A Priori Probabilities of Quantum Disentanglements

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  • 8 October 1998
Abstract
Zyczkowski, Horodecki, Sanpera, and Lewenstein (ZHSL) recently proposed a ``natural measure'' on the N-dimensional quantum systems (quant-ph/9804024). However, they expressed surprise when implementations led them to conclude that for N = 2 x 2, disentangled (separable) systems were more probable (0.632) in nature than entangled ones. We contend here that the original intuition of ZHSL has, in fact, a sound theoretical basis, and that the a priori probability of disentangled 2 x 2 systems should more properly be viewed as (considerably) less than 0.5. We arrive at this conclusion in two quite distinct ways, the first based on classical and the second, quantum considerations. Both approaches, however, replace the measure of ZHSL by ones based on the volume elements of monotone metrics, which in the classical case is equivalent to the adoption of the Jeffreys' prior of Bayesian theory. Only the quantum-theoretic analysis (which yields the smallest probabilities of disentanglement) avoids the use of more parameters than is most natural.

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