Dragomir Ž. Ðoković

Boundary of the set of separable states

Motivated by the separability problem in quantum systems 2⊗4, 3⊗3 and 2⊗2⊗2, we study the maximal (proper) faces of the convex body, S1, of normalized separable states in an arbitrary quantum system with finite-dimensional Hilbert space H=H1⊗H2⊗⋯⊗Hn. To any subspace V⊆H, we associate a face FV of S1 consisting of all states ρ∈S1 whose range is contained in V . We prove that FV is a maximal face if and only if V is a hyperplane. If V =|ψ⟩⊥, where |ψ⟩ is a product vector, we prove that Dim FV=d2−1−∏(2di−1), where di=Dim Hi and d=∏di. We classify the maximal faces of S1 in the cases 2⊗2 and 2⊗3. In particular, we show that the minimum and the maximum dimension of maximal faces is 6 and 8 for 2⊗2, and 20 and 24 for 2⊗3. The boundary, ∂S1, of S1 is the union of all maximal faces. When d>6, it is easy to show that there exist full states on ∂S1, i.e. states ρ∈∂S1 such that all partial transposes of ρ (including ρ itself) have rank d. Ha and Kye have recently constructed explicit such states in 2⊗4 and 3⊗3. In the latter case, they have also constructed a remarkable family of faces, depending on a real parameter b>0, b≠1. Each face in the family is a nine-dimensional simplex, and any interior point of the face is a full state. We construct suitable optimal entanglement witnesses for these faces and analyse the three limiting cases b=0,1,∞.

Orthogonal product bases of four qubits

An orthogonal product basis (OPB) of a finite-dimensional Hilbert space is an orthonormal basis of consisting of product vectors . We show that the problem of constructing the OPBs of an n-qubit system can be reduced to a purely combinatorial problem. We solve this combinatorial problem in the case of four qubits and obtain 33 multiparameter families of OPBs. Each OPB of four qubits is equivalent, under local unitary operations and qubit permutations, to an OPB belonging to at least one of these families.

Turyn-Type Sequences: Classification, Enumeration, and Construction

Turyn-type sequences, , are quadruples of -sequences , with lengths , respectively, where the sum of the nonperiodic autocorrelation functions of and twice that of is a δ-function (i.e., vanishes everywhere except at 0). Turyn-type sequences are known to exist for all even n not larger than 36. We introduce a definition of equivalence to construct a canonical form for in general. By using this canonical form, we enumerate the equivalence classes of for . We also construct the first example of Turyn-type sequences .

The checkerboard family of entangled states of two qutrits

By modifying the method of Bruß and Peres, we construct two new families of entangled two qutrit states. For all density matrices ρ in these families we have ρij = 0 for i + j odd. The first family depends on 27 independent real parameters and includes both PPT and NPT states. The second family consists of PPT entangled states. The number of independent real parameters of this family is ≥ 11

Factorization of hermitian matrix polynomials with constant signature

Abstract Gohberg, Lancaster, and Rodman have shown that if a polynomial A(t), with complex hermitian matrices as coefficients, has nonzero determinant and is such that A(λ) has a constant signature for all real λ for which det A(λ) ≠ 0, then A(t) admits a factorization A(t)= B ∗ (t)DB(t) , where B(t) is a polynomial with complex matrices as coefficients and D is a nonsingular complex hermitian matrix (which may be assumed to be diagonal with diagonal entries ±1). We give a new, short, and simple proof of this theorem and extend it to Laurent polynomial matrices with a suitably defined involution.

Length filtration of the separable states

We investigate the separable states ρ of an arbitrary multi-partite quantum system with Hilbert space H of dimension d. The length L(ρ) of ρ is defined as the smallest number of pure product states having ρ as their mixture. The length filtration of the set of separable states, S, is the increasing chain ∅⊊S1′⊆S2′⊆⋯, where Si′={ρ∈S:L(ρ)≤i}. We define the maximum length, Lmax=maxρ∈SL(ρ), critical length, Lcrit, and yet another special length, Lc, which was defined by a simple formula in one of our previous papers. The critical length indicates the first term in the length filtration whose dimension is equal to Dim S. We show that in general d≤Lc≤Lcrit≤Lmax≤d2. We conjecture that the equality Lcrit=Lc holds for all finite-dimensional multi-partite quantum systems. Our main result is that Lcrit=Lc for the bipartite systems having a single qubit as one of the parties. This is accomplished by computing the rank of the Jacobian matrix of a suitable map having S as its range.

Generalization of Mirsky’s theorem on diagonals and eigenvalues of matrices☆

Article history: Received 17 June 2012 Accepted 20 June 2012 Available online 19 July 2012 Submitted by V. Sergeichuk AMS classification: 15A18 15A29

On singular values and similarity classes of matrices

Abstract Let s be the map M n → R n , where M n is the space of n × n complex matrices, which assigns to each matrix A the n -tuple of its singular values arranged in decreasing order. We are interested in describing the image s (Δ) of a similarity class of matrices Δ ⊂ M n . The problem is completely solved when Δ consists of cyclic matrices and in a number of other cases. We tabulate descriptions of s (Δ) for all similarity classes Δ ⊂ M n when 2⩽ n ⩽4.