Uncertainty Relations in Hydrodynamics

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.

The Heisenberg uncertainty principle (HUP) plays an important role in quantum mechanics. It is necessary to demonstrate this principle experimentally for undergraduates to understand HUP clearly. In this paper, we originally employ the quantum quadrature operators to demonstrate HUP in undergraduate laboratories. Starting from the quantization of electromagnetic field, we briefly review the definition of quadrature operators and demonstrate HUP via the variances of quadrature operators. And the scheme of HUP demonstration with homodyne detection is introduced. More importantly, we develop a tabletop experimental apparatus for the demonstration of HUP in undergraduate laboratories. Our proposal focuses on the basic quantum mechanics concepts, experimental principles, and instruction as well as experimental techniques. It provides a way to help undergraduates, especially the ones who will go on graduate study in quantum technology in the future, make connections between abstract quantum mechanics concepts and concrete experiments, which makes the concepts better understood.

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.

If nature allowed nonlocal correlations other than those predicted by quantum mechanics, would that contradict some physical principle? Various approaches have been put forward in the past two decades in an attempt to single out quantum nonlocality. However, none of them can explain the set of quantum correlations arising in the simplest scenarios. In [Sci. Adv. 5, eaav8370] we have shown that generalized uncertainty relations, as well as a specific notion of locality, give rise to both familiar and new characterizations of quantum correlations. In particular, we have identified a condition, relativistic independence, which states that uncertainty relations are local in the sense that they cannot be influenced by other experimenters’ choices of measuring instruments. We proved that theories with nonlocal correlations stronger than the quantum ones do not satisfy this notion of locality, and therefore, they either violate the underlying generalized uncertainty relations or allow experimenters to nonlocally tamper with the uncertainty relations of their peers. In this talk I will discuss several implications of this work to multiplicative Bell inequalities, causality and networks. Finally, I will support the claim that locally consistent indeterminism should be understood as the enabler of most quantum phenomena (and possibly all of them).

Unification theory of classical statistical uncertainty relation and quantum uncertainty relation and its applications

After more than ninety years of discussions over the uncertainty principle, there is still no universal agreement on what the principle states. The Robertson uncertainty relation (incorporating standard deviations) is given as the mathematical expression of the principle in most quantum mechanics textbooks. However, the uncertainty principle is not merely a statement of what any of the several uncertainty relations affirm. It is suggested that a better approach would be to present the uncertainty principle as a statement about the probability distributions of incompatible variables and the resulting restrictions on quantum states.

The uncertainty principle in quantum mechanics is a fundamental relation with different forms, including Heisenberg's uncertainty relation and Schrodinger's uncertainty relation. We discuss the gen...

In quantum mechanics it is well known that observables are represented by hermitian matrices (or operators). Uncertainty relations are represented as some kinds of trace inequalities satisfied by two observables and one density matrix (or operator). By releasing the hermitian restriction on observables, we obtained non-hermite uncertainty relations. In this paper we give several non-hermitian extensions of Heisenberg or Schrödinger type q uncertainty relations for generalized skew information under some conditions.

Heisenberg’s uncertainty principle (HUP) is a central concept in quantum mechanics that reveals inherent limitations in the measurement of microscopic particles and profoundly affects the understanding of the physical world. This paper reviews the physical significance of HUP and its applications and research progress in modern physics. The complexity of measurements in quantum systems is revealed by deriving the uncertainty relation and analysing the wavefunction collapse and the classicalisation of quantum states. This paper summarises the current experimental verification of the principle of measurement uncertainty in different quantum systems and discusses the research progress in improving measurement accuracy through quantum entanglement. Recent studies have shown that controlling quantum entangled states can effectively reduce measurement disturbances and thus improve the measurement accuracy of quantum systems without violating the principle of mismeasurement. Despite the many advances made, there are still challenges to further improve the measurement accuracy in complex quantum systems. Future research will continue to explore the potential for applications in this area.

