Multiparticle Entanglement 1

Transcription

1 Foundations of Physics, Vol. 9, No. 4, 999 Multiparticle Entanglement H. A. Carteret, N. Linden, 3, 5 S. Popescu, 3, 4 and A. Sudbery Received September 8, 998 The Greenberger± Horne± Zeilinger state is the most famous example of a state with multiparticle entanglement. In this article we describe a group theoretic framework we have been developing for understanding the entanglement in general states of two or more quantum particles. As far as entanglement is concerned, two states of n spin-/ particles are equivalent if they are on the same orbit of the group of local rotations (U() n ). We consider both pure and mixed states and calculate the number of independent parameters needed to describe such states up to this equivalence. We describe how the entanglement of states in a given equivalence class may be characterized by the stability group of the action of the group of local rotations on any of the states in the class. We also show how to calculate invariants under the group of local actions for both pure and mixed states. In the case of mixed states we are able to explicitly exhibit sets of invariants which allow one to determine whether two generic mixed states are equivalent up to local unitary transformations.. INTRODUCTION It was the pioneering paper of Greenberger, Horne, and Zeilinger ( ) which opened up the study of multiparticle entanglement and introduced the now famous GHZ state. It was clear from Ref. that correlations among more than two particles present novel and highly nontrivial features not present Dedicated to Professor Greenberger on his 65th birthday. Department of Mathematics, University of York, Heslington York YO 5DD, United Kingdom. 3 Isaac Newton Institute for Mathematical Sciences, Clarkson Road, Cambridge CB3 EH, United Kingdom. 4 BRIMS, Hewlett± Packard Laboratories, Filton Road, Stoke Gifford, Bristol BS 6QZ, United Kingdom. 5 To whom correspondence should be addressed; n.linden@newton.cam.ac.uk /99/4-57$6./ Ñ 999 Plenum Publishing Corporation

2 58 Carteret, Linden, Popescu, and Sudbery in states of two particles. It is the aim of this article to describe some aspects of the general structure of multiparticle entanglement, a study inspired by the original paper of GHZ. It is a pleasure to dedicate it to Professor Greenberger on the occasion of his 65th birthday. The existence of nonlocal correlations among remote quantum systems, discovered by Bell in 964, ( ) is one of the most fascinating quantum phenomena. While for a long time these correlations were considered a curiosity, it has emerged recently that entanglement is the key ingredient in quantum computation ( 3) and communication ( 4) and plays an important ( 5, 6) role in cryptography. It has become clear that entanglement is a resource which may be manipulated (for example, by concentration, ( 7) dilution, or purification, ) ) and transformed from one form into another. ( 8± Traditionally, starting with Bell, the example which was most studied was that of nonlocal correlations between two remote quantum particles. However, it is now clear that the correlations among more than two remote particles present new features, for example, the correlations (, ) generated by the GHZ state. Nevertheless, at present we have only glimpses of the complete picture of multiparticle entanglement. It is the aim of this article to describe some aspects of this picture and put forward a general framework in which multiparticle entanglement can be (, 3) investigated. The key element in our approach is to note that two states which can be transformed one into another by local unitary operations are equivalent as far as their nonlocal properties are concerned. 6 This leads us to investigate the properties of the space of n particles under local unitary transformations. We will restrict ourselves to spin-/ particles, and consider both pure states and more general states described by density matrices. With a slight abuse of language we refer to these latter states as ``mixed, although, as will be clear, when we discuss mixed states we include the limiting case of pure states as well. While, of course, we could have simply treated general mixed states first and treated pure states as a special case, mathematically we have found it more convenient to deal with pure states separately. In both cases we will see that the general structure is as follows. Each particular state belongs to an equivalence class comprised of all states which can be obtained from it by acting on it with local unitary operators; all states in a class are equivalent as far as nonlocality is concerned. Thus the space of states decomposes completely into equivalence classes, or ``orbits, under the action of the group of local unitary transformations. An arbitrary state of n spin-/ particles may be described by n + real 6 It is possible to consider more general local operations; see Sec. 5.

3 Multiparticle Entanglement 59 parameters if the state is pure and n real parameters if the state is mixed. 7 Some of these parameters (or functions of them) specify the equivalence class to which the state belongs. These parameters ( or functions) are invariants under local transformations. The remainder describe where the state is situated inside the equivalence class± ± they do change under local transformations. For many purposes, only parameters describing nonlocal properties are significant; an example is that any good measure of entanglement must be invariant under local transformations and thus it (8, 4± 7) should be a function of nonlocal parameters only. Here and henceforth we refer to parameters which are invariant under local transformations as invariants. Invariants are also relevant in discussions of Bell inequalities, ( 8, 9) teleportation, etc. ( ) Before describing our general framework, it is worth pointing out why pure states of two particles are technically so much simpler to deal with than states of more particles. This is because there is a simple way to identify the invariant parameters± ± the Schmidt decomposition. Indeed, let e iä e j, i, j =,, be some arbitrary basis vectors in the Hilbert space of the two particles; then a general pure state of two particles is given by Y = + a ij e iä e j ( ) i, j However, by choosing some appropriate basis vectors for each particle f i and g i, the double sum in () can be reduced to a single sum, Y = + i b i f iä g i ( ) where the b i can be taken to be real and positive. The Schmidt coefficients b i are manifestly invariant under local transformations. Indeed, local unitary transformations can change only the Schmidt vectors, and not the Schmidt coefficients. Y = + i b i f iä g i Y = + i b i f iä g i ( 3) It is only for pure states of two particles that such a simple decomposition exists in general. () As we describe below, instead of simply trying to find something which formally resembles the Schmidt decomposition for multiparticle states, we should try to follow its spirit, not its form. That is, we find representations which separate local and nonlocal parameters. 7 For convenience we always consider nonnormalized pure states, and thus the norm and phase also appear as parameters.

