Characterization of Equivariant Maps and Application to Entanglement Detection
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Annales Henri Poincar´ e
Characterization of Equivariant Maps and Application to Entanglement Detection Ivan Bardet, Benoˆıt Collins and Gunjan Sapra Abstract. We study equivariant linear maps between finite-dimensional matrix algebras, as introduced in Collins et al. (Linear Algebra Appl 555:398–411, 2018). These maps satisfy an algebraic property which makes it easy to study their positivity or k-positivity. They are therefore particularly suitable for applications to entanglement detection in quantum information theory. We characterize their Choi matrices. In particular, we focus on a subfamily that we call (a, b)-unitarily equivariant. They can be seen as both a generalization of maps invariant under unitary conjugation as studied by Bhat (Banach J Math Anal 5(2):1–5, 2011) and as a generalization of the equivariant maps studied in Collins et al. (2018). Using representation theory, we fully compute them and study their graphical representation and show that they are basically enough to study all equivariant maps. We finally apply them to the problem of entanglement detection and prove that they form a sufficient (infinite) family of positive maps to detect all k-entangled density matrices.
1. Introduction Due to their crucial role in numerous tasks in quantum processing and quantum computation, it is of great importance to decide whether a certain density matrix on a bipartite system is entangled or not [3,4]. However, this problem, referred to as entanglement detection, is known to be a computationally hard one in quantum information theory [5,6]. In the last two decades, lots of effort have been accomplished in order to determine necessary and sufficient conditions for a density matrix to be entangled. For instance, one such criterion is the k-extendibility hierarchy [7], which provides a sequence of tests to check that the density matrix is separable, that ultimately detects all entangled states. Another appealing method is the positive map criterion [8], which gives an operational interpretation of the Hahn–Banach theorem applied to the convex set of separable density matrices. The Horodecki’s Theorem thus states that a
I. Bardet et al.
Ann. Henri Poincar´e
density matrix ρ is entangled if and only if there exists a positive map Φ such that (i ⊗ Φ)(ρ) is not positive semi-definite, where Φ only acts on one of the two subsystems. Necessarily, this positive map is not completely positive. The most well-known example of such map is the transpose map, and it leads to the positive partial transpose (PPT) criterion [9]. However, because of the complex geometrical structure of the set of separable density matrices, an infinite number of maps that one does not know how to describe efficiently would be necessary to detect all entangled states (see for instance [10]). The goal of this article is to propose a family of maps, with increasing complexity, which suffice to detect any entanglement. Their main interest lies in that it is rather easy to check if they are positive or not. Compared to the k-extendibility hierarchy, th
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