs. f7,8].) In fact, it can be easily shown [6] that a m~imum violation of the Bell inequality (1) always entails a confliction of the GHZ type. Before analysing the N-particle case, it is worth noting an implica~on of Eq. (8). Indeed, from that equation it follows that, for fixed Cp,the eigenvalue h does not determine the individual values of cx and p but, rather, the value of their product cup. This means that, whenever 1cr.1 f I j3 I, if the state vector lt1’,a > = Q I zl, z2, 2,) + pe’+I -zl, -z2, - z,} happens to be an eigenvector of (2) with associated eigenvalue h, the same is true for tbe (linearly independent) vector l?@~y$,> = j3 I z,, z,, z,) + ae’” I -zI, -z2, -z,>. This degeneracy is due to the very structure of the state vector (7), and will be called here a tidal degeneracy. Notice that the rriuial degeneracy is removed when h corresponds to the largest eigenvalue since, as we have seen, in this case we have I cx I = 1 j3 I = l/J’z. This type of degeneracy is to be distinguished from the nontrivial degeneracy which occurs when eigenvectors with different relative signs for the z’s are associated with the same eigenvalue. As already noted, for I AI > 2 the relative signs of z,, z2 and z3 in (7) are uniquely determined by A so that any eigenvector of the Bell operator (2) violating inequality (1) (excepting the nondegenerate eigenvector corresponding to the largest eigenvalue) is only trivially degenerate. Turning to the N-particle case, one could equally prove that an N-particle state giving the largest violation of a Bell inequality has to be maximally entangled. Properly speaking, such an iv-particle state will in general consist of a superposition of maximally entangled states. That this requirement has to be met for those states yielding the maximum violation follows in a rather straightforward way when one considers the correlation function P(c#J!, +2,..., &,; p) that quantum mechanics predicts for a general (pure) state of the form

(14) with

and where I zi> represents the eigenvector of the spin operator along the z-axis for the ith particle fi= 1,2,..., N), with eigenvalue zi = f 1. As before, and without loss of generality, we assume that each

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Letters A 212 (1996) 123-129

127

particle is subjected to a spin measurement along a direction lying in the x-y plane, with azimuthal angle c&. The quantum prediction for P(& , &, . . . , &; q) is given by

xcos(~z,.z~,....zN-zl~l

-zz42-

***-zN4N)*

(15)

where b,,, **,..., .ZN = e-&-Z2 . .. . . -:N - %,.Z*. . .. . ZN’ and where the asterisk indicates that, for each pair of combinations zI, z2,. . . , zN and - zI, - zZ, . . . , - zN, the summation runs over either the indices zI, z2,. . . , zN or -zl, -z2..... -zN. By examining Eq. (15). one finds that for the function P(c#J,, c&, . . . , &,; ?P> to take on its extreme values f 1 it is necessary that I u_ ,,,,, z~ I = I a- z,,-z *,,,,, -+ I, for all zlr z2,. . . , zN. This is because, if this condition is fulfilled, then we have 2 CZf,,Z2r,, ,, zNI uz,+,~,_,zNI 2 = 1, and thus expression (15) may attain the value f 1 for a suitable choice of the relative phases c$~,.~,,, , zN, namely, for c#J~,,~~., zN= z, 4, z,&+nlr,where n=O, fl. +2 ,... . It thus follows that a state vector of the form +zz42+... IP)=

C’ z,.z**....

with

K,,zz

,....

ZN I c,,.z*

W( 21, ZZ....,

c,,,Z2....,LJ~(zI.

Z2~-.4)

(16)

z‘q

. . . . . ZN I 2 =

1,

z,>> = -j+(

and

I zl, z2,...,zN)*ei~r~~~+LZ~2+~~~+ZN~N~l-z,,

-z2,....

-z,>),

(17)

will yield the value f 1 for PC&, c#~,. . . , c#+; P), and then it will be able to violate maximally a Bell inequality built up from correlation functions involving spin measurements in the x-y plane. State vector (13) can be regarded as the simplest instance of Eq. (16). In this case only one term appears because either of the extreme eigenvalues A = 4 or A = - 4 of the Bell operator (12) is nondegenerate. In general the number of terms appearing in expansion (16) will be equal to the dimensionality of the (nontiuiully) degenerate subspace corresponding to the maximum eigenvalue of the relevant Bell operator. The Bell operator B, for the Hardy inequality (1) takes the general form [61 Bn=aa(C)a(4?)

. ..+&T)

+ W(C)@(W

. ..+#$)

+p+,B)a.(+,B)...+NB) +su(~1”)~(~26)...,(~N6),

(18)

where, as before, a, /3, y. and 6 are sign factors with a&i3 = - 1. As was mentioned, the state vectors violating inequality (1) will be those eigenvectors of (18) with eigenvalues A such that 2 < I A 1 < 4. Naturally, the four parameters +p, &‘-?,c#+?‘, and C#I”are not all independent. Indeed, for each value of i, there are the following possibilities [61: (i> &a = 99. 4: = @‘; (ii> @’ = $p, 4” = 4r; and (iii) 4: = 9:. 67 = 48. In any caSe the most general eigenvector for the Bell operator (18) is one of the form (16) with I !P( zI, z2,. . . , z,)> given by IT(z,,

z2 ,... ,~~))=a,,,,

*,...,

zNIzl, z~,....z~)+P~,,~~ .._.,~Nei’~l.z2.-...~“I-zI. -z2.-..,-zN)9 (1%

where a z,,Z2.... , zN and P,,., *,.. , zN are real numbers with a:,+. . , zN+ P;,z2 (1.1,zN= 1. This is so b-u=, as it stands,thestatevector(l6)with IWz,, z2,..., z,)) given by (19) turns out to be the most general (pure> state for N spin-4 particles and then it will be always possible for any eigenvector of the Bell operator (18) (and, in fact, of any Bell operator involving spin measurements in the x-y plane) to be arranged so as to fit in with the form displayed by such Eqs. (16) and (19). Also generally, for any given eigenvalue A the summation in (16) will extend over all those (nonrriuiully> degenerate eigenvectors (19) associated with that given eigenvalue. Of