4 53 Carteret, Linden, Popescu, and Sudbery As an important result, we will find that for large n, most of the parameters describe nonlocal properties. This is opposite to the case of small n± ± for example, for two spins in pure states, of the eight real parameters which describe a generic ( unnormalized) state, only two, the unique independent Schmidt coefficient and the norm, are nonlocal ( thus, in fact, there is only one physical nonlocal parameter). Finally, we note that among pure two-particle states, some are, in some sense, special. Such states are the direct products and the singlet-like states. We show that the special nature of these states is determined by their invariance properties. Namely, for these special states there are more local actions which leave them unchanged than in the case of generic states; in other words, these special states have larger ``stability groups than generic states. For example, in the case of a singlet, Y = Ï ( e Ä e e Ä e ) ( 4) where e and e represent spin polarized ``up or ``down, say, along the z axis, identical rotations of the two spins leave the state unchanged. Furthermore, such enhanced invariance properties are in fact common for all states in an equivalence class, and thus characterize the class itself. To find the ``special equivalence classes, we therefore have to study their invariance properties. We argue that these ``special classes describe fundamentally different types of entanglement, while a generic class represents a combination of different types of entanglement. The approach of using symmetry properties to investigate entanglement has also been used by Fivel. ( ) In the next section we show how to calculate how many parameters are needed to describe (pure and mixed) states up to local unitary transformations. In Sec. 3 we show how to construct explicit invariants for pure and mixed states. In the case of mixed states we are able to to characterize generic orbits of the group of local rotations, both by giving an explicit parametrization of the orbits and by finding a finite set of polynomial invariants which separate the orbits. Thus given two density matrices we can compute explicitly whether they are on the same orbit or not. 8 In Sec. 4 we discuss how the entanglement of states in a given equivalence class may be characterized by the stability group of the action of local rotations on any of the states in the class. This gives a way of understanding what 8 Other authors have also discussed the use of invariants in discussing entanglement (3, 4) and applied invariant theory to quantum codes. ( 5)

5 Multiparticle Entanglement 53 ``special types of entanglement can occur. In Sec. 5 we discuss invariance under more general types of local transformations.. THE NUMBER OF NONLOCAL PARAMETERS.. Pure States The space of pure states of n spin-/ particles is the n-fold tensor product C n = C Ä... Ä C, and the group of local transformations is the n-fold product U() n = U() U() [each copy of U( ) acting on a different spin, i.e., on the corresponding copy of C ]. The equivalence classes are orbits under the action of the local transformations group. Hence, the space of orbits is 3 3 C n U()... U() ( 5) this is the main mathematical object we are investigating. The number of parameters needed to describe the position of a state on its orbit is the dimension of the orbit. Not all orbits have the same dimension. As noted above, there are ``special orbits± ± singular orbits± ± which have higher invariance, i.e., lower dimension. The total number of parameters ( n complex parameters = n + real parameters) describing the space of states minus the number of parameters describing a generic orbit ( the dimension of the orbit) gives the number of parameters describing the location of the orbit in the space of orbits, i.e., the number of parameters describing the nonlocal properties of the states. An important initial question is how many parameters are needed to describe the space of orbits of the action of U() n on the space of states, i.e., the number of parameters which describe inequivalent states. To do this it will be convenient to find the ( real) dimension of a general orbit; the number of parameters is then found by subtracting this number from n +. A lower bound on this number can be obtained by a simple argument of counting parameters. Each of the n copies of the local unitary group U( ) is described by four real parameters. Thus there can be no more than 4n parameters describing local properties of the states, and hence at least n + 4n nonlocal parameters ( i.e., invariants under local transformations). One can immediately see the important result that, for large n, almost all parameters have nonlocal significance. We note ( see below) that the case of large n is quite different from the most commonly studied case of

6 53 Carteret, Linden, Popescu, and Sudbery n =, where of a total of eight real parameters ( six if one excludes the norm and overall phase), there is only one nonlocal parameter. The above bound is, in general, not satisfied. The reason is that not all 4n parameters describing the local transformations lead to independent effects. For example, changing the phase of any particular spin has the same effect as changing the phase of any other. Hence, at least, the group of local transformations reduces from U( ) n to U() 3 SU( ) n, which has dimension 3n +. This leads to a better lower bound on the number of nonlocal parameters of n + (3n + ). In fact, as we show below, for three or more spin-/ particles, this bound is reached. Only in the case of two spin-/ particles is the bound not satisfied, because the number of parameters describing independent local transformations is fewer ( and, correspondingly, the number of nonlocal parameters larger); this is related to the fact that generic two spin pure states have a nontrivial stability group.... Dimension of a General Orbit To find the dimension of a general orbit it is simplest to work infinitesimally. Thus, in general, associated with the action of each element of a Lie algebra of a Lie group K which acts on a space V (in this case, the Hilbert space of states), there is a vector field: take an element T of a basis for the Lie algebra, the action of the group element k = exp iet Î K on an element v Î V induces an action on functions from V to C ; and the vector field, X T, associate with the Lie algebra element T is found by differentiating: X T f ( v) def = e f ( eiet v) ) e = ( 6) The linear span of tangent vectors at the point v associated with the whole Lie algebra forms the tangent space to the orbit at the point v and so the number of linearly independent tangent vectors at this point gives the dimension of the orbit.... A Single Spin Let us start with the case of a single spin. Obviously in the case of a single spin, there is no nonlocality, however, this case helps to fix notation and to illustrate the general formalism which we use for more spins. The space of pure states has real dimension four ( complex dimension two). It is also clear that the action of a unitary operator on a vector cannot change its norm, so that the dimension of the space of orbits must be at

