Entanglement, Particle Identity and the GNS Construction: A Unifying Approach
A novel approach to entanglement, based on the Gelfand-Naimark-Segal (GNS) construction, is introduced. It considers states as well as algebras of observables on an equal footing. The conventional approach to the emergence of mixed from pure ones based on taking partial traces is replaced by the more general notion of the restriction of a state to a subalgebra. For bipartite systems of nonidentical particles, this approach reproduces the standard results. But it also very naturally overcomes the limitations of the usual treatment of systems of identical particles. This GNS approach seems very general and can be applied for example to systems obeying para- and braid- statistics including anyons.
In spite of the numerous efforts to achieve a satisfactory understanding of entanglement for systems of identical particles, there is no general agreement on the appropriate generalization of concepts valid for non-identical constituents Tichy et al. (2011). That is because many concepts are usually only discussed in the context of quantum systems for which the Hilbert space is a simple tensor product with no additional structure. An example is the Hilbert space of two non identical particles. In this case the partial trace for to obtain the reduced density matrix has a good physical meaning: it corresponds to observations only on the subsystem .
In contrast, the Hilbert space of a system of identical bosons (fermions) is given by the symmetric (antisymmetric) -fold tensor product of the single-particle spaces. The consequence is that any multi-particle state contains intrinsic correlations between subsystems due to quantum indistinguishability. This, in turn, forces a departure from the straightforward application of entanglement-related concepts like singular value decomposition (SVD), Schmidt rank, or entanglement entropy.
In this context, Schliemann et al. Schliemann et al. (2001) have introduced an analogue of the Schmidt rank, the ‘Slater rank’ to study entanglement in two-fermion systems, using a new version of the SVD adapted to deal with antisymmetric matrices. The extension of these ideas to the boson case was worked out in Paskauskas and You (2001) and Li et al. (2001). These approaches also have not found general acceptance.
The problems arising in the interpretation of the Slater rank and the von Neumann entropy of the reduced density matrix (obtained by partial tracing) for these systems have been analyzed in Ghirardi et al. (2002); Ghirardi and Marinatto (2004).
Numerous other proposals for the treatment of identical particles have recently been put forward. But, as a closer look at the literature on the subject Adhikari et al. (2009); Amico et al. (2008); Bañuls et al. (2009); Benatti et al. (2012); Cavalcanti et al. (2007); Eckert et al. (2002); Ghirardi et al. (2002); Ghirardi and Marinatto (2004); Horodecki et al. (2009); Lévay et al. (2005); Li et al. (2001); Paskauskas and You (2001); Sasaki et al. (2011); Schliemann et al. (2001); Shi (2003); Tichy et al. (2011); Wiseman and Vaccaro (2003); Zanardi (2002); Zander et al. (2012); Cavalcanti et al. (2007); Grabowski et al. (2011) reveals, it is apparent that there is no consensus yet as to what the proper formalism should be.
In this paper, we propose an approach to the study of entanglement based on the theory of operator algebras. The foundational results of Gelfand, Naimark and Segal on the representation theory of -algebras, abbreviated as the GNS-constructionGelfand and Neumark (1943); Segal (1947); Haag (1996) are used in order to obtain a generalized notion of entanglement. In particular, the notion of partial trace is replaced by the much more general notion of restriction of a state to a subalgebraBarnum et al. (2004). This will allow us to treat entanglement of identical and non-identical particles on an equal footing, without the need to resort to different separability criteria according to the case under study.
In order to display the usefulness of our approach, several explicit examples will be worked out. In particular we show that the GNS-construction gives zero for the von Neumann entropy of a fermionic or a bosonic state containing the least possible amount of correlations. We believe that this settles an issue that has caused a lot of confusion regarding the use of von Neumann entropy as a measure of entanglement for identical particles Paskauskas and You (2001); Wiseman and Vaccaro (2003); Ghirardi and Marinatto (2004); Amico et al. (2008); Plastino et al. (2009).
