Uncertainty relation in quantum mechanics with quantum group symmetry

Physical systems powering motion and creating structure in a fixed amount of time dissipate energy and produce entropy. Whether living, synthetic or engineered, systems performing these dynamic functions must balance dissipation and speed. Here, we show that rates of energy and entropy exchange are subject to a speed limit—a time–information uncertainty relation—imposed by the rates of change in the information content of the system. This uncertainty relation bounds the time that elapses before the change in a thermodynamic quantity has the same magnitude as its s.d. From this general bound, we establish a family of speed limits for heat, dissipated/chemical work and entropy depending on the experimental constraints on the system and its environment. In all of these inequalities, the timescale of transient dynamical fluctuations is universally bounded by the Fisher information. Moreover, they all have a mathematical form that mirrors the Mandelstam–Tamm version of the time–energy uncertainty relation in quantum mechanics. These bounds on the speed of arbitrary observables apply to transient systems away from thermodynamic equilibrium, independent of the physical constraints on the stochastic dynamics or their function. A time–information uncertainty relation in thermodynamics has been derived, analogous to the time–energy uncertainty relation in quantum mechanics, imposing limits on the speed of energy and entropy exchange between a system and external reservoirs.

A brief review of various existing mathematical formulations of the uncertainty relations in quantum mechanics, containing variances of two or more non-commuting operators, is given. In particular, inequalities for the products of higher-order moments of a coordinate and a momentum are considered, as well as inequalities making the uncertainty relations more accurate when additional information about a quantum system is available (for example, the correlation coefficient or the degree of mixing of a quantum state characterized by the trace of the squared statistical operator). The special cases of two, three, and four operators are discussed in detail.

Heisenberg uncertainty relation in quantum mechanics sets the limit on the measurement precision of non-commuting observables in one system, which prevents us from measuring them accurately at the same time. However, quantum entanglement between two systems allows us to infer through Einstein-Podolsky-Rosen correlations two conjugate observables with precision better than what is allowed by Heisenberg uncertainty relation. With the help of the newly developed SU(1,) interferometer, we implement a scheme to jointly measure information encoded in multiple non-commuting observables of an optical field with a signal-to-noise ratio improvement of about 20% over the classical limit on all measured quantities simultaneously. This scheme can be generalized to the joint measurement of information in arbitrary number of non-commuting observables.

Inspired by string theory, Heisenberg's uncertainty principle can be generalized to include the photon-electron gravitational interaction, which leads to the Generalized Uncertainty Principle (GUP). Although GUP considers gravitational uncertainty at the minimum fundamental length scale in physics, it does not consider the effects of spacetime curvature on quantum mechanical uncertainty relations. The Extended Uncertainty Principle (EUP) is a generalization of Heisenberg's Uncertainty Principle that, unlike the GUP, applies to large length scales. GEUP is also a linear combination of EUP and GUP that creates minimal uncertainty on large length scales. The Einstein-Gauss-Bonnet theory (EGB) can be considered as one of the most promising candidates for modified gravity. In this paper, by using GUP, EUP, and GEUP, we intend to obtain the Hawking temperature of a four-dimensional EGB black hole in the asymptotically flat and (Anti)-de Sitter spacetime. We show that coupling constant, cosmological constant, mass, and radius significantly affect Hawking temperature and decrease or increase Hawking temperature depending on the chosen horizons.

Uncertainty measures must not depend on the choice of origin of the measurement scale; it is therefore argued that quantum-mechanical uncertainty relations, too, should remain invariant under changes of origin. These points have often been neglected in dealing with angle observables. Known measures of location and uncertainty for angles are surveyed. The angle variance angv {o} is defined and discussed. It is particularly suited to the needs of quantum theory, because of its affinity to the Hilbert space metric, and its use of the basic sine and cosine operators. Corresponding uncertainty relations involving azimuthal or phase angles are indicated, and their relevance to study and definition of coherent states is briefly reviewed.

Developing quantum machines which can outperform their classical counterparts, thereby achieving quantum supremacy or quantum advantage, is a major aim of the current research on quantum thermodynamics and quantum technologies. Here, we show that a fast-modulated cyclic quantum heat machine operating in the non-Markovian regime can lead to significant heat current and power boosts induced by the anti-Zeno effect. Such boosts signify a quantum advantage over almost all heat machines proposed thus far that operate in the conventional Markovian regime, where the quantumness of the system-bath interaction plays no role. The present effect owes its origin to the time-energy uncertainty relation in quantum mechanics, which may result in enhanced system-bath energy exchange for modulation periods shorter than the bath correlation-time.

A general procedure to construct criteria for identifying genuine multipartite continuous variable entanglement is presented. It relies on the proper definition of adequate global operators describing the multipartite system, the positive partial transpose criterion of separability, and quantum mechanical uncertainty relations. As a consequence, each criterion encountered consists of a single inequality that is nicely computable and experimentally feasible, and that when violated is sufficient condition for genuine multipartite entanglement. Additionally we show that the previous work of van Loock and Furusawa [Phys. Rev. A, 67, 052315 (2003)] is a special case of our result that includes strongest criteria to detect entanglement.