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Letters A 212 (1996) 123-129

coursethe cotm+ms a,,,,*,..., and P,,,,,.. , + (or, rather, their product), as well as the phase factor zN

4 z,.z2..,,,zN will depend on the actual value of A. So, for the case in which 1A 1attains its maximum value 4, (i.e., when the inequality is maximally violated), we must have I#J,,,,,,,,,, zN= z14, + z2& + . . . +z,,& + mr, andla *,,Q. . . . . ZNI = I P,,.,, ,_.., ZN1 = vfi, for all Zl? zz.**** zN (see Eq. (17)). It is easy to show, however, that this latter requirement should be fulfilled not only by the maximum possible eigenvalue of the relevant Bell operator but also by its largest one. For this purpose, let us consider the expectation value (!P I B,., IV > of the operator (18)for the state vector(161 with IWz,, z~,.... zN)) given by (19). This expectation value is

[A,(zI,

z2 ,...,

z,)+B,(z,,

z2r....zN)+Cy(z19 z2r....zN)+4(z,,

z2 ,...

rzN)],

(20)

where, for example, CY= y cos(& *,,, , zN- z14: - z2 c,b: - . . . -z&,$1. Clearly, as the product factorises out in each of the terms involved in Eq. (20), it follows that the largest cr Pz,.,, . . . . . ZN e&%alig for B, must fulfil I a_ ,.,,, zNI = I & L ,.,,* N I = l/ fi, for all zl, z2,. . . , zN. In view of Eq. (20) this conclusion holds irrespective of the numb& ‘of correlation functions ( A,, B, , Cy , D, , . . . ) involved. Furthermore, the structure of Fqs. (19) and (20) remains unchanged for the general case in which the spin measurements are carried out along arbitrary directions (in fact, for this case, we have only to redefine the meaning of the ) zi) in (19) as denoting states of spin-up ( zi = + 11 or -down (izi = - 11 for particle i along some appropriate z-axis which, in general, will differ from one particle to the other). Thus, we have demonstrated that a state of N spin- 4 particles (N > 31 giving the largest violation of a Bell inequality must generally consist of a superposition of maximally entangled states. This conclusion applies in particular to those states giving the maximum possible violation. This is achieved when each of the correlation functions attains an appropriate extremum value f 1. So, as a direct (“all or nothing”) contradiction with local realism arises just at the level of perfect correlations, it follows that any state leading to such a contradiction should in general involve a superposition of maximally entangled states. In fact, as we have seen, any state vector yielding the value f 1 for the correlation function PC&, 42,. . . , &; !P) must necessarily assume the form of Eqs. (16) and (17). We conclude by noting that, as expected, this function factorises for a general product state of the form I*)= lTP,)o l?P2)@... 0 l*p,r), with IPi)=az!l zi> +a_,il-~i), a,;= I~zileiezi, and Ia,,12+ Iu_~,I~ =l,i=1,2 ,..., N. Indeed, by making use of the identity 2

C”

cos( z,y, + z2y2 + . . . +z,y,)

= 2N cos y1 cos y2. ..cos yN,

(21)

Z,..?I,.... ZN

one can readily see that expression (15) takes the form

P($,, 42.....&;

9) =2 Nu~ cos((PI+q,) Cos(~2+.1)2)*-CoS(~N+r]N)~

(22)

where qi = o,, - 8_:,, and a0 is a constant with value a, = I a,, I I uz2I . . . I uzNI X I u_,, I I u-Z2 I . . . I u_~~I. Note that a, Q 2-N and then, as it should be, I P I G 1. Obviously, a, reaches its maximum value whenever I u,,I = I u_~, I = l/ fi, for all i. In any case, it is apparent from (221 that for a product state the outcome of a spin measurement for any one of the particles becomes completely uncorrelated with respect to the outcomes corresponding to the other particles, and then such a state will be unable to yield a violation of Bell’s inequality.

The author would like to acknowledge informative discussions with Gregorio Renter0 on the subject of quantum nonlocality.

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References [1] [2] [3] [4] [5] [6] [7] [8]

G. Kar, Phys. Lett. A 204 (1995) 99. J.F. Clauser. MA. Home, A. Shimony and R.A. Holt, Phys. Rev. Lett. 23 (1%9) 880. S.L. Braunstein, A. Mann and M. Revzen. Phys. Rev. Lett. 68 (1992) 3259. D.M. Greenberger, M.A. Horne and A. Zeilinger, in: Bell’s theorem, quantum theory, and conceptions of the universe, ed. M. Kafatos (Kluwer, Dordrecht, 1989) p. 69. D.M. Greenberger, M.A. Home, A. Shimony and A. Zeilinger, Am. J. Phys. 58 (1990) 1131. L. Hardy, Phys. Lett. A 160 (1991) 1. J.L. Cereceda, Found. Phys. 25 (1995) 925. N.D. Mermin. Phys. Today 43. No. 6 (1990) 9; Am. J. Phys. 58 (1990) 731; Phys. Rev. Lett. 65 (1990) 3373.