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Advances in Mathematical Physics
Volume 2012 (2012), Article ID 463610, 7 pages
Werner State Structure and Entanglement Classification
1Department of Mathematical Sciences, Lebanon Valley College, Annvile, PA 17003, USA
2Department of Physics, Lebanon Valley College, Annvile, PA 17003, USA
Received 28 September 2011; Accepted 16 January 2012
Academic Editor: B. G. Konopelchenko
Copyright © 2012 David W. Lyons et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We present applications of the representation theory of Lie groups to the analysis of structure and local unitary classification of Werner states, sometimes called the decoherence-free states, which are states of n quantum bits left unchanged by local transformations that are the same on each particle. We introduce a multiqubit generalization of the singlet state and a construction that assembles these qubits into Werner states.
Quantum entanglement, a feature of quantum theory named by Schrödinger  and employed by Bell [2, 3] in the rejection of local realism, has come to be seen as a resource for quantum information processing tasks including measurement-based quantum computation, teleportation, and some forms of quantum cryptography. Driven by applications to computation and communication, entanglement of composite systems of quantum bits, or qubits, is of particular interest.
The problem of entanglement is to understand nonlocal properties of states and to answer operational questions such as when two given states can be interconverted by local operations on individual subsystems. This inspires the mathematical problem of classifying orbits of the local unitary group action on the space of states.
The goal of this paper is to address these questions for the Werner states, which are defined to be those states invariant under the action of any particular single-qubit unitary operator acting on all qubits. Werner states have found a multitude of uses in quantum information science. Originally introduced in 1989 for two particles  to distinguish between classical correlation and the Bell inequality satisfaction, Werner states have found use in the description of noisy quantum channels , as examples in nonadditivity claims  and in the study of deterministic purification . In what may prove to be a practical application to computing in noisy environments, Werner states lie in the decoherence-free subspace for collective decoherence [8–10]. A recent example of how analysis of state structure can be useful is the work of Migdał and Banaszek  on protecting information against the loss of a qubit using Werner states.
We apply the representation theory of Lie groups, in particular the Clebsch-Gordan decomposition of representations of on tensor products and the representation theory of on polynomials in three variables, to obtain structural theorems and local unitary classification for Werner states. We summarize recent results for the special cases of pure Werner states  and symmetric Werner states  in Section 3. We present new results for the general case of mixed Werner states in Section 4. We introduce a generalization of the singlet state and use these states to construct Werner states.
2. Local Unitary Group Action
Let denote the local unitary (LU) group for -qubit states. An LU operator acts on an -qubit density matrix (i.e., a positive semidefinite matrix with trace 1) by In this notation, the Werner states are defined to be the set of density matrices such that for all in . We will write to denote the subgroup of the LU group .
The set of -qubit density matrices is a convex set inside of the vector space , where is the 4-dimensional real vector space of Hermitian matrices. A convenient basis for is , where is the identity matrix, and , , and are the Pauli matrices. Every element (whether or not is positive or has trace 1) of can be uniquely written in the form , where is a multi-index with for , and denotes with real coefficients .
Sitting inside is the space of pure states, which are the rank 1 density matrices of the form , where is a vector in the Hilbert space of pure -qubit states. We will use the computational basis vectors for , where is a multi-index with for . The expansion of a pure state vector in the computational basis has the form , where the coefficients are complex. Note that we use the same multi-index notation for “mod 4” multi-indices for tensors of the Pauli matrices in , and for “mod 2” multi-indices for computational basis vectors in . The distinction will be clear from context.
3. Pure and Symmetric Werner States
In , we prove a structure theorem for the pure Werner states based on the following geometric construction. Begin with a circle with an even number of marked points, labeled 1 through in order around the circle, say clockwise. Let be a partition of into two-element subsets. Each two-element subset determines a chord connecting and . We impose the condition that no two chords coming from may intersect. For each chord in , let be a singlet state in the two qubits at the ends of , and define the state to be the product of singlet states , over all in . We call states of the form “nonintersecting chord diagram states.” Figure 1 illustrates the two possibilities for 4 qubits.
We show that any linear combination of chord diagram states is a Werner state, and conversely, any pure Werner state can be written uniquely as a linear combination of nonintersecting chord diagram states. Further, these linear combinations are unique representatives of their LU equivalence class, up to a phase factor. Representation theory and combinatorics enter the story in the proof that the nonintersecting chord diagram states span the space of pure Werner states. The Werner states are the trivial summand in the decomposition into irreducible submodules of the -space . The dimension of the trivial summand is equal to the Catalan number when the number of qubits is even, and the dimension of this space is zero when is odd. The nonintersecting chord diagrams with nodes are one of the well-known sets enumerated by the Catalan numbers . Together with an argument that the nonintersecting chord diagram states are linearly independent, the fact that these two numbers agree establishes that the chord diagram states form a linear basis for the space of pure Werner states.
