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Entanglement Witness and Bell Operator by Hilbert-Schmidt Decomposition

Y. Ben-Aryeh, A. Mann

Published May 26, 2019• 0 citations
OpenAlex

Abstract

A new method is developed for obtaining entanglement witnesses (EW) and Bell operators for n-qubit state. Observing that both EW and Bell operators must have nonzero overlap with the given state we suggest constructing them using some of the terms that appear in the Hilbert-Schmidt (HS) decomposition of the state (not necessarily with the same coefficients). We illustrate the method for several 3 and 4 qubits states, and show improvement of the entanglement region compared with known results. Since the Bell operator and the EW may be restricted to few HS terms, separability/nonseparability of the quantum state can be checked by measuring only these terms.

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A new method is developed for obtaining entanglement witnesses (EW) and Bell operators for n-qubit state. Observing that both EW and Bell operators must have nonzero overlap with the given state we suggest constructing them using some of the terms that appear in the Hilbert-Schmidt (HS) decomposition of the state (not necessarily with the same coefficients). We illustrate the method for several 3 and 4 qubits states, and show improvement of the entanglement region compared with known results. Since the Bell operator and the EW may be restricted to few HS terms, separability/nonseparability of the quantum state can be checked by measuring only these terms.
The use of entanglement witness (EW) for non-full separability and the Bell operator for non-local hidden-variables (LHV) model are analyzed by relating them to the Hilbert-Schmidt (HS) decomposition of n-qubits states and these methods are applied explicitly to some 3 and 4 qubits states. EW for non-full separability (fs) is given by fs parameter minus operator G where the choice of G in the HS decomposition leads to the fs parameter and to the condition for non-separability by using criterions which are different from those used for genuine entanglement. We analyze especially entangled states with probability p mixed with white noise with probability 1-p and find the critical value p(crit.) above which the system is not fully separable. As the choice of EW might not be optimal we add to the analysis of EW explicit construction of fully separable density matrix and find the critical value p below which the system is fully separable. If the two values for p coincide we conclude that this parameter gives the optimal result. Such optimal results are obtained in the present work for some 3 and 4 qubits entangled states mixed with white noise. The use of partial-transpose (PT ) (say relative to qubit A) gives also p value above which the system is not fully separable. The use of EW gives better results (or at least equal) than those obtained by PT .
Using the Hilbert–Schmidt (HS) decomposition we suggest new possible choices of Bell operators and entanglement witnesses (EWs) for [Formula: see text] ([Formula: see text]) qubits systems for (full/bi) separability. The latter give upper bounds for (full/bi) separability. Also using the HS decomposition, we find explicitly (full/bi) separable forms for some qubits states which give lower bounds for (full/bi) separability. When the lower bounds and upper bounds coincide it means that the EW is optimal. In the case of full separability, the positive transpose method can sometimes give optimal results. As concrete examples, we give results for the GHZ(3), [Formula: see text] and cluster [Formula: see text] states.
Separability and entanglement for n-qubits systems are quantified by using Hilbert-Schmidt (HS) decompositions in which the density matrices are decomposed into various terms representing certain one qubit, two-qubits,and larger qubits measurements. The present method is more general than previous methods for bipartite systems as it can be used for quantification of entanglement for large n-qubits systems.We demonstrate the use of such method by analyzing three qubits GHZ states, and three qubits general Bell states produced by a certain multiplications of Braid operators, operating on the computational basis of states. Quantum correlations are obtained by measuring all qubits of these systems, while a measurement of a part of the system gives only classical correlations. Quantification of entanglement for these systems is given by the use of HS decompositions.
We introduce an experimental procedure for the detection of quantum entanglement of an unknown quantum state with a small number of measurements. The method requires neither a priori knowledge of the state nor a shared reference frame between the observers and can thus be regarded as a perfectly state-independent entanglement witness. The scheme starts with local measurements, possibly supplemented with suitable filtering, which essentially establishes the Schmidt decomposition for pure states. Alternatively we develop a decision tree that reveals entanglement within few steps. These methods are illustrated and verified experimentally for various entangled states of two and three qubits.
We study the connection between the Hilbert-Schmidt measure of entanglement (that is the minimal distance of an entangled state to the set of separable states) and entanglement witness in terms of a generalized Bell inequality which distinguishes between entangled and separable states. A method for checking the nearest separable state to a given entangled one is presented. We illustrate the general results by considering isotropic states, in particular two-qubit and two-qutrit states---and their generalizations to arbitrary dimensions---where we calculate the optimal entanglement witnesses explicitly.
Characterizing entanglement is central for quantum information science. Special observables which indicate entanglement, so-called entanglement witnesses, are a widely used tool for this task. The construction of these witnesses typically relies on the observation that quantum states with a high fidelity to some entangled target state are entangled, too. We introduce a general method to construct entanglement witnesses based on the Schmidt decomposition of observables. The method works for two-particle and multiparticle systems and is strictly stronger than fidelity-based constructions. The resulting witnesses can also be used to quantify entanglement and to characterize its dimensionality. Finally, we present experimentally relevant examples, where our approach improves entanglement detection significantly.