N. V. Brazovskaja, V. Ye Brazovsky
Some aspects of application of the Uncertainty Principle in the range of interaction radiation with matter surveyed. The procedure of adjustment is proposed at calculation of values of an electromagnetic energy in a quantum theory of a field.
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Minimization of the expectation value of energy under the constraints imposed by the uncertainty principle can be a convenient method of solving quantum-mechanical problems.
The two-slit experiment with quantum particles provides many insights into the behaviour of quantum mechanics, including Bohr's complementarity principle. Here we analyze Einstein's recoiling slit version of the experiment and show how the inevitable entanglement between the particle and the recoiling slit as a which-way detector is responsible for complementarity. We derive the Englert-Greenberger-Yasin duality from this entanglement, which can also be thought of as a consequence of sum-uncertainty relations between certain complementary observables of the recoiling slit. Thus, entanglement i…
String theory, quantum geometry, loop quantum gravity and black hole physics all indicate the existence of a minimal observable length on the order of Planck length. This feature leads to a modification of Heisenberg uncertainty principle. Such a modified Heisenberg uncertainty principle is referred as gravitational uncertainty principle(GUP) in literatures. This proposal has some novel implications on various domains of theoretical physics. Here, we study some consequences of GUP in the spirit of Quantum mechanics. We consider two problem: a particle in an one-dimensional box and momentum spa…
We use geometric concepts originally proposed by Anandan and Aharonov to show that the Farhi-Gutmann time optimal analog quantum search evolution between two orthogonal quantum states is characterized by unit efficiency dynamical trajectories traced on a projective Hilbert space. In particular, we prove that these optimal dynamical trajectories are the shortest geodesic paths joining the initial and the final states of the quantum evolution. In addition, we verify they describe minimum uncertainty evolutions specified by an uncertainty inequality that is tighter than the ordinary time-energy u…
We study classical and quantum phases in the adiabatic Born-Oppenheimer context. These include a classical astronomical case, the general dual description of the phases, a new "Paradox" connected to scattering Berry phase and its resolution and various elaboration of topological/geometrical/non-abelian phases.
How much of the uncertainty in predicting measurement outcomes for non-commuting quantum observables is genuinely quantum mechanical? We provide a natural decomposition of the total entropic uncertainty of two non-commuting observables into a classical component, and an intrinsically quantum mechanical component. We show that the total quantum component in a state is never lower or upper bounded by any state-independent quantities, but instead admits "purity-based" lower bounds that generalize entropic formulations such as the Maassen-Uffink relation. These relations reveal a non-trivial inter…
A numerical illustration of a universally valid Heisenberg uncertainty relation, which was proposed recently, is presented by using the experimental data on spin-measurements by J. Erhart, et al.[ Nature Phys. {\bf 8}, 185 (2012)]. This uncertainty relation is closely related to a modified form of the Arthurs-Kelly uncertainty relation which is also tested by the spin-measurements. The universally valid Heisenberg uncertainty relation always holds, but both the modified Arthurs-Kelly uncertainty relation and Heisenberg's error-disturbance relation proposed by Ozawa, which was analyzed in the o…
The uncertainty principle limits quantum states such that when one observable takes predictable values there must be some other mutually unbiased observables which take uniformly random values. We show that this restrictive condition plays a positive role as the enabler of non-classical dynamics in an interferometer. First we note that instantaneous action at a distance between different paths of an interferometer should not be possible. We show that for general probabilistic theories this heavily curtails the non-classical dynamics. We prove that there is a trade-off with the uncertainty prin…
The sum of entropic uncertainties for the measurement of two non-commuting observables is not always reduced by the amount of entanglement (quantum memory) between two parties, and in certain cases may be impacted by quantum correlations beyond entanglement (discord). An optimal lower bound of entropic uncertainty in the presence of any correlations may be determined by fine-graining. Here we express the uncertainty relation in a new form where the maximum possible reduction of uncertainty is shown to be given by the extractable classical information. We show that the lower bound of uncertaint…
Building on a model recently proposed by F. Calogero, we postulate the existence of a coherent, long--range universal tremor affecting any stable and confined classical dynamical system. Deriving the characteristic fluctuative unit of action for each classical interaction, we obtain in all cases its numerical coincidence with the Planck action constant. We therefore suggest that quantum corrections to classical dynamics can be simulated by suitable classical stochastic fluctuations.
