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The separability and entanglement of quantum mixed states in $\Cb^2 \otimes \Cb^3 \otimes \Cb^N$ composite quantum systems are investigated. It is shown that all quantum states $\rho$ with positive partial transposes and rank $r(\rho)\leq…
This paper considers theoretical analysis of recovering a low rank matrix given a few expansion coefficients with respect to any basis. The current approach generalizes the existing analysis for the low-rank matrix completion problem with…
A systematic method for generating bound entangled states in any bipartite system, with ranks ranging from five to full rank, is presented. These states are constructed by mixing separable states with UPB (Unextendible Product Basis) -…
In a recent paper (quant-ph/0102133) Chen, Liang, Li and Huang suggest a necessary and sufficient separability criterion, which is supposedly practical in judging the separability of any mixed state. In this note we briefly recapitulate…
In nonadaptive group testing, the main research objective is to design an efficient algorithm to identify a set of up to $t$ positive elements among $n$ samples with as few tests as possible. Disjunct matrices and separable matrices are two…
The description of the complex separability structure of quantum states in terms of partially ordered sets has been recently put forward. In this work, we address the question of how to efficiently determine these structures for unknown…
For a given Hamiltonian $H$ on a multipartite quantum system, one is interested in finding the energy $E_0$ of its ground state. In the separability approximation, arising as a natural consequence of measurement in a separable basis, one…
We introduce a sequence of numerical tests that can determine the entanglement or separability of a state even when there is not enough information to completely determine its density matrix. Given partial information about the state in the…
We exhibit a two-parameter family of bipartite mixed states $\rho_{bc}$, in a $d\otimes d$ Hilbert space, which are negative under partial transposition (NPT), but for which we conjecture that no maximally entangled pure states in $2\otimes…
This paper provides the best bounds to date on the number of randomly sampled entries required to reconstruct an unknown low rank matrix. These results improve on prior work by Candes and Recht, Candes and Tao, and Keshavan, Montanari, and…
In this contribution we solve the following problem. Let H_{nm} be a Hilbert space of dimension nm, and let A be a positive semidefinite self-adjoint linear operator on H_{nm}. Under which conditions on the spectrum has A a positive partial…
A confluence of numerical and theoretical results leads us to conjecture that the Hilbert-Schmidt separability probabilities of the 15- and 9-dimensional convex sets of complex and real two-qubit states (representable by 4 x 4 density…
We treat 3-qubits states with maximally disordered subsystems, by using Hilbert-Schmidt decompositions.By using unfolding methods, the tensors are converted into matrices and by applying singular values decompositions to these matrices the…
We discuss the subject of Unextendible Product Bases with the orthogonality condition dropped and we prove that the lowest rank non-separable positive-partial-transpose states, i.e. states of rank 4 in 3 x 3 systems are always locally…
We use an algebraic approach to construct minimal decompositions of symmetric tensors with low rank. This is done by using Apolarity Theory and by studying minimal sets of reduced points apolar to a given symmetric tensor, namely, whose…
The Separability Problem is approached from the perspective of Ellipsoidal Classification. A Density Operator of dimension N can be represented as a vector in a real vector space of dimension $N^{2}- 1$, whose components are the projections…
The detection of multipartite entanglement in arbitrary dimensional systems is investigated. We derive useful $k$-separability criteria of mixed $n$-partite ($n\geq 3$) quantum states to detect $k$-nonseparable $n$-partite quantum states.…
We give necessary and sufficient conditions for a pair of (generalized) functions $\rho_1(\mathbf{r}_1)$ and $\rho_2(\mathbf{r}_1,\mathbf{r}_2)$, $\mathbf{r}_i\in X$, to be the density and pair correlations of some point process in a…
Low-rank matrix completion (LRMC) problems arise in a wide variety of applications. Previous theory mainly provides conditions for completion under missing-at-random samplings. This paper studies deterministic conditions for completion. An…
It is known that realignment crierion is necessary but not a sufficient criterion for lower as well as higher dimensional system. In this work, we first consider a two-qubit system and derived the necessary and sufficient condition based on…