Two-qubit separability probabilities and Beta functions

Abstract

Due to recent important work of Życzkowski and Sommers ͓J. Phys. A 36, 10115 ͑2003͒; 36, 10083 ͑2003͔͒, exact formulas are available ͑in terms of both the Hilbert-Schmidt and the Bures metrics͒ for the ͑n 2 −1͒-dimensional and ͓n͑n −1͒ /2−1͔-dimensional volumes of the complex and real n ϫ n density matrices. However, no comparable formulas are available for the volumes ͑and, hence, probabilities͒ of various separable subsets of them. We seek to clarify this situation for the Hilbert-Schmidt metric for the simplest possible case of n = 4, that is, two-qubit systems. Making use of the density matrix ͑͒ parametrization of Bloore ͓J. Phys. A 9, 2059 ͑1976͔͒, we are able to reduce each of the real and complex volume problems to the calculation of a one-dimensional integral, the single relevant variable being a certain ratio of diagonal entries, = 11 44 / 22 33 . The associated integrand in each case is the product of a known Jacobian ͑highly oscillatory near =1͒ and a certain unknown univariate function, which our extensive numerical ͑quasi Monte Carlo͒ computations indicate is very closely proportional to an ͑incomplete͒ Beta function B ͑a , b͒, with a =1/2, b = ͱ 3 in the real case, and a =2 ͱ 6/5,b =3/ ͱ 2 in the complex case. Assuming the full applicability of these specific incomplete Beta functions, we undertake separable-volume calculations.

Extended studies of separability functions and probabilities and the relevance of Dyson indices

“…(These proposals are strongly consistent [17, Fig. 4] with the numerical results generated in [11].) This chain of reasoning, then, immediately suggests the further proposition that the separability function in the quaternionic case is exactly proportional to the fourth power of that for the real case (and, obviously, the square of that for the complex case).…”

“…So, we have evolved the following general strategy [11,17]. We integrate over the larger set (off-diagonal or Euler-angle parameters), while enforcing separability conditions, leaving us with separability functions that are functions of only the smaller set of parameters (diagonal entries or eigenvalues).…”

“…Quaternionic two-qubit Hilbert-Schmidt analysis Due to the "curse of dimensionality" [35,36], we must anticipate that for the same number of sample ("low-discrepancy" Tezuka-Faure (TF) [37,38]) points generated in the quasi-Monte Carlo integration procedure employed in [11] and here, our numerical estimates of the quaternionic separability function will be less precise than the estimates were for the complex, and a fortiori, real cases. (An interesting, sophisticated alternative approach to computing the Euclidean volume of convex bodies involves a variant of simulated annealing [39] (cf.…”

“…In our extensive numerical (quasi-Monte Carlo integration) investigation [11] of the 9-dimensional and 15-dimensional convex sets of real and complex 4 × 4 density matrices, we had formulated ansätze for the two associated separability functions (S HS real (µ) and S HS complex (µ)), proposing that they were proportional to certain (independent) incomplete beta functions [32],…”

“…For several years now, elaborating upon an idea proposed in [1], we have been pursuing the problem of deriving (hypothetically exact) formulas for the proportion of states of qubitqubit [2] and qubit-qutrit [3] systems that are separable (classically-correlated) in nature [4,5,6,7,8,9,10,11]. Of course, any such proportions will critically depend upon the measure that is placed upon the quantum systems.…”