Quantity in Quantum Mechanics and the Quantity of Quantum Information

The paper interprets the concept “operator in the separable complex Hilbert space” (particalry, “Hermitian operator” as “quantity” is defined in the “classical” quantum mechanics) by that of “quantum information”. As far as wave function is the characteristic function of the probability (density) distribution for all possible values of a certain quantity to be measured, the definition of quantity in quantum mechanics means any unitary change of the probability (density) distribution. It can be represented as a particular case of “unitary” qubits. The converse interpretation of any qubits as referring to a certain physical quantity implies its generalization to non-Hermitian operators, thus neither unitary, nor conserving energy. Their physical sense, speaking loosely, consists in exchanging temporal moments therefore being implemented out of the space-time “screen”. “Dark matter” and “dark energy” can be explained by the same generalization of “quantity” to non-Hermitian operators only secondarily projected on the pseudo-Riemannian space-time “screen” of general relativity according to Einstein's “Mach’s principle” and his field equation.

Experimental semi-autonomous eigensolver using reinforcement learning

The characterization of observables, expressed via Hermitian operators, is a crucial task in quantum mechanics. For this reason, an eigensolver is a fundamental algorithm for any quantum technology. In this work, we implement a semi-autonomous algorithm to obtain an approximation of the eigenvectors of an arbitrary Hermitian operator using the IBM quantum computer. To this end, we only use single-shot measurements and pseudo-random changes handled by a feedback loop, reducing the number of measures in the system. Due to the classical feedback loop, this algorithm can be cast into the reinforcement learning paradigm. Using this algorithm, for a single-qubit observable, we obtain both eigenvectors with fidelities over 0.97 with around 200 single-shot measurements. For two-qubits observables, we get fidelities over 0.91 with around 1500 single-shot measurements for the four eigenvectors, which is a comparatively low resource demand, suitable for current devices. This work is useful to the development of quantum devices able to decide with partial information, which helps to implement future technologies in quantum artificial intelligence.

Coherent states attached to the quantum disc algebra and their associated polynomials

The one-variable [Formula: see text]-coherent states attached to the [Formula: see text]-disc algebra are constructed and used to obtain the [Formula: see text]-Bargmann–Fock realization of its Fock representation. Then, this realization is used to obtain the [Formula: see text]-continuous Hermite polynomials as well as continuous and discrete [Formula: see text]-Hermite polynomials by using a pair of Hermitian canonical conjugate operators and two pairs of the non-Hermitian conjugate operators, respectively. Besides, we introduce a two-variable family of [Formula: see text]-coherent states attached to the Fock representation space of the [Formula: see text]-disc algebra and its opposite algebra and obtain their simultaneous [Formula: see text]-Bargmann–Fock realization. For an appropriate non-Hermitian operator, the latter realization is served to obtain the well-known little [Formula: see text]-Jacobi polynomials used in constructing the [Formula: see text]-disc polynomials.

Dissipative flow equations

We generalize the theory of flow equations to open quantum systems focusing on Lindblad master equations. We introduce and discuss three different generators of the flow that transform a linear non-Hermitian operator into a diagonal one. We first test our dissipative flow equations on a generic matrix and on a physical problem with a driven-dissipative single fermionic mode. We then move to problems with many fermionic modes and discuss the interplay between coherent (disordered) dynamics and localized losses. Our method can also be applied to non-Hermitian Hamiltonians.

Time Observables in a Timeless Universe

Time in quantum mechanics is peculiar: it is an observable that cannot be associated to an Hermitian operator. As a consequence it is impossible to explain dynamics in an isolated system without invoking an external classical clock, a fact that becomes particularly problematic in the context of quantum gravity. An unconventional solution was pioneered by Page and Wootters (PaW) in 1983. PaW showed that dynamics can be an emergent property of the entanglement between two subsystems of a static Universe. In this work we first investigate the possibility to introduce in this framework a Hermitian time operator complement of a clock Hamiltonian having an equally-spaced energy spectrum. An Hermitian operator complement of such Hamiltonian was introduced by Pegg in 1998, who named it "Age". We show here that Age, when introduced in the PaW context, can be interpreted as a proper Hermitian time operator conjugate to a "good" clock Hamiltonian. We therefore show that, still following Pegg's formalism, it is possible to introduce in the PaW framework bounded clock Hamiltonians with an unequally-spaced energy spectrum with rational energy ratios. In this case time is described by a POVM and we demonstrate that Pegg's POVM states provide a consistent dynamical evolution of the system even if they are not orthogonal, and therefore partially undistinguishables.

Time-Dependent Pseudo-Hermitian Hamiltonians and a Hidden Geometric Aspect of Quantum Mechanics

A non-Hermitian operator H defined in a Hilbert space with inner product ⟨ · | · ⟩ may serve as the Hamiltonian for a unitary quantum system if it is η -pseudo-Hermitian for a metric operator (positive-definite automorphism) η . The latter defines the inner product ⟨ · | η · ⟩ of the physical Hilbert space H η of the system. For situations where some of the eigenstates of H depend on time, η becomes time-dependent. Therefore, the system has a non-stationary Hilbert space. Such quantum systems, which are also encountered in the study of quantum mechanics in cosmological backgrounds, suffer from a conflict between the unitarity of time evolution and the unobservability of the Hamiltonian. Their proper treatment requires a geometric framework which clarifies the notion of the energy observable and leads to a geometric extension of quantum mechanics (GEQM). We provide a general introduction to the subject, review some of the recent developments, offer a straightforward description of the Heisenberg-picture formulation of the dynamics for quantum systems having a time-dependent Hilbert space, and outline the Heisenberg-picture formulation of dynamics in GEQM.

Geometric Cauchy-Schwarz inequality

In this paper, using the generalized geometric commutator and geomutatorto develop the geometric anticommutator and anti-geomutator, then we strictly prove geometric Cauchy-Schwarz inequality in quantum method as a generalization of the Cauchy-Schwarz inequality based on the Hermitian operator. It certainly demonstrates a fact that environment variable is unavoidable for the revision of the Cauchy-Schwarz inequality for the quantum mechanics.

FUZZY SOFT HERMITIAN OPERATORS

In this paper, we define the fuzzy soft hermitian operator, which is a special type of fuzzy soft linear operators in fuzzy soft Hilbert spaces. Moreover, related theorems including fuzzy soft point spectrum theorem and more are introduced. Furthermore, an example in favor of the fuzzy soft hermitian operator and another one against it are investigated.