Invariance of Bipartite Separability and PPT-Probabilities over Casimir Invariants of Reduced States

TL;DR

This paper investigates how separability and PPT probabilities of bipartite quantum states remain invariant over Casimir invariants of reduced states, revealing new uniformity properties across different quantum systems.

Contribution

It provides evidence that separability and PPT probabilities are invariant over quadratic and cubic Casimir invariants in bipartite quantum systems, extending previous findings.

Findings

Separability probability is uniform over the generalized Bloch radius in qubit-qutrit systems.

Invariance of separability probabilities over quadratic and cubic Casimir invariants.

PPT-probability invariances hold over partial Casimir invariants in higher-dimensional systems.

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.