Absolute Non-Violation of a Three-Setting Steering Inequality by Two-Qubit States

TL;DR

This paper investigates two-qubit states that do not violate a three-setting steering inequality under any global unitary transformation, characterizing their spectrum and properties, and relating steerability to quantum teleportation.

Contribution

It provides a necessary and sufficient spectral condition for states that preserve non-violation under all global unitaries, and links steerability with teleportation and purity measures.

Findings

States with certain spectra do not violate the steering inequality under any global unitary.

A relation between steerability and quantum teleportation is established.

An analytic estimate of non-violating states' size in terms of purity is derived.

Abstract

Steerability is a characteristic nonlocal trait of quantum states lying in between entanglement and Bell nonlocality. A given quantum state is considered to be steerable if it violates a suitably chosen steering inequality. A quantum state which otherwise satisfies a certain inequality can violate the inequality under a global change of basis i.e, if the state is transformed by a nonlocal unitary operation. Intriguingly there are states which preserve their non-violation(pertaining to the said inequality) under any global unitary operation. The present work explores the effect of global unitary operations on the steering ability of a quantum state which live in two qubits. We characterize such states in terms of a necessary and sufficient condition on their spectrum. Such states are also characterized in terms of some analytic characteristics of the set to which they belong. Looking back at steerability the present work also provides a relation between steerability and quantum teleportation together with the derivation of a result related to the optimal violation of steering inequality . An analytic estimation of the size of such non-violating states in terms of purity is also obtained. Interestingly the estimation in terms of purity also gives a necessary and sufficient condition in terms of bloch parameters of the state. Illustrations from some signature class of quantum states further underscore our observations.