Uncertainty Relation In Quantum Mechanics Research Articles

Possible enhancements of collective flow anisotropy induced by uncertainty relation for fluid element

In the stochastic formulation of viscous hydrodynamics, the velocity of a fluid element fluctuates satisfying a similar relation to the quantum-mechanical uncertainty relation. Using a non-relativistic toy model, we show that the presence of a velocity fluctuation increases the local anisotropy of the momentum distributions of produced hadrons and, consequently, the collective flow parameters such as v 2 are enhanced.

Hawking temperature for 4D-Einstein-Gauss-Bonnet black holes from uncertainty principle

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.

Experimental investigation of the relation between measurement uncertainties and non-local quantum correlations

Bell’s inequalities are defined by sums of correlations involving non-commuting observables in each of the two systems. Violations of Bell’s inequalities are only possible because the precision of any joint measurement of these observables will be limited by quantum mechanical uncertainty relations. In this paper we explore the relation between the local measurement uncertainties and the magnitude of the correlations by preparing polarization entangled photon pairs and performing joint measurements of non-commuting polarization components at different uncertainty trade-offs. The change in measurement visibility reveals the existence of a non-trivial balance between the measurement uncertainties where the probabilities of a specific pair of measurement outcomes approaches zero because of the particular combination of enhancement and suppression of the experimentally observed correlations. The occurrence of these high-contrast results shows that the quantum correlations between the photons are close to their maximal value, confirming that the Cirel’son bound of Bell’s inequality violations is defined by the minimal uncertainties that limit the precision of joint measurements.

From Rényi Entropy Power to Information Scan of Quantum States.

In this paper, we generalize the notion of Shannon’s entropy power to the Rényi-entropy setting. With this, we propose generalizations of the de Bruijn identity, isoperimetric inequality, or Stam inequality. This framework not only allows for finding new estimation inequalities, but it also provides a convenient technical framework for the derivation of a one-parameter family of Rényi-entropy-power-based quantum-mechanical uncertainty relations. To illustrate the usefulness of the Rényi entropy power obtained, we show how the information probability distribution associated with a quantum state can be reconstructed in a process that is akin to quantum-state tomography. We illustrate the inner workings of this with the so-called “cat states”, which are of fundamental interest and practical use in schemes such as quantum metrology. Salient issues, including the extension of the notion of entropy power to Tsallis entropy and ensuing implications in estimation theory, are also briefly discussed.

Time–information uncertainty relations in thermodynamics

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-enhanced finite-time Otto cycle

We use fast periodic control to realize finite-time Otto cycles exhibiting quantum advantage. Such periodic modulation of the working medium - bath interaction Hamiltonian during the thermalization strokes can give rise to non-Markovian anti-Zeno dynamics, and corresponding reduction in the thermalization times. Faster thermalization can in turn significantly enhance the power output in engines, or equivalently, the rate of refrigeration in refrigerators. This improvement in performance of dynamically controlled Otto thermal machines arises due to the time-energy uncertainty relation of quantum mechanics.

Uncertainty and trade-offs in quantum multiparameter estimation

Uncertainty relations in quantum mechanics express bounds on our ability to simultaneously obtain knowledge about expectation values of non-commuting observables of a quantum system. They quantify trade-offs in accuracy between complementary pieces of information about the system. In quantum multiparameter estimation, such trade-offs occur for the precision achievable for different parameters characterizing a density matrix: an uncertainty relation emerges between the achievable variances of the different estimators. This is in contrast to classical multiparameter estimation, where simultaneous optimal precision is attainable in the asymptotic limit. We study trade-off relations that follow from known tight bounds in quantum multiparameter estimation. We compute trade-off curves and surfaces from Cramér–Rao type bounds which provide a compelling graphical representation of the information encoded in such bounds, and argue that bounds on simultaneously achievable precision in quantum multiparameter estimation should be regarded as measurement uncertainty relations. From the state-dependent bounds on the expected cost in parameter estimation, we derive a state-independent uncertainty relation between the parameters of a qubit system.

Anti-Zeno quantum advantage in fast-driven heat machines

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.

