CLASSIFICATION OF MAXIMALLY ENTANGLED STATES OF SPIN 1/2 PARTICLES

Transcription

1 CLASSIFICATION OF MAXIMALLY ENTANGLED STATES OF SPIN 1/ PARTICLES S. Ghosh, G. Kar, and A. Roy Physics and Applied Mathematics Unit Indian Statistical Institute 03, B. T. Road Calcutta India. E mail : res9603@isical.ac.in Abstract A simple geometrical proof is been given in order to show that any two maximally entangled states of three spin 1/ particles are locally unitarily connected. And assuming a condition, which ensures that a typical plane in a Hilbert space of n spin 1/ particles contains at least one product state, it is shown that any two maximally entangled states of spin 1/ particles are locally unitarily connected. PACS No. : Bz 1

2 1 Introduction Entanglement is the basic property that is manifested in different quantum behaviours, and makes a sharp distinction between classical and quantum worlds. It has been shown that entanglement of quantum mechanical states plays the key role in showing the violations of Bell s inequalities ([1], [], [3]) and in showing Hardy s nonlocality ([4], [5], [6], [7], [8]) both of them being the criteria proving the fact that the locality and reality assumptions of EPR ([9]), when applied to any model (i.e., hidden variable model) in quantum mechanics, lead to a contradiction. Also entanglement is the basic feature in showing teleportation ([10]), quantum cryptography ([11]), superdense coding ([1]), entanglement enhanced classical communication ([13]), entanglement enhanced communication complexity ([14]), and quantum computational speedups ([15]). The entanglement property of states occurs only when we consider composite quantum systems. A state of a composite quantum mechanical system is said to be entangled if this state can not be represented as a tensor product of states of the individual subsystems (constituting the composite system), in any orthogonal basis of the individual subsystems. It is known that ([16]), the product states of two spin 1/ particles (in fact one can take any product state of n spin 1/ particles) do not violate Bell s inequality (it is true for any choice of observables), while the singlet state (or, a state which is locally unitarily connected to it) violates the same with maximum probality (in comparision to other nonsinglet states), for a specific choice of observables. And for the same choice of observables, any entangled pure state of two spin 1/ particles, which is neither a product state nor a state which is locally unitarily connected to the singlet state, violates Bell s inequality with positive probability, but less than the above maximum one ([3]). So there is a direct relation between entanglement of pure states and violation of Bell s inequality by this state. And from this point of view, the singlet state is said to be maximally entangled. Thus any pure state of two spin 1/ particles, which is locally unitarily connected to the singlet state, is again a maximally entangled state. The maximally entangled states of two spin 1/ particles (i.e., states which are locally unitarily connected to the singlet state) do not satisfy Hardy s nonlocality criteria, while any non-maximally entangled (pure) state of two spin 1/ particles does satisfy the above criteria ([17]). The GHZ state of three spin 1/ particles ([18]), and any state, which is

3 locally unitarily connected to the GHZ state, satisfy the Hardy s nonlocality criteria ([8]). 1 It is also known that the GHZ state contradicts the locality and reality assumptions of EPR with 100% probability ([18]). And in this respect, the GHZ state may be called a maximally entangled state of three spin 1/ particles. So one may think of how to define maximally entangled states of n spin 1/ particles (where n ) and how to classify them. Maximally entangled states are those (pure) states each of which has no single particle property. That is, the single particle reduced density matrices are maximal mixture states, i.e., 1 I. Thus, if ρ is the density matrix of a quantum mechanical pure state of n spin 1/ particles (where n is a positive integer greater than 1), and if ρ 1, ρ,..., ρ n be the single particle reduced density matrices of first, second,..., n-th particle respectively, then above state (whose density matrix is ρ) will be a maximally entangled state if ρ i = 1 I in some chosen basis in H i (i =1,,...,n), where I is the identity matrix. Schlienz and Mahler ([0]) provided an elaborate mathematical proof that every maximally entangled state of three spin 1/ particles is locally unitarily connected to the GHZ state, using some invariants of three and two particle properties. This result is also valid for maximally entangled states of two spin 1/ particles 3, but the result has not been extended for n spin 1/ particles for n 4, as the method of Schlienz and Mahler, in that case, is almost impossible to apply. In this paper we shall provide a geometrical proof (which is much more simple than that of Schlienz and Mahler) to show that the maximally entangled states of three spin 1/ particles are locally unitarily connected. And assuming a criterion, we shall prove the above result for n spin 1/ particles, where n 4. The paper is arranged as follows. In section, we shall mention the algebraic equations to be satisfied by the coefficients of a maximally entangled state of n spin 1/ particles. In section 3, we shall describe a condition, whose validity will imply that the maximally 1 1 In fact it can be shown that ([19]) any GHZ like state of n spin 1/ particles : + z 1 +z... +z n + 1 z 1 z... z n, and any state, locally unitarily connected to it, satisfy Hardy s nonlocality criteria with a positive probability, whose maximum value is 1/ n for n 3. Thus ρ i is the density matrix of the i-th spin 1/ particle and it operates on the two dimensional Hilbert space H i of the i-th particle. 3 Thus every maximally entangled state of two spin 1/ particles is locally unitarily connected to the singlet state. 3