Uncertainty relations in classical and quantum theories of electromagnetism
本文推导了位置空间与动量（波矢）空间中方差受限制的尖锐不确定关系，并在经典光束理论、相干光束量子理论和单光子量子理论中均得到相同形式 $\Delta r \Delta k \ge 5/2$。
It is generally believed that classical regime emerges as a limiting case of quantum theory. Exploring such quantum-classical correspondences in a more transparent manner is central to the deeper understanding of foundational aspects and has attracted a great deal of attention - starting from the early days of quantum theory. While it is often highlighted that quantum to classical transition occurs in the limit hbar tending to zero, several objections have been raised about its suitability in some physical contexts. Ehrenfest's theorem is another widely discussed classical limit - however, its…
Even though mutually unbiased bases and entropic uncertainty relations play an important role in quantum cryptographic protocols they remain ill understood. Here, we construct special sets of up to 2n+1 mutually unbiased bases (MUBs) in dimension d=2^n which have particularly beautiful symmetry properties derived from the Clifford algebra. More precisely, we show that there exists a unitary transformation that cyclically permutes such bases. This unitary can be understood as a generalization of the Fourier transform, which exchanges two MUBs, to multiple complementary aspects. We proceed to pr…
The quantum component in uncertainty relation can be naturally characterized by the quantum coherence of a quantum state, which is of paramount importance in quantum information science. Here, we experimentally investigate quantum uncertainty relations construed with relative entropy of coherence, $l_1$ norm of coherence and coherence of formation. In stead of quantum state tomographic technology, we employ the classical shadow algorithm for the detection of lower bounds in quantum uncertainty relations. With an all optical setup, we prepare a family of quantum states whose purity can be fully…
What does it take for real-deterministic c-valued (i.e., classical, commuting) variables to comply with the Heisenberg uncertainty principle? Here, we construct a class of real-deterministic c-valued variables out of the weak values obtained via a non-perturbing weak measurement of quantum operators with a post-selection over a complete set of state vectors basis, which always satisfies the Kennard-Robertson-Schrödinger uncertainty relation. First, we introduce an auxiliary global random variable and couple it to the imaginary part of the weak value to transform the incompatibility between the…
Our knowledge of quantum mechanics can satisfactorily describe simple, microscopic systems, but is yet to explain the macroscopic everyday phenomena we observe. Here we aim to shed some light on the quantum-to-classical transition as seen through the analysis of uncertainty relations. We employ entropic uncertainty relations to show that it is only by the inclusion of imprecision in our model of macroscopic measurements that we can prepare a system with two simultaneously well-defined quantities, even if their associated observables do not commute. We also establish how the precision of measur…
Classical probability density distributions with uncertainty relations for ground states of simple non-relativistic quantum-mechanical systems
本文通过将海森堡不确定关系纳入经典概率密度分布，显著改善了经典与量子基态分布的吻合度，且优于WKB方法，并适用于激发态。
The uncertainty principle sets lower bound on the uncertainties of two incompatible observables measured on a particle. The uncertainty lower bound can be reduced by considering a particle as a quantum memory entangled with the measured particle. In this paper, we consider a tripartite scenario in which a quantum state has been shared between Alice, Bob, and Charlie. The aim of Bob and Charlie is to minimize Charlie's lower bound about Alice's measurement outcomes. To this aim, they concentrate their correlation with Alice in Charlie's side via a cooperative strategy based on local operations …
We present a unified view of the Berry phase of a quantum system and its entanglement with surroundings. The former reflects the nonseparability between a system and a classical environment as the latter for a quantum environment, and the concept of geometric time-energy uncertainty can be adopted as a signature of the nonseparability. Based on this viewpoint, we study their relationship in the quantum-classical transition of the environment, with the aid of a spin-half particle (qubit) model exposed to a quantum-classical hybrid field. In the quantum-classical transition, the Berry phase has …