7 Multiparticle Entanglement 533 least one (in fact we will soon see that it is precisely one). However, the group U( ) has dimension four so that the set of vector fields associated to an arbitrary basis for the Lie algebra cannot be linearly independent. In the representation of U() acting on C a convenient Hermitian basis for the Lie algebra is s x =, s y = i i, s z =, = ( 7) Now take a pure state Y = a Î C b ( 8) which for mathematical convenience we consider to be nonnormalized. The infinitesimal change in Y under a tranformation in the direction s x : dy = ies x Y = ie Y = ieb ( 9) iea So that under a group tranformation close to the identity, Y = a ƒ Y + dy = b a+ ieb ( ) b + iea We now write everything in terms of real variables: so that a= c + id, b = c + id ( ) Y = c d c d and dy = e d c d c = e c d c d ( ) Thus there is an induced action on a function f (v) = f ( c, d, c, d ): f ( c, d, c, d ) ƒ f ( c ed, d + ec, c ed, d + ec ) ( 3)

8 534 Carteret, Linden, Popescu, and Sudbery Differentiating with respect to e, we find f = e ) d + c e= d c + c d c d f ( 4) We write the vector field associated with this Lie algebra element s x as d + c d + c c d c d = u x. $, where u x = d c d c ( 5) In a similar way we may find the vectors u y, u z, and u associated with transformations by s y, s z, and : u y = c d c d, u z = d c d c, u = d c d c ( 6) It is not too difficult to check that only three of these four vectors are linearly independent at any given point. Indeed, (d d + c c ) u x + ( c d d c ) u y + ( c + d c d ) u z (c + d + c + d ) u = ( 7) Thus the dimension of the orbit is three and so there is one parameter ( the norm) which describes the different orbits. Obviously the physical states have norm one; the other states are not physical...3. Two Spins In a similar way we may analyze the case of two spins. A general pure state may be written + Y = a ij e iä e j ( 8) i, j =

9 Multiparticle Entanglement 535 where {e, e } is a general basis of C ; once again we take Y to be nonnormalized. In the representation of U() on C 4 we may use the following basis for the eight Lie algebra elements: s x Ä, s yä, s zä, Ä, Ä s x, Ä s y, Ä s z, Ä ( 9) One sees that the element Ä appears twice, so that in fact there are only seven different Lie algebra elements to consider. Similar calculations to the previous subsection show that only six of these vectors are linearly independent for general values of the a ij. Thus the dimension of the generic orbit is six and therefore the number of parameters describing the different orbits is two. This confirms the wellknown result that any pure state of two spins is equivalent, under local rotations, to one of the form..4. Three or More Spins N( cos we Ä e + sin we Ä e ) ( ) A computation similar to the one in the above subsections shows that in the case of 3 spin-/ particles the dimension of a generic orbit is, and hence the number of real nonlocal parameters ( including the norm) is 6 ( = 3 + ). By brute force one can show that any 3 spin-/ particle state is equivalent, up to local transformations to 9 N cos ae Ä (cos be Ä e + sin be Ä e ) + N sin a cos ce Ä (sin be Ä e cos be Ä e ) + N sin a sin ce Ä (cos de Ä e + e ig sin de Ä e ) ( ) The six invariant parameters are the norm N ( which, of course, is one for physical states) and the five nonlocal parameters a, b, c, d, and g. A systematic way of finding the invariants is given in the next section. It is interesting to note that in this case all the = parameters describing the local transformations U( ) 3 SU( ) 3 are actually independent. In fact it is possible to show that for three or more spins in pure states, all the local transformations are independent, and so the number of nonlocal parameters is n + 3n. 9 This result was found independently by J. Schlienz. ( 6)

10 c c 536 Carteret, Linden, Popescu, and Sudbery This may be proved by induction by showing that a generic state of n particles has no stability group ( see Sec. 4) under the action of U( ) 3 SU( ) n. The steps in the inductive proof are (a) showing that if there is no stability group for a generic state of n particles, then there is no stability group for a generic state of n particles, and ( b) showing that there is no stability group for n = 3. Using the Schmidt decomposition, an arbitrary state of n particles may be written as Y ( n) = ac ( n ) Ä u + bw ( n ) Ä u ( ) ( n ) where u and u are orthogonal states of the nth particle, and c and w ( n ) are states of the remaining particles and a and b may be taken to be real. Suppose now that this state has a stability group which includes actions on particle n. The only possibilities for transformations on particle n are u ƒ e ih u ( 3) u ƒ e ih u for some phase h, since the direction of the vectors in the Schmidt decomposition are uniquely specified ( we are considering generic states so that aþ b). In order for Y ( n) is to be invariant, there must be compensating local transformations on the other particles such that w ( n ) ƒ e ih w ( n ) ( 4) (n ) ƒ e ih c (n ) First, note that we cannot induce this transformation of w ( n ) and ( n ) by an overall change of phase of one of the (n ) spins because this leads simply to an equal phase change of w ( n ) ( n and c ). We can thus restrict attention to actions of SU( ) n on w (n ) ( n and c ). However, the assumption of the induction, namely, that a generic state of n particles has no stability group is equivalent to the fact that one cannot modify the phase of w ( n ) ( n ) or c by an action of SU() n. For if we could change the phase of w ( n ), say, by such an action, we could construct an action of the full group U( ) 3 SU() n which leaves w ( n ) invariant; i.e., w ( n ) would have a stability group contrary to the assumption. We thus conclude that if there is no stability group for a generic state of n particles, then there is no stability group for a generic state of n particles.

