Bounds on probability of transformations between multi-partite pure states

Abstract

For a tripartite pure state of three qubits, it is well known that there are two inequivalent classes of genuine tripartite entanglement, namely the GHZ-class and the W-class. Any two states within the same class can be transformed into each other with stochastic local operations and classical communication (SLOCC) with a non-zero probability. The optimal conversion probability, however, is only known for special cases. Here, we derive new lower and upper bounds for the optimal probability of transformation from a GHZ-state to other states of the GHZ-class. A key idea in the derivation of the upper bounds is to consider the action of the LOCC protocol on a different input state, namely $1/\sqrt{2} [\ket{000} - \ket{111}]$, and demand that the probability of an outcome remains bounded by 1. We also find an upper bound for more general cases by using the constraints of the so-called interference term and 3-tangle. Moreover, we generalize some of our results to the case where each party holds a higher-dimensional system. In particular, we found that the GHZ state generalized to three qutrits, i.e., $\ket{\mathrm{GHZ}_3} = 1/\sqrt{3} [ \ket{000} + \ket{111} + \ket{222} ] $, shared among three parties can be transformed to {\it any} tripartite 3-qubit pure state with probability 1 via LOCC. Some of our results can also be generalized to the case of a multipartite state shared by more than three parties.