Characterization of nonlocal resources under global unitary action

Abstract

The task of device independent secure key distribution requires preparation and subsequently distribution of nonlocal resources. For secured practical implementation, one needs to take two initially uncorrelated quantum systems and perform a unitary on the composite system to generate the nonlocal resource, which is supposed to violate a Bell inequality (say, Bell-CHSH). States which do not violate Bell-CHSH inequality, but violate it when transformed by a global unitary, can be deemed useful for the preparation of nonlocal resource. One may then start from a state which is Bell-CHSH local (take for example, a pure product state) and apply an appropriate global unitary on it which results in a Bell-CHSH non-local state. However, an intriguing fact is the existence of useless states from which no Bell-CHSH non-local resource can be generated with a global unitary. This is due to the purity preserving nature of unitary operators which bound the amount of correlation in the set of final states depending on the purity of the initial (possibly uncorrelated) states. The present work confirms the existence of such a set, pertaining to two qubit systems. The set exhibits counter intuitive features by containing within it some entangled states which remain Bell-CHSH local on the action of any unitary. From practical perspective, this work draws a line between useful and useless states for the task of preparing nonlocal resource using global unitary transformations. Furthermore through an analytic characterization of the set we lay down a generic prescription through which one can operationally identify the useful states. It has also been shown that our prescription remains valid for any linear Bell inequality.