Two-Qubit Separability Probabilities and Beta Functions

Due to recent important work of Zyczkowski and Sommers (quant-ph/0302197 and quant-ph/0304041), exact formulas are available (both in terms of the Hilbert-Schmidt and Bures metrics) for the (n^2-1)-dimensional and (n(n-1)/2-1)-dimensional volumes of the complex and real n x n density matrices. However, no comparable formulas are available for the volumes (and, hence, probabilities) 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, the two-qubit systems. Making use of the density matrix (rho) parameterization of Bloore (J. Phys. A 9, 2059 [1976]), 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, nu = (rho_{11} rho_{44})/{rho_{22} rho_{33})$. The associated integrand in each case is the product of a known (highly oscillatory near nu=1) jacobian and a certain unknown univariate function, which our extensive numerical (quasi-Monte Carlo) computations indicate is very closely proportional to an (incomplete) beta function B_{nu}(a,b), with a=1/2, b=sqrt{3}in the real case, and a=2 sqrt{6}/5, b =3/sqrt{2} in the complex case. Assuming the full applicability of these specific incomplete beta functions, we undertake separable volume calculations.Comment: 17 pages, 4 figures, paper is substantially rewritten and reorganized, with the quasi-Monte Carlo integration sample size being greatly increase

Advances in delimiting the Hilbert-Schmidt separability probability of real two-qubit systems

We seek to derive the probability--expressed in terms of the Hilbert-Schmidt (Euclidean or flat) metric--that a generic (nine-dimensional) real two-qubit system is separable, by implementing the well-known Peres-Horodecki test on the partial transposes (PT's) of the associated 4 x 4 density matrices). But the full implementation of the test--requiring that the determinant of the PT be nonnegative for separability to hold--appears to be, at least presently, computationally intractable. So, we have previously implemented--using the auxiliary concept of a diagonal-entry-parameterized separability function (DESF)--the weaker implied test of nonnegativity of the six 2 x 2 principal minors of the PT. This yielded an exact upper bound on the separability probability of 1024/{135 pi^2} =0.76854$. Here, we piece together (reflection-symmetric) results obtained by requiring that each of the four 3 x 3 principal minors of the PT, in turn, be nonnegative, giving an improved/reduced upper bound of 22/35 = 0.628571. Then, we conclude that a still further improved upper bound of 1129/2100 = 0.537619 can be found by similarly piecing together the (reflection-symmetric) results of enforcing the simultaneous nonnegativity of certain pairs of the four 3 x 3 principal minors. In deriving our improved upper bounds, we rely repeatedly upon the use of certain integrals over cubes that arise. Finally, we apply an independence assumption to a pair of DESF's that comes close to reproducing our numerical estimate of the true separability function.Comment: 16 pages, 9 figures, a few inadvertent misstatements made near the end are correcte

A Comparative Study of Two Real Root Isolation Methods

Recent progress in polynomial elimination has rendered the computation of the real roots of ill-conditioned polynomials of high degree (over 1000) with huge coefficients (several thousand digits) a critical operation in computer algebra. To rise to the occasion, the only method-candidate that has been considered by various authors for modification and improvement has been the Collins-Akritas bisection method [1], which is a based on a variation of Vincent’s theorem [2]. The most recent example is the paper by Rouillier and Zimmermann [3], where the authors present “... a new algorithm, which is optimal in terms of memory usage and as fast as both Collins and Akritas’ algorithm and Krandick variant ...” [3] In this paper we compare our own continued fractions method CF [4] (which is directly based on Vincent’s theorem) with the best bisection method REL described in [3]. Experimentation with the data presented in [3] showed that, with respect to time, our continued fractions method CF is by far superior to REL, whereas the two are about equal with respect to space

Real Algebraic Strategies for MetiTarski Proofs

Abstract. MetiTarski [1] is an automatic theorem prover that can prove inequalities involving sin, cos, exp, ln, etc. During its proof search, it generates a series of subproblems in nonlinear polynomial real arithmetic which are reduced to true or false using a decision procedure for the theory of real closed fields (RCF). These calls are often a bottleneck: RCF is fundamentally infeasible. However, by studying these subproblems, we can design specialised variants of RCF decision procedures that run faster and improve MetiTarski’s performance.

Optimal static pricing for a tree network

We study the static pricing problem for a network service provider in a loss system with a tree structure. In the network, multiple classes share a common inbound link and then have dedicated outbound links. The motivation is from a company that sells phone cards and needs to price calls to different destinations. We characterize the optimal static prices in order to maximize the steady-state revenue. We report new structural findings as well as alternative proofs for some known results. We compare the optimal static prices versus prices that are asymptotically optimal, and through a set of illustrative numerical examples we show that in certain cases the loss in revenue can be significant. Finally, we show that static prices obtained using the reduced load approximation of the blocking probabilities can be easily obtained and have near-optimal performance, which makes them more attractive for applications.Massachusetts Institute of Technology. Center for Digital BusinessUnited States. Office of Naval Research (Contract N00014-95-1-0232)United States. Office of Naval Research (Contract N00014-01-1-0146)National Science Foundation (U.S.) (Contract DMI-9732795)National Science Foundation (U.S.) (Contract DMI-0085683)National Science Foundation (U.S.) (Contract DMI-0245352

A Comparative Study of Two Real Root Isolation Methods

Recent progress in polynomial elimination has rendered the computation of the real roots of ill-conditioned polynomials of high degree (over 1000) with huge coefficients (several thousand digits) a critical operation in computer algebra.&#x0D; To rise to the occasion, the only method-candidate that has been considered by various authors for modification and improvement has been the Collins-Akritas bisection method [1], which is a based on a variation of Vincent’s theorem [2].&#x0D; The most recent example is the paper by Rouillier and Zimmermann [3], where the authors present “... a new algorithm, which is optimal in terms of memory usage and as fast as both Collins and Akritas’ algorithm and Krandick variant ...” [3]&#x0D; In this paper we compare our own continued fractions method CF [4] (which is directly based on Vincent’s theorem) with the best bisection method REL described in [3]. Experimentation with the data presented in [3] showed that, with respect to time, our continued fractions method CF is by far superior to REL, whereas the two are about equal with respect to space.</jats:p

Improving the Performance of the Continued Fractions Method Using New Bounds of Positive Roots

In this paper we compare four implementations of the Vincent-Akritas-Strzebonski Continued Fractions (VAS-CF) real root isolation method using four different (two linear and two quadratic complexity) bounds on the values of the positive roots of polynomials. The quadratic complexity bounds were included to see if the quality of their estimates compensates for their quadratic complexity. Indeed, experimentation on various classes of special and random polynomials revealed that the VAS-CF implementation using LMQ, the Quadratic complexity variant of our Local Max bound, achieved an overall average speed-up of 40% over the original implementation using Cauchy's linear bound

Improving the Performance of the Continued Fractions Method Using New Bounds of Positive Roots

In this paper we compare four implementations of the Vincent-AkritasStrzebo´nski Continued Fractions (VAS-CF) real root isolation method using four different (two linear and two quadratic complexity) bounds on the values of the positive roots of polynomials. The quadratic complexity bounds were included to see if the quality of their estimates compensates for their quadratic complexity. Indeed, experimentation on various classes of special and random polynomials revealed that the VAS-CF implementation using LMQ, the Quadratic complexity variant of our Local Max bound, achieved an overall average speed-up of 40 % over the original implementation using Cauchy’s linear bound.</jats:p