4 entangled states of n spin 1/ particles are locally unitarily connected, where n. For n =,3, it will be shown in section 4 that the above condition is satisfied. In section 5, we shall show that the Schmidt like states of three spin 1/ particles are not locally unitarily connected even if the single particle reduced density matrices satisfy conditions similar to those of maximally entangled states. In section 6, we shall describe elaborately the system of algebraic equations to be satisfied in order to show that the maximally entangled states of n spin 1/ particles are locally unitarily connected, but we have failed to show the existence of solution(s) of this system of equations. Section 7, provides discussions. Criteria of maximal entanglement Let us now consider the following pure state of n spin-1/ particles, where n : Ψ = (i 1,i,...,i n) S n v (i 1,i,...,i n) u 1i1 u 1i... u 1in, (1) where v (i 1,i,...,i n) S n (i 1,i,...,i n) =1. () Here S = {1, }, S n = {1, } {1,}...ntimes; v (i1,i,...,i n) C for all (i 1,i,...,i n ) S n and u j1, u j are two orthonormal states in the two dimensional Hilbert space H j of the j-th spin 1/ particle. Let ρ = Ψ Ψ. (3) Let us now consider the single particle reduced density matrices ρ 1, ρ,..., ρ n of ρ : ρ 1 = (i,i 3,...,i u n) S n 1 ni n... u i ρ u i... u nin, ρ = (i 1,i 3,...,i u n) S n 1 ni n... u 3i3 u 1i1 ρ u 1i1 u 3i3... u nin,... ρ n = (i 1,i,...,i n 1 ) S u n 1 n 1i n 1... u 1i1 ρ u 1i1... u n 1in 1. (4) The state Ψ will be a maximally entangled state if and only if (by definition), ρ j = 1 I, (5) corresponding to the j-th spin 1/ particle, j =1,,...,n. One can write I = u j1 u j1 + u j u j for j =1,,...,n. And so Ψ will be a maximally entangled state if and only 4