Ii The Basic Idea
ii.1 Preliminary Remarks
A vector state of a quantum system is usually described by a vector in a Hilbert space (pure case). More generally, a state is a density matrix , a linear map satisfying (normalization), (self-adjointness) and positivity . For pure states the additional condition is required, which amounts to the assertion that is of the form for some normalized vector in .
Now, given that the expectation value of an observable is defined by , we can equivalently regard as a linear functional on a unital (-) algebra of observables with unity (we consider only unital algebras). That is, , for . The normalization and positivity conditions above then take the form and (for any ). Such a positive linear functional of unit norm is called a state on the algebra .
As already mentioned, in the bipartite case , the definition of involves a partial trace operation. It is therefore natural to ask for a characterization of this operation in terms of the notion of states on an algebra. For this purpose consider the subalgebra consisting of all operators of the form , for an observable on . Defining a new state as the restriction of to the subalgebra , one easily checks that for any observable of the equality holds.
As we show below, an algebraic description of the quantum system where the basic objects are a -algebra and a state on the algebra provides the solution to some of the problems that appear when does not have the form of a ‘simple tensor product’.
ii.2 The GNS Construction
The basic idea of the GNS-construction is that given an algebra of observables and a state on this algebra, we can construct the Hilbert space on which the algebra of observables acts on. The key steps are: (a) Using , we can endow itself with an inner product. So it becomes an ‘inner product’ space . (b) This inner product may be degenerate in the sense that the norm of some non-null elements of may be zero. (c) If we remove these null vectors by taking the quotient of by the null space of zero norm vectors to get , then we have a well-defined positive definite inner product on . Hence we get a well-defined Hilbert space (after completion). (d) The algebra of observables acts naturally on this Hilbert space in a simple manner.
We now make this set of ideas more precise.
From the mathematical point of view, the algebra of observables is a -algebra. This guarantees that one has enough structure to perform all the tasks in the list above.
A -algebra is a (complete normed) algebra , together with an antilinear involution , such that the basic property is satisfied for all in . The prototypical example of a -algebra is the algebra of all bounded operators on a Hilbert space , with the involution given by the adjoint: . Here we will only be interested in unital algebras, that is, we assume the existence of a unit for the algebra.
Given a state on a -algebra , we can obtain a representation of on a Hilbert space as follows. Since is an algebra, it is in particular a vector space. When elements are regarded as elements of a vector space we write them as . We then set . This is almost a scalar product, , but there could be a null space of zero norm vectors: Schwarz inequality shows that is a left ideal:
It also shows that
We denote the elements of the quotient space by , where . It has a well-defined scalar product
(it is independent of the choice of from because of (1)) and no nontrivial null vectors. Moreover we have a representation of on . (To show this, use ).
This representation is in general reducible. We decompose into a direct sum of irreducible spaces: . Let be the corresponding orthogonal projections and define
and also that
One then obtains
where is a density matrix on , given by
This follows from
as has zero matrix elements between different irreducible subspaces.
This is a crucial fact. Since is a pure state, it shows that is pure if and only if the representation is irreducible, a well-known result. In particular, the von Neumann entropy of , , is zero if and only if is irreducible. The latter is a property that depends on both the algebra and the state .
Consider now a (unital) subalgebra of and let denote the restriction to of a pure state on ?. We can apply the GNS-construction to the pair and use the von Neumann entropy of to study the entropy which arises from the restriction.
ii.3 Example 1:
In order to illustrate the above GNS-construction, consider the algebra of complex matrices. Denoting by the matrix with one on its entry and zero elsewhere, we can write any as .
It is readily checked that the map given by defines a state on the algebra, as long as . The vector space is generated by the four vectors () and the null space is generated by those such that . Since
the explicit form of the null vector condition is:
For the only solution to this equation is , implying that there are no null vectors: . Therefore, in this case the GNS-space is given by . The matrices act on this space as :
The matrix of for instance is:
when we order the basis as .
