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Mathematical Results in Quantum Mechanics (QMath)

RELATIVE ENTROPIES AND ENTANGLEMENT MONOTONES

Nilanjana Datta

Published May 1, 2011• 1 citation
Publisher
DOI
10.1142/9789814350365_0001

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The formalism of functions of commuting nilpotent variables allows to describe multi-qubit pure states and their entanglement. The family of states defined by the generalized trigonometric η-functions is specially interesting from mathematical and physical point of view (it covers the set of physically interesting states, including: Werner states, cluster Werner states, GHZ states etc.). We analyze the behavior of two recently proposed symmetric entanglement monotones on the trigonometric states.
We present a method to construct entanglement measures for pure states of multipartite qubit systems. The key element of our approach is an antilinear operator that we call comb in reference to the hairy-ball theorem. For qubits (i.e. spin 1/2) the combs are automatically invariant under SL (2, ℂ). This implies that the filters obtained from the combs are entanglement monotones by construction. We give alternative formulae for the concurrence and the 3-tangle as expectation values of certain antilinear operators. As an application we discuss inequivalent types of genuine four-, five- and six-qubit entanglement.
Abstract In this paper, we introduce two notions of a relative operator (α, β)-entropy and a Tsallis relative operator (α, β)-entropy as two parameter extensions of the relative operator entropy and the Tsallis relative operator entropy. We apply a perspective approach to prove the joint convexity or concavity of these new notions, under certain conditions concerning α and β. Indeed, we give the parametric extensions, but in such a manner that they remain jointly convex or jointly concave. Significance Statement. What is novel here is that we convincingly demonstrate how our techniques can be used to give simple proofs for the old and new theorems for the functions that are relevant to quantum statistics. Our proof strategy shows that the joint convexity of the perspective of some functions plays a crucial role to give simple proofs for the joint convexity (resp. concavity) of some relative operator entropies.