Invariance of bipartite separability and PPT-probabilities over Casimir invariants of reduced states

Abstract

Milz and Strunz ({\it J. Phys. A}: {\bf{48}} [2015] 035306) recently studied the probabilities that two-qubit and qubit-qutrit states, randomly generated with respect to Hilbert-Schmidt (Euclidean/flat) measure, are separable. They concluded that in both cases, the separability probabilities (apparently exactly $\frac{8}{33}$ in the two-qubit scenario) hold {\it constant} over the Bloch radii ($r$) of the single-qubit subsystems, jumping to 1 at the pure state boundaries ($r=1$). Here, firstly, we present evidence that in the qubit-qutrit case, the separability probability is uniformly distributed, as well, over the {\it generalized} Bloch radius ($R$) of the qutrit subsystem. While the qubit (standard) Bloch vector is positioned in three-dimensional space, the qutrit generalized Bloch vector lives in eight-dimensional space. The radii variables $r$ and $R$ themselves are the lengths/norms (being square roots of {\it quadratic} Casimir invariants) of these ("coherence") vectors. Additionally, we find that not only are the qubit-qutrit separability probabilities invariant over the quadratic Casimir invariant of the qutrit subsystem, but apparently also over the {\it cubic} one--and similarly the case, more generally, with the use of random induced measure. We also investigate two-qutrit ($3 \times 3$) and qubit-{\it qudit} ($2 \times 4$) systems--with seemingly analogous {\it positive-partial-transpose}-probability invariances holding over what have been termed by Altafini, the {\it partial} Casimir invariants of these systems.