Zyczkowski，Horodecki，Sanpera和Lewenstein（ZHSL）最近提出了N维量子系统的“自然测量”（quant-ph/9804024），但当它导致他们得出结论时表示惊讶，对于N = 2 x 2，解纠缠（可分离）系统在自然界中比纠缠系统更有可能（0.632）。然而，我们认为，ZHSL的（拒绝）的直觉，事实上，一个健全的理论基础，和先验的概率解开2 × 2系统应该更恰当地被视为（相当）小于0.5。我们以两种截然不同的方式得出这个结论，第一种是基于经典的考虑，第二种是基于量子的考虑。然而，这两种方法都用基于单调度量的体积元素的方法（全部或部分）取代ZHSL（乘积）度量，这在经典情况下相当于采用贝叶斯理论的Jeffreys先验。只有量子理论分析（产生最小的解纠缠概率）使用了可能的最小参数数N^2 - 1，而不是N^2 + N - 1（尽管这种“过度参数化”，正如伯德最近指出的，应该是可以避免的）。然而，尽管进行了大量的计算，我们仍然无法获得这些概率的精确估计，并且需要额外的（可能是超级计算机）分析（特别是对于高维量子系统，例如我们也在这里研究的2 x 3）。
英文摘要：
Zyczkowski, Horodecki, Sanpera, and Lewenstein (ZHSL) recently proposed a ``natural measure'' on the N-dimensional quantum systems (quant-ph/9804024), but expressed surprise when it led them to conclude that for N = 2 x 2, disentangled (separable) systems are more probable (0.632) in nature than entangled ones. We contend, however, that ZHSL's (rejected) intuition 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 (in whole or part) the ZHSL (product) measure by ones based on the volume elements of monotone metrics, which in the classical case amounts to adopting the Jeffreys' prior of Bayesian theory. Only the quantum-theoretic analysis (which yields the smallest probabilities of disentanglement) uses the minimum number of parameters possible, N^2 - 1, as opposed to N^2 + N - 1 (although this "over-parameterization", as recently indicated by Byrd, should be avoidable). However, despite substantial computation, we are not able to obtain precise estimates of these probabilities, and the need for additional (possibly supercomputer) analyses is indicated (particularly so, for higher-dimensional quantum systems, such as the 2 x 3 we also study here).
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We reexamine a recent analysis in which, using the volume of the associated quantum steering ellipsoid (QES) as a measure, we sought to estimate the probability that a two-qubit state is separable. In the estimation process, we, in effect, sought to attach to states random with respect to Hilbert-Schmidt (HS) measure, the corresponding QES volumes. However, a study of the relations between HS and Bures ensembles and their well-supported separability probabilities of $\frac{8}{33}$ and $\frac{25}{341}$, respectively, now lead us to explore as a possible alternative measure, the QES volume divid…
The statistical problem of the distribution of $s$ quanta of equal energy $ε_0$ and total energy $E$ among $N$ distinguishable particles is resolved using the conventional theory based on Boltzmann's principle of equal probabilities of configurations of particles distributed among energy levels and the concept of average state. In particular, the probability that a particle is in the \k{appa}-th energy level i.e. contains \k{appa} quanta, is given by $p(κ)=\displaystyle \frac{\displaystyle \binom{N+s-κ-2}{N-2}}{\displaystyle \binom{N+s-1}{N-1}} \;\;\; ; \;\;\; κ= 0, 1, 2, \cdots, s$ In this co…
Employing the volume of quantum steering ellipsoids (QSEs) as a measure on the fifteen-dimensional convex set of two-qubit states, we estimate the ratio of the integral of the measure over the separable states to its integral over all (separable and entangled) states to be 0.0288. This can be contrasted with the considerably larger separability ratios (probabilities) of $\frac{8}{33} = \frac{2^3}{3 \cdot 11} \approx 0.242424$ and $\frac{25}{341}=\frac{5^2}{11 \cdot 31} \approx 0.0733138$ that various forms of evidence point to with the use of the prominent Hilbert-Schmidt and Bures measures, r…
