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arXiv:quant-ph/9912089 · Journal of Modern Optics

Separability of entangled q-bit pairs

Berthold‐Georg Englert, NASSER METWALLY

Published Oct 1, 2000• 42 citations
arXivHTMLPDF

Abstract

The state of an entangled q-bit pair is specified by 15 numerical parameters that are naturally regarded as the components of two 3-vectors and a 3 × 3 dyadic. There are easy-to-use criteria to check whether a given pair of 3-vectors plus a dyadic specify a 2-q-bit state; and if they do, whether the state is entangled; and if it is, whether it is a separable state. Some progress has been made in the search for analytical expressions for the degree of separability. We report, in particular, the answer in the case of vanishing 3-vectors.

DOI
10.1080/09500340008235144

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The state of an entangled q-bit pair is specified by 15 numerical parameters that are naturally regarded as the components of two 3-vectors and a 3 × 3 dyadic. There are easy-to-use criteria to check whether a given pair of 3-vectors plus a dyadic specify a 2-q-bit state; and if they do, whether the state is entangled; and if it is, whether it is a separable state. Some progress has been made in the search for analytical expressions for the degree of separability. We report, in particular, the answer in the case of vanishing 3-vectors.
The state of an entangled q-bit pair is specified by 15 numerical parameters that are naturally regarded as the components of two 3-vectors and a 3 × 3-dyadic.There are easy-to-use criteria to check whether a given pair of 3-vectors plus a dyadic specify a 2-q-bit state; and if they do, whether the state is entangled; and if it is, whether it is a separable state.Some progress has been made in the search for analytical expressions for the degree of separability.We report, in particular, the answer in the case of vanishing 3-vectors.
Abstract We review some of the important properties of the states of qubit pairs. They are specified by 15 numerical parameters that are naturally regarded as the components of two 3-vectors and a 3 × 3dyadic. There are six classes of families of locally equivalent states in a straightforward scheme for classifying all 2-qubit states; four of the classes consist of two subclasses each. Easy-to-use criteria enable one to check whether a given pair of 3-vectors plus a 3× 3-dyadic specify a 2-qubit state; and if they do, whether the state is entangled; and if it is, whether it is a separable state. We remark on the Hill-Wootters concurrence, discuss the properties of the fundamental Lewenstein-Sanpera decompositions of 2-qubit states, and report a number of examples for which the optimal decomposition is known explicitly.
The separability problem is one of the basic and emergent problems in the present and future quantum information processing. The latter focuses on information and computing based on quantum mechanics and uses quantum bits as its basic information units. In this paper, we present an overview of the progress in the separability problem in bipartite systems, more specifically in two quantum bits systems from the criterion based on the inequalities of Bell in 196419641964 to the recent criteria of separability in 2018.
The purpose of this paper is to obtain a sufficient and necessary condition as a criteria to test whether an arbitrary multipartite state is entangled or not. Based on the tensor expression of a multipartite pure state, the paper shows that a state is separable iff $| \mathbf{C}(ρ)| $ =0 for pure states and iff C(ρ)C(ρ)C(ρ) vanishes for mixed states.