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Quadratic Entanglement Criteria for Qutrits

Krzysztof Rosołek, Marcin Wieśniak, Lukas Knips

Published Mar 1, 2020• 0 citations
OpenAlex

Abstract

The problem of detecting non-classical correlations of states of many qudits is incomparably more involved than in the case of qubits.One reason is that for qubits we have a convenient description of the system by the means of the well-studied correlation tensor allowing to encode the complete information about the state in mean values of dichotomic measurements.The other reason is the more complicated structure of the state space, where, for example, different Schmidt ranks or bound entanglement comes into play.We demonstrate that for three-dimensional quantum subsystems we are able to formulate nonlinear entanglement criteria of the state with existing formalisms.We also point out where the idea for constructing these criteria fails for higher-dimensional systems, which poses well-defined open questions.

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The problem of detecting non-classical correlations of states of many qudits is incomparably more involved than in the case of qubits.One reason is that for qubits we have a convenient description of the system by the means of the well-studied correlation tensor allowing to encode the complete information about the state in mean values of dichotomic measurements.The other reason is the more complicated structure of the state space, where, for example, different Schmidt ranks or bound entanglement comes into play.We demonstrate that for three-dimensional quantum subsystems we are able to formulate nonlinear entanglement criteria of the state with existing formalisms.We also point out where the idea for constructing these criteria fails for higher-dimensional systems, which poses well-defined open questions.

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The problem of detecting non-classical correlations of states of many qudits is incomparably more involved than in the case of qubits.One reason is that for qubits we have a convenient description of the system by the means of the well-studied correlation tensor allowing to encode the complete information about the state in mean values of dichotomic measurements.The other reason is the more complicated structure of the state space, where, for example, different Schmidt ranks or bound entanglement comes into play.We demonstrate that for three-dimensional quantum subsystems we are able to formulate nonlinear entanglement criteria of the state with existing formalisms.We also point out where the idea for constructing these criteria fails for higher-dimensional systems, which poses well-defined open questions.
The problem of detecting non-classical correlations of states of many qudits is incomparably more involved than in the case of qubits.One reason is that for qubits we have a convenient description of the system by the means of the well-studied correlation tensor allowing to encode the complete information about the state in mean values of dichotomic measurements.The other reason is the more complicated structure of the state space, where, for example, different Schmidt ranks or bound entanglement comes into play.We demonstrate that for three-dimensional quantum subsystems we are able to formulate nonlinear entanglement criteria of the state with existing formalisms.We also point out where the idea for constructing these criteria fails for higher-dimensional systems, which poses well-defined open questions.
If only limited control over a multiparticle quantum system is available, a viable method to characterize correlations is to perform random measurements and consider the moments of the resulting probability distribution. We present systematic methods to analyze the different forms of entanglement with these moments in an optimized manner. First, we find the optimal criteria for different forms of multiparticle entanglement in three-qubit systems using the second moments of randomized measurements. Second, we present the optimal inequalities if entanglement in a bipartition of a multiqubit system shall be analyzed in terms of these moments. Finally, for higher-dimensional two-particle systems and higher moments, we provide criteria that are able to characterize various examples of bound entangled states, showing that detection of such states is possible in this framework.
If only limited control over a multiparticle quantum system is available, a viable method to characterize correlations is to perform random measurements and consider the moments of the resulting probability distribution. We present systematic methods to analyze the different forms of entanglement with these moments in an optimized manner. First, we find the optimal criteria for different forms of multiparticle entanglement in three-qubit systems using the second moments of randomized measurements. Second, we present the optimal inequalities if entanglement in a bipartition of a multi-qubit system shall be analyzed in terms of these moments. Finally, for higher-dimensional two-particle systems and higher moments, we provide criteria that are able to characterize various examples of bound entangled states, showing that detection of such states is possible in this framework.
We present sufficient criteria for the entanglement of three-qubit states. For some special families of states, the criteria are also necessary for the entanglement. They are formulated as simple sets of inequalities for the mean values of certain observables defined as tensor products of Pauli matrices. The criteria are good indicators of the entanglement in the vicinity of three-qubit GHZ and W states and enjoy the capability of detecting the entangled states with positive partial transpositions. Furthermore, they improve the best known result for the case of W state mixed with the white noise. The efficiency of the criteria is illustrated through several examples.
Abstract Chapter 17 deals with the description of high-dimensional quantum systems and entanglement between them. We start with the generalized Bloch decomposition and present several useful options for the choice of operator bases, including the generalized Gell-Mann basis, the polarization-operator basis, and the Weyl-operator basis. We apply these bases for the example of isotropic two-qudit states and for the detection of bound entanglement for systems of two qutrits. We then turn to the problem of detecting entanglement in high-dimensional systems. For this purpose we review mutually unbiased bases and employ them for entanglement detection. We then discuss entanglement detection using entropic uncertainty relations, before we show how to obtain lower bounds on pure-state fidelities, on the entanglement of formation, and on the Schmidt number from measurement in pairs of mutually unbiased bases
We present a development of a geometric approach to entanglement indicators. The method is applied to detect genuine multiqubit entanglement. The criteria are given in the form of nonlinear conditions imposed on correlation tensors. Thus they involve directly observable quantities, and in some cases require only few specific measurements to find multiqubit entanglement. The nonlinearity of each of the criteria allows detection of entanglement in wide classes of states. In contrast to entanglement witnesses, which in the space of Hermitian operators define a hyperplane, these conditions define a geometric figure encapsulating the nonfully entangled states within it.