  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.

Quantum Physics