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Invariance of bipartite separability and PPT-probabilities over Casimir invariants of reduced states. (English) Zbl 1348.81111

Summary: S. Milz and W. T. Strunz [J. Phys. A, Math. Theor. 48, No. 3, Article ID 035306, 16 p. (2015; Zbl 1315.81018)] 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.

MSC:

81P40 Quantum coherence, entanglement, quantum correlations
22E70 Applications of Lie groups to the sciences; explicit representations

Citations:

Zbl 1315.81018
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References:

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