A further remark on uncertainty relations

As pointed out earlier, the commutation relation 1/i[A, B]=C for quantum mechanical observablesA, B andC by itself does not imply the uncertainty relation ΔA·ΔB≧1/2¦¯C¦. In this note,A, B andC are assumed to be generators of a unitary representation of a suitable Lie group, such that is implied by the group structure. This assumption is then sufficient to yield.     

Generalized uncertainty relations

The standard uncertainty relations (UR) in quantum mechanics are typically used for unbounded operators (like the canonical pair). This implies the need for the control of the domain problems. On the other hand, the use of (possibly bounded) functions of basic observables usually leads to more complex and less readily interpretable relations. In addition, UR may turn trivial for certain states if the commutator of observables is not proportional to a positive operator. In this letter we consider a generalization of standard UR resulting from the use of two, instead of one, vector states. The possibility to link these states to each other in various ways adds additional flexibility to UR, which may compensate some of the above-mentioned drawbacks. We discuss applications of the general scheme, leading not only to technical improvements, but also to interesting new insight.     

Entropic uncertainty relations

New entropic uncertainty relations for angle-angular momentum and position-momentum, derived recently by Partovi, are related to older relations of a similar type, which were proved by Bialynicki-Birula and Mycielski. Significantly improved lower bounds are obtained in both cases.     

Causality of invariant form factors

It is proved that the form factors for inelastic electron scattering by a polarized nucleon are causal.Translated from Izvestiya Vysshikh Uchebnykh, Fizika, No. 7, pp. 39–43, July, 1976.The authors axe grateful to A. N. Tavkhelidze for helpful remarks and suggestions.     

测不准关系和最小不确定态

利用力学量算符的厄密性和希尔伯特状态矢量模的非负性，重新给出了关于测不准关系的数 学证明.简单的证明过程不仅揭示了测不准关系某些经常被忽视的特征，而且还可以直接给 出最小不确定态的充要条件.在此基础上，我们提出了对易子为非零常数的任意一对力学量 的最小不确定态问题，并且采用玻色型产生和湮没算符给出了它们的压缩态的明显表达式. 关键词： 海森堡测不准原理 最小不确定态 压缩态     

Generalized thermodynamic uncertainty relations

We analyze an ensemble in which energy (E), temperature (T) and multiplicity (N) can all fluctuate and with the help of nonextensive statistics we propose a relation connecting all fluctuating variables. It generalizes Lindhard’s thermodynamic uncertainty relations known in literature.     

Quantum electrodynamic uncertainty relations

The canonical commutation relations between the electromagnetic fields E and B are stated in a form which is of use in the electrical engineering of magnetometers and electrometers. The result does not depend on gauge.     

Entropic formulation of uncertainty relations

We review the recent investigations on the improved formulation of uncertainty relations which employ the information-theoretic entropy rather than variance as a measure of uncertainty. We show that this formulation also brings out clearly the relation between the overall uncertainty and the quantum mechanical interference due to measurements. Lecture delivered at the International Symposium on Theoretical Physics, Bangalore, November 1984.     

Coherent states and uncertainty relations

The most general form of the minimality-preserving Hamiltonians is found and the uncertainty products ΔqΔp are calculated for some quadratic quantum systems     

Characterizing entanglement via uncertainty relations

We derive a family of necessary separability criteria for finite-dimensional systems based on inequalities for variances of observables. We show that every pure bipartite entangled state violates some of these inequalities. Furthermore, a family of bound entangled states and true multipartite entangled states can be detected. The inequalities also allow us to distinguish between different classes of true tripartite entanglement for qubits. We formulate an equivalent criterion in terms of covariance matrices. This allows us to apply criteria known from the regime of continuous variables to finite-dimensional systems.     

Uncertainty principle and uncertainty relations

It is generally believed that the uncertainty relation q p1/2, where q and p are standard deviations, is the precise mathematical expression of the uncertainty principle for position and momentum in quantum mechanics. We show that actually it is not possible to derive from this relation two central claims of the uncertainty principle, namely, the impossibility of an arbitrarily sharp specification of both position and momentum (as in the single-slit diffraction experiment), and the impossibility of the determination of the path of a particle in an interference experiment (such as the double-slit experiment).The failure of the uncertainty relation to produce these results is not a question of the interpretation of the formalism; it is a mathematical fact which follows from general considerations about the widths of wave functions.To express the uncertainty principle, one must distinguish two aspects of the spread of a wave function: its extent and its fine structure. We define the overall widthW and the mean peak width w of a general wave function and show that the productW w is bounded from below if is the Fourier transform of . It is shown that this relation expresses the uncertainty principle as it is used in the single- and double-slit experiments.     

Thermodynamic uncertainty relations and irreversibility

The Cramér-Rao lower bound for the minimum variance of an unbiased estimator is derived from the second law of thermodynamics. The inequality is in the form of a uncertainty relation for conjugate thermodynamic variables where the minimum uncertainty occurs for reversible processes in which the conjugate variables are completely negatively correlated. An upper bound on the probability for arbitrarily large deviations in the energy is given in terms of the difference in entropies at the initial temperature of the body and the final equilibrium temperature of the medium.     

A note on entropic uncertainty relations of position and momentum

We consider two entropic uncertainty relations of position and momentum recently discussed in the literature. By a suitable rescaling of one of them, we obtain a smooth interpolation for both high-resolution and low-resolution measurements, respectively. Because our interpolation has never been mentioned in the literature before, we propose it as a candidate for an improved entropic uncertainty relation of position and momentum. Up to now, we have neither been able to falsify nor prove the new inequality. In our opinion, it is a challenge to do either one.     

Quantum fisher metric and uncertainty relations

The connection between the recently discovered quantum-mechanical analogue of the Fisher information metric and the uncertainty relations is further investigated, with special emphasis on the role of the quantum counterpart of the Cramér-Rao inequality.     

Wigner-Yanase skew information and uncertainty relations

The Wigner-Araki-Yanase theorem puts a limitation on the measurement of observables in the presence of a conserved quantity, and the notion of Wigner-Yanase skew information quantifies the amount of information on the values of observables not commuting with the conserved quantity. We demonstrate that the statistical idea underlying the skew information is the Fisher information in the theory of statistical estimation. A quantum Cramér-Rao inequality and a new uncertainty relation in terms of the skew information are established, which shed considerable new light on the relationships between quantum measurement and statistical inference. The result is applied to estimating the evolution speed of quantum states.     

Probability representation and state-extended uncertainty relations

The new inequality recently found by Trifonov and called the state-extended inequality is considered in the tomographic-probability representation of quantum mechanics. The Trifonov uncertainty relations are expressed in terms of optical tomograms and can be checked in experiments on homodyne detection of the photon states.     

Entropic uncertainty relations for angular distributions

A new entropic uncertainty relation for simultaneous measurements of two angles ? and θ and two corresponding angular momentum operators L_z and L² is derived. Step function techniques are introduced to complete the proof.