Narrow Results

The relation between scatters of time and energy is well known in Classical and Quantum Physics, but the interpretations of such relation are not well understood. What is the meaning of a time scatter in a measurement? What is the meaning of the correlated energy gap? Why should they combine to give a value greater or equal to a theoretical lower bound? In this paper, we discuss uncertainty in measurement from the computational point of view and introduce the archetype of a generic stochastic oracle to a Turing machine.

In this paper, we investigate the representation of the Wigner algebra by using the q-calculus like approach. For this algebra, the coherent state is constructed and the uncertainty relation for the ground state is discussed. Following Wigner's approach, the Wigner deformation of the fermion algebra is obtained and its Fock space is shown to have finite dimension.

In this paper, we study some optical properties of the superposition of q-deformed coherent states. We study the effect of the deformation on the non-classicality of q-coherent states as well as some superposition of such states.

A philosophically consistent axiomatic approach to classical and quantum mechanics is given. The approach realizes a strong formal implementation of Bohr's correspondence principle. In all instances, classical and quantum concepts are fully parallel: the same general theory has a classical realization and a quantum realization. Extending the ''probability via expectation'' approach of Whittle to noncommuting quantities, this paper defines quantities, ensembles, and experiments as mathematical concepts and shows how to model complementarity, uncertainty, probability, nonlocality and dynamics in these terms. The approach carries no connotation of unlimited repeatability; hence it can be applied to unique systems such as the universe. Consistent experiments provide an elegant solution to the reality problem, confirming the insistence of the orthodox Copenhagen interpretation on that there is nothing but ensembles, while avoiding its elusive reality picture. The weak law of large numbers explains the emergence of classical properties for macroscopic systems.

We obtained the uncertainty relation in squeezed states for a time-dependent oscillator. The uncertainty relation in coherent states is same as that of the number states with n=0. However, the uncertainty relation in squeezed states does not satisfy this property and depends on squeezing parameter c. For instance, the uncertainty relation is ℏ/2 which is the minimum value as far as quantum mechanics permits for c=1, same as that in coherent state for c=±∞, and infinity for c=-1. If the time-dependency of the Hamiltonian for the system vanishes, the uncertainty relation in squeezed states will no longer depend on c and becomes the same as that in number state with n=0, like the uncertainty relation in coherent states.

Uncertainty relations for the time-dependent singular oscillator in the number state and in the coherent state are investigated. We applied our developement to the Caldirola–Kanai oscillator perturbed by a singularity. For this system, the variation (Δx) decreased exponentially while (Δp) increased exponentially with time both in the number and in the coherent states. As k → 0 and χ → 0, the number state uncertainty relation in the ground state becomes 0.583216ℏ which is somewhat larger than that of the standard harmonic oscillator, ℏ/2. On the other hand, the uncertainty relation in all excited states become smaller than that of the standard harmonic oscillator with the same quantum number n. However, as k → ∞ and χ → 0, the uncertainty relations of the system approach the uncertainty relations of the standard harmonic oscillator, (n+1/2)ℏ.

We address the question, does a system A being entangled with another system B, put any constraints on the Heisenberg uncertainty relation (or the Schrödinger–Robertson inequality)? We find that the equality of the uncertainty relation cannot be reached for any two noncommuting observables, for finite dimensional Hilbert spaces if the Schmidt rank of the entangled state is maximal. One consequence is that the lower bound of the uncertainty relation can never be attained for any two observables for qubits, if the state is entangled. For infinite-dimensional Hilbert space too, we show that there is a class of physically interesting entangled states for which no two noncommuting observables can attain the minimum uncertainty equality.

We put forward a measure for evaluating quantum speed limit for arbitrary mixed states of open systems by means of trace distance. Compared with some present measures, it can provide an optimal bound to the speed of the evolution. The dynamical crossover from no speedup region to speedup region happens during the spontaneous decay of an atom. The evolution is characteristic of the alternating behavior between quantum acceleration and deceleration in the strong coupling case. Under the condition of detuning, the evolution can be initially accelerated and then decelerated to a normal process either in the weak or strong coupling regime. In accordance with the uncertainty relation, we demonstrate that the potential capacity for quantum speedup evolution is closely related to the energy feedback from the reservoir to the system. The negative decay rate for the evolution results in the speedup process where the photons previously emitted by the atom are reabsorbed at a later time. The values of the spontaneous decay rate become positive after a long enough time, which results in the evolution with no speedup potential.

In this paper, we extend existing variance-based sum uncertainty relations for pure states to those for mixed states by a mathematical approach. Furthermore, we show that the monotonicity of the standard deviation of observables induces a variance-based sum uncertainty relation. Finally, the multiobservable case is also discussed.