In , we consider the case of pure and mixed Werner states that are invariant under permutations of qubits, also called symmetric states. Given nonnegative integers , , with , we identify the monomial in three variables with the matrix where , the symmetrizing operator sums all the permutations of the products of copies of for , and is a normalization factor. This establishes a correspondence between mixed symmetric states (not necessarily Werner states) and real polynomials in three variables. Using the representation theory of , we show that the symmetric Werner states correspond to polynomials that are linear combinations of for some . Further, any two such states are local unitarily inequivalent.
Now we turn to the general case of mixed Werner states.
4. The General Case of Mixed Werner States
We begin with the construction of a family of density matrices that generalize the singlet state.
Given an -qubit binary string , let denote the pure state where and is the cyclic permutation of given by , for , and is a normalizing factor so that , whenever (notice that , so is not a state). For example, Let denote the density matrix where is a normalizing factor so that . Observe that is the density matrix of the singlet state , so that the states are -qubit generalizations of the singlet.
Next we form products of states to make Werner states. (It is perhaps nontrivial to show that the and the diagram states constructed from them below are indeed Werner states. This can be done with straightforward calculations, but at the expense of technical overhead. We refer the interested reader to our paper  which gives details on the action of the Lie algebra of the local unitary group on density matrices. One can show that the generators of the Lie algebra of the Werner stabilizer group annihilate .) As with the case of pure Werner states, we utilize diagrams. This time we consider diagrams consisting of points labeled on a circle, with nonintersecting polygons that have vertices in the given set of points. Again, there is a Catalan number of such -vertex diagrams . (There is a one-to-one correspondence between these “nonintersecting polygon” diagrams and the “nonintersecting chord” diagrams on points in our pure Werner states analysis of the previous section. Given a nonintersecting chord diagram with vertices, rename the vertices and then glue each pair for .)
Given an -vertex nonintersecting polygon diagram , we construct a state , where the tensor has positions specified by elements of the partition and denotes the state in qubit positions in . Figure 2 shows an example.
Here is our main conjecture.
The states form a basis for the space of Werner states (in the larger space of real linear combinations of Pauli tensors).
Again, representation theory says that we have the right dimension: the 1-qubit density matrix representation space decomposes into irreducible -submodules as follows: where the summand is isomorphic to the adjoint representation. The complexification is isomorphic to , which is 2-qubit pure state space. In general, the complexification of -qubit density matrix space is isomorphic to -qubit pure state space, as spaces. Thus the real dimension of the trivial summand for -qubit density matrices is equal to the complex dimension of the trivial summand for -qubit pure states, which is the Catalan number . This establishes that we only need to show that the diagram states are independent in order to prove the conjecture.
We conclude with a conjecture regarding a precise statement about the stabilizer subgroup of the local unitary group for our constructed Werner states. The full stabilizer of a Werner state , that is, the set of all local unitary transformations that fix , could be larger than the subgroup of the unitary group. For example, a diagram state is stabilized by the subgroup where denotes the subgroup that consists of elements in qubits in , and all other coordinates are the identity.
In , we give a criterion on the diagrams that appear in the expansion of a pure Werner state for when the stabilizer subgroup of a pure Werner state is precisely the subgroup of the local unitary group, and not larger. The criterion is that for any bipartition of the set of qubits, there must be a diagram (with nonzero coefficient) in the expansion of the given state that has a chord with one end in each of the sets of the partition. Our final conjecture is a generalization of this idea to the general mixed Werner case.
Consider a poset lattice of partitions of (we consider all partitions, with and without crossing polygons), where if is a subdivision of . The -gon is the least element at the bottom, and the all-singleton diagram is the greatest element at the top of this lattice. The noncrossing polygon diagram lattice is a sublattice. There is a corresponding lattice of subgroups of local unitary group , where is less than or equal to in the partial order if is a subgroup of . The subgroup is the least element at the bottom and at the top. A diagram corresponds to the subgroup defined above. We conjecture that where “glb” denotes the greatest lower bound in the lattice. This would give a picture criterion for when a Werner state has the Werner stabilizer (and not a larger one).
5. Summary and Outlook
We have surveyed known results on the structure and local unitary equivalence classification of Werner states for the special cases of pure states and symmetric states. We have presented a diagram-based construction for the general case of the mixed Werner states that generalizes the “sums of products of singlets” construction known for pure states. Finally, we conjecture that the general construction will prove to be a basis for the Werner states and that this basis will lead to local unitary classification and a precise analysis of stabilizer subgroups.
This work has been supported by the National Science Foundation Grant no. PHY-0903690.
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