From c-Numbers to q-Numbers

Roger Cooperman, A.E. Petrunina V.V.

George Jaroszkiewicz

Introduction In this chapter we apply temporal discretization to mechanical systems described by anticommuting variables rather than the commuting variables normally used in classical mechanics (CM). The anticommuting numbers representing such variables are called Grassmann(ian) numbers by mathematicians, after Hermann G. Grassmann (1809–1877), who did some pioneering work in linear algebra. On that account we shall refer to anticommuting numbers as g-numbers , in contrast to c-numbers ( c for commuting 1 ) when we refer to ordinary real or complex variables. The significant difference between g-numbers and c-numbers is that, where as any two c numbers x and y satisfy the commutative multiplication rule xy = + yx , any two g-numbers θ and ϕ satisfy the anticommutation rule θ ϕ = –ϕ θ . G-numbers and c-numbers can be multiplied together in any order, i.e., g-numbers and c-numbers commute. The product zθ of a c-number z and a g-number θ is a g-number. The values of classical variables in ordinary CM are c-numbers, so we may refer to such variables as c-variables. It is possible to construct forms of CM where the classical variables are g-numbers, in which case we shall refer to such variables as g-variables. G-variables should not be thought of as quantized versions of c-variables. They are just as classical as c-variables but much less familiar in terms of their applications to the real world. G-variables should not be confused with Dirac's q-numbers either.

Lajos Diósi

Lucas Daiha, Roberto Rivelino

We employ an algebraic procedure based on quantum mechanics to propose a `quantum number theory' (QNT) as a possible extension of the `classical number theory'. We built our QNT by defining pure quantum number operators ($q$-numbers) of a Hilbert space that generate classical numbers ($c$-numbers) belonging to discrete Euclidean spaces. To start with this formalism, we define a 2-component natural $q$-number $\textbf{N}$, such that $\mathbf{N}^{2} \equiv N_{1}^{2} + N_{2}^{2}$, satisfying a Heisenberg-Dirac algebra, which allows to generate a set of natural $c$-numbers $n \in \mathbb{N}$. A probabilistic interpretation of QNT is then inferred from this representation. Furthermore, we define a 3-component integer $q$-number $\textbf{Z}$, such that $\mathbf{Z}^{2} \equiv Z_{1}^{2} + Z_{2}^{2} + Z_{3}^{2}$ and obeys a Lie algebra structure. The eigenvalues of each $\textbf{Z}$ component generate a set of classical integers $m \in \mathbb{Z}\cup \frac{1}{2}\mathbb{Z}^{*}$, $\mathbb{Z}^{*} = \mathbb{Z} \setminus \{0\}$, albeit all components do not generate $\mathbb{Z}^3$ simultaneously. We interpret the eigenvectors of the $q$-numbers as `$q$-number state vectors' (QNSV), which form multidimensional orthonormal basis sets useful to describe state-vector superpositions defined here as qu$n$its. To interconnect QNSV of different dimensions, associated to the same $c$-number, we propose a quantum mapping operation to relate distinct Hilbert subspaces, and its structure can generate a subset $W \subseteq \mathbb{Q}^{*}$, the field of non-zero rationals. In the present description, QNT is related to quantum computing theory and allows dealing with nontrivial computations in high dimensions.

Lajos Diósi

Travis S. Humble

A system and method for generating random numbers. The system may include a random number generator (RNG), such as a quantum random number generator (QRNG) configured to self-correct or adapt in order to substantially achieve randomness from the output of the RNG. By adapting, the RNG may generate a random number that may be considered random regardless of whether the random number itself is tested as such. As an example, the RNG may include components to monitor one or more characteristics of the RNG during operation, and may use the monitored characteristics as a basis for adapting, or self-correcting, to provide a random number according to one or more performance criteria.