Formulas for Rational-Valued Separability Probabilities of Random Induced Generalized Two-Qubit States

Previously, a formula, incorporating a5F4hypergeometric function, for the Hilbert-Schmidt-averaged determinantal momentsρPTnρk/ρkof4×4density-matrices (ρ) and their partial transposes (|ρPT|), was applied withk=0to the generalized two-qubit separability probability question. The formula can, furthermore, be viewed, as we note here, as an averaging over “induced measures in the space of mixed quantum states.” The associated induced-measure separability probabilities (k=1,2,…) are found—viaa high-precision density approximation procedure—to assume interesting, relatively simple rational values in the two-re[al]bit (α=1/2), (standard) two-qubit (α=1), and two-quater[nionic]bit (α=2) cases. We deduce rather simple companion (rebit, qubit, quaterbit, …) formulas that successfully reproduce the rational values assumed forgeneral k. These formulas are observed to share certain features, possibly allowing them to be incorporated into a single master formula.

Basins of measures on inverse limit spaces for the induced homeomorphism

Let f:X→X be continuous and onto, where X is a compact metric space. Let $Y:=\invlim {X,f}$ be the inverse limit and F:Y →Y the induced homeomorphism. Suppose that μ is an f-invariant measure, and let m be the measure induced on Y by (μ,μ,…). We show that B is a basin of μ if and only if π−11(B) is a basin of m. From this it follows that if μ is an SRB measure for f on X, then the induced measure m on Y is an inverse-limit SRB measure for F. Conversely, if m is an inverse-limit SRB measure for F on Y, then the induced measure μ on X is an SRB measure for f.

A NOTE ON SIMULTANEOUS AND MULTIPLICATIVE DIOPHANTINE APPROXIMATION ON PLANAR CURVES

Let $\mathbb C$ be a non-degenerate planar curve. We show that the curve is of Khintchine-type for convergence in the case of simultaneous approximation in $\mathbb R^2$ with two independent approximation functions; that is if a certain sum converges then the set of all points (x,y) on the curve which satisfy simultaneously the inequalities ||qx|| < ψ1(q) and ||qy|| < ψ2(q) infinitely often has induced measure 0. This completes the metric theory for the Lebesgue case. Further, for multiplicative approximation ||qx|| ||qy|| < ψ(q) we establish a Hausdorff measure convergence result for the same class of curves, the first such result for a general class of manifolds in this particular setup.

Effective Action and Measure in Matrix Model of IIB Superstrings

We calculate an effective action and measure induced by the integration over the auxiliary field in the matrix model recently proposed to describe IIB superstrings. It is shown that the measure of integration over the auxiliary matrix is uniquely determined by locality and reparametrization invariance of the resulting effective action. The large-N limit of the induced measure for string coordinates is discussed in detail. It is found to be ultralocal and, thus, is possibly irrelevant in the continuum limit. The model of the GKM type is considered in relation to the effective action problem.