Measurement Uncertainty Relations (original) (raw)

Error and unsharpness in approximate joint measurements of position and momentum

In recent years, novel quantifications of measurement error in quantum mechanics have for the first time enabled precise formulations of Heisenberg’s famous but often challenged measurement uncertainty relation. This relation takes the form of a trade-off for the necessary errors in joint approximate measurements of position and momentum and other incompatible pairs of observables. Much work remains to be done to obtain a better understanding of the new error measures and their suitability. To this end we review here some of these error measures and associated measurement uncertainty relations. We investigate the properties and suitability of these measures, give examples to show how they can be computed in specific cases, and compare their relative strengths as criteria for “good” approximations.

Experimental Test of Heisenberg's Measurement Uncertainty Relation Based on Statistical Distances

Physical review letters, 2016

Incompatible observables can be approximated by compatible observables in joint measurement or measured sequentially, with constrained accuracy as implied by Heisenberg's original formulation of the uncertainty principle. Recently, Busch, Lahti, and Werner proposed inaccuracy trade-off relations based on statistical distances between probability distributions of measurement outcomes [P. Busch et al., Phys. Rev. Lett. 111, 160405 (2013); P. Busch et al., Phys. Rev. A 89, 012129 (2014)]. Here we reformulate their theoretical framework, derive an improved relation for qubit measurement, and perform an experimental test on a spin system. The relation reveals that the worst-case inaccuracy is tightly bounded from below by the incompatibility of target observables, and is verified by the experiment employing joint measurement in which two compatible observables designed to approximate two incompatible observables on one qubit are measured simultaneously.

Classical information theoretic view of physical measurements and generalized uncertainty relations

2013

Uncertainty relations are discussed in detail not only for free particles but also for bound states within the framework of classical information theory. Uncertainty relation for simultaneous measurements of two physical observables is defined in this framework for generalized dynamic systems governed by a Sturm-Liouville-type equation of motion. In the first step, the reduction of Kennard-Robertson type uncertainties because of boundary conditions with a mean-square error is discussed quantitatively with reference to the information entropy. Several concrete examples of generalized uncertainty relations are given. Then, by considering disturbance effects, a universally valid uncertainty relation is investigated for the generalized equation of motion with a certain boundary condition. Necessary conditions for violation (reduction) of the Heisenberg-type uncertainty relation are discussed in detail. The reduction of the generalized uncertainty relation because of the boundary condition is discussed by reanalyzing experimental data for measured electron densities in a hydrogen molecule encapsulated in a fullerene C 60 cage.

Colloquium: Quantum root-mean-square error and measurement uncertainty relations

Recent years have witnessed a controversy over Heisenberg’s famous error-disturbance relation. Here we resolve the conflict by way of an analysis of the possible conceptualizations of measurement error and disturbance in quantum mechanics. We discuss two approaches to adapting the classic notion of root-mean-square error to quantum measurements. One is based on the concept of noise operator; its natural operational content is that of a mean deviation of the values of two observables measured jointly, and thus its applicability is limited to cases where such joint measurements are available. The second error measure quantifies the differences between two probability distributions obtained in separate runs of measurements and is of unrestricted applicability. We show that there are no nontrivial unconditional joint-measurement bounds for state-dependent errors in the conceptual framework discussed here, while Heisenberg-type measurement uncertainty relations for state-independent errors have been proven.

Fine-grained uncertainty relations for several quantum measurements

Quantum Information Processing, 2014

We study fine-grained uncertainty relations for several quantum measurements in a finitedimensional Hilbert space. The proposed approach is based on exact calculation or estimation of the spectral norms of corresponding positive matrices. Fine-grained uncertainty relations of the state-independent form are derived for an arbitrary set of mutually unbiased bases. Such relations are extended with a recent notion of mutually unbiased measurements. The case of so-called mutually biased bases is considered in a similar manner. We also discuss a formulation of fine-grained uncertainty relations in the case of generalized measurements. The general approach is then applied to two measurements related to state discrimination. The case of three rank-one projective measurements is further examined in details. In particular, we consider fine-grained uncertainty relations for mutually unbiased bases in three-dimensional Hilbert space.

Preparation and measurement: two independent sources of uncertainty in quantum mechanics

1999

In the Copenhagen interpretation the Heisenberg uncertainty relation is interpreted as the mathematical expression of the concept of complementarity, quantifying the mutual disturbance necessarily taking place in a simultaneous or joint measurement of incompatible observables. This interpretation has already been criticized by Ballentine a long time ago, and has recently been challenged in an experimental way. These criticisms can be substantiated by using the generalized formalism of positive operator-valued measures, from which a new inequality can be derived, precisely illustrating the Copenhagen concept of complementarity. The different roles of preparation and measurement in creating uncertainty in quantum mechanics are discussed.

The Theory of Quantum Uncertainties and Quantum Measurements

  1. We shall discuss what modern interpretations say about the Heisenberg's uncertainties. These interpretations explain that a quantity begins to 'lose' meaning when a conjugate property begins to 'acquire' definite meaning. We know that a quantity losing meaning means that it has no fixed value and has an uncertainty . In this paper we look deeper into this interpretation and the outcome reveals more evidence of the quantity losing meaning. Newer insights appear. 2) We consider two extreme cases of hypothetical processes nature undergoes, without interference by a measurement: One, a system collapses to an energy eigenstate under the influence of a Hamiltonian instantaneously at a time ttt. This is thus what would happen if we would measure the system's energy. Next, when a particle becomes localised to a point at a time t_0t_0t_0 under the influence of a Hamiltonian. This is thus what would happen if we would measure the system's position. We shall prove th...

Heisenberg's Uncertainty Relation (Compendium entry)

2008

The term Heisenberg uncertainty relation is a name for not one but three distinct trade-off relations which are all formulated in a more or less intuitive and vague way in Heisenberg's seminal paper of 1927 [1]. These relations are expressions and quantifications of three fundamental limitations of the operational possibilities of preparing and measuring quantum mechanical systems which are stated here informally with reference to position and momentum as a paradigmatic example of canonically conjugate pairs of quantities: ... (A) It is impossible to prepare states in ...