11 Multiparticle Entanglement 537 Finally, in order to demonstrate part ( b) of the induction proof, we note that it is clearly not possible to introduce an overall phase into a generic two particle state ( ) by actions of SU() 3 SU( ). By the previous argument this means that a generic three particle state has no stability group, as required. Thus the proof by induction is complete. Therefore a generic n particle state has no invariance group under U( ) 3 SU( ) n and so the dimension of a generic orbit is (3n + ) and hence the number of local parameters is n + 3n... Mixed States We now turn to considering to states described by density matrices, i.e., we consider states which may be mixed, in general. We show that for n>, of the n real parameters describing density matrices of n spin- / particles, n 3n are invariant under local transformations, U( ) n. For an arbitrary set of n particles, the number of nonlocal parameters is P r d r å r d r + n, where d r is the dimension of the state space of the r th particle.... One Spin To fix notation, it will be convenient to consider the case of a one-particle density matrix first. The space of pure states of a single spin-/ particle is C and thus a density matrix is a 3 complex matrix which is hermitian, positive, and with trace one and may, therefore, be described by three real parameters. A particularly convenient representation of such a matrix is r = ( + a i s i) ( 5) where a i, i =,, 3, are real and s i are the Pauli matrices. Under a local transformation by a unitary matrix U, r is transformed as r ƒ UrU ² ( 6) Every unitary matrix may be written as a product of an element of SU( ) and a phase transformation, represented by a unitary matrix e iw. The latter element clearly leaves any density matrix invariant under the transformation ( 6), so that when considering the action ( 6) we may restrict attention to elements of SU( ). As in the case of pure states, in order to find the number of invariants it will be more convenient to find the dimension of a generic orbit under the action of SU( ) and to work infinitesimally.

12 538 Carteret, Linden, Popescu, and Sudbery As with pure states we may calculate the three vector fields X i associated with the Lie algebra elements of SU(), s i, as X i = e ijk a j a k ( 7) We note that at generic values of a, a, a 3, only two of these tangent vectors are linearly independent since a X + a X + a 3 X 3 = ( 8) Thus the dimension of the generic orbit is two, and therefore of the three parameters describing a generic density matrix, two are noninvariant, leaving only one invariant parameter, as one expects, since only the single independent eigenvalue of r is invariant under local transformations.... Two or More Spins We now turn to the case of two-particle density matrices. Such a density matrix has 5 real parameters, and the maximum dimension that a generic orbit could have is 6 [corresponding to two copies of SU() ] if all the tangent vectors corresponding to a basis of the Lie algebra were independent. We show that the tangent vectors do indeed span six dimensions and, thus, that there are nine nonlocal parameters. We may write a density matrix as r = 4( Ä + a. s Ä + Ä b. s + R ij s iä s j) ( 9) The action of a Lie algebra element of the subgroup SU() acting on the first component of the tensor product is d ( ) r = ( a k g m e mki s iä + R kj g m e mki s iä s j) ( 3) with a similar expression coming from the action of a Lie algebra element of the subgroup SU( ) acting on the second component of the tensor product. The vector fields corresponding to the six basis elements s k Ä, Ä s k are X k = Y k = e kim a i + R ij a m R mj ( 3) e kim b i b m + R ji R jm

13 Multiparticle Entanglement 539 Consider that set X k first: one can see that these three are linearly independent at generic points by considering the coefficients of / a i, since a linear relation would have to be of the form a k X k =, but one can see that this relation will not hold for nonzero a s by looking at the coefficients of the partial derivatives with respect to R ij. Similarly by considering the coefficients of the partial derivatives with respect to b, b, b 3, one sees that Y, Y, Y 3 are linearly independent. Finally, we note that the coefficients of the partial derivatives with respect to b, b, b 3 are zero for X, X, X 3 and the coefficients of the partial derivatives with respect to a, a, a 3 are zero for Y, Y, Y 3, so that there can be no linear relation at all between the six vector fields X, X, X 3, Y, Y, Y 3. Thus the dimension of the orbit of a generic density matrix is 6, and the number of nonlocal parameters, 5 6 = 9. In general, we can consider a system of n particles with individual state spaces of dimensions d,..., d n. An analysis similar to that given above shows that, of these parameters, only P r d r å r d r + n are nonlocal invariants. 3. EXPLICIT FORMS OF THE NONLOCAL INVARIANTS For some purposes one might wish to know whether or not two states are on the same orbit, i.e., are equivalent. In principle, one can take the ideas of the previous section further to find invariants of the orbits. Consider any function on the space of states. If it is invariant under the action of the group then, in particular it is invariant under infinitesimal group transformations. Thus it must be annihilated by the vector fields associated with the infinitesimal group transformations. Therefore in order to find a set of infinitesimal invariants, one has to solve a set of simultaneous partial differential equations; the number of such equations is the number of linearly independent vectors associated with the Lie algebra, as in the previous section. Let us label the Lie algebra elements of local transformations {T i }, i =... N, where for pure states of n spin-/ particles, N = 3n +, and for mixed states of n spin-/ particles, N = 3n ( corresponding to the dimension of the appropriate local transformation group). Then the vector fields X T i are derived as in Eq. ( 6) and an invariant function satisfies X T i f =, i =... N ( 3) a set of N simultaneous linear partial differential equations. The method of characteristics allows one to solve the problem in principle, subject to

