Werner state structure and entanglement classification

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 {\em 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 into Werner states.


Introduction
Quantum entanglement, a feature of quantum theory named by Schrödinger [1] and employed by Bell [2] 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 n 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 article 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 n qubits. Werner states have found a multitude of uses in quantum information science. Originally introduced in 1989 for two particles [3] to distinguish between classical correlation and Bell inequality satisfaction, Werner states have found use in the description of noisy quantum channels [4], as examples in nonadditivity claims [5], and in the study of deterministic purification [6]. 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 [7,8,9]. A recent example of how analysis of state structure can be useful is the work of Migda l et al. [10] on protecting information against the loss of a qubit using Werner states.
We apply the representation theory of Lie groups, in particular, Clebsch-Gordan decomposition of representations of SU(2) on tensor products and the representation theory of SO(3) 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 [11] and symmetric Werner states [12] in Section 3 below. 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.
In this notation, the Werner states are defined to be the set of density matrices ρ such that ρ = g ⊗n ρ(g † ) ⊗n for all g in SU(2). We will write ∆ to denote the subgroup ∆ = {(g, g, . . . , g) : g ∈ SU(2)} of the LU group G.
The set of n-qubit density matrices is a convex set inside of the vector space V ⊗n , where V is the 4-dimensional real vector space of 2 × 2 Hermitian matrices. A convenient basis for V is {σ 0 , σ 1 , σ 2 , σ 3 }, where σ 0 is the 2 × 2 identity matrix, and σ 1 = σ x , σ 2 = σ y , and σ 3 = σ z are the Pauli matrices.
Every element ρ (whether or not ρ is positive or has trace 1) of V ⊗n can be uniquely written in the form ρ = I s I σ I , where I = i 1 i 2 . . . i n is a multiindex with i k = 0, 1, 2, 3 for 1 ≤ k ≤ n, and σ I denotes with real coefficients s I .
Sitting inside V ⊗n is the space of pure states, which are the rank 1 density matrices of the form |ψ ψ|, where |ψ is a vector in the Hilbert space H = (C 2 ) ⊗n of pure n-qubit states. We will use the computational basis vectors The expansion of a pure state vector |ψ in the computational basis has the form |ψ = I c I |I , where the coefficients c I are complex. Note that we use the same multiindex notation I for "mod 4" multiindices for tensors of Pauli matrices in V ⊗n , and for "mod 2" multiindices for computational basis vectors in H. The distinction will be clear from context.

Pure and Symmetric Werner States
In [11], we prove a structure theorem for pure Werner states based on the following geometric construction. Begin with a circle with an even number n = 2m of marked points, labeled 1 through n in order around the circle, say clockwise. Let P be a partition of {1, 2, . . . , n} into two-element subsets. Each two-element subset {a, b} determines a chord connecting a and b. We impose the condition that no two chords coming from P may intersect. For each chord C in P, let |s C be a singlet state 1 √ 2 (|01 −|10 ) in the two qubits at the ends of C, and define the state |s P to be the product |s P = C∈P |s C of singlet states |s C , over all C in P. We call states of the form |s P "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 non-intersecting 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 non-intersecting chord diagram states span the space of pure Werner states. The Werner states are the trivial summand in the  when the number of qubits n = 2m is even, and the dimension of this space is zero when n is odd. The non-intersecting chord diagrams with n = 2m nodes are one of the well-known sets enumerated by the Catalan numbers [13]. Together with an argument that the non-intersecting 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 [12], we consider the case of pure and mixed Werner states that are invariant under permutations of qubits, also called symmetric states. Given nonnegative integers n 1 , n 1 , n 3 with n 1 +n 2 +n 3 ≤ n, we identify the monomial x n 1 y n 2 z n 3 in three variables with the matrix where n 0 = n − n 1 − n 2 − n 3 , the symmetrizing operator Sym sums all the permutations of the products of n k copies of σ k for k = 0, 1, 2, 3, and α 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 SO(3), we show that symmetric Werner states correspond to polynomials that are linear combinations of (x 2 + y 2 + z 2 ) m for some m ≤ ⌊n/2⌋. Further, any two such states are local unitarily inequivalent. Now we turn to the general case of mixed Werner states.

The General Case of Mixed Werner States
We begin with the construction of a family of density matrices C n that generalize the singlet state.
Given an n-qubit binary string I, let C(I) denote the pure state where ω = e 2πi n and π is the the cyclic permutation of {1, 2, . . . , n} given by 1 → n, k → k − 1 for 2 ≤ k ≤ n, and α is a normalizing factor so that |C(I)| = 1, whenever C(I) = 0 (notice that C(00) = 0, so is not a state). For example, Let C n denote the density matrix where β is a normalizing factor so that tr(C n ) = 1. Observe that C 2 is the density matrix |s s| of the singlet state |s = 1 √ 2 (|01 − |10 ), so that the C n states are n-qubit generalizations of the singlet.
Next we form products of C k states to make Werner states 1 . As with the case of pure Werner states, we utilize diagrams. This time we consider diagrams consisting of n points labeled 1, 2, . . . , n on a circle, with nonintersecting polygons that have vertices in the given set of n points. Again, there are a Catalan number 1 n+1 2n n of such n-vertex diagrams 2 [13]. Given an n-vertex non-intersecting polygon diagram D, we construct a state It is perhaps nontrivial to show that the C n and the diagram states constructed from them 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 [14] 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 ∆ = {(g, g, . . . , g) : g ∈ SU (2)} annihilate C n .
2 There is a one-to-one correspondence between these "non-intersecting polygon" diagrams and the "non-intersecting chord" diagrams on 2n points in our pure Werner states analysis of the previous section. Given a non-intersecting chord diagram with 2n vertices, rename the vertices 1, 1 ′ , 2, 2 ′ , . . . , n, n ′ and then glue each pair of jj ′ for 1 ≤ j ≤ n. where the tensor has positions specified by elements of the partition D and C U denotes the state C |U | in qubit positions in U. Figure 2 shows an example.
Here is our main conjecture.
Conjecture. The states ρ D 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 V decomposes into irreducible SU(2)-submodules as where the R 3 summand is isomorphic to the adjoint representation. The complexification V C is isomorphic to C 1 ⊕ C 3 ≈ C 2 ⊗ C 2 , which is 2-qubit pure state space. In general, the complexification (V ⊗n ) C of n-qubit density matrix space V ⊗n is isomorphic to 2n-qubit pure state space, as SU(2) spaces. Thus the real dimension of the trivial summand for n-qubit density matrices is equal to the complex dimension of the trivial summand for 2n-qubit pure states, which is the Catalan number 1 n+1 2n n . 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 ρ, that is, the set Stab ρ = {g ∈ G : ρ = gρg † } of all local unitary transformations that fix ρ, could be larger than the subgroup ∆ of the unitary group. For example, a diagram state ρ D is stabilized by the subgroup where ∆ U denotes the subgroup that consists of elements (g, g, . . . , g) in qubits in U, and all other coordinates are the identity.
In [11], 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 {1, 2, . . . , n} (we consider all partitions, with and without crossing polygons) where D ≤ D ′ if D ′ is a subdivision of D. The n-gon is the least element at the bottom and the all-singleton diagram is the greatest element at the top of this lattice. The non-crossing polygon diagram lattice is a sublattice. There is a corresponding lattice of subgroups of local unitary group G, where H is less than or equal to K in the partial order if H is a subgroup of K. The subgroup ∆ is the least element at the bottom and LG at the top. A diagram D corresponds to the subgroup ∆ D defined above. We conjecture that where "glb" denotes 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).

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 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.