Unextendible product bases and the construction of inseparable states

Let H[N] denote the tensor product of n finite dimensional Hilbert spaces H(r). A state |phi> of H[N] is separable if |phi> is the tensor product of states in the respective product spaces. An orthogonal unextendible product basis is a finite set B of separable orthonormal states |phi(k)> such that the non-empty space B9perp), the set of vectors orthogonal to B, contains no separable projection. Examples of orthogonal UPB sets were first constructed by Bennett et al [1] and other examples appear, for example, in [2] and [3]. If F denotes the set of convex combinations of the projections |phi(k)><phi(k)|, then F is a face in the set S of separable densities. In this note we show how to use F to construct families of positive partial transform states (PPT) which are not separable. We also show how to make an analogous construction when the condition of orthogonality is dropped. The analysis is motivated by the geometry of the faces of the separable states and leads to a natural construction of entanglement witnesses separating the inseparable PPT states from S.Comment: to appear in Lin. Alg. App

Complete Separability and Fourier representations of n-qubit states

Necessary conditions for separability are most easily expressed in the computational basis, while sufficient conditions are most conveniently expressed in the spin basis. We use the Hadamard matrix to define the relationship between these two bases and to emphasize its interpretation as a Fourier transform. We then prove a general sufficient condition for complete separability in terms of the spin coefficients and give necessary and sufficient conditions for the complete separability of a class of generalized Werner densities. As a further application of the theory, we give necessary and sufficient conditions for full separability for a particular set of $n$-qubit states whose densities all satisfy the Peres condition

Mutually Unbiased Bases, Generalized Spin Matrices and Separability

A collection of orthonormal bases for a complex dXd Hilbert space is called mutually unbiased (MUB) if for any two vectors v and w from different bases the square of the inner product equals 1/d: || ^{2}=1/d. The MUB problem is to prove or disprove the the existence of a maximal set of d+1 bases. It has been shown in [W. K. Wootters, B. D. Fields, Annals of Physics, 191, no. 2, 363-381, (1989)] that such a collection exists if d is a power of a prime number p. We revisit this problem and use dX d generalizations of the Pauli spin matrices to give a constructive proof of this result. Specifically we give explicit representations of commuting families of unitary matrices whose eigenvectors solve the MUB problem. Additionally we give formulas from which the orthogonal bases can be readily computed. We show how the techniques developed here provide a natural way to analyze the separability of the bases. The techniques used require properties of algebraic field extensions, and the relevant part of that theory is included in an Appendix

Generalized Circulant Densities and a Sufficient Condition for Separability

In a series of papers with Kossakowski, the first author has examined properties of densities for which the positive partial transpositrionm (PPT) property can be readily checked. These densities were also investigated from a different perspective by Baumgartner, Hiesmayr and Narnhofer. In this paper we show how the support of such densities can be expressed in terms of lines in a finite geometry and how that same structure lends itself to checking the necessary PPT condition and to a novel sufficient condition for separability.Comment: 15 page

Classicality in discrete Wigner functions

Gibbons et al. [Phys. Rev. A 70, 062101(2004)] have recently defined a class of discrete Wigner functions W to represent quantum states in a Hilbert space with finite dimension. We show that the only pure states having non-negative W for all such functions are stabilizer states, as conjectured by one of us [Phys. Rev. A 71, 042302 (2005)]. We also show that the unitaries preserving non-negativity of W for all definitions of W form a subgroup of the Clifford group. This means pure states with non-negative W and their associated unitary dynamics are classical in the sense of admitting an efficient classical simulation scheme using the stabilizer formalism.Comment: 10 pages, 1 figur

Separability and Fourier representations of density matrices

Using the finite Fourier transform, we introduce a generalization of Pauli-spin matrices for $d$-dimensional spaces, and the resulting set of unitary matrices $S(d)$ is a basis for $d\times d$ matrices. If $N=d_{1}\times d_{2}\times...\times d_{b}$ and H^{[ N]}=\bigotimes H^{% [ d_{k}]}, we give a sufficient condition for separability of a density matrix $\rho$ relative to the $H^{[ d_{k}]}$ in terms of the $L_{1}$ norm of the spin coefficients of $\rho >.$ Since the spin representation depends on the form of the tensor product, the theory applies to both full and partial separability on a given space $H^{[ N]}$% . It follows from this result that for a prescribed form of separability, there is always a neighborhood of the normalized identity in which every density matrix is separable. We also show that for every prime $p$ and $n>1$ the generalized Werner density matrix $W^{[ p^{n}]}(s)$ is fully separable if and only if $s\leq (1+p^{n-1}) ^{-1}$

