Detection of a mixed bipartite entangled state in arbitrary dimension via a structural physical approximation of partial transposition

Abstract

It is very crucial to know that whether the quantum state generated in the experiment is entangled or not. In the literature, this topic was studied extensively and researchers proposed different approaches for the detection of mixed bipartite entangled state in arbitrary dimension. Proceeding in this line of research, we also propose three different criteria for the detection of mixed bipartite negative partial transpose (NPT) entangled state in arbitrary dimension. Our criteria is based on the method of structural physical approximation (SPA) of partial transposition (PT). We have shown that the proposed criteria for the detection of NPT entangled state can be realized experimentally. Two of the proposed criteria is given in terms of the concurrence of the given state in arbitrary dimension so it is essential to find out the concurrence. Thus, we provide new lower and upper bound of concurrence of the quantum state under investigation in terms of average fidelity of two quantum states and hence these bounds can be realized experimentally. Moreover, we have shown how to perform SPA map on qutrit-qubit system and then explicitly calculated the matrix elements of the density matrix describing the SPA-PT of the qutrit-qubit system. We then illustrate our criteria for the detection of entanglement by considering a class of qubit-qubit system and a class of qutrit-qubit system.

Structured negativity: A physically realizable measure of entanglement based on structural physical approximation

“…Except for two-qubit system, we don't have any closed formula for bipartite d 1 ⊗ d 2 dimensional mixed state. For higher dimensional mixed states, we can estimate the amount of entanglement through the lower bound of the concurrence [21,33,37]. Negativity: Negativity is another measure of entanglement based on the negative eigenvalues of the partially transposed matrix.…”

“…As we have mentioned in Sec.II, that there does not exist any closed formula for concurrence of arbitrary dimensional bipartite mixed state but neverthless, there exist lower bound of the concurrence for arbitrary d ⊗ d dimensional system [21,33,37]. One such lower bound of concurrence obtained in [33] which may be given by…”

“…This implies that the SPA map transform a positive but not completely positve maps to a completely positive map. A lot of progress has been made in both theoretical and experimental aspects of SPA [19][20][21][22][23][24]. Initially a conjecture that SPA leads to separable states has been posed [25].…”

Guaranteeing Completely Positive Quantum Evolution

“…The problem of converting a NCP map to a CP map has been studied under structural physical approximation (SPA), which was first introduced by Horodecki and Ekert, and expanded upon by several authors [20,[25][26][27][28][29][30][31][32][33][34]. SPAs are often used to approximate the partial transpose map and detect entanglement [35][36][37][38][39][40][41][42][43][44][45][46][47][48][49].…”

Physical realization of realignment criteria using structural physical approximation

“…In order to make it experimentally implementable, partial transposition map have been approximated to a completely positive map using the method of SPA [12]. A lot of work have been done on the structural physical approximation of partial transposition (SPA-PT) [12,19,[22][23][24]. The SPA-PT have been used to detect and quantify entanglement [12,25] but till now it can only be used to detect and quantify NPTES only.…”