We investigate the system of a particle moving on a half line x >= 0 under the general walls at x = 0 that are permitted quantum mechanically. These quantum walls, characterized by a parameter L, are shown to be realized as a limit of regularized potentials. We then study the classical aspects of the quantum walls, by seeking a classical counterpart which admits the same time delay in scattering with the quantum wall, and also by examining the WKB-exactness of the transition kernel based on the regularized potentials. It is shown that no classical counterpart exists for walls with L < 0, and t…
Since the uncertainty about an observable of a system prepared in a quantum state is usually described by its variance, when the state is mixed, the variance is a hybrid of quantum and classical uncertainties. Besides that, complementarity relations are saturated only for pure, single-quanton, quantum states. For mixed states, the wave-particle quantifiers never saturate the complementarity relation and can even reach zero for a maximally mixed state. So, to fully characterize a quanton it is not sufficient to consider its wave-particle aspect; one has also to regard its correlations with othe…
Recent theoretical and experimental studies have given raise to new aspects in quantum measurements and error-disturbance uncertainty relations. After a brief review of these issues, we present an experimental test of the error-disturbance uncertainty relations in photon polarization measurement. Using generalized, strength-variable measurement of a single photon polarization state, we experimentally evaluate the error and disturbance in the measurement process and demonstrate the validity of recently proposed uncertainty relations.
Asymmetry and tighter uncertainty relations for Rényi entropies via quantum-classical decompositions of resource measures
本文提出一种基于量子-经典分解的资源度量方法，利用非对易性和态混合度刻画不确定性，从而导出对任意离散可观测量有效的更强Rényi熵不确定性关系。
Quantum Coherence as a Thermodynamic Resource Beyond the Classical Uncertainty Bound
研究非平衡系统中量子相干在热力学不确定性界限里的作用，通过结合量子克拉美 - 罗不等式与戴森级数展开，推导出广义TUR，以三能级量子微波激射器为例证明稳态相干可超经典极限，为连接经典与量子不确定性关系提供统一框架。
We review Bohr's reasoning in the Bohr-Einstein debate on the photon box experiment. The essential point of his reasoning leads us to an uncertainty relation between the proper time and the rest mass of the clock. It is shown that this uncertainty relation can be derived if only we take the fundamental point of view that the proper time should be included as a dynamic variable in the Lagrangian describing the system of the clock. Some problems and some positive aspects of our approach are then discussed.
Separation of measurement uncertainty into quantum and classical parts based on Kirkwood-Dirac quasiprobability and generalized entropy
本文提出将量子测量总不确定度加性分解为量子与经典部分的方法，利用广义Kirkwood-Dirac拟概率和广义熵量化，并证明其满足合理要求，且真实量子不确定度是证明量子情境性的充要条件。
An exact uncertainty principle, formulated as the assumption that a classical ensemble is subject to random momentum fluctuations of a strength which is determined by and scales inversely with uncertainty in position, leads from the classical equations of motion to the Schrodinger equation. Thus there is an exact formulation of the uncertainty principle which precisely captures the essence of what is "quantum" about quantum mechanics.
In the present contribution we discuss the role of experimental limitations in the classical limit problem. We studied some simple models and found that Quantum Mechanics does not re-produce classical mechanical predictions, unless we consider the experimental limitations ruled by uncertainty principle. We have shown that the discrete nature of energy levels of integrable systems can be accessed by classical measurements. We have defined a precise limit for this procedure. It may be used as a tool to define the classical limit as far as the discrete spectra of integrable systems are concerned.…
In the present contribution we discuss the role of experimental limitations in the classical limit problem. We studied some simple models and found that Quantum Mechanics does not re-produce classical mechanical predictions, unless we consider the experimental limitations ruled by uncertainty principle. We have shown that the discrete nature of energy levels of integrable systems can be accessed by classical measurements. We have defined a precise limit for this procedure. It may be used as a tool to define the classical limit as far as the discrete spectra of integrable systems are concerned.
