- Email: [email protected]

2 1 October 1996

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PHYSICS

ELSJZVJER

LETTERS

A

Physics Letters A 222 ( 1996) 21-25

Teleportation, Bell’s inequalities and inseparability Ryszard Horodecki a,1, Michal Horodecki b, Pawel Horodecki ’ a Institute of Theoretical Physics and Astrophysics, University of Gdarisk, 80-952 Gdatisk, Poland ’ Department of Mathematics and Physics, University of Gdan’sk, 80-952 Gdan’sk, Poland c Faculty of Applied Physics and Mathematics, Technical University of Gdarisk, 80-952 Gdan’sk, Poland

Received 3 July 1996; accepted for publication Communicated

13 August 1996

by P.R. Holland

Abstract Relations between teleportation, Bell’s inequalities and inseparability are investigated. It is shown that any mixed two spin+ state which violates the Bell-CHSH inequality is useful for teleportation. The result is extended to any Bell’s inequalities constructed from the expectation values of products of spin operators. It is also shown that there exist inseparable states which are not useful for teleportation within the standard scheme. PACS: 03.65.B~

1. Introduction Recently aspect

Bennett

of quantum

involves

et al. [ 1] have discovered inseparability

the separation

and quantum

a new

- teleportation.

This

of an input state into classical

parts from which the state can be recon-

structed with perfect fidelity F = 1. The basic idea is to use a pair of particles in a singlet state shared by sender (Alice) [2]

and receiver

noticed

be useful

(Bob).

Quite recently

that the pairs in a mixed

for (imperfect)

teleportation.

Popescu

state could still There was the

what value of the fidelity of the transmission of an unknown state can ensure us of the nonclassiquestion

cal character

of the state forming

the quantum

chan-

nel. It has been shown [ 2,3] that the purely classical

channel can give at most F = 3 (see also Ref. [4] in this context). Then Popescu raised basic questions concerning a possible relation between teleportation,

’ E-mail: [email protected].

Bell’s inequalities and inseparability: “What is the exact relation between Bell’s inequalities violation and teleportation? Is every mixed state that cannot be expressed as a mixture of product states useful for teleportation?’ [ 21 2 . The problem is rather complicated, as these questions concern the mixed states which apparently possess the ability to behave classically in some respect but quantum mechanically in others [ 21. Fortunately for 2 x 2 systems two basic questions concerning the violation of Bell’s inequalities and the inseparability of mixed states have been solved completely. In particular, in Ref. [ 51 the effective criterion for the violation of Bell’s inequalities has been obtained. Quite recently the problem of inseparability has been investigated in detail by Peres [ 61 and the present authors [ 71. In particular, the necessary [ 61 and sufficient [ 71

*We call a state inseparable if it cannot be written as convex combination of product states.

0375-960 l/96/$12.00 Copyright 0 1996 Elsevier Science B.V. All rights reserved. Pfl SO375-9601(96>00639-1

22

R. Horodecki et al./ Physics Letters A 222 (1996) 21-25

condition for separability of mixed states for 2 x 2 systems has been provided. The main purpose of the present Letter is to present the effective criterion for teleportation via mixed two spin-4 states and discuss it in the context of Bell’s inequalities and inseparability. Using the results contained in Refs. [5-71 we will show further that if a mixed two spin-i state violates any Bell’s inequality constructed from the expectation values of the products of spin operators (in particular if it violates the original Bell-CHSH one), then it is also useful for teleportation. We will also demonstrate that there are inseparable states which are not useful for teleportation within the standard scheme 3

2. Maximal scheme

fidelity for the standard

We start with the representation Hilbert-Schmidt space

teleportation

11 I 1. One of the particles is given to Bob while the other one and a third particle in an unknown state $ are subjected to Alice’s joint measurement. The latter is given by a family of projectors Pk = ]tf/k)(Jfkl,

where the $k constitute $,;, = -$(et

$,;,

(

- [email protected]+IC3s~

1,2,3,

(2)

the so-called Bell basis

@et F e2 @ e2>,

=-$j(e,@[email protected]~).