14 54 Carteret, Linden, Popescu, and Sudbery being able to perform the integrals which arise. Unfortunately, one can easily see that the problem becomes very difficult, even for pure states of two spins, for in this case one has to solve six simultaneous partial differential equations. Fortunately there are other ways of finding invariants under the local actions. 3.. Pure States We first discuss a few examples and then discuss the general case One and Two Spins In the case of one spin, with general state one can easily see that the expression + Y = a i e i ( 33) i = + a i a* i ( 34) i = ( i.e., the norm of the state) is invariant under local unitary transformations. In the case of two spins, with general state Y = å i, j = a ij e iä e j, the norm of the state is invariant and given by a similar expression: I = + i, i, j, j = + a ij a* i j d ii d j j = a ij a* ij ( 35) i, j = There is, however a second, quartic, expression which is functionally independent of I which is also clearly invariant, since the indices have been contracted with the invariant tensor d: I = + = + a ik a* i m a jm a* j k d ii d j j d kk d mm a ik a* im a jm a* jk = Trace( ( aa ² ) ) ( 36) As we saw earlier, only six of the seven vector fields are linear independent in this case.

15 Multiparticle Entanglement 54 In the familiar form of the Schmidt coefficients, Eq. (), I = N ( 37) I = N 4 (cos 4 w + sin 4 w) Since we know that in the case of two spins there can be only two invariants, any further invariants must be able to be written in terms of I and I. For example, consider I 3 = + i, j, k, m, n, p = a ik a* im a jn a* jk a pm a* pn = Trace( ( aa ² ) 3 ) ( 38) By noting, for example, that the 3 matrix aa ² is hermitian and satisfies a quadratic equation ( by the Cayley Hamilton theorem), one may show that I 3 = (3I I I 3 ) ( 39) In a similar way one may see that all higher-order invariants are of the form and are expressible in terms of I and I General Case of Pure States I N = Trace( ( aa ² ) N ), N> 3 ( 4) A generic state of n spin-/ particles can be written + Ä Ä Ä Y = a i i... i n e i e i... e in i, i,..., i n = Then a general polynomial expression in the coefficients is + c i... k n... a i i... i n a j j... j n... a* k k... k n... ( 4) If the polynomial ( 4) has equal numbers of a and a* and all the indexes of a are contracted with those of a*, each index being contracted with an index located on the same slot ( i.e., if c i... k n... are appropriate products of d s), then the polynomial is manifestly invariant. For example, in the case of three spins with generic state + Y = a ijk e iä e jä e k ( 4) i, j, k =

16 54 Carteret, Linden, Popescu, and Sudbery there is one quadratic invariant, the norm, there are the quartic invariants ( in addition to the square of the norm), J = + J = + J 3 = + a ijk a* ijm a pqm a* pqk a ikj a* imj a pmq a* pkq ( 43) a kij a* mij a mpq a* kpq and so on, the different invariants arising by contracting indices in different ways. Furthermore, one can prove that all invariant polynomials are constructed in this way. The proof of this theorem ( not given here) is based on the fact that all polynomial functions of k vectors in C, invariant under U( ) are polynomials in the inner product of the vectors. ( 7) A key issue is the following. There are infinitely many polynomial invariants, and it is known that this infinite set separates the orbits, i.e., it allows one to distinguish whether two states are equivalent up to local transformations. However, to be of practical use, we need to be able to construct from these polynomials a finite complete set of invariants for arbitrary numbers of spins. General arguments from invariant theory tell us that such a set, or basis, exists. In other words, any polynomial in the complete infinite set we have described may be written as a polynomial in the elements of this basis. However, as far as we are aware, there are no general results describing such a set for general n. There are, however, some useful mathematical tools available. For example, given any set of polynomials, there is a algorithmic procedure of determining the relations between them using the theory of Grobner bases. (8) There is also a simple formula, the Molien formula, for the generating function of the number of linearly independent invariants at each order. ( 8) However, constructing a basis for general n is a key task for the future. Surprisingly, however, mixed states turn out to be more straightforward to deal with than pure states, as we will see in the next section. 3.. Mixed States In order to exhibit explicit invariants for density matrices, it is convenient to make use of a property of the action of the unitary matrices ( 6).

17 Multiparticle Entanglement 543 For example, the density matrix for a single spin may be written, as before, r = ( + a i s i) ( 44) We note that fiche effect of the transformations (6) is to act on the vector a by rotation by an orthogonal matrix, i.e., an element of SO ( 3)± ± this follows from the fact that a i s i is the representative of a Lie algebra element and the conjugation action ( 6) is the adjoint action of the group on its Lie algebra. Explicitly, the orthogonal matrix O under which a transforms is related to the unitary matrix U by O ij a j s i = Ua. s U ² ( 45) We may thus find a way of exhibiting the invariant under local transformations: I = a i a j d ij = a, where we have used the fact that SO ( 3) has an invariant tensor d ij. We note that this invariant may also be expressed as I= Tr( r ). Let us now turn to the case of n> spin-/ particles and explicitly identify a set of invariant parameters which characterize generic orbits. To be explicit, consider the case of three spin-/ particles with a density matrix which may be written as r = 8 ( Ä Ä + a i s iä Ä + b i Ä s iä + c i Ä Ä s i + R ij s iä s jä + S ij s iä Ä s j + T ij Ä s iä s j + Q ijk s iä s jä s k) ( 46) The three spin case has all the features of the general case. The action by a local unitary transformation on the first component in the tensor product induces the following transformations on components of r: a i ƒ L ij a j, R ij ƒ L ik R kj, S ij ƒ L ik S kj, Q ijk ƒ L im Q mjk, where L ij is an orthogonal matrix, and the other components of r do not change. Local actions on the second and third components of the tensor product induce similar transformations of b, R, T, Q and c, S, T, Q, respectively, by orthogonal matrices M and N independent of L. We may fix a canonical point on a generic orbit as follows: first, let us define X ii = Q ijk Q i jk, Y j j = Q ijk Q ij k, Z kk = Q ijk Q ijk ( 47) and perform unitary transformations on particles,, and 3 so as to move to a point on the orbit in which X, Y, and Z are diagonal; this is possible