This representation is clearly reducible, the subspaces spanned by being invariant. Hence is the direct sum of two isomorphic irreducible representations, the corresponding decomposition of being
Next we decompose into pure states. The unity of is just so that
gives the decomposition of into irreducible subspaces. The norms of the two components are and by (3). So
with as in (4). It follows that
so that is not pure. It has von Neumann entropy
If we choose , from (11) we see that , since it is spanned by elements of the form
that is, by linear combinations of and . Accordingly, the GNS-space is generated by and . In this case the representation of is irreducible and given by matrices :
The state is pure with zero entropy. A similar situation is found for .
ii.4 Example 2: Bell State
Let and consider the state vector
Then can be thought of as a state on the algebra of linear operators on . This algebra is isomorphic to the matrix algebra and is generated by elements of the form (), with and the Pauli matrices.
In this context, the entanglement of is understood in terms of correlations between “local” measurements performed separately on subsystems and . Measurements performed on correspond to the restriction of to the subalgebra generated by elements of the form .
We now study this case. We set . In order to construct the GNS-space , we first notice that , so that in this case there are no nontrivial null states, that is, . We obtain , with basis vectors and inner product .
The action of on is given, as explained above, by linear operators :
The RHS can be explicitly computed using the identity . One then finds that the GNS space splits into the sum of two invariant subspaces:
They are spanned by
The corresponding projections are
Thus, the representation of as a density matrix on the GNS-space is:
The von Neumann entropy computed via the GNS-construction is therefore
reproducing the standard result, as expected.
Iii Systems of Identical Particles
Let be the Hilbert space of a one-particle system. The group acts on by the representation and the algebra of observables is given by a -representation of the group algebra on . Its elements are of the form
where is a complex function on and the Haar measure Balachandran et al. (2010).
The elements span the matrix algebra . We can understand (32) in terms of matrix elements
which are just the integrals of the functions .
Consider now a fermionic system with single-particle space . The Hilbert space of this system is the Fock space , where is the space of antisymmetric -tensors in . Let denote an orthonormal basis for . Then, the set provides an orthonormal basis for .
We can alternatively consider the canonical anticommutation relations (CAR) algebra , and obtain all basis vectors by repeated application of creation operators to the vacuum vector :
A self-adjoint operator on can be made to act on in a way that preserves the antisymmetric character of the vectors by considering combinations of the form
The map is a Lie algebra homomorphism. We further comment on this important point below.
At the group level, we may consider exponentials of such operators, of the form We then see that the operators of the form
act properly on .
The map is an isomorphism from into . This is also important as discussed below.
These constructions are most conveniently expressed in terms of a coproduct Balachandran et al. (2010). In fact, an approach based on Hopf algebras (as explained in Balachandran et al. (2010)), has the great advantage that para- and braid-statistics can be automatically included. In our present case, the construction of the observable algebra corresponds to the following simple choice for the coproduct: , (), linearly extended to all of . This choice fixes the form of (36). At the Lie algebra level, it reduces to (35).
Physically, the existence of such a coproduct is very important, since it allows us to homomorphically represent the one-particle observable algebra on the -particle sector. In a many particle system, the choice to perform observations of only one-particle observables corresponds to restricting the full-algebra of observables to the homomorphic image of the one-particle observable algebra obtained by using the coproduct.
Now, if for some reason we perform only partial one-particle observations (for instance, if at the one-particle level, we decide to measure -or have access to- only the spin degrees of freedom, or only the position), the one-particle observable algebra will have to be restricted accordingly and hence its homomorphic image at the -particle level will be a subalgebra of the original algebra.
As we will see in what follows, the GNS approach covers all such cases whether or not particles are identical. In particular it merits to reemphasize that it covers observations of particles obeying para- and braid- statistics, including anyons.
iii.1 Example 3: Two Fermions, .