(3)

with et, e2 being the standard basis in C2. Then using two bits Alice sends to Bob the number of outcome k and Bob applies some unitary transformation uk obtaining in this way his particle in a state ek. Then the fidelity of a transmission of the unknown state is given by the formula [ 2,4,9]

of the state in the 3 = 3

[email protected]+r

Q = $

k=O,

o+ C t,d~~C3f~,, 1 ll,Ilt=1 (1)

where Q acts on the Hilbert space 3-1= Xi @‘Hz = C*@ C*, Z stands for the identity operator, {a,,)$t are the standard Pauli matrices, r, s are vectors in R3, r - u = Cl, r-pi. The coefficients tnnl = Tr( PO, @ (T,,) form a real matrix which we shall denote by T. Note that the representation appears to be very convenient in the investigation of some aspects of inseparability of the mixed states. Indeed, all the parameters fall into two different classes: the first (r and s) describes the local behaviour of the state, the second (T matrix) is responsible for correlations. It is compatible with the fact that the mean value of the Bell-CHSH observable depends only on the correlation parameters T of the state Q [5]. Let us briefly recall the standard teleportation scheme. It involves two particle sources producing pairs in a given mixed state e which forms the quantum channel (originally formed by a pure singlet state 3 By the standard teleportation scheme we mean here that Alice uses the Bell operator basis [ 81 in her measurement while Bob is allowed to apply any unitary transformation.

s s

dM(6)

cPkTr(ekP&

(4)

k

where the integral is taken over all 4 belonging to the Bloch sphere with uniform distribution M, pk = Tr [(Pk 63 Z) (P$ @ e)] denotes the probability of the kth outcome. Now the task is to find such Uk’s as produce the highest fidelity (a choice of a quadruple of uk’s we shall call strategy). Par this purpose, let us compute the integral (4). The output state & is given by ek=~TT1.2[(pk~~k)(p~~e)(pk~(I:)I.

(5)

Here the partial trace is taken over the states of the unknown particle and Alice’s one. Putting P4 = i (I + a * a) one obtains

(6)

Here the Tk’S and r, s, T correspond to the Pk’s and e, respectively, via formula ( 1) (we have: To = diag(-1,-1,-l), Tl = diag(-l,l,l) T2 = diag( 1, -1, 1), Ts = diag( 1,1,--l), rk = Sk = 0, for k = 0, 1,2,3.); the Ok’s are rotations in R3 obtained from the uk’s by uii * dJ+ = (o+i-i> *u

(7)

23

R. Horodecki et al./ Physics Letters A 222 (1996) 21-25

(0 is tations group tribute

here determined uniquely as the O+( 3) is a homomorphic image [ IO]). Omitting the terms which to the integral (4) and using the

(a,Aa)dM(a)

= $TrA

I

group of roof the U(2) do not conformula (8)

S

Example. Consider a pure state of the form I$) = aei @ e2 - be2 8 ei.

(13)

One obtains &,,

2 a3 - b3 = -~ 3 a-b ’

which is compatible

(14) with Ref. [ 41.

one obtains (9) Now we shall maximize F under all strategies. Clearly, as -Ti is a rotation we see that the maxima of the terms in the above formula do not depend on k any longer so that F max =m;x$(l

- $TrTO).

where the maximum that we have

(IO)

is taken over all rotations. Note

F,,,,, d i( 1 + $Tr\/TTT).

(11)

Of course, we need to derive the expression for 3,,,,, only if the Iatter is greater than 5, which is the upper bound for the classical teleportation [ 2,3]. If Fm,, > $ we say that the state forming the quantum channel is useful for teleportation. Clearly, F,,,,, can exceed 5 only if Tr &% > 1. Now basing on the results contained in Ref. [ 91 one can see that the latter condition implies det T < 0. But then the inequality ( 11) passes into an equality. Consequently, defining the function N( e> := Tr v?? one has the following theorem. Theorem 1. Any mixed spin-i state is useful for (standard) teleportation iff N(Q) > 1. Then the fidelity amounts to 3 max = i[l

-t @J(e)].

