Theoretical proposal for the experimental realization of realignment operation

TL;DR

This paper proposes a theoretical framework for experimentally realizing the realignment operation, enabling entanglement detection in bipartite and tripartite quantum states using measurable moments and SWAP operators.

Contribution

It introduces a method to implement the realignment operation via SWAP operators and develops entanglement criteria for bipartite and three-qubit states based on moments.

Findings

First realigned moment expressed as SWAP expectation value

Higher order moments estimated from the first moment

Entanglement detection criteria for bipartite and tripartite states

Abstract

Realignment operation has a significant role in detecting bound as well as free entanglement. Just like partial transposition, it is also based on permutations of the matrix elements. However, the physical implementation of realignment operation is not known yet. In this letter, we address the problem of experimental realization of realignment operation and to achieve this aim, we propose a theoretical proposal for the same. We first show that after applying the realignment operation on a bipartite state, the resulting matrix can be expressed in terms of the partial transposition operation along with column interchange operations. We observed that these column interchange operations forms a permutation matrix which can be implemented via SWAP operator acting on the density matrix. This mathematical framework is used to exactly determine the first moment of the realignment matrix experimentally. This has been done by showing that the first moment of the realignment matrix can be expressed as the expectation value of a SWAP operator which indicates the possibility of its measurement. Further, we have provided an estimation of the higher order realigned moments in terms of the first realigned moment and thus pave a way to estimate the higher order moments experimentally. Next, we develop moments based entanglement detection criteria that detect positive partial transpose entangled states (PPTES) as well as negative partial transpose entangled states (NPTES). Moreover, we define a new matrix realignment operation for three-qubit states and have devised an entanglement criteria that is able to detect three-qubit fully entangled states. We have developed various methods and techniques in the detection of bipartite and tripartite entangled states that may be realized in the current technology.