Generalized concurrence and partial transpose for pure continuous variable systems of arbitrary degrees of freedom using Lagrange's identity and wedge product

Concurrence, introduced by Hill and Wootters [Phys. Rev. Lett. 78, 5022 (1997)], provides an important measure of entanglement for a general pair of qubits that is strictly positive for entangled states and vanishing for all separable states. We present an extension of concurrence to general continuous variable pure states of multiple degrees of freedom by generalizing the Lagrange's identity and wedge product framework proposed by Bhaskara et al. [Quantum Inf. Process. 16, 118 (2017)] for pure discrete variable systems in arbitrary dimensions. A family of faithful entanglement measures, of which concurrence is a member, is constructed that admit necessary and sufficient conditions for separability across arbitrary bipartitions, which is shown as a particular invariance with connections to the partial transpose, uncovering an inherent geometry of entanglement. This framework may be useful for the further extensions to mixed states and entanglement in quantum field theories.Comment: 8 pages, 2 figure

The Minimum Rank Problem: a counterexample

We provide a counterexample to a recent conjecture that the minimum rank of every sign pattern matrix can be realized by a rational matrix. We use one of the equivalences of the conjecture and some results from projective geometry. As a consequence of the counterexample, we show that there is a graph for which the minimum rank over the reals is strictly smaller than the minimum rank over the rationals. We also make some comments on the minimum rank of sign pattern matrices over different subfields of $\mathbb R$.Comment: 4 pages, 1 figur

Geometry of good sets in n-fold Cartesian product

We propose here a multidimensional generalisation of the notion of link introduced in our previous papers and we discuss some consequences for simplicial measures and sums of function algebras.Comment: 17 pages, no figures, no table