Classical statistical average values are generally generalized to average values of quantum mechanics, it is discovered that quantum mechanics is direct generalization of classical statistical mechanics, and we generally deduce both a new general continuous eigenvalue equation and a general discrete eigenvalue equation in quantum mechanics, and discover that a eigenvalue of quantum mechanics is just an extreme value of an operator in possibility distribution, the eigenvalue f is just classical observable quantity. A general classical statistical uncertain relation is further given, the general…
We show that three principle means of treating privacy amplification in quantum key distribution, private state distillation, classical privacy amplification, and via the uncertainty principle, are equivalent and interchangeable. By adapting the security proof based on the uncertainty principle, we construct a new protocol for private state distillation which we prove is identical to standard classical privacy amplification. Underlying this approach is a new characterization of private states, related to their standard formulation by the uncertainty principle, which gives a more physical under…
The upper bound of the fine-grained uncertainty relation is different for classical physics, quantum physics and no-signaling theories with maximal nonlocality (supper quantum correlation), as was shown in the case of bipartite systems [J. Oppenheim and S. Wehner, Science 330, 1072 (2010)]. Here, we extend the fine-grained uncertainty relation to the case of tripartite systems. We show that the fine-grained uncertainty relation determines the nonlocality of tripartite systems as manifested by the Svetlichny inequality, discriminating between classical physics, quantum physics and super quantum…
Indeterminacy associated with probing of a quantum state is commonly expressed through spectral distances (metric) featured in the outcomes of repeated experiments. Here we express it as an effective amount (measure) of distinct outcomes instead. The resulting $μ$-uncertainties are described by the effective number theory [1] whose central result, the existence of a minimal amount, leads to a well-defined notion of intrinsic irremovable uncertainty. We derive $μ$-uncertainty formulas for arbitrary set of commuting operators, including the cases with continuous spectra. The associated entropy-l…
The standard quantum teleportation scheme is deconstructed, and those aspects of it that appear remarkable and "non-classical" are identified. An alternative teleportation scheme, involving only classical states and classical information, is then formulated, and it is shown that the classical scheme reproduces all of these remarkable aspects, including those that had seemed non-classical. This leads to a re-examination of quantum teleportation, which suggests that its significance depends on the interpretation of quantum states.
As an undergraduate exercise, in an article (2012 Am. J. Phys. $\bf{80}$ 780-14), quantum and classical uncertainties for dimensionless variables of position and momentum were evaluated in three potentials: infinite well, bouncing ball, and harmonic oscillator. While original quantum uncertainty products depend on ${\rm{\hbar}}$ and the number of states ($n$), a dimensionless approach makes the comparison between quantum uncertainty and classical dispersion possible by excluding ${\rm{\hbar}}$. But the question is whether the uncertainty still remains dependent on quantum number $n$. In the ab…
The uncertainty principle, which bounds the uncertainties involved in obtaining precise outcomes for two complementary variables defining a quantum particle, is a crucial aspect in quantum mechanics. Recently, the uncertainty principle in terms of entropy has been extended to the case involving quantum entanglement. With previously obtained quantum information for the particle of interest, the outcomes of both non-commuting observables can be predicted precisely, which greatly generalises the uncertainty relation. Here, we experimentally investigated the entanglement-assisted entropic uncertai…
We analytically show that it is possible to perform coherent imaging by using the classical correlation of two beams obtained by splitting incoherent thermal radiation. A formal analogy is demonstrated between two such classically correlated beams and two entangled beams produced by parametric down-conversion. Because of this analogy, the classical beams can mimic qualitatively all the imaging properties of the entangled beams, even in ways which up to now were not believed possible. A key feature is that these classical beams are spatially correlated both in the near-field and in the far-fiel…
Mechanics can be founded on a principle relating the uncertainty delta-q in the trajectory of an observable particle to its motion relative to the observer. From this principle, p.delta-q=const., p being the q-conjugated momentum, mechanical laws are derived and the meaning of the Lagrangian and Hamiltonian functions are discussed. The connection between the presented principle and Hamilton's Least Action Principle is examined. Wave mechanics and Schrodinger equation appear without additional assumptions by choosing the representation for delta-q in the case the motion is not trajectory descri…
We studied the Szilard engine under the effect of generalized uncertainty principle (GUP). In the classical Szilard engine, the work done by the engine is reduced by the GUP effect via a modified ideal gas law. In the quantum Szilard engine, the correction comes from the shifted eigen energy due to the nonlinear momentum dependence. We studied its effect on both bosonic and fermionic molecules.