5 if (i,i 3,...,i v n) S n 1 (i 1,i,...,i n)v(i 1,i,...,i n) = 1 δ i 1,i, 1 (i 1,i 3,...,i v n) S n 1 (i 1,i,...,i n)v(i 1,i,...,in) = 1δ i,i, (i,i 3,...,i v n) S n 1 (i 1,i,...,i n)v(i 1,i,...,i n) = 1 δ i 1,i, 1... (i 1,i,...,i n 1 ) S v n 1 (i 1,i,...,i n)v(i 1,i,...,i = n) 1δ i n,i, n (6) where i 1,i 1,...,i n,i n S. It is easy to check that the states (1/ )( + z 1 z z 1 +z ), (1/ )( + z 1 z + z 1 +z ) are maximally entangled states of two spin 1/ particles, the GHZ state (1/ )( + z 1 + z + z 3 z 1 z z 3 ) is a maximally entangled state, where σ z ±z i =(±1) ±z i for i =1,,3. It is obvious that any state Φ, which is locally unitarily connected to the maximally entangled state Ψ (given in equation (1)), is again a maximally entangled state. 3 Assumption for maximal entanglement The converse of the result in the last paragraph of the previous section is as follows : If Ψ and Φ are two maximally entangled states of n spin 1/ particles, they are locally unitarily connected. We shall show in the next section that this result is true for n = and 3. And in this section, we shall describe here a conjecture in order to prove that the above result is true for n 4. Let us consider the maximally entangled state Φ to be the following GHZ like state : Φ = 1 + z z n + 1 z 1... z n. (7) Let Ψ be the maximally entangled state given in equation (1), where the coefficients satisfy the equations () and (6). 4 And our aim is to show that the two states Φ and Ψ are locally unitarily connected under a certain assumption. 5 Let us consider the following 4 In fact equation (6) implies equation (). 5 If χ is any other maximally entangled state of n spin 1/ particles, it will be again locally unitarily connected to Φ, and hence χ will be locally unitarily connected to Ψ. 5

6 orthogonal transformations of the bases u j1 = cosθ j w j1 +sinθ j e iɛ j w j, u j = sinθ j e iɛ j w j1 +cosθ j w j, (8) where { w j1, w j } is an orthonormal basis in H j, j =1,. Then from equation (1) we get that Ψ = 1 w 11 χ 3...n (θ 1,ɛ 1 ) + 1 w 1 χ 3...n (θ 1,ɛ 1 ), (9) where and χ 3...n (θ 1,ɛ 1 ) = cosθ 1 Ψ 3...n sinθ 1 e iɛ 1 Ψ 3...n, χ 3...n (θ 1,ɛ 1 ) = sinθ 1 e iɛ 1 Ψ 3...n +cosθ 1 Ψ 3...n, Ψ 3...n = (i,i 3,...,i n) S n 1 v(1,i,...,i n) u i... u nin, Ψ 3...n = (i,i 3,...,i n) S n 1 v(,i,...,i n) u i... u nin. (10) (11) Equations in (6) guarantee that the normalized states Ψ 3...n and Ψ 3...n are orthogonal in the (total) Hilbert space H... H n of the (n 1) number of spin 1/ particles, namely nd, 3rd,..., n-th particle. And so χ 3...n (θ 1,ɛ 1 ), χ 3...n (θ 1,ɛ 1 ) are any two orthonormal states in the two dimensional closed subspace P 3...n of H... H n, 6 generated by Ψ 3...n and Ψ 3...n. We shall now assume the following. Conjecture (1) : P 3...n contains a product state. Regarding the product states, we have the following results. Result (1) : If H 1 and H be two Hilbert spaces of dimensions n and m respectively, the set of all product states of H 1 H will form a (n+m 1) dimensional (complex) manifold ; and so every (closed) subspace (of H 1 H ) of dimension higher than n m (n+m 1) will always contain at least one product state of H 1 H ([1]). As corollaries to the Result (1), we have the following results. Result (.1) : If P is any closed subspace of the four dimensional Hilbert space of two spin 1/ particles and if dimension of P is higher than one, it will always contain at least one product state. Result (.) : Any hyperplane in a four dimensional Hilbert space of two spin 1/ particles always contains at least one product state ([]). 6 Thus P 3...n is a hyperplane in H... H n. 6