Gravitational Space-Time Quantization for Charged Wormholes and the Diophantine Uncertainty Relation

This research work proceeds from the assumption, which was still considered by Einstein, that the quantization of gravity does not require additional external procedures: quantum phenomena can be a consequence of the properties of the universal gravitational interaction, which maps any physical field upon the space-time geometry. Therefore, an attempt is made in this research work to reduce the quantization of physical fields in GRT to the space-time quantization. Three reasons for quantum phenomena are considered: Partition of space-time into a set of unconnected Novikov’s R- and T-domains impenetrable for light paths; the set is generated by the invariance of Einstein’s equations with respect to dual mappings; The existence of electric charge quanta of wormholes, which geometrically describe elementary particles in GRT. This gives rise to a discrete spectrum of their physical and geometric parameters governed by Diophantine equations. It is shown that the fundamental constants (electric charge, rest masses of an electron and a proton) are interconnected arithmetically; The existence of the so-called Diophantine catastrophe, when fluctuations in the values of physical constants tending to zero lead to fluctuations in the number of electric charges and the number of nucleons at the wormhole throats, which tend to infinity, so that the product of the increments of these numbers by the increment of physical constants forms a relation equivalent to the uncertainty relation in quantum mechanics. This suggests that space-time cannot but fluctuate, and, moreover, its fluctuations are bounded from below, so that all processes become chaotic, and the observables become averaged over this chaos.

Uncertainty Relations for Coherence**Supported by the National Natural Science Foundation of China under Grant No. 11875317, the National Center for Mathematics and Interdisciplinary Sciences, and Chinese Academy of Sciences under Grant No. Y029152K51

Quantum mechanical uncertainty relations are fundamental consequences of the incompatible nature of noncommuting observables. In terms of the coherence measure based on the Wigner-Yanase skew information, we establish several uncertainty relations for coherence with respect to von Neumann measurements, mutually unbiased bases (MUBs), and general symmetric informationally complete positive operator valued measurements (SIC-POVMs), respectively. Since coherence is intimately connected with quantum uncertainties, the obtained uncertainty relations are of intrinsically quantum nature, in contrast to the conventional uncertainty relations expressed in terms of variance, which are of hybrid nature (mixing both classical and quantum uncertainties). From a dual viewpoint, we also derive some uncertainty relations for coherence of quantum states with respect to a fixed measurement. In particular, it is shown that if the density operators representing the quantum states do not commute, then there is no measurement (reference basis) such that the coherence of these states can be simultaneously small.

Comment on “Quantum reciprocity relations for fluctuations of position and momentum”

The authors of this paper [M. Pandit et al., Phys. Rev. A 100, 012131 (2019)] proposed an alternative formulation of the uncertainty relation in quantum mechanics in terms of the Lipschitz constants. They evaluated these constants in various cases and decided that the product of the square roots of Lipschitz constants for position and momentum probability distributions is bounded below by $0.3/\ensuremath{\hbar}$. In this comment, I invalidate this claim by giving an example of wave functions for which the value of the product of the Lipschitz constants for position and momentum does not obey this bound and tends to zero.

Understanding two-photon double ionization of helium from the perspective of the characteristic time of dynamic transitions

By using the B-spline numerical method, we investigate a two-photon double-ionization (TPDI) process of helium in a high-frequency laser field with its carrier frequency ranging from 1.6 to 3.0 a.u. and the pulse duration ranging from 75 to 160 attoseconds. We found that there exists a characteristic time tc for a TPDI process, such that the pattern of energy distribution of two ionized electrons presents a peak or two, depending respectively on whether the pulse duration is shorter or longer than tc. Especially, when the pulse duration is larger than tc, the TPDI spectrum shows a double-peak structure which is attributed to the fact that most of the electron–electron Coulomb interaction energy is acquired by a single electron during their oscillation around the nucleus before the two electrons leave, and hence the double-peak structure cannot be identified as a signal of sequential ionization. Additionally, if the carrier frequency is less than the ionization energy of He+, i.e. that carrier frequency is in the so called correlated region, tc is not a fixed value, and it increases as the carrier frequency decreases; while if the carrier frequency is greater than the ionization energy of He+, i.e. the carrier frequency is in the so called non-correlated region, tc is fixed at about 105 attoseconds. We further found that, for a helium-like ion in its ground state, the characteristic time for when the carrier frequency is larger than the ionization energy of the second electron has a key relation with the Coulomb interaction energy between the two electrons, which can be expressed as , a type of quantum mechanical uncertainty relation between time and energy. In addition, this relation can be attributed to the existence of a minimal evolution time from the ground state to a double ionization state with two electrons carrying different energies. These results may shed light on a deeper understanding of many-electron quantum dynamical processes.