Keeping the same notation as above, put . We focus our attention on the two-fermion space , with basis
being an orthonormal basis for . The algebra of observables for the two-fermion system is the matrix algebra generated by (). It is isomorphic to .
Now, acts on through the defining representation (), so that one particle observables are given - at the two fermion level - by the action of on . This action is given by the restriction of the operators to the space of antisymmetric vectors. Let 3 be the defining (or fundamental) representation on . Then the restriction can be obtained from the decomposition of the representation. The span this representation.
Choice 1 for :
Let be any two-fermion state vector. If we take to be the full algebra of one-particle observables acting on , then and the GNS-representation corresponding to the pair is irreducible, the state remaining unchanged upon restriction. This is just the fact that the representation of is irreducible and corresponds to the fact that, for , all two-fermion vector states have Slater rank 1.
We hence get zero for the von Neumann entropy.
Notice, however, that the von Neumann entropy computed by partial trace is equal to 1 for all choices of (cf. Ghirardi and Marinatto (2004)), in disagreement with the GNS-approach.
Choice 2 for :
The situation changes drastically if we make a different choice for the subalgebra of . Let us, for the sake of concreteness, choose to be given by those one-particle observables pertaining only to the one-particle states and . In this case, will be the five dimensional algebra generated by and , the unit matrix.
As a physical illustration for the meaning of , let us think of as quarks. Observables acting just on and amounts to the isospin group acting just on and in the representation. And the group algebra is also generated by .
We now explicitly perform the GNS-construction for the particular choice
Let denote the corresponding state:
Now notice that independently of . The same holds true for so that both are null vectors: .
For , there are no more linearly independent null vectors. We can see this from
Therefore, the null space is two-dimensional and the GNS-space is the three-dimensional space with basis , where .
Since if , we immediately recognize that, in terms of irreducibles, . Call and the corresponding projections. After noting that , we obtain
The corresponding ‘weights’ are computed using the inner product of . We obtain
The result for the entropy as a function of is therefore
It is readily checked that at additional null states appear as compared to Case 1. Thus, for the null vectors are spanned by
The GNS space is two-dimensional and irreducible. It is spanned by
Since acts nontrivially on this space, and the smallest nontrivial representation of is its two-dimensional IRR, this representation is irreducible. Hence is pure with zero entropy. For completeness we note that the projector to is .
For instead, all of are null vectors. So is one-dimensional and spanned by . Clearly is pure with zero entropy.
The decomposition of as a direct sum of irreducible subspaces as we change the value of is, therefore, as follows:
This result should be contrasted against the fact that the representation, when regarded as a representation space for acting on , and splits as .
iii.2 Example 4: Two Fermions,
Consider, in the spirit of Eckert et al. (2002), a one-particle space describing fermions with two degrees of freedom which we call external (e.g. ‘left’ and ‘right’) and two degrees of freedom which we call internal e.g. ‘spin 1/2’). Here it is convenient to use a description in terms of fermionic creation/annihilation operators , with standing for ‘left’, for ‘right’ and for spin up and down, respectively. A basis for is then given by the vectors , and , with .
Again we consider a -dependent state vector, this time given by
At the two-particle level, the full observable algebra is the matrix algebra . From this algebra we pick the subalgebra of one-particle observables corresponding to measurements at the left location. This is the six-dimensional algebra generated by
For we readily find that a basis of null vectors of are and . The GNS Hilbert space is hence four-dimensional and spanned by the vectors and .
Let be the GNS representation of on . We can evidently find the decomposition of into irreducible subspaces under by computing the Casimir operator and the highest weight vectors of the Lie algebra given by the representation . We find , with spanned by and and spanned by and . The two representations are isomorphic.
We can find the components of into by writing
As for their normalization, using (49),
Hence the restriction of to can be written in terms of pure states as
This gives the following result for entropy:
Consider first . In this case,
The null vectors are obtained by solving for . That shows that
The quotient space is and is isomorphic to above:
For , when , we find instead a isomorphic to above: