Critical look at the time-energy uncertainty relations
K. Urbanowski
备注：
16 pages, talk given at 3rd Jagiellonian Symposium on Fundamental and Applied Subatomic Physics, 2019, Krakow, Poland. arXiv admin note: substantial text overlap with arXiv:1810.11462
The Heisenberg and Mandelstam-Tamm time-energy uncertainty relations are analyzed. The conlusion resulting from this analysis is that within the Quantum Mechanics of Schrödinger and von Neumann, the status of these relations can not be considered as the same as the status of the position-momentum uncertainty relations, which are rigorous. The conclusion is that the time--energy uncertainty relations can not be considered as universally valid.
A generalized Cauchy-Schwarz inequality is derived and applied to uncertainty relation in quantum mechanics. We see a modification in the uncertainty relation and minimum uncertainty wave packet.
Minimum uncertainty states and squeezed states from the sum uncertainty relation
Indian Institute of Science Education and Research, Pune（印度科学教育研究院，浦那）；Department of Physics, Institute of Science, BHU（贝拿勒斯印度教大学理学院物理系）；Indian Institute of Science Education and Research, Berhampur（印度科学教育研究院，贝尔汉普尔）
We present a new uncertainty relation by defining a measure of uncertainty based on skew information. For bipartite systems, we establish uncertainty relations with the existence of a quantum memory. A general relation between quantum correlations and tight bounds of uncertainty has been presented.
Uncertainty relations are fundamental in quantum mechanics. Here I propose state-independent variance-based uncertainty relations for two or more arbitrary observables in finite dimensional spaces. The uncertainty relations provide near-optimal lower bounds in some typical examples, and are useful for entanglement detection.
The Robertson's formulation of the uncertainty relation is the most widely accepted form of the Heisenberg uncertainty relation (HUR). It gets modified when we consider it for entangled particles. But this formulation does not consider the measurement process itself. There are reformulation of the uncertainty relations called the generalized uncertainty relations by including the measurement process into the uncertainty relation. HUR gets modified for the case of entangled particles. Here it is shown that the limit of the Generalized Uncertainty Relation (GUR) also reduces for entangled partic…
We investigate the product form uncertainty relations of variances for $n\,(n\geq 3)$ quantum observables. In particular, tight uncertainty relations satisfied by three observables has been derived, which is shown to be better than the ones derived from the strengthened Heisenberg and the generalized Schrödinger uncertainty relations, and some existing uncertainty relation for three spin-half operators. Uncertainty relation of arbitrary number of observables is also derived. As an example, the uncertainty relation satisfied by the eight Gell-Mann matrices is presented.
Uncertainty relations describe the lower bound of product of standard deviations of observables. By revealing a connection between standard deviations of quantum observables and numerical radius of operators, we establish a universal uncertainty relation for any $k$ observables, of which the formulation depends on the even or odd quality of $k$. This universal uncertainty relation is tight at least for the cases $k=2n$ and $k=3$. For two observables, the uncertainty relation is exactly a simpler reformulation of Schrödinger's uncertainty principle.
Uncertainty principle is one of the fundamental principles of quantum mechanics. In this work, we derive two uncertainty equalities, which hold for all pairs of incompatible observables. We also obtain an uncertainty relation in weak measurement which captures the limitation on the preparation of pre- and post-selected ensemble and hold for two non-Hermitian operators corresponding to two non-commuting observables.
Uncertainty relation is one of the fundamental building blocks of quantum theory. Nevertheless, the traditional uncertainty relations do not fully capture the concept of incompatible observables. Here we present a stronger Schrödinger-like uncertainty relation, which is stronger than the relation recently derived by L. Maccone and A. K. Pati [Phys. Rev. Lett. 113 (2014) 260401]. Furthermore, we give an additive uncertainty relation which holds for three incompatible observables, which is stronger than the relation newly obtained by S. Kechrimparis and S. Weigert [Phys. Rev. A 90 (2014) 062118]…
Analyzing general uncertainty relations one can find that there can exist such pairs of non-commuting observables $A$ and $B$ and such vectors that the lower bound for the product of standard deviations $ΔA$ and $ΔB$ calculated for these vectors is zero: $ΔA\,\cdot\,ΔB \geq 0$. Here we discuss examples of such cases and some other inconsistencies which can be found performing a rigorous analysis of the uncertainty relations in some special cases. As an illustration of such cases matrices $(2\times 2)$ and $(3 \times 3)$ and the position--momentum uncertainty relation for a quantum particle in …
Entropic uncertainty relations play an important role in both fundamentals and applications of quantum theory. Although they have been well-investigated in quantum theory, little is known about entropic uncertainty in generalized probabilistic theories (GPTs). The current study explores two types of entropic uncertainty relations, preparation and measurement uncertainty relations, in a class of GPTs which can be considered generalizations of quantum theory. Not only a method for obtaining entropic preparation uncertainty relations but also an entropic measurement uncertainty relation similar t…
不确定关系是量子理论的一个显著特征，它对物理性质可以同时确定的精度施加了内在的限制。不确定关系的现代工作采用\emph{entropic measures}来量化与测量非对易可观测量相关的知识的缺乏。然而，使用熵作为量词没有根本的理由;任何表征测量结果不确定性的函数关系都定义了不确定性关系。从一个非常合理的假设下，仅仅重新标记的测量结果的不变性，我们表明，舒尔凹函数是最一般的不确定性量词。然后，我们发现了一个细粒度的不确定性关系，给出了两个概率向量之间的优序，\textcolor{black}{significantly extending a majorization-based uncertainty relation first introduced in [M. H. Partovi, Phys. Rev. A \textbf{84}, 052117 (2011)].}这样的向量型不确定性关系生成一个无限的家庭不同的标量不确定性关系通过应用任意的不确定性量词。因此，我们的关系是普适的，并且抓住了量子理论中不确定性的本质。
We discuss some applications of various versions of uncertainty relations for both discrete and continuous variables in the context of quantum information theory. The Heisenberg uncertainty relation enables demonstration of the EPR paradox. Entropic uncertainty relations are used to reveal quantum steering for non-Gaussian continuous variable states. Entropic uncertainty relations for discrete variables are studied in the context of quantum memory where fine-graining yields the optimum lower bound of uncertainty. The fine-grained uncertainty relation is used to obtain connections between uncer…
Uncertainty relation lies at the heart of quantum mechanics, characterizing the incompatibility of non-commuting observables in the preparation of quantum states. An important question is how to improve the lower bound of uncertainty relation. Here we present a variance-based sum uncertainty relation for $N$ incompatible observables stronger than the simple generalization of the uncertainty relation for two observables derived by Maccone and Pati [Phys. Rev. Lett. {\bf113}, 260401 (2014)]. Further comparisons of our uncertainty relation with other related ones for spin-$\frac{1}{2}$ and spin-$…
Uncertainty relations play a significant role in drawing a line between classical physics and quantum physics. Since the introduction by Heisenberg, these relations have been considerably explored. However, the effect of quantum entanglement on uncertainty relations was not probed. Berta et al. [Nature Physics 6, 659-662 (2010)] removed this gap by deriving a conditional-entropic uncertainty relation in the presence of quantum entanglement. In the same spirit, using inferred-variance, we formulate uncertainty relations in the presence of entanglement for general two-qubit systems and arbitrary…
Commutator-based entropic uncertainty relations in multidimensional position and momentum spaces are derived, twofold generalizing previous entropic uncertainty relations for one-mode states. They provide optimal lower bounds and imply the multidimensional variance-based uncertainty principle. The article concludes with an open conjecture.
This work completes the program started in \cite{bb1,bb2,bb3} to derive the Heisenberg uncertainty relation for relativistic particles. Sharp uncertainty relations for massive relativistic particles with spin 0 and spin 1 are derived. The main conclusion is that the uncertainty relations for relativistic bosons are markedly different from those for relativistic fermions. The uncertainty relations for bosons are based on the energy density. It is shown that the uncertainty relations based on the time-component of the four-current, as was have done in \cite{bb3} for electrons, are untenable beca…
We have obtained the uncertainty relations for arbitrary states of the hydrogen atom. It is shown that the minimal value of the uncertainty relation is attained for the circular Rydberg states.
We study the sum uncertainty relations based on variance and skew information for arbitrary finite N quantum mechanical observables. We derive new uncertainty inequalities which improve the exiting results about the related uncertainty relations. Detailed examples are provided to illustrate the advantages of our uncertainty inequalities.
The uncertainty relation lies at the heart of quantum theory and behaves as a non-classical constraint on the indeterminacies of incompatible observables in a system. In the literature, many experiments have been devoted to the test of the uncertainty relations which mainly focus on the pure states. In this work we test the novel majorization uncertainty relations of three incompatible observables using a series of mixed states with adjustable mixing degrees, and compare the compactness of various entropy uncertainty relations. The experimental results confirm that for general mixed quantum sy…
The canonical Robertson-Schrödinger 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-Schrödinger uncertainty relation and …
It brings into attention briefly the genuine significance of uncertainty relations and of their extrapolations for which conventional(usual) doctrine promotes unjustified ideas.
The uncertainty relation is a distinguishing feature of quantum theory, characterizing the incompatibility of noncommuting observables in the preparation of quantum states. Recently, many uncertainty relations were proposed with improved lower bounds and were deemed capable of incorporating multiple observables. Here we report an experimental verification of seven uncertainty relations of this type with single-photon measurements. The results, while confirming these uncertainty relations, show as well the relative stringency of various uncertainty lower bounds.
We study the uncertainty relation in the product form of variances and obtain some new uncertainty relations with weight, which are shown to be tighter than those derived from the Cauchy Schwarz inequality.
Uncertainty relations take a crucial and fundamental part in the frame of quantum theory, and are bringing on many marvelous applications in the emerging field of quantum information sciences. Especially, as entropy is imposed into the uncertainty principle, entropy-based uncertainty relations lead to a number of applications including quantum key distribution, entanglement witness, quantum steering, quantum metrology, and quantum teleportation. Herein, the history of the development of the uncertainty relations is discussed, especially focusing on the recent progress with regard to quantum-me…
We study sum uncertainty relations for arbitrary finite $N$ quantum mechanical observables. Some uncertainty inequalities are presented by using skew information introduced by Wigner and Yanase. These uncertainty inequalities are nontrivial as long as the observables are mutually noncommutative. The relations among these new and existing uncertainty inequalities have been investigated. Detailed examples are presented.
Uncertainty principle plays a vital role in quantum physics. The Wigner-Yanase skew information characterizes the uncertainty of an observable with respect to the measured state. We generalize the uncertainty relations for two quantum channels to arbitrary N quantum channels based on Wigner-Yanase skew information. We illustrate that these uncertainty inequalities are tighter than the existing ones by detailed examples. Especially, we also discuss the uncertainty relations for N unitary channels, which could be regarded as variance-based sum uncertainty relations with respect to any pure state…
Taking advantage of coherent light beams, we experimentally investigate the variancebased uncertainty relations and the optimal majorization uncertainty relation for the two-dimensional quantum mechanical system.Different from most of the experiments which devoted to record each individual quantum, we examine the uncertainty relations by measuring an ensemble of photons with two polarization degree of freedom characterized by the Stokes parameters which allow us to determine the polarization density matrix with high precision. The optimality of the recently proposed direct-sum majorization unc…
We investigate the uncertainty principle for two successive projective measurements in terms of Rényi entropy based on a single quantum system. Our results cover a large family of the entropy (including the Shannon entropy) uncertainty relations with a lower optimal bound. We compare our relation with other formulations of the uncertainty principle in a two-spin observables measured on a pure quantum state of qubit. It is shown that the low bound of our uncertainty relation has better tightness.
