Narrow Results

We obtain the most general ensemble of qubits for which it is possible to design a universal Hadamard gate. These states, when geometrically represented on the Bloch sphere, give a new trajectory. We further consider some Hadamard "type" operations and find ensembles of states for which such transformations hold. The unequal superposition of a qubit and its orthogonal complement is also investigated.

We detect a certain pattern of behavior of separability probabilities $p(r_A,r_B)$ for two-qubit systems endowed with Hilbert–Schmidt (HS), and more generally, random induced measures, where $r_A$ and $r_B$ are the Bloch radii ($0\le r_A,r_B\le 1$) of the qubit reduced states ($A,B$). We observe a relative repulsion of radii effect, that is $p(r_A,r_A)<p(r_A,1-r_A)$, except for rather narrow “crossover” intervals $[{\tilde{r}}_A,\frac{1}{2}]$. Among the seven specific cases we study are, firstly, the “toy” seven-dimensional $X$-states model and, then, the fifteen-dimensional two-qubit states obtained by tracing over the pure states in $4\times K$-dimensions, for $K=3,4,5$, with $K=4$ corresponding to HS (flat/Euclidean) measure. We also examine the real (two-rebit) $K=4$, the $X$-states $K=5$, and Bures (minimal monotone)–for which no nontrivial crossover behavior is observed–instances. In the two $X$-states cases, we derive analytical results; for $K=3,4$, we propose formulas that well-fit our numerical results; and for the other scenarios, rely presently upon large numerical analyses. The separability probability crossover regions found expand in length (lower ${\tilde{r}}_A$) as $K$ increases. This report continues our efforts [P. B. Slater, arXiv:1506.08739] to extend the recent work of [S. Milz and W. T. Strunz, J. Phys. A48 (2015) 035306.] from a univariate ($r_A$) framework — in which they found separability probabilities to hold constant with $r_A$ — to a bivariate ($r_A,r_B$) one. We also analyze the two-qutrit and qubit–qutrit counterparts reported in Quantum Inform. Process. 15 (2016) 3745 in this context, and study two-qubit separability probabilities of the form $p(r_A,\frac{1}{2})$. A physics.stack.exchange link to a contribution by Mark Fischler addressing, in considerable detail, the construction of suitable bivariate distributions is indicated at the end of the paper.

We describe the action of the symplectic group on the homogeneous space of squeezed states (quantum blobs) and extend this action to the semigroup. We then extend the metaplectic representation to the metaplectic (or oscillator) semigroup and study the properties of such an extension using Bargmann-Fock space. The shape geometry of squeezing is analyzed and noncommuting elements from the symplectic semigroup are proposed to be used in simultaneous monitoring of noncommuting quantum variables — which should lead to fractal patterns on the manifold of squeezed states.

We present a very general geometrico-dynamical description of physical or more abstract entities, called the general tension-reduction (GTR) model, where not only states, but also measurement-interactions can be represented, and the associated outcome probabilities calculated. Underlying the model is the hypothesis that indeterminism manifests as a consequence of unavoidable uctuations in the experimental context, in accordance with the hidden-measurements interpretation of quantum mechanics. When the structure of the state space is Hilbertian, and measurements are of the universal kind, i.e., are the result of an average over all possible ways of selecting an outcome, the GTR-model provides the same predictions of the Born rule, and therefore provides a natural completed version of quantum mechanics. However, when the structure of the state space is non-Hilbertian and/or not all possible ways of selecting an outcome are available to be actualized, the predictions of the model generally differ from the quantum ones, especially when sequential measurements are considered. Some paradigmatic examples will be discussed, taken from physics and human cognition. Particular attention will be given to some known psychological effects, like question order effects and response replicability, which we show are able to generate non-Hilbertian statistics. We also suggest a realistic interpretation of the GTR-model, when applied to human cognition and decision, which we think could become the generally adopted interpretative framework in quantum cognition research.

We show that the extended Bloch representation of quantum mechanics also applies to infinite-dimensional entities, to the extent that the number of (possibly infinitely degenerate) outcomes of a measurement remains finite, which is always the case in practical situations.

We postulate bulk universal quantum computing (QC) cannot be achieved without surmounting the quantum uncertainty principle, an inherent barrier by empirical definition in the regime described by the Copenhagen interpretation of quantum theory - the last remaining hurdle to bulk QC. To surmount uncertainty with probability 1, we redefine the basis for the qubit utilizing a unique form of M-Theoretic Calabi-Yau mirror symmetry cast in an LSXD Dirac covariant polarized vacuum with an inherent ‘Feynman synchronization backbone’. This also incorporates a relativistic qubit (r-qubit) providing additional degrees of freedom beyond the traditional Block 2-sphere qubit bringing the r-qubit into correspondence with our version of Relativistic Topological Quantum Field Theory (RTQFT). We present a 3^rd generation prototype design for simplifying bulk QC implementation.