Quantum Mechanical Uncertainty Research Articles

Frequency Fluctuations in Tunable and Nonlinear Microwave Cavities

We present a model for how frequency fluctuations comparable to the total cavity linewidth may arise in tunable and nonlinear microwave cavities, and how these fluctuations affect the measurement of scattering matrix elements. Applying this model to the specific case of a two-sided cavity, we obtain closed-form expressions for the average scattering matrix elements in several important cases. A key signature of our model is the subtle deformation of the trajectories swept out by scattering matrix elements in the complex plane. Despite this signature, the fluctuating and non-fluctuating models are qualitatively similar enough to be mistaken for one another. In the case of tunable cavities we show that if one fails to account for these fluctuations then one will find damping rates that appear to depend on the tuning parameter, which is a common observation in such systems. In the case of a Kerr cavity, we show that there exists a fundamental lower bound to the scale of these frequency fluctuations in the steady state, imposed by quantum mechanical uncertainty, which can appreciably affect the apparent damping rates of the cavity as the strength of the nonlinearity approaches the single-photon level. By using the model we present as a fitting function for experimental data, however, one can extract both the true damping rates of the cavity and the effective scale of these frequency fluctuations over the scattering measurement's bandwidth. Lastly, we compare this new method for observing frequency fluctuations to other methods, one of which we extend beyond the regime of small fluctuations.

An 8.4 Gbps real-time quantum random number generator based on quantum phase fluctuation

Based on the fundamental uncertainty of quantum mechanics, quantum random number generators can generate truly random number which is critically important for many applications, such as information security. However, while the generation rate of offline quantum random number generators could be up to tens of Gbps or more, that of real-time quantum random number generators is only 3.2 Gbps due to the relatively low post-processing speed, and the application of real-time quantum random number generators is limited. We propose an efficient and robust high-speed real-time quantum random number generation scheme based on quantum phase fluctuation of a distributed feedback laser, with improved post-processing techniques. Firstly, the quantum signal-to-noise ratio is increased by analyzing and quantifying quantum noise. Secondly, a time-interleaved analog-to-digital converter scheme is designed to achieve a real-time sampling rate of 8 GSa/s. Finally, the random bits are extracted using the minimum entropy estimation and Toeplitz-hashing randomness extraction. Experimental results show that the proposed scheme could achieves a real-time generation rate up to 8.4 Gbps, with an estimated offline generation rate of 82.32 Gbps.

Quantum correlations between light and the kilogram-mass mirrors of LIGO.

The measurement of minuscule forces and displacements with ever greater precision is inhibited by the Heisenberg uncertainty principle, which imposes a limit to the precision with which the position of an object can be measured continuously, known as the standard quantum limit1-4. When light is used as the probe, the standard quantum limit arises from the balance between the uncertainties of the photon radiation pressure applied to the object and of the photon number in the photoelectric detection. The only way to surpass the standard quantum limit is by introducing correlations between the position/momentum uncertainty of the object and the photon number/phase uncertainty of the light that it reflects5. Here we confirm experimentally the theoretical prediction5 that this type of quantum correlation is naturally produced in the Laser Interferometer Gravitational-wave Observatory (LIGO). We characterize and compare noise spectra taken without squeezing and with squeezed vacuum states injected at varying quadrature angles. After subtracting classical noise, our measurements show that the quantum mechanical uncertainties in the phases of the 200-kilowatt laser beams and in the positions of the 40-kilogram mirrors of the Advanced LIGO detectors yield a joint quantum uncertainty that is a factor of 1.4 (3 decibels) below the standard quantum limit. We anticipate that the use of quantum correlations will improve not only the observation of gravitational waves, but also more broadly future quantum noise-limited measurements.

Direct experimental test of forward and reverse uncertainty relations

The canonical Robertson-Schr\"{o}dinger uncertainty relation provides a loose bound for the product of variances of two non-commuting observables. Recently, several tight forward and reverse uncertainty relations have been proved which go beyond the traditional uncertainty relations. Here we experimentally test multifold of state-dependent uncertainty relations for the product as well as the sum of variances of two incompatible observables for photonic qutrits. These uncertainty relations are independent of any optimization and still tighter than the Robertson-Schr\"{o}dinger uncertainty relation and other ones found in current literatures. For the first time, we also test the state-dependent reverse uncertainty relations for the sum of variances of two incompatible observables, which implies another unique feature of preparation uncertainty in quantum mechanics. Our experimental results not only foster insight into a fundamental limitation on preparation uncertainty but also may contribute to the study of upper limit of precession measurements for incompatible observables.

