Bounds on localizable information via semidefinite programming

Abstract

We investigate the so-called localizable information of bipartite states and a parallel notion of information deficit. Localizable information is defined as the amount of information that can be concentrated by means of classical communication and local operations where only maximally mixed states can be added for free. The information deficit is defined as difference between total information contents of the state and localizable information. We consider a larger class of operations, the so-called PPT operations, which in addition preserve maximally mixed state (PPT-PMM operations). We formulate the related optimization problem as a semidefinite program with suitable constraints. We then provide bound for fidelity of transition of a given state into product pure state on Hilbert space of dimension d. This allows to obtain a general upper bound for localizable information (and also for information deficit). We calculated the bounds exactly for Werner states and isotropic states in any dimension. Surprisingly it turns out that related bounds for information deficit are equal to relative entropy of entanglement (in the case of Werner states, regularized one). We compare the upper bounds with lower bounds based on simple protocol of localization of information.