arXiv subjects
Publications and source records attributed to Leonid Gurvits.
The main topic of this paper is various "hyperbolic" generalizations of the Edmonds-Rado theorem on the rank of intersection of two matroids. We prove several results in this direction and pose a few questions. We also give generalizations of the Obreschkoff theorem and recent results of J. Borcea and B. Shapiro.
This paper continues research initiated in quant-ph/0201022 . The main subject here is the so-called Edmonds' problem of deciding if a given linear subspace of square matrices contains a nonsingular matrix . We present a deterministic polynomial time algorithm to solve this problem for linear subspaces satisfying a special matroids motivated property, called in the paper the Edmonds-Rado property . This property is shown to be very closely related to the separability of bipartite mixed states . One of the main tools used in the paper is the Quantum Permanent introduced in quant-ph/0201022 .
We show that for an m-partite quantum system, there is a ball of radius 2^{-(m/2-1)} in Frobenius norm, centered at the identity matrix, of separable (unentangled) positive semidefinite matrices. This can be used to derive an epsilon below which mixtures of epsilon of any density matrix with 1 - epsilon of the maximally mixed state will be separable. The epsilon thus obtained is exponentially better (in the number of systems) than existing results. This gives a number of qubits below which NMR with standard pseudopure-state preparation techniques can access only unentangled states; with parameters realistic for current experiments, this is 23 qubits (compared to 13 qubits via earlier results). A ball of radius 1 is obtained for multipartite states separable over the reals.
For finite-dimensional bipartite quantum systems, we find the exact size of the largest balls, in spectral $l_p$ norms for $1 \le p \le \infty$, of separable (unentangled) matrices around the identity matrix. This implies a simple and intutively meaningful geometrical sufficient condition for separability of bipartite density matrices: that their purity $\tr ρ^2$ not be too large. Theoretical and experimental applications of these results include algorithmic problems such as computing whether or not a state is entangled, and practical ones such as obtaining information about the existence or nature of entanglement in states reached by NMR quantum computation implementations or other experimental situations.
Classical matching theory can be defined in terms of matrices with nonnegative entries. The notion of Positive operator, central in Quantum Theory, is a natural generalization of matrices with nonnegative entries. Based on this point of view, we introduce a definition of perfect Quantum (operator) matching . We show that the new notion inherits many "classical" properties, but not all of them . This new notion goes somewhere beyound matroids . For separable bipartite quantum states this new notion coinsides with the full rank property of the intersection of two corresponding geometric matroids . In the classical situation, permanents are naturally associated with perfects matchings. We introduce an analog of permanents for positive operators, called Quantum Permanent and show how this generalization of the permanent is related to the Quantum Entanglement. Besides many other things, Quantum Permanents provide new rational inequalities necessary for the separability of bipartite quantum states . Using Quantum Permanents, we give deterministic poly-time algorithm to solve Hidden Matroids Intersection Problem and indicate some "classical" complexity difficulties associated with the Quantum Entanglement. Finally, we prove that the weak membership problem for the convex set of separable bipartite density matrices is NP-HARD.