Wigner Functions and Separability for Finite Systems

A discussion of discrete Wigner functions in phase space related to mutually unbiased bases is presented. This approach requires mathematical assumptions which limits it to systems with density matrices defined on complex Hilbert spaces of dimension p^n where p is a prime number. With this limitation it is possible to define a phase space and Wigner functions in close analogy to the continuous case. That is, we use a phase space that is a direct sum of n two-dimensional vector spaces each containing p^2 points. This is in contrast to the more usual choice of a two-dimensional phase space containing p^(2n) points. A useful aspect of this approach is that we can relate complete separability of density matrices and their Wigner functions in a natural way. We discuss this in detail for bipartite systems and present the generalization to arbitrary numbers of subsystems when p is odd. Special attention is required for two qubits (p=2) and our technique fails to establish the separability property for more than two qubits.Comment: Some misprints have been corrected and a proof of the separability of the A matrices has been adde

Monocytes induce STAT3 activation in human mesenchymal stem cells to promote osteoblast formation

A major therapeutic challenge is how to replace bone once it is lost. Bone loss is a characteristic of chronic inflammatory and degenerative diseases such as rheumatoid arthritis and osteoporosis. Cells and cytokines of the immune system are known to regulate bone turnover by controlling the differentiation and activity of osteoclasts, the bone resorbing cells. However, less is known about the regulation of osteoblasts (OB), the bone forming cells. This study aimed to investigate whether immune cells also regulate OB differentiation. Using in vitro cell cultures of human bone marrow-derived mesenchymal stem cells (MSC), it was shown that monocytes/macrophages potently induced MSC differentiation into OBs. This was evident by increased alkaline phosphatase (ALP) after 7 days and the formation of mineralised bone nodules at 21 days. This monocyte-induced osteogenic effect was mediated by cell contact with MSCs leading to the production of soluble factor(s) by the monocytes. As a consequence of these interactions we observed a rapid activation of STAT3 in the MSCs. Gene profiling of STAT3 constitutively active (STAT3C) infected MSCs using Illumina whole human genome arrays showed that Runx2 and ALP were up-regulated whilst DKK1 was down-regulated in response to STAT3 signalling. STAT3C also led to the up-regulation of the oncostatin M (OSM) and LIF receptors. In the co-cultures, OSM that was produced by monocytes activated STAT3 in MSCs, and neutralising antibodies to OSM reduced ALP by 50%. These data indicate that OSM, in conjunction with other mediators, can drive MSC differentiation into OB. This study establishes a role for monocyte/macrophages as critical regulators of osteogenic differentiation via OSM production and the induction of STAT3 signalling in MSCs. Inducing the local activation of STAT3 in bone cells may be a valuable tool to increase bone formation in osteoporosis and arthritis, and in localised bone remodelling during fracture repair

An introduction to quantum computing algorithms

In 1994 Peter Shor [65] published a factoring algorithm for a quantum computer that finds the prime factors of a composite integer N more efficiently than is possible with the known algorithms for a classical com­ puter. Since the difficulty of the factoring problem is crucial for the se­ curity of a public key encryption system, interest (and funding) in quan­ tum computing and quantum computation suddenly blossomed. Quan­ tum computing had arrived. The study of the role of quantum mechanics in the theory of computa­ tion seems to have begun in the early 1980s with the publications of Paul Benioff [6]' [7] who considered a quantum mechanical model of computers and the computation process. A related question was discussed shortly thereafter by Richard Feynman [35] who began from a different perspec­ tive by asking what kind of computer should be used to simulate physics. His analysis led him to the belief that with a suitable class of "quantum machines" one could imitate any quantum system