Quantum information-theoretic approach had been identified as a way to understand the foundations of quantum mechanics as early as 1950 due to Shannon. However, there hasn’t been enough advancement or rigorous development of the subject. In the following paper we try to find the relationship between a general quantum mechanical observable and von Neumann entropy. We find that the expectation values and the uncertainties of the observables have bounds which depend on the entropy. The results also show that von Neumann entropy is not just the uncertainty of the state but also encompasses the information about expectation values and uncertainties of any observable which depend on the observers choice for a particular measurement. Also a reverse uncertainty relation is derived for $n$ quantum mechanical observables.

The normalized even and odd $q$-cat states corresponding to Arik–Coon $q$-oscillator on the noncommutative complex plane ${ℂ}_{q^{-1}}$ are constructed as the eigenstates of the lowering operator of a $q$-deformed $\mathrm{\text{su}}(1,1)$ algebra with the left eigenvalues. We present the appropriate noncommutative measures in order to realize the resolution of the identity condition by the even and odd $q$-cat states. Then, we obtain the $q$-Bargmann–Fock realizations of the Fock representation of the $q$-deformed $\mathrm{\text{su}}(1,1)$ algebra as well as the inner products of standard states in the $q$-Bargmann representations of the even and odd subspaces. Also, the Euler’s formula of the $q$-factorial and the Gaussian integrals based on the noncommutative $q$-integration are obtained. Violation of the uncertainty relation, photon antibunching effect and sub-Poissonian photon statistics by the even and odd $q$-cat states are considered in the cases $0<q<1$ and $q>1$.

The set of maximally fermion–boson entangled Bell super-coherent states is introduced. A superposition of these states with separable bosonic coherent states, represented by points on the super-Bloch sphere, we call the Bell-based super-coherent states. Entanglement of bosonic and fermionic degrees of freedom in these states is studied by using displacement bosonic operator. It acts on the super-qubit reference state, representing superposition of the zero and the one super-number states, forming computational basis super-states. We show that the states are completely characterized by displaced Fock states, as a superposition with non-classical, the photon added coherent states, and the entanglement is independent of coherent state parameter $\alpha$ and of the time evolution. In contrast to never orthogonal Glauber coherent states, our entangled super-coherent states can be orthogonal. The uncertainty relation in the states is monotonically growing function of the concurrence and for entangled states we get non-classical quadrature squeezing and representation of uncertainty by ratio of two Fibonacci numbers. The sequence of concurrences, and corresponding uncertainties $\hslash F_n/F_{n+1}$, in the limit $n\to \infty$, convergent to the Golden ratio uncertainty $\hslash /\phi$, where $\phi =\frac{1+\sqrt{5}}{2}$ is found.

The goal of this paper is to calculate exactly the average of uncertainty product of two bounded observables and to establish its typicality over the whole set of finite dimensional quantum pure states. Here we use the uniform ensembles of pure and isospectral states as well as the states distributed uniformly according to the measure induced by the Hilbert-Schmidt norm. Firstly, we investigate the average uncertainty of an observable over isospectral density matrices. By letting the isospectral density matrices be of rank-one, we get the average uncertainty of an observable restricted to pure quantum states. These results can help us check how large is the gap between the uncertainty product and any lower bounds obtained for the uncertainty product. Although our method in the present paper cannot give a tighter lower bound of uncertainty product for bounded observables, it can help us drop any one that is not substantially tighter than the known one.

The Dark Energy became now a commonly-accepted paradigm of cosmology, but its physical essence remains absolutely unknown, and its numerical values are drastically different in the early and modern Universe. The Dark Energy is usually introduced in the contemporary literature either by postulating some additional terms in the Lagrangians or by employing the empirical equations of state. In the present work, we try to look at this problem from a more general point of view, namely, employing the quantum-mechanical uncertainty relation between the time and energy in the Mandelstam–Tamm form, which is appropriate for the long-term evolution of quantum systems. This leads us to the time-dependent effective Lambda-term, decaying as 1/t. The corresponding cosmological model possesses a number of quite appealing features: (1) While in the standard cosmology there are a few very different expansion stages (governed by the Lambda-term, radiation, dust-like matter, and Lambda-term again), our model provides a universal description of the entire evolution of the Universe by the same “quasi-exponential” function. (2) As follows from the analysis of causal structure, the present-day cosmological horizon comprises a single domain developing from the Big Bang. Therefore, the problems of homogeneity and isotropy of the matter, the absence of topological defects, etc. should be naturally resolved. (3) At last, our model naturally explains the observed approximately flat 3D space, i.e., solution with zero curvature is formed “dynamically”, starting from the arbitrary initial conditions.

Floquet theory together with the resonating averages method (RAM) were used to solve the Schrödinger equation of periodically-time-dependent harmonic oscillators. The approach gives another way of solving this equation, and lead to identical analytical solutions to those published in the literature based on different other methods.

This article is a short review on our recent results on uncertainty relations with skew informations. In final section, we give the problem on uncertainty relations with skew informations.

We discuss properties of L²-eigenfunctions of Schrödinger operators and elliptic partial differential operators. The focus is set on unique continuation principles and equidistribution properties. We review recent results and announce new ones.