(12)

Now, if N(e) > 1 then there exist rotations 01 and 02 such that OlTO2 is diagonal with tii < 0 for i = 1,2,3. Then the best strategy is given by unitaries uk = Urrk where U is determined (up to an irrelevant phase factor) by 0 = 0102 via formula (7).

3. Relation between Bell’s inequalities teleportation

and

Note that N( @) is a function of the correlation parameters T only. This allows one to establish a relation between teleportation and Bell’s inequality due to Clauser, Horne, Shimony and Holt [ 1 l] (BellCHSH). As one knows, the necessary and sufficient condition for violating the Bell-CHSH inequality involves a real valued function M( Q) = maxi>j (Ui + u; ) where ni are eigenvalues of the matrix TtT [ 51. Then the inequality M(Q) 6 1 is equivalent to the BellCHSH one. NOW as r*i < 1 for i = 1,2,3 [ 121 and N(Q) = xi=, fi we obtain the relation N(e)

b M(e).

(15)

Note that for any state which violates the Bell-CHSH inequality we have M(Q) > 1. Then, according to the relation (15) and Theorem 1 we obtain the estimate FIn?-lX2 ill

+

$Wdl

> 3.

As the maxima1 mean value of the Bell-CHSH servable is B,,, = 2dm we have also &lax 2 i(l

+ I$$,,).

(16) ob-

(17)

The inequalities (16), ( 17) are valid for an arbitrary mixed two spin-$ state which violates the Bell-CHSH inequality and they tell us that any such state is useful for teleportation. Now we shall see that even a stronger statement is valid. For this purpose consider generalized BellCHSH inequalities, i.e. all the Bell’s inequalities which can be constructed from the expectations of products of spin operators a - u @ b - CT,where a and b are unit vectors [ 131. Of course, the expectations (or correlation functions) E(a,b)

rTr([email protected]*u)=(a,Tb)

(18)

R. Horodecki er al./ Physics Letrers A 222 (1996) 21-25

24

depend only on the T matrix. Hence the generalized Bell-CHSH inequalities can be violated only if N(e) > 1. Indeed, if N(e) < 1, there always exists some separable state that has the same T matrix as the state Q (see Ref. [9]). In this way we have obtained the following theorem. Theorem 2. Every mixed two spin-i state which violates any generalized Bell-CHSH inequality is useful for teleportation.

4. Inseparability

and teleportation

Let us now turn back to the question: “Is every mixed state that cannot be expressed as a mixture of product states useful for teleportation?‘. Generally, the problem is rather complicated as it requires one to obtain the maximum of the fidelity over all possible teleportation procedures. Here we will see that within the standard teleportation scheme the answer is “no”. For this purpose consider the following class of the states [ 141, e=PII$I)wII

-tPZI~2)(~2l~

(19)

where II/Q) = ael @ et + be2 @ e2,

(20)

/#Q) = set @ e2 + be:! @ et,

(21)

with a, b > 0, {ei} being standard basis in C2, 0 < (pt - ~2)~ < (a* - b2)2. The above states have interesting properties. First, note that as M(Q) = 1 + (Pl -P2J2 - (a2 - b2)2 < 1 they do not violate the Bell-CHSH inequality. In addition, it is possible to choose the parameters pr and a so that the maximal absolute value of the expectation of products of spin operators is arbitrarily close to zero. Then it follows from Theorem 1 that many of the states ( 19) are not useful for teleportation. But what can we say about the above states in the context of the inseparability? As it was mentioned in the introduction, the effective criterion for inseparability of the states of 2 x 2 systems has been found [ 6,7]. Namely a two spin-i state is inseparable if and only if its partial transposition is not a positive operator. The matrix elements of partial transposition $2 of a state Q is given by

where e nlP,nv = (e,, @ e,lele,

@ fy).