Correlation measure equivalence in dynamic causal structures of quantum gravity

We prove an equivalence transformation between the correlation measure functions of the causally-unbiased quantum gravity space and the causally-biased standard space. The theory of quantum gravity fuses the dynamic (nonfixed) causal structure of general relativity and the quantum uncertainty of quantum mechanics. In a quantum gravity space, the events are causally nonseparable and all time bias vanishes, which makes it no possible to use the standard causally-biased entropy and the correlation measure functions. Since a corrected causally-unbiased entropy function leads to an undefined, obscure mathematical structure, in our approach the correction is made in the data representation of the causally-unbiased space. Here we prove that the standard causally-biased entropy function with a data correction can be used to identify correlations in dynamic causal structures. As a corollary, all mathematical properties of the causally-biased correlation measure functions are preserved in the causally-unbiased space. The equivalence transformation allows us to measure correlations in a quantum gravity space with the stable, well-defined mathematical background and apparatus of the causally-biased functions of quantum Shannon theory.

On the Justification and Validity of the Kennard Inequality

In 1927, Earle Hesse Kennard derived an inequality describing Heisenberg’s uncertainty principle. Since then, we have traditionally been using the standard deviation as the measure of uncertainty in quantum mechanics. But Jan Hilgevoord asserts that the standard deviation is neither a natural nor a generally adequate measure of quantum uncertainty. Specifically, he asserts that the standard deviations are inadequate to use as the quantum uncertainties in the single- and double-slit diffraction experiments. He even tells that from these examples it will become clear that the standard deviation is the wrong concept to express the uncertainty principle generally and that the Kennard relation has little to do with the uncertainty principle. We will investigate what are adequate as the measures of quantum uncertainty. And, beyond that, we will investigate the effects of multiplying the two uncertainties; namely, characteristics which is hiding in deep interior of the Kennard inequality. Through investigations we’ll come to naturally realize that his assertions were wrong. All of our discussions will help raise understanding of the Heisenberg uncertainty principle. Our discussions will afford us an opportunity to think about the essence of the Fourier transform. The aim of this paper is to draw conclusions about whether the Kennard inequality is justified or not.

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.

Theory of quantum gravity information processing

The theory of quantum gravity is aimed to fuse general relativity with quantum theory into a more fundamental framework. The space of quantum gravity provides both the non-fixed causality of general relativity and the quantum uncertainty of quantum mechanics. In a quantum gravity scenario, the causal structure is indefinite and the processes are causally non-separable. Here, we provide a model for the information processing structure of quantum gravity. We show that the quantum gravity environment is an information resource-pool from which valuable information can be extracted. We analyze the structure of the quantum gravity space and the entanglement of the space-time geometry. We study the information transfer capabilities of quantum gravity space and define the quantum gravity channel. We reveal that the quantum gravity space acts as a background noise on the local environment states. We characterize the properties of the noise of the quantum gravity space and show that it allows the separate local parties to simulate remote outputs from the local environment state, through the process of remote simulation.

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.

Theoretical model of quantum random number generator based on phase fluctuation

The security of cryptography algorithms is based on the security of the key and the secret key is essentially random number. Pseudo-random numbers based on the complexity of the algorithm can not guarantee the true randomness theoretically. The uncertainty of quantum mechanics provides a perfect source of entropy for the true random number generator. In this paper, we analyze the real-time quantum random number scheme based on quantum phase fluctuation given by YANG et al. On the basis of this scheme, a theoretical model of quantum random number optical interference with two delay coils is designed, and the randomness of the theoretical model is analyzed. The theoretical model obtained has the characteristics of random normal distribution.