Bohr and Heisenberg suggested that the thermodynamical quantities of temperature and energy are complementary in the same way as position and momentum in quantum mechanics. Roughly speaking their idea was that a definite temperature can be attributed to a system only if it is submerged in a heat bath, in which case energy fluctuations are unavoidable. On the other hand, a definite energy can be assigned only to systems in thermal isolation, thus excluding the simultaneous determination of its temperature. Rosenfeld extended this analogy with quantum mechanics and obtained a quantitative uncertainty relation in the form ΔU Δ(1/T) ≥ k, where k is Boltzmann's constant. The two “extreme” cases of this relation would then characterize this complementarity between isolation (U definite) and contact with a heat bath (T definite). Other formulations of the thermodynamical uncertainty relations were proposed by Mandelbrot (1956, 1989), Lindhard (1986), and Lavenda (1987, 1991). This work, however, has not led to a consensus in the literature. It is shown here that the uncertainty relation for temperature and energy in the version of Mandelbrot is indeed exactly analogous to modern formulations of the quantum mechanical uncertainty relations. However, his relation holds only for the canonical distribution, describing a system in contact with a heat bath. There is, therefore, no complementarily between this situation and a thermally isolated system.

The main aim of this paper is to develop an alternative approach within the phase space quantum mechanics for the pseudoharmonic oscillator potential, which allows us to derive analytical expressions for the Wigner distribution function and the use of Weyl’s tranforms enables us the determination of the general relation for the moments associated with higher-order position and momentum operators. We see that adding the dimensionless parameter α to the key transformation makes it possible to improve the general analytical formula for energy levels, which explicitly depends on the quantum number (state) n and the angular quantum number l, and to approach the expectation value of the squared-momentum p^2_r with a fairly good value, and so the energy eigenvalues. These latter are evaluated numerically for several diatomic molecular systems. Additionally, we verify the Heisenberg-Born-Jordan-Dirac commutation relation and Heisenberg’s uncertainty principle.

Theoretical formulation of finite-dimensional discrete phase spaces: I. Algebraic structures and uncertainty principles

Some inequalities satisfied by the quantities of information of Fisher and Shannon

This paper has been motivated by a recent paper by Dey [Phys. Rev. D 91, 044024 (2015)] on the known Arik-Coon $q$ oscillator. We construct $q$ coherent, even and odd $q$-cat states in Fock representation for the Biedenharn-Macfarlane $q$ oscillator with $q>1$ and study their nonclassical properties. The $q$-coherent states minimize the Heisenberg uncertainty relation between the generalized position and momentum operators as well as the $x$ and $y$ components of a $q$-deformed $\text{su}(1,1)$ algebra in the Schwinger boson representation. The latter is also minimized by the even and odd $q$-cat states. We show that, contrary to the undeformed harmonic oscillator, the squeezing effect in both position and momentum operators can be exhibited by odd $q$-cat states. It is also violated by even $q$-cat states. Furthermore, it is shown that the antibunching effect and sub-Poissonian or super-Poissonian statistics can simultaneously appear by each of the even or odd $q$-cat states. Finally, a unitary Fock representation of the $q$-deformed $\text{su}(1,1)$ algebra is obtained by the $q$-deformed Bargmann-Fock realization.

We show that there is a close relationship between quantum mechanics and ordinary probability theory. The main difference is that in quantum mechanics the probability is computed in terms of an amplitude function, while in probability theory a probability distribution is used. Applying this idea, we then construct an amplitude model for quantum mechanics on phase space. In this model, states are represented by amplitude functions and observables are represented by functions on phase space. If we now postulate a conjugation condition, the model provides the same predictions as conventional quantum mechanics. In particular, we obtain the usual quantum marginal probabilities, conditional probabilities and expectations. The commutation relations and uncertainty principle also follow. Moreover Schrodinger's equation is shown to be an averaged version of Hamilton's equation in classical mechanics.

The uncertainty relation is considered to be one of key features of quantum mechanics which distinguishes quantum and classical systems. Recently, we developed a new formulation of the uncertainty relation based on the generalized scheme of variational principle, the stochastic variational method (SVM). In this method, the uncertainty relation is related to the nondifferentiability of observables and thus can be obtained even in classical stochastic systems. This new formulation resolves the famous paradox in quantum mechanics, the angular uncertainty relation without introducing artificial assumptions. In this paper, we show that the fluctuations of position and momentum for a nonrelativistic viscous fluid element satisfies the uncertainty relation analogous to the corresponding quantum mechanical one. Such a fluctuation is sensitive to the temperature gradient at the freezeout surface and can affect the collective flow anisotropy in relativistic heavy‐ion collisions.

Quantum group and quantum symmetry