A Unified Model of Dark Energy Based on the Mandelstam-Tamm Uncertainty Relation

It is commonly recognized now that Dark Energy (Lambda-term) is of crucial importance both at the early (inflationary) stage of cosmological evolution and at the present time. However, little is known about its nature and origin till now. In particular, it is still unclear if Lambda-term is a new fundamental constant or represents just an effective contribution from the underlying field theory. Here, we show that a quite promising and universal approach to this problem might be based on the Mandelstam-Tamm uncertainty relation of quantum mechanics. As a result, we get the effective Lambda-term that is important throughout the entire history of the Universe. Besides, such an approach requires a substantial reconsideration of some other cosmological parameters, e.g., the age of the Universe.

Loss-tolerant quantum dense metrology with SU(1,1) interferometer.

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.

Generalized Uncertainty Relations in Quantum Mechanics and the Principles of Completeness in Physics

Generalized Uncertainty Relations in Quantum Mechanics and the Principles of Completeness in Physics

New class of entropy-power-based uncertainty relations

We use the concept of entropy power to introduce a new one-parameter class of information-theoretic uncertainty relations. This class constitutes an infinite hierarchy of uncertainty relations, which allows to determine the shape of the underlying information-distribution function by measuring the relevant entropy powers. The efficiency of such uncertainty relations in quantum mechanics is illustrated with two examples: superpositions of two squeezed states and the Lévy-type heavy-tailed wave function. Improvement over both the variance-based and Shannon entropy based uncertainty relations is demonstrated in both these cases.

Tunneling time in attosecond experiments, intrinsic-type of time. Keldysh, and Mandelstam–Tamm time

Tunneling time in attosecond and strong-field experiments is one of the most controversial issues in current research, because of its importance to the theory of time, the time operator and the time–energy uncertainty relation in quantum mechanics. In Kullie (2015 Phys. Rev. A 92 052118) we derived an estimation of the (real) tunneling time, which shows an excellent agreement with the time measured in attosecond experiments, our derivation is found by utilizing the time–energy uncertainty relation, and it represents a quantum clock. In this work, we show different aspects of the tunneling time in attosecond experiments, we discuss and compare the different views and approaches, which are used to calculate the tunneling time, i.e. Keldysh time (as a real or imaginary quantity), Mandelstam–Tamm time, the classical view of the time measurement and our tunneling time relation(s). We draw some conclusions concerning the validity and the relation between the different types of the tunneling time with the hope that they will help to answer the question put forward by Orlando et al (2014 J. Phys. B 47 204002, 2014 Phys. Rev. A 89 014102): tunneling time, what does it mean? However, as we will see, the important question is a more general one: how to understand the time and the measurement of the time of a quantum system? In respect to our result, the time in quantum mechanics can be, in more general fashion, classified in two types, intrinsic dynamically connected, and external dynamically not connected to the system, and consequently (perhaps only) classical Newtonian time remains as a parametric type of time.

Systematic construction of genuine-multipartite-entanglement criteria in continuous-variable systems using uncertainty relations

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.

Controlling of Entropic Uncertainty in Qubits System Under the Generalized Amplitude Damping Channel via Weak Measurements

We study the effect of weak measurements on the entropic uncertainty in two-qubit system under the generalized amplitude damping channel. Our results show that, the entropic uncertainty in qubits system can be reduced under weak measurements by choosing appropriate measuring strength, which provides a new method to break through the restriction of uncertainty relation in quantum mechanics.

Role of information theoretic uncertainty relations in quantum theory

Uncertainty relations based on information theory for both discrete and continuous distribution functions are briefly reviewed. We extend these results to account for (differential) Rényi entropy and its related entropy power. This allows us to find a new class of information-theoretic uncertainty relations (ITURs). The potency of such uncertainty relations in quantum mechanics is illustrated with a simple two-energy-level model where they outperform both the usual Robertson–Schrödinger uncertainty relation and Shannon entropy based uncertainty relation. In the continuous case the ensuing entropy power uncertainty relations are discussed in the context of heavy tailed wave functions and Schrödinger cat states. Again, improvement over both the Robertson–Schrödinger uncertainty principle and Shannon ITUR is demonstrated in these cases. Further salient issues such as the proof of a generalized entropy power inequality and a geometric picture of information-theoretic uncertainty relations are also discussed.