Quantum Cylindric Set Algebras

David Finkelstein

Paul B. Slater

We investigate relationships between two forms of Hilbert-Schmidt two-re[al]bit and two-qubit functions''--those recently advanced by Lovas and Andai (arXiv:1610.01410), and those earlier presented by Slater ({\it J. Phys. A} {\bf{40}} [2007] 14279). In the Lovas-Andai framework, the independent variable $\varepsilon \in [0,1]$ is the ratio $\sigma(V)$ of the singular values of the $2 \times 2$ matrix $V=D_2^{1/2} D_1^{-1/2}$ formed from the two $2 \times 2$ diagonal blocks ($D_1, D_2$) of a randomly generated $4 \times 4$ density matrix $D$. In the Slater setting, the independent variable $\mu$ is the diagonal-entry ratio $\sqrt{\frac{d_ {11} d_ {44}}{d_ {22} d_ {33}}}$--with, importantly, $\mu=\varepsilon$ or $\mu=\frac{1}{\varepsilon}$ when both $D_1$ and $D_2$ are themselves diagonal. Lovas and Andai established that their two-rebit function $\tilde{\chi}_1 (\varepsilon )$ ($\approx \varepsilon$) yields the previously conjectured Hilbert-Schmidt separability probability of $\frac{29}{64}$. We are able, in the Slater framework (using cylindrical algebraic decompositions [CAD] to enforce positivity constraints), to reproduce this result. Further, we similarly obtain its new (much simpler) two-qubit counterpart, $\tilde{\chi}_2(\varepsilon) =\frac{1}{3} \varepsilon ^2 \left(4-\varepsilon ^2\right)$. Verification of the companion conjecture of a Hilbert-Schmidt separability probability of $\frac{8}{33}$ immediately follows in the Lovas-Andai framework. We obtain the formulas for $\tilde{\chi}_1(\varepsilon)$ and $\tilde{\chi}_2(\varepsilon)$ by taking $D_1$ and $D_2$ to be diagonal, allowing us to proceed in lower (7 and 11), rather than the full (9 and 15) dimensions occupied by the convex sets of two-rebit and two-qubit states. The CAD's themselves involve 4 and 8 variables, in addition to $\mu=\varepsilon$. We also investigate extensions of these analyses to rebit-retrit and qubit-qutrit ($6 \times 6$) settings.

Norman D. Megill, Mladen Pavičić

A Sicard, J Ospina, M Vélez

An adaptation of Kieu's hypercomputational quantum algorithm (KHQA) is presented. The method that was used was to replace the Weyl–Heisenberg algebra by other dynamical algebra of low dimension that admits infinite-dimensional irreducible representations with naturally defined generalized coherent states. We have selected the Lie algebra , because this algebra possesses the necessary characteristics to realize the hypercomputation and also because such algebra has been identified as the dynamical algebra associated with many relatively simple quantum systems. In addition to an algebraic adaptation of KHQA over the algebra , we presented an adaptation of KHQA over some concrete physical referents: the infinite square well, the infinite cylindrical well, the perturbed infinite cylindrical well, the Pöschl–Teller potentials, the Holstein–Primakoff system and the Laguerre oscillator. We conclude that it is possible to have many physical systems within condensed matter and quantum optics in which it is possible to consider an implementation of KHQA.

David Finkelstein

L. Ortíz

We proof that if we have a thermal equilibrium state on Minkowski spacetime in two dimensions then we have a thermal equilibrium state on the cylindrical spacetime obtained from this Minkowski spacetime by making 2π-periodic the spatial direction. We perform this by using the algebraic approach to Quantum Field Theory.