7 Result (.3) : Any hyperplane in a four dimensional Hilbert space of two spin 1/ particles always contains at least one state diferent from the (maximally entangled) state of the form (1/ ) w 11 w 1 +(1/ ) w 1 w,where{ w j1, w j } are any orthonormal states in the two dimensional Hilbert space of j-th spin 1/ particle, j =1,. Result (.4) : The set of all product states in the n dimensional Hilbert space H 1... H n of n spin 1/ particles (H j being the two dimensional Hilbert space of the j-th spin 1/ particles) forms a ( + n 1 1) dimensional (complex) manifold, and so every subspace (of H 1... H n ) of dimension higher than n ( + n 1 1) = n 1 1 will always contain at least one product state. Result (.5) : An arbitrary hyperplane in a n dimensional Hilbert space H 1... H n of n spin 1/ particles (H j being the two dimensional Hilbert space of the j-th spin 1/ particles) may or may not contain any product state. Thus from the result (.5) we see that Conjecture (1) may or may not be true if n 4. But for the sake of our argument, we assume now that Conjecture (1) is true for n 4 also. So, for some real values of θ 1 and ɛ 1,wehave And let us assume that where χ 3...n (θ 1 ɛ 1 ) = w 1... w n1. (1) χ 3...n (θ 1 ɛ 1 ) = (i,i 3,...,i n) S n 1 w (i,i 3,...,i n) w i... w nin, (13) Then from equation (9) we have, where w (i,i 3,...,i n) S n 1 (i,i 3,...,i n) =1. (14) Ψ = (i 1,...,i n) S n d (i 1,...,i n) w 1i1... w nin, (15) d (1,1,...,1) = 1, d (1,i,...,i n) = 0 for all (i,...,i n ) (S n 1 {(1, 1,...,1)}), d (,i,...,i n) = 1 w (i,i 3,...,i n) for all (i,...,i n ) S n 1. As Ψ is a maximally entangled state, therefore the coefficients d (i1,i,...,i n) will satisfy the equations in (6), with v (i1,i,...,i n) replaced by d (i1,i,...,i n). One then easily gets that 7 (16)

8 w (,,...,) =1andw (i,i 3,...,i n) =0forallother(i,i 3,...,i n ) s in S n 1. 7 Hence we have χ 3...n (θ 1,ɛ 1 ) = e iα w 1 w... w n, (17) where α R. Thus equation (15) takes the following form Ψ = 1 w 11 w 1... w n1 + 1 e iα w 1 w... w n. (18) The last equation imedeately shows that the maximally entangled state Ψ of equation (1) is locally unitarily connented to the GHZ like state in equation (7). 4 The case for n =3 It is obvious from the general treatement of the previous section that the maximally entangled states of two spin 1/ particles are locally unitarily connected. The case for n = 3 is also clear from Result (.) of the previous section. In this section we shall give an alternative proof (using only the Result (.3)) to show that Conjecture (1) is true for n =3. We take an arbitrary maximally entangled state of three spin 1/ particles as (see equation (9)) : Ψ = 1 w 11 χ 3 (θ 1,ɛ 1 ) + 1 w 1 χ 3 (θ 1,ɛ 1 ). (19) Using Schmidt decomposition, the state χ 3 (θ 1,ɛ 1 ) can be written as (for at least one set of real values θ 1 and ɛ 1 ) χ 3 (θ 1,ɛ 1 ) =cosθ w 1 w 31 +sinθ w w 3, (0) for some θ ([0,π/] {π/4}). And assuming χ 3 (θ 1,ɛ 1 ) = (i,i 3 ) S c (i,i 3 ) w i w 3i3, (1) 7 As Ψ is a maximally entangled state, therefore each of the single particle reduced density matrices ρ 1,ρ,...,ρ n of Ψ must have 1/, 1/ as its eigen values. And so the coefficients v (i1,i,...,i n) (and so the coefficients d (i1,i,...,i n)) must satisfy the equations : { (i v 1,...,i j 1,i j+1,...,i n) S n 1 (i 1,...,i j 1,1,i j+1,...,i n) } { (i v 1,...,i j 1,i j+1,...,i n) S n 1 (i 1,...,i j 1,,i j+1,...,i n) } (i v 1,...,i j 1,i j+1,...,i n) S n 1 (i 1,...,i j 1,1,i j+1,...,i n) v(i 1,...,i j 1,,i j+1,...,i n) =1/4, where j =1,,...,n.And using equation (16), one easily gets that w (i,...,i j 1,1,i j+1,...,i n) =0foralli j Swhere j =,...,n.this shows (using equation (14)) that w (,...,) = 1 and all other w (i,...,i n) s are zero. 8

