Uncertainty measures and uncertainty relations for angle observables

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.

Quantum 1/f noise is the manifestation of the coherent and conventional quantum 1/f effects (Q1/fE). The conventional Q1/fE is a fundamental quantum fluctuation of physical cross sections σ and process rates Γ, caused by the bremsstrahlung (recoil) energy and momentum losses of charged particles, when they are scattered, or accelerated in any way. The closely related coherent Q1/fE is present in any current carried by many particles. It is caused by the energy spread characterizing any coherent state of the electromagnetic field oscillators. According to the Heisenberg's uncertainty principle, because an approximation of the phase or position variable is known, exact knowledge of the energy is precluded. This energy spread results in nonstationary energy values, or fluctuations in the energy of the oscillators. To find the spectral density of these inescapable basic fluctuations, which are known to characterize any quantum state, which is not an energy eigenstate, we use an elementary physical derivation based on Schrödinger's definition of coherent states, which can be supplemented by a rigorous derivation from a well-known quantum-electrodynamical branch-point propagator. The example of a simple harmonic oscillator is also useful for illustrating the uncertainty that arises due to Q 1/f Noise. Clearly illustrating the relation between the uncertainty principle and Q 1/f noise.

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.

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.

The commutation relations, uncertainty relations, and spectra of position and momentum operators were studied within the framework of quantum group symmetric Heisenberg algebras and their (Bargmann) Fock representations. As an effect of the underlying noncommutative geometry, a length and a momentum scale appear, leading to the existence of nonzero minimal uncertainties in the positions and momenta. The usual quantum mechanical behavior is recovered as a limiting case for not too small and not too large distances and momenta.

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.

The uncertainty principle states that the amount of uncertainty of a particle’s velocity multiplied by the amount of uncertainty of a particle’s position is a constant. In other words, the more certain about particle’s velocity, the less certain about particle’s position and vice versa. This paper will focus on the uncertainty measure is not a constant but a variable quantity. The uncertainty principle is situation based realism. In certain situations, it is more and in certain situations, it is less. The variable uncertainty principle is the total amount of uncertainty of position and velocity multiplied by the total amount of certainty of position and velocity is a constant. I will also discuss that the physical time is a complex quantity and the variable uncertainty principle actually depends on the complex physical time. The complex physical time has two parts - the real physical time and the imaginary physical time.

Modern science is eliminating the old, unrealistic approach to experimental errors. The Heisenberg principle of uncertainty has effected a fundamental change in the scientific attitude toward experimental errors and toward the degree of exactness in stating physical laws. Quantum physics has brought about the realization that a measurement or a mere observation at the quantum level perturbs the observed system in such a way that corrections cannot be made for the ``loading'' effect of the measuring instrument. There is something more basic and fundamental to our limitations in measurements than the lack of perfection in experimental planning and equipment. This article examines the types of uncertainty in measurements or observations. The interest then focuses on the fundamental importance of the relation between experimental uncertainty in measuring, observing, or in any way identifying a physical quantity, and the energy required to accomplish this in a certain allotted interval of time. This last relation has a direct bearing on the speed limitations of measurements.

We use a spectrum interpolation technique to improve the distributed strain measurement accuracy in a Rayleigh-scatter-based optical frequency domain reflectometry sensing system. We demonstrate that strain accuracy is not limited by the "uncertainty principle" that exists in the time-frequency analysis. Different interpolation methods are investigated and used to improve the accuracy of peak position of the cross-correlation and, therefore, improve the accuracy of the strain. Interpolation implemented by padding zeros on one side of the windowed data in the spatial domain, before the inverse fast Fourier transform, is found to have the best accuracy. Using this method, the strain accuracy and resolution are both improved without decreasing the spatial resolution. The strain of 3μϵ within the spatial resolution of 1cm at the position of 21.4m is distinguished, and the measurement uncertainty is 3.3μϵ.

As a very fundamental principle in quantum physics, uncertainty principle has been studied intensively via various uncertainty inequalities. A natural and fundamental question is whether an equality exists for the uncertainty principle. Here we derive an entropic uncertainty equality relation for a bipartite system consisting of a quantum system and a coupled quantum memory, based on the information measure introduced by Brukner and Zeilinger (Phys. Rev. Lett. 83:3354, 1999). The equality indicates that the sum of measurement uncertainties over a complete set of mutually unbiased bases on a subsystem is equal to a total, fixed uncertainty determined by the initial bipartite state. For the special case where the system and the memory are the maximally entangled, all of the uncertainties related to each mutually unbiased base measurement are zero, which is substantially different from the uncertainty inequality relation. The results are meaningful for fundamental reasons and give rise to operational applications such as in quantum random number generation and quantum guessing games. Moreover, we experimentally verify the measurement uncertainty relation in the presence of quantum memory on a five-qubit spin system by directly measuring the corresponding quantum mechanical observables, rather than quantum state tomography in all the previous experiments of testing entropic uncertainty relations.

One of the formulations of Heisenberg uncertainty principle, concerning so-called measurement uncertainty, states that the measurement of one observable modifies the statistics of the other. Here, we derive such a measurement uncertainty principle from two comprehensible assumptions: impossibility of instantaneous messaging at a distance (no-signaling), and violation of Bell inequalities (nonlocality). The uncertainty is established for a pair of observables of one of two spatially separated systems that exhibit nonlocal correlations. To this end, we introduce a gentle form of measurement which acquires partial information about one of the observables. We then bound disturbance of the remaining observables by the amount of information gained from the gentle measurement, minus a correction depending on the degree of nonlocality. The obtained quantitative expression resembles the quantum mechanical formulations, yet it is derived without the quantum formalism and complements the known qualitative effect of disturbance implied by nonlocality and no-signaling.

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.

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

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.