Heisenberg's uncertainty principle is formulated for a set of generalized measurements within the framework of majorization theory, resulting in a partial uncertainty order on probability vectors that is stronger than those based on quasientropic measures. The theorem that emerges from this formulation guarantees that the uncertainty of the results of a set of generalized measurements without a common eigenstate has an inviolable lower bound which depends on the measurement set but not the state. A corollary to this theorem yields a parallel formulation of the uncertainty principle for generalized measurements corresponding to the entire class of quasientropic measures. Optimal majorization bounds for two and three mutually unbiased bases in two dimensions are calculated. Similarly, the leading term of the majorization bound for position and momentum measurements is calculated which provides a strong statement of Heisenberg's uncertainty principle in direct operational terms. Another theorem provides a majorization condition for the least-uncertain generalized measurement of a given state with interesting physical implications.

Heisenberg’s uncertainty principle (HUP) is a central concept in quantum mechanics that reveals inherent limitations in the measurement of microscopic particles and profoundly affects the understanding of the physical world. This paper reviews the physical significance of HUP and its applications and research progress in modern physics. The complexity of measurements in quantum systems is revealed by deriving the uncertainty relation and analysing the wavefunction collapse and the classicalisation of quantum states. This paper summarises the current experimental verification of the principle of measurement uncertainty in different quantum systems and discusses the research progress in improving measurement accuracy through quantum entanglement. Recent studies have shown that controlling quantum entangled states can effectively reduce measurement disturbances and thus improve the measurement accuracy of quantum systems without violating the principle of mismeasurement. Despite the many advances made, there are still challenges to further improve the measurement accuracy in complex quantum systems. Future research will continue to explore the potential for applications in this area.

The stochastic variational method is applied to particle systems and continuum media. As a brief review of this method, we first discuss the application to particle Lagrangians and derive a diffusion-type equation and the Schrödinger equation with the minimum gauge coupling. We further extend the application of the stochastic variational method to Lagrangians of continuum media and show that the Navier–Stokes, Gross–Pitaevskii and generalized diffusion equations are derived. The correction term for the Navier–Stokes equation is also obtained in this method. We discuss the meaning of this correction by comparing it with the diffusion equation.

At the heart of the black hole information loss paradox and the firewall controversy lies the conflict between quantum mechanics and general relativity. Much has been said about quantum corrections to general relativity, but much less in the opposite direction. It is therefore crucial to examine possible corrections to quantum mechanics due to gravity. Indeed, the Heisenberg Uncertainty Principle is one profound feature of quantum mechanics, which nevertheless may receive correction when gravitational effects become important. Such generalized uncertainty principle [GUP] has been motivated from not only quite general considerations of quantum mechanics and gravity, but also string theoretic arguments. We examine the role of GUP in the context of black hole complementarity. We find that while complementarity can be violated by large N rescaling if one assumes only the Heisenberg’s Uncertainty Principle, the application of GUP may save complementarity, but only if certain N -dependence is also assumed. This raises two important questions beyond the scope of this work, i.e., whether GUP really has the proposed form of N -dependence, and whether black hole complementarity is indeed correct.

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.

We extend a class of recently derived thermodynamic uncertainty relations to vector-valued observables. In contrast to the scalar-valued observables examined previously, this multidimensional thermodynamic uncertainty relation provides a natural way to study currents in high-dimensional systems and to obtain relations between different observables. Our proof is based on the generalized Crámer–Rao inequality, which we interpret as a relation between physical observables and the Fisher information. This allows us to develop high-dimensional versions of both the original, steady state uncertainty relation and the more recently obtained generalized uncertainty relation for time-periodic systems. We apply the multidimensional uncertainty relation to obtain a new constraint on the performance of steady-state heat engines, which is tighter than previous bounds and reveals the role of heat-work correlations. As a second application, we show that the uncertainty relation is connected to a bound on the differential mobility. As a result of this connection, we find that a necessary condition for equality in the uncertainty relation is that the system obeys the equilibrium fluctuation–dissipation relation.