Patient observers and non-perturbative infrared dynamics in inflation

We have previously derived the effect of soft graviton modes on the quantum state of de Sitter using spontaneously broken asymptotic symmetries. In the present paper we prove that this effect can be reinterpreted in terms of Bogoliubov transformations acting on the quantum state. This also enables us to address the much discussed issues regarding the observability of infrared effects in de Sitter from a new perspective. While it is commonly agreed that infrared effects are not visible to a single sub-horizon observer at late times, we argue that the question is less trivial for a patient observer who has lived long enough to have a record of the state before the soft mode was created. Though classically there is no obstruction to measuring this effect locally, we give several indications that quantum mechanical uncertainties may censor the effect. We then apply our methods to find a non-perturbative description of the quantum state pertaining to the Page time of de Sitter, and derive with these new methods the probability distribution for the local quantum states of de Sitter and slow-roll inflation in the presence of long modes. Finally, we show that this formalism reproduces and generalizes the usual criterion for the presence of eternal inflation in general classes of slow-roll inflation.

Infinite Freedom of Space-Time for Zero-Energy-Entity in Quantum Mechanics

Zero-energy state is investigated by taking infinitesimal energy and observing its uncertainty in space-time, adopting quantum mechanics. In this paper, the uncertainty in conventional quantum mechanics is found to be interpreted as freedom in space-time, which results in possibility of time travel and space transition of the zero-energy state, which could be information or mind. The wave function of a physical system composed of multiple particles or wave-packets is examined and found that it can be arbitrarily changed by grouping by observers. It leads to an idea that even infinitesimal energy or wave-packets in a heavy physical system may separately exist and it has the infinite freedom of space-time.

State-Independent Uncertainty Relations and Entanglement Detection in Noisy Systems.

Quantifying quantum mechanical uncertainty is vital for the increasing number of experiments that reach the uncertainty limited regime. We present a method for computing tight variance uncertainty relations, i.e., the optimal state-independent lower bound for the sum of the variances for any set of two or more measurements. The bounds come with a guaranteed error estimate, so results of preassigned accuracy can be obtained straightforwardly. Our method also works for postive-operator-valued measurements. Therefore, it can be used for detecting entanglement in noisy environments, even in cases where conventional spin squeezing criteria fail because of detector noise.

Is patience a virtue? Cosmic censorship of infrared effects in de Sitter

While the accumulation of long wavelength modes during inflation wreaks havoc on the large scale structure of spacetime, the question of even observability of their presence by any local observer has lead to considerable confusion. Though, it is commonly agreed that infrared effects are not visible to a single sub-horizon observer at late times, we argue that the question is less trivial for a patient observer who has lived long enough to have a record of the state before the soft mode was created. Though classically, there is no obstruction to measuring this effect locally, we give several indications that quantum mechanical uncertainties censor the effect, rendering the observation of long modes ultimately forbidden.

Communication: Constant uncertainty molecular dynamics: A simple and efficient algorithm to incorporate quantum nature into a real-time molecular dynamics simulation.

We propose a novel molecular dynamics (MD) algorithm for approximately dealing with a nuclear quantum dynamics in a real-time MD simulation. We have found that real-time dynamics of the ensemble of classical particles acquires quantum nature by introducing a constant quantum mechanical uncertainty constraint on its classical dynamics. The constant uncertainty constraint is handled by the Lagrange multiplier method and implemented into a conventional MD algorithm. The resulting constant uncertainty molecular dynamics (CUMD) is applied to the calculation of quantum position autocorrelation functions on quartic and Morse potentials. The test calculations show that CUMD gives better performance than ring-polymer MD because of the inclusion of the quantum zero-point energy during real-time evolution as well as the quantum imaginary-time statistical effect stored in an initial condition. The CUMD approach will be a possible starting point for new real-time quantum dynamics simulation in condensed phase.

Uncertainty, joint uncertainty, and the quantum uncertainty principle

Historically, the element of uncertainty in quantum mechanics has been expressed through mathematical identities called uncertainty relations, a great many of which continue to be discovered. These relations use diverse measures to quantify uncertainty (and joint uncertainty). In this paper we use operational information-theoretic principles to identify the common essence of all such measures, thereby defining measure-independent notions of uncertainty and joint uncertainty. We find that most existing entropic uncertainty relations use measures of joint uncertainty that yield themselves to a small class of operational interpretations. Our notion relaxes this restriction, revealing previously unexplored joint uncertainty measures. To illustrate the utility of our formalism, we derive an uncertainty relation based on one such new measure. We also use our formalism to gain insight into the conditions under which measure-independent uncertainty relations can be found.

