Hierarchy, factorization law of two measurement-induced nonlocalities and their performances in quantum phase transition

Abstract

There are two measurement-induced nonlocalities, which are, respectively, defined via the trace norm (MIN-1) and Hilbert---Schmidt norm (MIN-2). We investigate the hierarchy relation and factorization law of them. Their performances in quantum phase transition have also been explored. For X-shape states, a rigorous hierarchy relation is established between two MINs. When two qubits, which are initially prepared in an X-shape state, interact independently with the corresponding multimode vacuum reservoirs, the evolutions of two MINs satisfy the factorization law. With quantum renormalization group method, it is found that two MINs can signify the criticality of the spin system while the position where the derivative of MIN-1 takes the minimum value is always larger than that where the derivative of MIN-2 takes the minimum value. Therefore, MIN-1 is more suitable to identify the critical point of quantum phase transition.