(23)

Now is easy to see that all the states (19) are inseparable4. In fact, one can show that the inseparability of the above states manifests itself via a “hidden” nonlocality (see Ref. [ 151) which can be revealed [ 141 by means of Gisin’s filtering method [ 161. Thus we have provided an example of states which are inseparable and nonlocal but still are not useful for teleportation within the standard scheme.

5. Conclusion In conclusion, we have considered the questions concerning possible relations between teleportation, violation of Bell’s inequalities and inseparability. In particular, we have obtained the maximal fidelity for the standard teleportation scheme with the quantum channel formed by any mixed two spin-3 state. This involves only the correlation parameters of the state. Then it was possible to compare the two different aspects of quantum inseparability: teleportation and Bell’s inequalities. More precisely, we have shown that if a mixed two spin-i state violates any generalized Bell-CHSH inequality (in particular if it violates the original Bell-CHSH one) then it is also useful for teleportation. We have also considered the states which are inseparable, but are not useful for the standard teleportation. Here the inseparability is due to the relation between the local and correlation parameters. Then there is the question: what would happen if we allowed Alice to use any projectors - not only the maximally entangled ones? In fact she may perform any generalized measurements. In the formula for the fidelity the local parameters could then also appear. It is not clear whether a higher fidelity can be obtained within such a generalized scheme. Thus, the problem of a relation between the widely understood Bell’s inequalities (e.g. involving nonstandard measurements [ 2,14-161) and J In Ref. [ 141 the states ( 19) were shown to be inseparable by means of the entropic criterion (see in this context Ref. [ 91) The latter appears to be equivalent to inseparability for the considered states. but this is not the case in general [ 6,7 1.

R. Horodecki et al./ Physics Letters A 222 (1996) 21-25

more general teleportation vestigations.

schemes needs further in-

Acknowledgement We would like to acknowledge sions with Nicolas Gisin.

stimulating

discus-

References [l] C. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres and [2]

[ 31 [4] [5] [ 61

W.K. Wootters, Phys. Rev. Lett. 70 (1993) 1895. S. Popescu, Phys. Rev. Len, 72 (1994) 797. S. Massar and S. Popescu. Phys. Rev. Lett. 74 (1995) 1259. N. Gisin, Phys. Lett. A 210 (1996) 157. R. Horodecki, P. Horodecki and M. Horodecki, Phys. Lett. A 200 ( 1995) 340. A. Peres, Separability criterion for density matrices, to appear in Phys. Rev. Len. 77 (available at e-print archive quantph/9604005).

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171 M. Horodecki, P Horodecki and R. Horodecki, On the necessary and sufficient conditions for separability of mixed quantum states, quant-ph/9605038. 181 S.L. Braunstein, A. Mann and M. Revzen, Phys. Rev. Len. 68 (1992) 3259. [9] R. Horodecki and M. Horodecki, Information-theoretic aspects of quantum inseparability of mixed states, Phys. Rev. A, in press. [ 101 M. Wiessblutch, Atoms and molecules (Academic Press, New York, 1978). [ 111 J.F. Clauser, M.A. Home, A. Shimony and R.A. Holt, Phys. Rev. I&t. 23 ( 1969) 80. [ 121 R. Horodecki and P Horodecki. Phys. L.&t. A 210 (1996) 227;

1131

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P Horodecki and R. Horodecki, Phys. Rev. Lett. 76 ( 1996) 2196. SM. Roy and V. Singh, J. Phys A 11 (1977) L167; 12 (1978) 1003. R. Horodecki, Phys. L&t. A 210 (1996) 223. S. Popescu, Phys. Rev. Lett. 74 (1995) 2619; N.D. Mermin, Hidden quantum nonlocality, preprint, Lecture given in Bielefeld at the Conf. on Quantum mechanics without observer (1995). N. Gisin, Phys. Lett. A 210 (1996) 151.

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