Entropy, energy and temperature–length inequality for Friedmann universes

In this paper, we continue the study of the physical consequences of our modified black hole entropy formula in expanding spacetimes. In particular, we apply the new formula to apparent horizons of Friedmann expanding universes with zero, negative and positive spatial curvature. As a first result, we found that apart from the static Einstein solution, the only Friedmann spacetimes with constant (zero) internal energy are the ones with zero spatial curvature. This happens because, in the computation of the internal energy [Formula: see text], the contribution due to the nonvanishing Hubble flow must be added to the usual Misner–Sharp energy giving, for zero curvature spacetimes, a zero value for [Formula: see text]. This fact does not hold when curvature is present. After analyzing the free energy [Formula: see text], we obtain the correct result that [Formula: see text] is stationary only for physical systems in isothermal equilibrium, i.e. a de Sitter expanding universe. This result permits us to trace back a physically reasonable hypothesis concerning the origin of the early and late times de Sitter phase of our universe. Finally, we deduce an interesting temperature–length inequality similar to the time–energy uncertainty of ordinary quantum mechanics but with temperature instead of time coordinate. Remarkably, this relation is independent of the gravitational constant [Formula: see text] and can thus be explored also in nongravitational contexts.

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.

A Non-local Reality: Is There a Phase Uncertainty in Quantum Mechanics?

A century after the advent of Quantum Mechanics and General Relativity, both theories enjoy incredible empirical success, constituting the cornerstones of modern physics. Yet, paradoxically, they suffer from deep-rooted, so-far intractable, conflicts. Motivations for violations of the notion of relativistic locality include the Bell's inequalities for hidden variable theories, the cosmological horizon problem, and Lorentz-violating approaches to quantum geometrodynamics, such as Horava-Lifshitz gravity. Here, we explore a recent proposal for a "real ensemble" non-local description of quantum mechanics, in which "particles" can copy each others' observable values AND phases, independent of their spatial separation. We first specify the exact theory, ensuring that it is consistent and has (ordinary) quantum mechanics as a fixed point, where all particles with the same values for a given observable have the same phases. We then study the stability of this fixed point numerically, and analytically, for simple models. We provide evidence that most systems (in our study) are locally stable to small deviations from quantum mechanics, and furthermore, the phase variance per value of the observable, as well as systematic deviations from quantum mechanics, decay as $\sim$ (Energy$\times$Time)$^{-2n}$, where $n \geq 1$. Interestingly, this convergence is controlled by the absolute value of energy (and not energy difference), suggesting a possible connection to gravitational physics. Finally, we discuss different issues related to this theory, as well as potential novel applications for the spectrum of primordial cosmological perturbations and the cosmological constant problem.

Independent simultaneous discoveries visualized through network analysis: the case of linear canonical transforms

We describe the structural dynamics of two groups of scientists in relation to the independent simultaneous discovery (i.e., definition and application) of linear canonical transforms. This mathematical construct was built as the transfer kernel of paraxial optical systems by Prof. Stuart A. Collins, working in the ElectroScience Laboratory in Ohio State University. At roughly the same time, it was established as the integral kernel that represents the preservation of uncertainty in quantum mechanics by Prof. Marcos Moshinsky and his postdoctoral associate, Dr. Christiane Quesne, at the Instituto de Fisica of the Universidad Nacional Autonoma de Mexico. We are interested in the birth and parallel development of the two follower groups that have formed around the two seminal articles, which for more than two decades did not know and acknowledge each other. Each group had different motivations, purposes and applications, and worked in distinct professional environments. As we will show, Moshinsky---Quesne had been highly cited by his associates and students in Mexico and Europe when the importance of his work started to permeate various other mostly theoretical fields; Collins' paper took more time to be referenced, but later originated a vast following notably among Chinese applied optical scientists. Through social network analysis we visualize the structure and development of these two major coauthoring groups, whose community dynamics shows two distinct patterns of communication that illustrate the disparity in the diffusion of theoretical and technological research.