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

Milz and Strunz (J Phys A 48:035306, 2015) 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 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 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 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 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- qudit ( $$2 \times 4$$ ) systems-with seemingly analogous positive partial transpose-probability invariances holding over what has been termed by Altafini the partial Casimir invariants of these systems. [ABSTRACT FROM AUTHOR]