Hilbert-Schmidt Separability Probabilities and Noninformativity of Priors

The Horodecki family employed the Jaynes maximum-entropy principle, fitting the mean (b_{1}) of the Bell-CHSH observable (B). This model was extended by Rajagopal by incorporating the dispersion (\sigma_{1}^2) of the observable, and by Canosa and Rossignoli, by generalizing the observable (B_{\alpha}). We further extend the Horodecki one-parameter model in both these manners, obtaining a three-parameter (b_{1},\sigma_{1}^2,\alpha) two-qubit model, for which we find a highly interesting/intricate continuum (-\infty < \alpha < \infty) of Hilbert-Schmidt (HS) separability probabilities -- in which, the golden ratio is featured. Our model can be contrasted with the three-parameter (b_{q}, \sigma_{q}^2,q) one of Abe and Rajagopal, which employs a q(Tsallis)-parameter rather than $\alpha$, and has simply q-invariant HS separability probabilities of 1/2. Our results emerge in a study initially focused on embedding certain information metrics over the two-level quantum systems into a q-framework. We find evidence that Srednicki's recently-stated biasedness criterion for noninformative priors yields rankings of priors fully consistent with an information-theoretic test of Clarke, previously applied to quantum systems by Slater.Comment: 26 pages, 12 figure

Volume of the quantum mechanical state space

The volume of the quantum mechanical state space over $n$-dimensional real, complex and quaternionic Hilbert-spaces with respect to the canonical Euclidean measure is computed, and explicit formulas are presented for the expected value of the determinant in the general setting too. The case when the state space is endowed with a monotone metric or a pull-back metric is considered too, we give formulas to compute the volume of the state space with respect to the given Riemannian metric. We present the volume of the space of qubits with respect to various monotone metrics. It turns out that the volume of the space of qubits can be infinite too. We characterize those monotone metrics which generates infinite volume.Comment: 17 page

A priori probability that a qubit-qutrit pair is separable

We extend to arbitrarily coupled pairs of qubits (two-state quantum systems) and qutrits (three-state quantum systems) our earlier study (quant-ph/0207181), 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 (35-d vs. 15-d) -- 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, u/(v Pi^3)= >.00124706, 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, 8/(11 Pi^2) = 0736881, 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.Comment: 9 pages, 3 figures, 2 tables, LaTeX, we utilize recent exact computations of Sommers and Zyczkowski (quant-ph/0304041) of "the Bures volume of mixed quantum states" to refine our conjecture

The oscillator model for the Lie superalgebra sh(2|2) and Charlier polynomials

We investigate an algebraic model for the quantum oscillator based upon the Lie superalgebra sh(2|2), known as the Heisenberg-Weyl superalgebra or "the algebra of supersymmetric quantum mechanics", and its Fock representation. The model offers some freedom in the choice of a position and a momentum operator, leading to a free model parameter gamma. Using the technique of Jacobi matrices, we determine the spectrum of the position operator, and show that its wavefunctions are related to Charlier polynomials C_n with parameter gamma^2. Some properties of these wavefunctions are discussed, as well as some other properties of the current oscillator model.Comment: Minor changes and some additional references added in version

A massive Feynman integral and some reduction relations for Appell functions

New explicit expressions are derived for the one-loop two-point Feynman integral with arbitrary external momentum and masses $m_1^2$ and $m_2^2$ in D dimensions. The results are given in terms of Appell functions, manifestly symmetric with respect to the masses $m_i^2$. Equating our expressions with previously known results in terms of Gauss hypergeometric functions yields reduction relations for the involved Appell functions that are apparently new mathematical results.Comment: 19 pages. To appear in Journal of Mathematical Physic

From presence to consciousness through virtual reality

Immersive virtual environments can break the deep, everyday connection between where our senses tell us we are and where we are actually located and whom we are with. The concept of 'presence' refers to the phenomenon of behaving and feeling as if we are in the virtual world created by computer displays. In this article, we argue that presence is worthy of study by neuroscientists, and that it might aid the study of perception and consciousness

Quantum communication and state transfer in spin chains

We investigate the time evolution of a single spin excitation state in certain linear spin chains, as a model for quantum communication. We consider first the simplest possible spin chain, where the spin chain data (the nearest neighbour interaction strengths and the magnetic field strengths) are constant throughout the chain. The time evolution of a single spin state is determined, and this time evolution is illustrated by means of an animation. Some years ago it was discovered that when the spin chain data are of a special form so-called perfect state transfer takes place. These special spin chain data can be linked to the Jacobi matrix entries of Krawtchouk polynomials or dual Hahn polynomials. We discuss here the case related to Krawtchouk polynomials, and illustrate the possibility of perfect state transfer by an animation showing the time evolution of the spin chain from an initial single spin state. Very recently, these ideas were extended to discrete orthogonal polynomials of q-hypergeometric type. Here, a remarkable result is a new analytic model where perfect state transfer is achieved: this is when the spin chain data are related to the Jacobi matrix of q-Krawtchouk polynomials. This case is discussed here, and again illustrated by means of an animation