In this article, we describe various aspects of categorification of the structures appearing in information theory. These aspects include probabilistic models both of classical and quantum physics, emergence of F-manifolds, and motivic enrichments.
From the perspective of Markovian piecewise deterministic processes (PDPs), we investigate the derivation of a kinetic uncertainty relation (KUR), which was originally proposed in Markovian open quantum systems. First, stationary distributions of classical PDPs are explicitly constructed. Then, a tilting method is used to derive a rate functional of large deviations. Finally, based on an improved approximation scheme, we recover the KUR. These classical results are directly extended to the open quantum systems. We use a driven two-level quantum system to exemplify the quantum results.
Heisenberg's uncertainty principle is often cited as an example of a "purely quantum" relation with no analogue in the classical limit where $\hbar \to 0$. However, this formulation of the classical limit is problematic for many reasons, one of which is dimensional analysis. Since $\hbar$ is a dimensionful constant, we may always work in natural units in which $\hbar = 1$. Dimensional analysis teaches us that all physical laws can be expressed purely in terms of dimensionless quantities. This indicates that the existence of a dimensionally consistent constraint on $Δx Δp$ requires the existenc…
A unified framework for classical and quantum uncertainty relations using stochastic representations
本研究提出基于随机表示的通用马尔可夫动力学统一方法，无需依赖确定性演化方程即可推导经典不确定关系，还可拓展至马尔可夫开放量子系统，得到更紧、更易物理实现的量子不确定关系，明确其为系统随机本质的固有属性。
Epistemic Boundaries and Quantum Uncertainty: What Local Observers Can (Not) Predict
本文探讨了量子不确定性的不可避免性，提出结合预测优势、无信号传输和可靠主体间性的不可行主张，证明真正预测优势存在根本限制，但违背主体间性假设可允许主观预测优势，从而揭示观察者间的认知边界。
It is proved that the width of a function and the width of the distribution of its values cannot be made arbitrarily small simultaneously. In the case of ergodic stochastic processes, an ensuing uncertainty relationship is demonstrated for the product of correlation length and variance. A closely related uncertainty principle is also established for the average degree of fourth-order coherence and the spatial width of modes of bosonic quantum fields. However, it is shown that, in the case of stochastic and quantum observables, certain non-classical states with sub-Poissonian statistics, such a…
Some fundamental aspects related with the construction of Robertson-Schrödinger like uncertainty principle inequalities are reported in order to provide an overall description of quantumness, separability and nonlocality of quantum systems in the noncommutative phase-space. Some consequences of the deformed noncommutative algebra are also considered in physical systems of interest.
A recent development of the studies on classical and quasi-classical properties of supersymmetric quantum mechanics in Witten's version is reviewed. First, classical mechanics of a supersymmetric system is considered. Solutions of the classical equations of motion are given and their properties are discussed in some detail. The corresponding quantum model is constructed by canonical quantization. The quantum model is analyzed by Feynman's path integral within a stationary-phase approximation. A quasi-classical quantization rule is derived, which is applicable when supersymmetry is exact or spo…
Thermodynamic uncertainty relation under continuous measurement and feedback with quantum-classical-transfer entropy
研究通过量子-经典转移熵量化连续测量获得的信息，揭示了电流精度受信息热力学成本约束，并展示反馈可提升精度同时抑制熵产。
Can Intense Quantum Light Beat Classical Uncertainty Relations?