18 a A 544 Carteret, Linden, Popescu, and Sudbery because the transformation on X depends solely on the transformation on particle, X ƒ LXL T, and similarly the transformations on Y and Z depend only on the transformations on and 3, respectively. Generically the diagonal entries of X, Y, and Z are distinct and we can arrange them in decreasing order ( X, Y, and Z are hermitian, positive matrices). The only remaining transformations which leave X, Y, and Z in these forms are local unitary transformations which induce orthogonal transformations in which L ij, M ij and N ij are one of the matrices diag(,, ), diag(,, ), diag(,, ). We may specify a canonical point on the generic orbit uniquely by specifying that all the components of a have the same sign, and similarly for b and c. This method works as long as X, Y, and Z have distinct eigenvalues and the components of a, b, and c are not zero at the canonical point on the orbit. The parameters which describe the generic orbits are the components of a, b, c, R, S, T, and Q at the canonical point on the orbit. We note that the number of parameters describing the canonical point are the 6 = 63 components of a, b, c, R, S, T, and Q minus the 33 3 = 9 constraints that the nondiagonal elements of X, Y, and Z are zero; thus the number of nonlocal parameters is 54 as given by the general formula. We note that the fact that the canonical point, as constructed, is unique means that all points on the same orbit will have the same canonical representative: conversely, if two density matrices r and r have the same canonical form, then U r U ² = r canonical = U r U ² for some U and U, so that r = (U ² U ) r ( U ² U ) ², and thus r and r are on the same orbit. Thus the canonical form given above allows one to determine explicitly whether or not two generic density matrices are equivalent up to local transformations. We now describe a finite set of polynomial invariants which separate generic orbits by finding a set which allows one to calculate the components of a, b, c, R, S, T, and Q at this canonical point. The complete infinite set of polynomial invariants is found by contracting the indices of a, b, c, R, S, T, and Q with the invariant tensors d ij and e ijk. However, we may find a finite set of invariants which separates generic orbits. First we note that tr(x), tr(x ), and tr(x 3 ) determine the diagonal elements l, l, and l 3 of X, and similarly for Y and Z. Now consider the three invariants A n = a T X n a, n =,, 3. We may write these three invariants in the following way: a A 3 4 l l l = l 4 l 4 l 4 3 a 3 A 6 ( 48)

19 Multiparticle Entanglement 545 where a, a, and a 3 are the components of a at the canonical point on the orbit. The Vandermonde matrix L on the left-hand side of Eq. ( 48) has determinant ( l l )(l l 3)(l 3 l ), and we may solve for a, a, and a 3 as long as det L is nonzero. Also, if the invariant A 9 = e ijk a i (Xa) j ( X a) k = a a a 3 det L ( 49) is nonzero, then we may determine the sign of the components of a; recall that, by definition, all the components of a have the same sign at the canonical point. The analogous expressions B 9, C 9 determine the values of b and c at the canonical point. The values of the components of R at the canonical point may be calculated from the following nine invariants: I r, s = ( X r a) i (Y s b) j R ij, r, s =,, 3 ( 5) These nine equations may be put together into a matrix form, I = ( ( LF ) Ä (MG) ) R ( 5) where I and R are column vectors with nine components and the matrix L is the Vandermonde matrix in Eq. ( 48), M is the analogous matrix with l i replaced by m i ( the diagonal elements of Y ), and F and G are diag(a, a, a 3 ) and diag( b, b, b 3 ), respectively. We note that det(lf ) = A 9 and det(mg) = B 9, so since we are assuming that these are nonzero, we may invert the matrix equation to find the components R ij. The components of S and T may be found in a similar way. Finally, we may use the 7 invariants I r, s, t = (X r a) i (Y s b) j (Z t c) k Q ijk ( 5) to find the components of Q at the canonical point on the orbit in terms of the I r, s, t ( there will, of course, be some relations between these components due to the constraints that X, Y, and Z are diagonal). Thus, by showing that the following set of polynomial invariants is sufficient to calculate the components of a generic density matrix at the canonical point, we have demonstrated that they characterize generic orbits: tr X r, tr Y r, tr Z r a T X r a, b T Y r b, c T Z r c a. ( Xa) Ù ( X a), b. (Yb ) Ù (Y b), c. ( Zc) Ù ( Z c) ( 53) ( X r a) i (Y s b) j R ij, (Y r b) i (Z s c) j T ij ( X s a) i (Z r c) j S ij, (X r a) i (Y s b) j (Z t c) k Q ijk the indices r, s, t range over the values,, 3.

