Structural physical approximation for the realization of the optimal singlet fraction with two measurements

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.

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“…where ( * ) denotes the complex conjugate, then the structural physical approximation of partial transpose of ρ 12 is given by [24]…”

Section: Preliminary Resultsmentioning

confidence: 99%

“…So the analysis of minimum eigenvalue of ρ out is needed. It has been found that SPA-PT method not only detect entanglement but also can be applied to estimate the optimal singlet fraction, entanglement negativity and concurrence of the two-qubit density matrix in an experiment [24,25]. H. T. Lim et.al.…”

Section: Introductionmentioning

confidence: 99%

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

Kumari

Adhikari

2019

Phys. Rev. A

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“…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].…”

Section: Structural Physical Approximationmentioning

confidence: 99%

Guaranteeing Completely Positive Quantum Evolution

Dilley,

Gonzales,

Byrd

2021

Preprint

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In open quantum systems, it is known that if the system and environment are in a product state, the evolution of the system is given by a linear completely positive (CP) Hermitian map. CP maps are a subset of general linear Hermitian maps, which also include non completely positive (NCP) maps. NCP maps can arise in evolutions such as non-Markovian evolution, where the CP divisibility of the map (writing the overall evolution as a composition of CP maps) usually fails. Positive but NCP maps are also useful as entanglement witnesses. In this paper, we focus on transforming NCP maps to CP maps through composition with the asymmetric depolarizing map. We use separate asymmetric depolarizing maps acting on the individual subsystems.Previous work have looked at structural physical approximation (SPA), which is a CP approximation of a NCP map using a mixture of the NCP map with a completely depolarizing map. We prove that the composition can always be made CP without completely depolarizing in any direction. It is often possible to depolarize less in some directions. We give the general proof by using the Choi matrix and an isomorphism from a maximally entangled two qudit state to qubits. We also give measures that describe the amount of disturbance the depolarization introduces to the original map. Given our measures, we show that asymmetric depolarization has many advantages over SPA in preserving the structure of the original NCP map. Finally, we give some examples. For some measures and examples, completely depolarizing (while not necessary) in some directions can give a better approximation than keeping the depolarizing parameters bounded by the required depolarization if symmetric depolarization is used.

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“…The minimum eigenvalue can be estimated by the procedure given in [16,17]. SPA-PT method not only can detect entangled states but also has many applications in quantum information processing tasks such as in estimating the optimal singlet fraction [18]. Criterion based on the method of SPA-PT have been given for the detection of mixed bipartite entangled state in arbitrary dimension [19].…”

Section: Introductionmentioning

confidence: 99%

Structural physical approximation of partial transposition makes possible to distinguish SLOCC inequivalent classes of three-qubit system

Kumari,

Adhikari

2022

Preprint

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Detection and classification of entanglement properties of a multi-qubit system is a topic of great interest. This topic has been studied extensively and thus we found different approaches for the detection and classification of multi-qubit entangled states. We have applied partial transposition operation on one of the qubit of the three-qubit system and then studied the entanglement properties of the three-qubit system, which is under investigation. Since the partial transposition operation is not a quantum operation so we have approximated partial transposition operation in such a way so that it represent a completely positive map. The approximated partial transposition operation is also known as structural physical approximation of partial transposition (SPA-PT). We have studied in detail the application of SPA-PT on a three qubit system and provided explicitly the matrix elements of the density matrix describing SPA-PT of a three qubit system. Moreover, we propose criterion to classify all possible stochastic local operations and classical communication(SLOCC) inequivalent classes of a pure as well as mixed three qubit state through SPA-PT map, which makes our criterion experimentally realizable. We have illustrated our criterion for detection and classification of three-qubit entangled states by considering few examples.

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Structural physical approximation for the realization of the optimal singlet fraction with two measurements

Cited by 14 publications

References 26 publications

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

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

Guaranteeing Completely Positive Quantum Evolution

Structural physical approximation of partial transposition makes possible to distinguish SLOCC inequivalent classes of three-qubit system

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