9 and using the fact that the state Ψ (in equation (19)) is a maximally entangled state, we have (as in the preceeding section), χ 3 (θ 1,ɛ 1 ) =e iβ ( sin θ w 1 w 31 + cos θ w w 3 ), () where β R. Using equations (0) and (), we get from equation (19) that Ψ = 1 w 11 w 1 w w 1 w w 3, (3) where w 11 = cos θ w 11 sin θ e iβ w 1, w 1 = sin θ w 11 + cos θ e iβ w 1. (4) Equation (3) assures that Conjecture (1) is true for n =3. 8 And so maximally entangled states of three spin 1/ particles are locally unitarily connected. 5 Three Particle State in Schmidt Form Let us cosider a pure state of three spin 1/ particles in the following Schmidt form : φ = c 1 φ 1 φ φ 3 +c φ 1 φ φ 3, (5) where c 1 and c are any given pair of complex numbers (with c 1 1/, c 1/ and c 1 + c =1), 9 and { φ j, φ j } is a set of mutually orthonormal spin 1/ states of the j-th spin 1/ particle (j =1,,3). It is easy to check that each of the single particle reduced density matrices of φ have their eigen values as c 1 and c. And any (pure) state of three spin 1/ particles, which is locally unitarily connected to ψ, possesses this property. So one may be tempted to assume that if φ and χ are any two (pure) states of three spin 1/ particles satisfying the above property, they are locally unitarily connected analogous to the case of maximally entangled states. But we shall show that this not true. 8 as χ 3 (θ 1,ɛ 1 ) = w 1 w 31 is a product state in the hyperplane P 3, and as for particle 1, the (orthonormalized) basis { w 11, w 1 } is obtained from the (orthonormalized) basis { u 11, u 1 } by a unitary transformation of the form given in equation (8), corresponding to some (real) values (say, (θ 1,ɛ 1 )) of the variables θ 1,ɛ 1 9 so c 1 c 9

10 Let us take two three particles states as φ = c 1 φ 1 φ φ 3 +c φ 1 φ φ 3, (6) and { +c ( 1 χ = c 1 ψ 1 ψ + 1 ) ψ 1 ψ ψ 3 1 c 1 c } 1/ ψ 1 ψ + c 1 c ψ 1 ψ ψ 3 (7) where c 1, c are two given complex numbers with the properties that c 1 1/, c 1/, c 1 + c =1and{ φ j, φ j }, { ψ j, ψ j } are two sets of mutually orthonormal spin 1/ states of the j-th spin 1/ particle (j =1,,3), they are being related by the following unitary transformations : ψ j = cos θ j φ j + sin θ j e iɛ j φ j, ψ j = sin θ j e iɛ j φ j + cos θ j φ j, (j =1,,3), θ 3 being not an integral multiple of π/. One can easily check that each of the single particle reduced density matrices of φ (and also of χ ) has c 1 and c as its eigen values. Then we have χ = { c 1 cos θ 3 + c sin θ 3 } 1/ ψ 1 φ 3 (8) where +{ c 1 sin θ 3 + c cos θ 3 } 1/ ψ 1 φ 3 (9) 10