We show that a proper expression of the uncertainty relation for a pair of canonically-conjugate continuous variables relies on entropy power, a standard notion in Shannon information theory for real-valued signals. The resulting entropy-power uncertainty relation is equivalent to the entropic formulation of the uncertainty relation due to Bialynicki-Birula and Mycielski, but can be further extended to rotated variables. Hence, based on a reasonable assumption, we give a partial proof of a tighter form of the entropy-power uncertainty relation taking correlations into account and provide exten…
Uncertainty relations provide fundamental limits on what can be said about the properties of quantum systems. For a quantum particle, the commutation relation of position and momentum observables entails Heisenberg's uncertainty relation. A third observable is presented which satisfies canonical commutation relations with both position and momentum. The resulting triple of pairwise canonical observables gives rise to a Heisenberg-type uncertainty relation for the product of three standard deviations. We derive the smallest possible value of this bound and determine the specific squeezed state …
Traditional forms of quantum uncertainty relations are invariably based on the standard deviation. This can be understood in the historical context of simultaneous development of quantum theory and mathematical statistics. Here, we present alternative forms of uncertainty relations, in both state dependent and state independent forms, based on the mean deviation. We illustrate the robustness of this formulation in situations where the standard deviation based uncertainty relation is inapplicable. We apply the mean deviation based uncertainty relation to detect EPR violation in a lossy scenario…
We prove a few novel state-dependent uncertainty relations for product as well the sum of variances of two incompatible observables. These uncertainty relations are shown to be tighter than the Roberson-Schrödinger uncertainty relation and other ones existing in the current literature. Also, we derive state dependent upper bound to the sum and the product of variances using the reverse Cauchy-Schwarz inequality and the Dunkl-Williams inequality. Our results suggest that not only we cannot prepare quantum states for which two incompatible observables can have sharp values, but also we have both…
We present the entropic uncertainty relations for multiple measurement settings in quantum mechanics. Those uncertainty relations are obtained for both cases with and without the presence of quantum memory. They take concise forms which can be proven in a unified method and easy to calculate. Our results recover the well known entropic uncertainty relations for two observables, which show the uncertainties about the outcomes of two incompatible measurements. Those uncertainty relations are applicable in both foundations of quantum theory and the security of many quantum cryptographic protocols…
We present an equivalence theorem to unify the two classes of uncertainty relations, i.e., the variance-based ones and the entropic forms, which shows that the entropy of an operator in a quantum system can be built from the variances of a set of commutative operators. That means an uncertainty relation in the language of entropy may be mapped onto a variance-based one, and vice versa. Employing the equivalence theorem, alternative formulations of entropic uncertainty relations stronger than existing ones in the literature are obtained for qubit system, and variance based uncertainty relations…
Recently, Maccone and Pati \cite{pa.1} derived new uncertainty relations which they claim to be stronger than Heisenberg-Robertson or Schrodinger uncertainty relations. Here we comment that their work is a special case of a more general uncertainty relation and we reexamine some of their conclusions.
The well-known Robertson-Schrödinger uncertainty relations have state-dependent lower bounds which are trivial for certain states. We present a general approach to deriving tight state-independent uncertainty relations for qubit measurements that completely characterise the obtainable uncertainty values. This approach can give such relations for any number of observables, and we do so explicitly for arbitrary pairs and triples of qubit measurements. We show how these relations can be transformed into equivalent tight entropic uncertainty relations. More generally, they can be expressed in term…
Uncertainty relation usually is one of the most important features in quantum mechanics, and is the backbone of quantum theory, which distinguishes from the rule in classical counterpart. Specifically, entropy-based uncertainty relations are of fundamental importance in the region of quantum information theory, offering one nontrivial bound of key rate towards quantum key distribution. In this work, we experimentally demonstrate the entropic uncertainty relations and coherence-based uncertainty relations in an all-optics platform. By means of preparing two kinds of bipartite initial states wit…
