Entanglement Universality of TGX States in Qubit-Qutrit Systems

TL;DR

This paper proves that all qubit-qutrit states are equivalent to a specific subset of X states called EPU-minimal TGX states, enabling simplified entanglement analysis using explicit formulas for I-concurrence.

Contribution

It establishes the universality of EPU-minimal TGX states for qubit-qutrit systems and provides explicit formulas for entanglement measurement.

Findings

All qubit-qutrit states are EPU-equivalent to EPU-minimal TGX states.

Explicit I-concurrence formulas are derived for these states.

Optimal decompositions are identified for minimal SGX states.

Abstract

We prove that all states (mixed or pure) of qubit-qutrit ($2\times 3$) systems have entanglement-preserving unitary (EPU) equivalence to a compact subset of true-generalized X (TGX) states called EPU-minimal TGX states which we give explicitly. Thus, for any spectrum-entanglement combination achievable by general states, there exists an EPU-minimal TGX state of the same spectrum and entanglement. We use I-concurrence to measure entanglement and give an explicit formula for it for all $2\times 3$ minimal TGX states (a more general set than EPU-minimal TGX states) whether mixed or pure, yielding its minimum average value over all decompositions. We also give a computable I-concurrence formula for a more general family called minimal super-generalized X (SGX) states, and give optimal decompositions for minimal SGX states and all of their subsets.