20 546 Carteret, Linden, Popescu, and Sudbery If two density matrices have different values of any of these invariants, they are not on the same orbit; if they have same value of all of these invariants, and if A 9, B 9, and C 9 are nonzero, then the density matrices are locally equivalent. We note that the number of independent components of a generic density matrix at the canonical point is equal to the number of functionally independent parameters calculated at the beginning of this article. However, the number of polynomial invariants needed to characterize the generic orbit is greater than this; this is related to the fact that the ring of invariants is nonpolynomial, i.e., that the geometry of the space of orbits is nontrivial. The procedure given above can be used for all n> : use the tensors of highest rank and rank one in the expression for r to fix a canonical point on the orbit; the polynomials which separate the generic orbits are the analogues of those used in the case n = 3. In the case of n = this method can be used but there is some redundancy in the description we have given: the matrices X ii = R ij R i j and Y j j = R ij R ij [ using the notation of (9) ] have the same eigenvalues and the matrix R ij is diagonal at the canonical point. In this case there are nine functionally independent invariants which specify the squares of the nonzero components of a, b, and R at the canonical point on a generic orbit: tr X n, a T X m a, and b T Y p b, where n, m, p take the values,, 3. Additional invariants are needed to specify the signs of the nonzero components. The five invariants a. ( Xa) Ù ( X a), b. ( Yb ) Ù (Y b), and ax r Rb, r =,, 3, are sufficient to determine these signs for generic orbits and hence separate these orbits. In fact, using slightly different arguments, one can show that, in this case, one can reduce the number of polynomial invariants to, namely tr X, tr X, det R, a T X r a, a T X r Rb, r =,, 3, and A 9, which are subject to a single relation expressing A 9 as a function of the other invariants. An important problem for the future is to find a systematic way of constructing a set of invariants which also separate nongeneric orbits. The general idea of investigating canonical points on orbits in the way we have described is also appropriate for higher spins, but the situation is somewhat more complicated. Consider the example of two particles of spin one in which case the unitary group under which r transforms ( by conjugation) is SU(3). However, the adjoint representation of SU(3) is not equivalent to SO ( 8) but to an eight-dimensional subgroup of it; this means that we cannot use SU( 3) transformations to bring 83 8 symmetric matrices to diagonal form. Thus the canonical form is rather more complicated than in the case of spin-/ particles.

21 Multiparticle Entanglement SPECIAL ENTANGLED STATES As discussed in Sec., a further important question that the group theoretic approach allows one to address is what types of entanglement can occur. One can do this by recalling that, by definition, any group G acts transitively on an orbit O and thus an orbit may be written O = G/H ( 54) where H is the stability group of any point on the orbit. Thus the space of states of n-spins breaks up into orbits each of which is characterized by its stability group. Each stability group is a subgroup of U( ) n, so the issue is then to find which subgroups occur as stability groups. A generic orbit will have a certain stability group, but there are also special cases are where an orbit has a larger symmetry group. If we denote by H Y the invariance group of the state Y, we will see that states with ``maximal symmetry are particularly interesting. By states of ``maximal symmetry, we mean those states Y for which there are no others which have an invariance group which contain H Y as a proper subgroup. One systematic way to analyze the space of states, in principle, is to use the infinitesimal methods in Sec.. Consider first the case of pure states of two spins. We found that of the eight generators of U( ), only six were linearly independent for generic states so that generic orbits have a twodimensional invariance group. However, there will be some values of the parameters describing the states for which the number of linearly independent vectors is smaller than six. Finding these points is a problem in linear algebra. Unfortunately the complexity of the calculation seems to make it impractical. An alternative approach is to make use of the fact that every stability group is a subgroup of U( ) n. One can make a list of subgroups of U( ) n and check which subgroups occur as stability groups. Goursat s theorem ( 9) gives a complete characterization of subgroups of any direct product of two groups and this enables one, in principle, to produce this list. The complete set of subgroups, even of U() 3 U() is considerable, once all discrete subgroups are taken into account. However, the example below shows that much progress in understanding the space of states can be made by considering only continuous subgroups in the first instance. For a more comprehensive study of this case including a discussion of discrete subgroups, see Ref. 3.

22 548 Carteret, Linden, Popescu, and Sudbery As an example, consider a ( fairly general) three-spin pure state of the form Y = ae Ä e Ä e + be Ä e Ä e + ce Ä e Ä e + de Ä e Ä e ( 55) In order to determine whether this state is invariant under any continuous ( connected) group, it suffices to check whether it is annihilated by any Lie algebra element. Since each copy of U() in the group U() 3 contains a U( ) subgroup corresponding to changing the global phase of the state, it suffices to consider SU() 3 3 U( ); thus the phase is counted only once. The most general Lie algebra element in this case is where T = a (s x) + a ( s x) + a 3 ( s x) 3 + b ( s y) + b (s y) + b 3 ( s y) 3 + c ( s z) + c ( s z) + c 3 ( s z) 3 + d 8 ( 56) ( s x) = s xä Ä, ( s x) = Ä s xä, etc. ( 57) and 8 is the identity element 8 = Ä Ä ( 58) By direct calculation one can check that if a, b, c, and d in (55) are all nonzero, then the state is not annihilated by any nonzero Lie algebra element so that the state is not invariant under any continuous ( connected) group. The special cases, where the state does have an invariance group, are interesting, however: consider first the case a =. If b, c, and d are all nonzero, then we find that the state is annihilated by the Lie algebra element with c = c = c 3 = d, with all other coefficients in T being zero; i.e., the state is invariant under U( ). If, however, a = b = and c and d are non zero with c Þ d, then we find that invariance is further enhanced and the state is invariant under U( ). If c = d, the state has yet further symmetry, namely, U( ) 3 SU( ), and one notices that the state is of the form of a singlet with respect to particles and 3 tensor product with a vector for particle ; we write this as singlet 3 Ä vector. The invariance group U() 3 SU() arises since a singlet is invariant under a ( diagonal) SU( ) and the state vector is invariant under U( ). The invariance group of the state cannot be increased by choosing special (nonzero) values of c and d so a state of the form singlet 3 Ä vector has maximal symmetry. If a = b = and one of c or d is also zero, we find that the symmetry is also enhanced with respect to the case where c and d are nonzero in this

