Inequalities for purity parameters for multipartite and single qudit states

TL;DR

This paper explores inequalities involving purity parameters for both multipartite and single qudit quantum states, analyzing their properties and relation to entanglement, with detailed studies on specific state classes.

Contribution

It reformulates a Minkowski-type trace inequality in terms of purity and examines its properties and implications for entanglement in qudit and two-qubit X-states.

Findings

Derived new purity inequalities for qudit and bipartite states

Analyzed the properties and implications of these inequalities

Explored the connection between purity inequalities and entanglement

Abstract

We analyze a recently found inequality for eigenvalues of the density matrix and purity parameter describing either a bipartite system state or a single qudit state. The Minkowski type trace inequality for the density matrices of the qudit states is rewritten in terms of the purities. The properties of the obtained inequality are discussed. The $X$-states of the two qubits and the single qudit are considered in detail. A study of the relation of the obtained purity inequalities with the entanglement is presented.