Concurrence in arbitrary dimensions

Entanglement plays central role in quantum information theory [1]. Pure state entanglement of bipartite systems is well understood in the sense that the relevant parameters for its optimal manipulation by local operations and classical communication (LOCC) have been identified and analyzed [2], [3]. Many efforts have also been devoted to the study of mixed-state entanglement. There, several possible entanglement measures have been proposed. Among these, entanglement of formation (EF ) [4], [5] attracts much of attention, as it is closely connected with (or, perhaps, equal to) the rate of production of mixed bipartite states out of pure ones by LOCC operations. It is, however, extremely difficult to evaluate EF , but for the analytical formula for a single copy of an arbitrary state of two qubits obtained by Wootters [6]. Despite efforts, not much progress has been recorded regarding generalization of Wootters’ result to the states in more than 2× 2 dimensions [7]. Wootters’ success in quantifying EF for two qubits can be attributed to associating EF with concurrence which is easier to calculate than EF . Concurrence, as introduced by Hill and Wootters [8], was defined via operation of spin flip. More recently, Rungta et al. [9] made an attempt to generalize the notion of concurrence to pure bipartite states in arbitrary dimensions by introducing operation of universal state inversion [10]. Their universal inverter generalizes spin flip to a transformation which brings pure state |ψ〉 into the maximally mixed state in the subspace orthogonal to |ψ〉. In the same way that the spin flip generates concurrence for a pair of qubits, the universal inverter generates a number which generalizes concurrence for joint pure states of pairs of quantum systems of arbitrary dimensions. Generalized in this way, concurrence measures entanglement of pure bipartite states in terms of the purity of their marginal density operators. As one knows [3], a complete characterization of quantum correlations in bipartite systems of many dimensions may require a quantity which, even for pure states, does not reduce to a single number [11]. Take, e.g., two pure states represented by vectors ψ = (|11〉+ |22〉) /2 and φ = aψ + b|33〉, with a = √x and b = √1− x, where x ≈ 0.2271 is a root of x [2 (1− x)]1−x = 1. The two states have the same entanglement EF of 1 ebit, nevertheless they have different Schmidt numbers and, consequently, it is impossible to locally convert one into the other. In this contribution, we argue than that a suitable generalization of spin flip to more dimensions should produce a multi-dimensional analogue of concurrence rather than a single number. Such a concurrence would then describe not only the amount of entanglement but also its structure, e.g., the size (the number of dimensions) of the entangled spaces on each side. The concurrence for pure states is thus a matrix acting on the antisymmetric subspace of the total Hilbert space of two systems. Having that, one can follow Wootters and generalize the concept to mixed states by introducing a matrix of preconcurrence. The elements of this matrix are matrices in their own right and at the end, the matrix is often difficult to analyze. At least partially, the difficulties can be associated with the matrix dependence on the choice of the local bases. Therefore, we also generalize the concept of concurrence in a somewhat different direction. We abandon the requirement for preconcurrence to be a second order object in the state’s ensemble. For this price we can define a fourth order object, biconcurrence matrix. It is independent of the local unitaries and allows us to reformulate separability problem in terms of the main diagonal of the matrix. Biconcurrence is a very simple function of ensemble of density matrix and has many symmetries. Therefore, the obtained necessary and sufficient separability condition seems to be the most promising one from algebraic point of view. The generalization of pre-concurrence which satisfies our criterion is presented in section II. Then, in section III we give an example to show how our multi-dimensional pre-concurrence can be used for analysis of separability