Test spaces and characterizations of quadratic spaces

Alexander Wilce

Luigi Accardi, Ameur Dhahri

We give a necessary and sufficient condition for the existence of a quadratic exponential vector with test function in L2(Rd)∩L∞(Rd). We prove the linear independence and totality, in the quadratic Fock space, of these vectors. Using a technique different from the one used by Accardi et al. [Quantum Probability and Infinite Dimensional Analysis, Vol. 25, p. 262, (2009)], we also extend, to a more general class of test functions, the explicit form of the scalar product between two such vectors.

Anatolij Dvurečenskij, Sylvia Pulmannová

Hugo J. Woerdeman

In this paper, we propose a sequence of tests which give a definitive test for checking $2\ifmmode\times\else\texttimes\fi{}M$ separability. The test is definitive in the sense that each test corresponds to checking membership in a cone, and that the closure of the union of all these cones consists exactly of all $2\ifmmode\times\else\texttimes\fi{}M$ separable states. Membership in each single cone may be checked via semidefinite programming, and is thus a tractable problem. This sequential test comes about by considering the dual problem, the characterization of all positive maps acting ${C}^{2\ifmmode\times\else\texttimes\fi{}2}\ensuremath{\rightarrow}{C}^{M\ifmmode\times\else\texttimes\fi{}M}.$ The latter in turn is solved by characterizing all positive quadratic matrix polynomials in a complex variable.

John Harding, Remi Salinas Schmeis

Koji Nagata, Masato Koashi, Nobuyuki Imoto

We derive tight quadratic inequalities for all kinds of hybrid separable-inseparable n-particle density operators on an arbitrary dimensional space. This methodology enables us to derive a tight quadratic inequality as tests for full n-partite entanglement in various Bell-type correlation experiments on the systems that may not be identified as a collection of qubits, e.g., those involving photons measured by incomplete detectors. It is also proved that when the two measured observables are assumed to precisely anticommute, a stronger quadratic inequality can be used as a witness of full n-partite entanglement.