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arXiv:1110.4108 · Physical Review A

Correlation-tensor criteria for genuine multiqubit entanglement

Wiesław Laskowski, Marcin Markiewicz, Tomasz Paterek, Marek Żukowski

Published Dec 6, 2011• 44 citations
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Abstract

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.

DOI
10.1103/physreva.84.062305

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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.
We introduce a general framework for detecting genuine multipartite entanglement and non-full-separability in multipartite quantum systems of arbitrary dimensions based on correlation tensors. Regarding genuine multipartite entanglement, our conditions are comparable to those of previous approaches in the case of qubits, while they show particular strength in the relatively unexplored case of higher-dimensional systems. In the case of non-full-separability, our conditions prove to be advantageous in situations where more than two-body correlations are relevant, where most previous conditions turned out to be weak. Moreover, they allow for the detection of fully bound entangled states. Finally, we also discuss experimentally friendly ways of implementing our conditions, which are based on directly measurable quantities.
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.
Existing methods for deciding whether a bipartite quantum state is separable or entangled typically fall into one of two categories: they are either complete but require access to an explicit density matrix followed by numerical optimization, or they can be evaluated directly by measuring the quantum system but are incomplete, in the sense that they cannot detect all forms of entanglement. In this work, we overcome both limitations in a unified framework. First, we bypass numerical optimization by deriving separability criteria in the form of universal bounds on tensor powers of separable states. We prove that these bounds are complete: every entangled state violates them for sufficiently large tensor powers. Second, we explicitly construct a corresponding complete family of nonlinear entanglement witnesses, which can detect all forms of entanglement without requiring an explicit density matrix. The witnesses we construct are moreover basis-independent, in the sense that they are invariant under conjugation by local unitaries. Altogether, our results expand the toolbox for entanglement detection in arbitrary local dimensions in a manifestly invariant way.
Complex forms of quantum entanglement can arise in two qualitatively different ways: either between many qubits or between two particles with higher-than-qubit dimension. While both the many-qubit frontier and the high-dimension frontier are well established, state-of-the-art quantum technology is becoming increasingly able to create and manipulate entangled states that simultaneously feature many particles and high dimension. Here, we investigate generic states that can be considered both genuinely high-dimensional and genuine multiparticle entangled. We consider a natural quantity that characterizes this key property. To detect it, we develop three different classes of criteria. These enable us both to probe the ultimate noise tolerance of this form of entanglement and to make detection schemes using sparse or even minimal measurement resources. The approach provides a simple way of benchmarking entanglement dimensionality in the multiparticle regime and general, platform-independent, detection methods that readily apply to experimental use.