Triangle inequalities in coherence measures and entanglement concurrence

TL;DR

This paper proves triangle inequalities for coherence measures and entanglement concurrence in quantum states, revealing a geometric-like property and providing an example to illustrate the results.

Contribution

It establishes new triangle inequalities for coherence and entanglement measures in rank-2 quantum states, enhancing understanding of their mathematical structure.

Findings

Triangle inequalities hold for coherence and entanglement measures in rank-2 states.

The inequalities resemble geometric triangle inequalities, showing a mathematical beauty.

An example demonstrates the application of the inequalities.

Abstract

We provide detailed proofs of triangle inequalities in coherence measures and entanglement concurrence. If a rank-$2$ state $\varrho$ can be expressed as a convex combination of two pure states, i.e., $\varrho=p_{1}|\psi_{1}\rangle\langle\psi_{1}|+p_{2}|\psi_{2}\rangle\langle\psi_{2}|$, a triangle inequality can be established as $\big{|}E(|\Psi_{1}\rangle)-E(|\Psi_{2}\rangle)\big{|}\leq E(\varrho)\leq E(|\Psi_{1}\rangle)+E(|\Psi_{2}\rangle)$, where $|\Psi_{1}\rangle=\sqrt{p_{1}}|\psi_{1}\rangle$ and $|\Psi_{2}\rangle=\sqrt{p_{2}}|\psi_{2}\rangle$, $E$ can be considered either coherence measures or entanglement concurrence. This inequality displays mathematical beauty for its similarity to the triangle inequality in plane geometry. An illustrative example is given after the proof.