11 1 ψ 1 = { c 1 cos θ 3 + c sin θ 3 } [c 1 cos θ 3 ψ 1/ 1 ψ c 1 c sin θ 3 e iɛ { 3 ψ 1 ψ c sin θ 3 1 c 1 } 1/ e iɛ 3 c c ψ 1 ψ ψ 1 = + c 1 cos θ 3 ψ 1 ψ ], (30) 1 { c 1 sin θ 3 + c cos θ 3 } 1/ [c 1 sin θ 3 e iɛ3 ψ 1 ψ + c { 1 c cos θ 3 ψ 1 ψ +c cos θ 3 1 c 1 } 1/ c c ψ 1 ψ + c 1 sin θ 3 e iɛ 3 ψ 1 ψ ]. (31) As here ψ 1 ψ 1 (= (1/)( c 1 c ) sin θ 3 e iɛ 3 ) 0, therefore the above state χ can not be locally unitarily connected to the state φ. Thus we see that the above assumption regarding states like φ, χ, etc., is not valid. They must satisfy some other conditions. 6 An Open Problem Conjecture (1) will be valid, if for some real values of θ 1 and ɛ 1, χ 3...n (θ 1,ɛ 1 ) (in equation (10)) is a product state. And this will be valid if cos θ 1 v (1,i,...,i j 1,1,i j+1,...i n) sin θ 1 e iɛ 1 v (,i,...,i j 1,1,i j+1,...i n) = µ j {cos θ 1 v (1,i,...,i j 1,,i j+1,...i n) sin θ 1 e iɛ 1 v (,i,...,i j 1,,i j+1,...i n)}, (3) where µ j s are some complex numbers for j =,3,...,n. 10 So Conjecture (1) will be valid if one can solve (or, even, one can show the existence of solution of) the set of (n 1) n number of equations in (3) for θ 1 and ɛ 1, where the (arbitrary) coefficients v (i1,...,i n) s satisfy the 4n number of equations in (6). 10 It should be noted that µ j is same for all (i,...,i j 1,i j+1,...,i n ) S n 11

12 7 Discussions We have provided here a simple proof that maximally entangled states of three spin 1/ particles are locally unitarily connected, and assuming that a (certain) hyperplane in the Hilbert space of (n 1) spin 1/ particles contains a product state, we have proved that maximally entangled states of n spin 1/ particles are also locally unitarily connected. The question of equivalence class(es) of non-maximally entangled states of n spin 1/ particles, for n 3, is still an unsolved problem where the equvalence relation being the local unitary connectedness of states. Actually solution of this problem would be directly connected to measure(s) of entanglement. But we have proved that Schmidt like states of multiple spin 1/ particles (we have done it for three spin 1/ particles, but it can be similarly extended) are not locally unitarily connected, if we demand that each of the single particle reduced density matrices has its eigen values as the squares of the moduli of the coefficients in the Schmidt form. 1

13 References 1. J. S. Bell, Physics 1 (1964) J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Phys. Rev. Lett. 3 (1969) N. Gisin, Phys. Lett. A 154 (1991) L. Hardy, Phys. Rev. Lett. 71 (1993) S. Goldstein, Phys. Rev. Lett. 7 (1994) G. Kar, Phys. Lett. A 8 (1997) S. Ghosh and G. Kar, Phys. Lett. A 40 (1998) S. Ghosh, G. Kar, and D. Sarkar, Phys. Lett. A 43 (1998) A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47 (1935) C. H. Bennett, G. Brassad, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wooters, Phys. Rev. Lett. 70 (1993) C. H. Bennett and G. Brassad, in Proceedings of IEEE International Conference on Computers, Systems and Signal Processing, Bangalore, India (IEEE, New York, 1984), p C. H. Bennett and S. J. Wiesner, Phys. Rev. Lett. 69 (199) C. H. Bennett, C. A. Fuchs, and J. A. Smolin, Entanglement- Enhanced Classical Communication on a Noisy Quantum Channel, in Quantum Communication, Computing and Measurement, edited by O. Hirota, A. S. Holevo and C. M. Caves (Plenum, New York, 1997). 14. R. Cleve and H. Buhrman, LANL e print quant ph/ D. Deutsch, Proc. R. Soc. London. A 400 (1985) 97. D. Deutsch, Proc. R. Soc. London. A 45 (1989)

14 16. S. L. Braunstein, A. Mann, and M. Revzen, Phys. Rev. Lett. 68 (199) J. L. Cereceda, A simple proof of the converse of Hardy s theorem (private communication). 18. D. M. Greenberger, M. Horne, and A. Zeilinger, Bell s theorem, Quantum Theory, and Conceptions of the Universe, ed. by M. Kafatos, Kluwer, Dordrecht (1980) p S. Ghosh, Ph.D. Thesis (in preparation). 0. J. Schlienz and G. Mahler, Phys. Lett. A 4 (1996) M. Lewenstein and A. Sanpera, Phys. Rev. Lett. 80 (1998) 61.. A. Sanpera, R. Tarrach, and G. Vidal, LANL e print quant ph/