该研究探讨了高强度量子光在超越经典不确定性关系中的潜力，揭示了非经典光子态中量子优势的强度标度及纠缠与光子统计的相互作用。
The limits of knowledge in classical physics resemble the quantum uncertainty relation
本文证明经典物理中观测信息获取受热力学限制，其形式类似量子不确定性关系，并引入探针质量概念，为经典测量资源理论奠基。
Old Rules in a New Game: Mapping Uncertainty Quantification to Quantum Machine Learning
本文针对量子机器学习黑箱问题，将经典不确定性量化方法映射到量子领域，通过理论构建与实证评估证明在设计量子模型时引入不确定性意识的必要性。
The generalized uncertainty principle is often used to modify various thermodynamics systems by regarding the greater-than-equal relation as an approximate relation. We give a method to improve this approximation and compare the differences between the original and improved methods during the evaporation of black hole from two aspects of positive and negative parameters. Finally, we prove the rationality of the improved method and give some guiding opinions.
Time-energy uncertainty relation for neutrino oscillations: historical development, applications and future prospects
本文综述了中微子振荡领域时间-能量不确定关系的研究进展，分析其物理意义，并探究引力与非相对论效应对中微子振荡标准条件的影响。
Quasiprobability Thermodynamic Uncertainty Relation
本文推导了量子热力学不确定性关系的扩展形式，利用Terletsky-Margenau-Hill准概率量化动态涨落，揭示了负值和非经典增强的逃逸率对超越经典限制的输出耗散比的必要性，且该关系不依赖基矢并强于量子相干性。
The Generalized Uncertainty Principle (GUP) has been directly applied to the motion of (macroscopic) test bodies on a given space-time in order to compute corrections to the classical orbits predicted in Newtonian Mechanics or General Relativity. These corrections generically violate the Equivalence Principle. The GUP has also been indirectly applied to the gravitational source by relating the GUP modified Hawking temperature to a deformation of the background metric. Such a deformed background metric determines new geodesic motions without violating the Equivalence Principle. We point out her…
Entropic uncertainty and quantum non-classicality of Unruh-Dewitt detectors in relativity
本文研究相对论背景下 Unruh 热效应对一对 Unruh-Dewitt 探测器熵不确定性与量子 discord 的影响，发现加速会导致量子性损失和不确定性增大，且二者呈反向关联。
Quantum theory has found a new field of applications in the realm of information and computation during the recent years. This paper reviews how quantum physics allows information coding in classically unexpected and subtle nonlocal ways, as well as information processing with an efficiency largely surpassing that of the present and foreseeable classical computers. Some outstanding aspects of classical and quantum information theory will be addressed here. Quantum teleportation, dense coding, and quantum cryptography are discussed as a few samples of the impact of quanta in the transmission of…
A generalisation of the classical covariance for quantum mechanical observables has previously been presented by Gibilisco, Hiai and Petz. Gibilisco and Isola has proved that the usual quantum covariance gives the sharpest inequalities for the determinants of covariance matrices. We introduce a new generalisation of the classical covariance which gives better inequalities than the classical one furthermore it has a direct geometric interpretation.
We derive universal codes for simultaneous transmission of classical messages and entanglement through quantum channels, possibly under attack of a malignant third party. These codes are robust to different kinds of channel uncertainty. To construct such universal codes, we invoke and generalize properties of random codes for classical and quantum message transmission through quantum channels. We show these codes to be optimal by giving a multi-letter characterization of regions corresponding to the capacity of compound quantum channels for simultaneously transmitting and generating entangleme…
We explore the quantization of classical models with position-dependent mass (PDM) terms constrained to a bounded interval in the canonical position. This is achieved through the Weyl-Heisenberg covariant integral quantization by properly choosing a regularizing function $Π(q,p)$ on the phase space that smooths the discontinuities present in the classical model. We thus obtain well-defined operators without requiring the construction of self-adjoint extensions. Simultaneously, the quantization mechanism leads naturally to a semi-classical system, that is, a classical-like model with a well-def…
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