Zyczkowski, Horodecki, Sanpera and Lewenstein (ZHSL) recently proposed a `natural measure' on the N -dimensional quantum systems, but expressed surprise when it led them to conclude that for N = 2 × 2, disentangled (separable) systems are more probable (0.632±0.002) 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 × 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, that is N 2 -1, as opposed to N 2 + N -1 (although this `over-parametrization', 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 × 3 ones, which we also study here).
Zyczkowski, Horodecki, Sanpera and Lewenstein (ZHSL) recently proposed a natural measure' on the N -dimensional quantum systems, but expressed surprise when it led them to conclude that for N = 2 × 2, disentangled (separable) systems are more probable (0.632±0.002) 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 × 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, that is N 2 -1, as opposed to N 2 + N -1 (although this over-parametrization', 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 × 3 ones, which we also study here).
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).
We report a concise answer--in the case of 2 x 2 systems--to the fundamental quantum-information-theoretic question as to "the volume of separable states" posed by Zyczkowski, Horodecki, Sanpera and Lewenstein (Phys. Rev. A, 58, 883 [1998]). We proceed by applying the Mathematica command FindSequenceFunction to a series of conjectured Hilbert-Schmidt generic 2 x 2 (rational-valued) separability probabilities p(a), a = 1, 2,...,32, with a = 1 indexing standard two-qubit systems, and a = 2, two-quater(nionic)bit systems. These 32 inputted values of p(a)--as well as 32 companion non-inputted values for the half-integers, a = 1/2 (two-re[al]bit) systems), 3/2,..., 63/2, are advanced on the basis of high-precision probability-distribution-reconstruction computations, employing 7,501 determinantal moments of partially transposed 4 x 4 density matrices. The function P(a) given by application of the command fully reproduces both of these 32-length sequences, and an equivalent outcome is obtained if the half-integral series is the one inputted. The lengthy expression (containing six hypergeometric functions) obtained for P(a) is, then, impressively condensed (by Qing-Hu Hou and colleagues), using Zeilberger's algorithm. For generic (9-dimensional) two-rebit systems, P(1/2) = 29/64, (15-dimensional) two-qubit systems, P(1) = 8/33, (27-dimensional) two-quaterbit systems, P(2) = 26/323, while for generic classical (3-dimensional) systems, P(0)=1.
We extend to arbitrarily coupled pairs of qubits (two-state quantum systems) and qutrits (three-state quantum systems) our earlier study (Slater 2002 Quantum Inf. Process. 1 387), which was concerned with the simplest instance of entangled quantum systems, pairs of qubits. As in that analysis—again on the basis of numerical (quasi-Monte Carlo) integration results, but now in a still higher-dimensional space (35D versus 15D)—we examine a conjecture that the Bures/SD (statistical distinguishability) probability that arbitrarily paired qubits and qutrits are separable (unentangled) has a simple exact value, , where u = 2 20 × 3 3 × 5 × 7 and v = 19 × 23 × 29 × 31 × 37 × 41 × 43 (the product of consecutive primes). This is considerably less than the conjectured value of the Bures/SD probability, , in the qubit–qubit case. Both of these conjectures, in turn, rely upon ones to the effect that the SD volumes of separable states assume certain remarkable forms, involving 'primorial' numbers. We also estimate the SD area of the boundary of separable qubit–qutrit states, and provide preliminary calculations of the Bures/SD probability of separability in the general qubit–qubit–qubit and qutrit–qutrit cases.
Paralleling our recent computationally-intensive work for the case N=4 (quant-ph/0308037), we undertake the task for N=6 of computing to high numerical accuracy, the formulas of Sommers and Zyczkowski (quant-ph/0304041) for the (N^2-1)-dimensional volume and (N^2-2)-dimensional hyperarea of the (separable and nonseparable) N x N density matrices, based on the Bures (minimal monotone) metric. At the same time, we estimate the unknown volumes and hyperareas based on a number of other monotone metrics of interest. Additionally, we estimate -- but with perhaps unavoidably diminished accuracy -- all these volume and hyperarea quantities, when restricted to the ``small'' subset of 6 x 6 density matrices that are separable (classically correlated) in nature. The ratios of separable to separable plus nonseparable volumes, then, yield corresponding estimates of the ``probabilities of separability''. We are particularly interested in exploring the possibility that a number of the various 35-dimensional volumes and 34-dimensional hyperareas, possess exact values -- which we had, in fact, conjectured to be the case for the qubit-qubit systems (N=4), with the ``silver mean'', sqrt{2}-1, appearing to play a fundamental role as regards the separable states.
We demonstrate --- using the case of the two-dimensional quantum systems --- that the "natural measure on the space of density matrices describing N-dimensional quantum systems" proposed by Zyczkowski et al (quant-ph/9804024) does not belong to the class of normalized volume elements of monotone metrics on the quantum systems. Such metrics possess the statistically important property of being decreasing under stochastic mappings (coarse-grainings). We do note that the proposed measure (and certain evident variations upon it) exhibit quite specific monotonicity properties, but not of the form required for membership in the class.
In a celebrated paper ([Phys. Rev. A 58, 883 (1998)]), K. Zyczkowski, P. Horodecki, A. Sanpera,and M. Lewenstein proved for the frst time a very interesting theorem that the volume of separable quantum states is nonzero. Inspired by their ideas, we obtain a general analytical lower bound of the volume of separable states (VOSS) for arbitrary N-dimensional system. Our results give quite simple and computable suffcient conditions for separability. Moreover, for bipartite system, an upper bound of the VOSS is also presented.