23 Multiparticle Entanglement 549 case the symmetry is U( ) 3 and such a state also has maximal symmetry in the sense that no state has a symmetry group of which this is a subset. The state is of the form w Ä w Ä w 3 (i.e., it is a direct product state). In the case that a = b = c =, the generators may be taken to be ( s z) + 8, ( s z) 8 and ( s z) 3 8, for example. One also finds a similar structure among the states with a = and c = or a = and d =, namely, invariance group of U( ), unless the state is one of the special ones with maximal symmetry, namely either direct products with invariance U( ) 3 or of the form singletä vector with invariance SU( ) 3 U( ). The cases of the sets of states with b = or c = have a similar structure to those with a =. The case of d = is different, however. If d = and a, b, and c are all nonzero, one calculates that the state is annihilated by ( s z) ( s z) only; the state is invariant under U(). If d = and a = but b and c are nonzero, the invariance is enhanced to U( ), in general, or SU( ) 3 U() when b = c, in which case the state is of the form singlet Ä vector 3, a state of maximal symmetry. When d = and b = but a and c are nonzero, the invariance is enhanced to U() 3 ; the state is a direct product. Perhaps the most interesting case is when d = and c = but a and b are nonzero, in which case one finds, for all values of a and b, that the state is invariant under U( ). However, although there are a number of states with this symmetry, thought of as an abstract group, as described above, the way that the group acts on the states is quite different in the case d = c = than, for example, d = a =. In the case d = c =, the generators are (s z) (s z) and (s z) ( s z) 3, corresponding to correlation between spins and and between spins and 3. In the case of d = a =, the invariance group arises since any vector in C is invariant under U( ) and a generic two-particle state is also invariant under U( ). Among those states with d = c =, there are some which have larger symmetry groups than U( ). If a = or b =, then the invariance group is U() 3 ; the state is a direct product. However, the case a = b, while not having further continuous symmetry, is picked out by the fact that only this state has a discrete symmetry of Z corresponding to the operation of (, ) simultaneously flipping all spins. This is the famous GHZ state. 5. INVARIANTS UNDER MORE GENERAL TRANSFORMATIONS While local unitary transformations are important, to get a deeper understanding of entanglement, one needs to take into account not only local unitary transformations and classical communication between

24 55 Carteret, Linden, Popescu, and Sudbery observers but also measurements on the particles. In this context we gave some preliminary results in Ref. 3 which we extend here. To be explicit, let us consider two particles with density matrix r; the case of more particles uses the same ideas and follows straightforwardly. Under a succession of rounds of local unitary transformations, measurements, and classical communication, the state transforms to r ƒ AÄ BrA ² Ä B ² tr(aä BrA ² Ä B ² ) ( 59) where we may write A as A = U A f a, n A, where U A is unitary, and f a, n A = n( + a n. s ), where a and n are real parameters with < a< and < n< /( + a) and n is a unit vector. Similarly B = U B f b, m B where U B is unitary and f b, m B = m( + bm. s ) where < b< and < m< /( + b) and m is a unit vector. Thus in terms of these expressions, r transforms to r ƒ U A f a, n A Ä U B f b, m B rf a, n A U ² A Ä f b, m B t( r; a, n; b, m) U ² B ( 6) with the normalization t( r; a, n; b, m) = tr[ f a, n A f a, n A Ä f b, m B Consider now the ``time-reversed matrix (3, 33) given by f b, m B r] ( 6) rä = s yä s yr*s yä s y ( 6) where complex conjugation is performed in the basis in which s z diagonal. It may be shown that is rä ƒ U A f a, n A Ä U B f b, m B rä f a, n t( r; a, n; b, m) A U ² AÄ f b, m B U ² B ( 63) Therefore the product rrä transforms as rrä ƒ m n ( a )( b ) t ( r; a, n; b, m) U A f a, n A Ä U B f b, m B rrä f a, n A U ² AÄ f b, m B U ² B ( 64)

25 Multiparticle Entanglement 55 Now we note that the following expressions have particularly simple transformation properties: tr [ ( rrä ) M ] ƒ 3 [m n ( a )( b ) ] t( r; a, n; b, m) 4 tr [ [ tr ( rrä ) ] N ] ƒ 3 [m n ( a )( b ) ] t( r; a, n; b, m) 4 tr [ [ tr ( rrä ) ] P ] ƒ 3 [m n ( a )( b ) ] t( r; a, n; b, m) 4 M N P tr [ ( rrä ) M ] tr [ [tr ( rrä )] N ] ( 65) tr [ [tr ( rrä ) ] P ] Thus any ratios of the above expressions with homogeneity zero will be invariant under ( 59). Similar ideas may be used for larger numbers of spins to generate an infinite class of invariants in these cases as well. The time reversal operation is rä = s yä s y... s yä s y r*s yä s y... s yä s y ( 66) and the analogues of ( 65) are constructed by forming polynomials in rrä with the indices contracted in all possible ways ( with indices associated with particle one always contracted with indices on particle one, etc.). 6. CONCLUSION In their seminal paper, Greenberger, Horne, and Zeilinger opened up the study of multiparticle entanglement. In this article we have reviewed and extended the general framework for understanding multiparticle entanglement which we have been developing. This framework is by no means complete. For example, one has to consider actions taken on a large number of copies of the state, and not only on a single copy as considered here. Nevertheless, the general framework we have outlined may be extended to these more general local actions, and it is also clear that any ``measure of entanglement must be a function of the invariants described here. ACKNOWLEDGMENTS We thank Serge Massar and Graeme Segal for very useful discussions. We are also very grateful to the Leverhulme and Newton Trusts for the financial support given to N.L.