Joint Quantum Measurements with Minimum Uncertainty (original) (raw)

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...

Towards a new uncertainty principle: quantum measurement noise

Physics Letters A, 1991

Two generalizations of a known approach to the joint measurement of position and momentum to the joint measurement of more general pairs of observables are compared. They weaken the restrictions on “noisy” measurements that prevented the above method from being more generally usable, in two different ways: additive object-dependent noise versus object-independent non-additive noise. In the latter approach a lower bound for the amount of noise in a joint measurement of incompatible observables is found, not as a consequence of the usual Heisenberg scatter principle, but of a new “inaccuracy principle”. Physically realizable examples are given.

Heisenberg uncertainty for qubit measurements

2014

Reports on experiments recently performed in Vienna [Erhard et al., Nature Phys. 8, 185 (2012)] and Toronto [Rozema et al., Phys. Rev. Lett. 109, 100404 (2012)] include claims of a violation of Heisenberg's error-disturbance relation. In contrast, we have presented and proven a Heisenberg-type relation for joint measurements of position and momentum [Phys. Rev. Lett. 111, 160405 (2013)]. To resolve the apparent conflict, we formulate here a general trade-off relation for errors in qubit measurements, using the same concepts as we did in the position-momentum case. We show that the combined errors in an approximate joint measurement of a pair of ±1-valued observables A,B are tightly bounded from below by a quantity that measures the degree of incompatibility of A and B. The claim of a violation of Heisenberg is shown to fail because it is based on unsuitable measures of error and disturbance. Finally we show how the experiments mentioned may directly be used to test our error inequality.

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.

Violation of Heisenberg’s Measurement-Disturbance Relationship by Weak Measurements

2012

While there is a rigorously proven relationship about uncertainties intrinsic to any quantum system, often referred to as ''Heisenberg's uncertainty principle,''Heisenberg originally formulated his ideas in terms of a relationship between the precision of a measurement and the disturbance it must create. Although this latter relationship is not rigorously proven, it is commonly believed (and taught) as an aspect of the broader uncertainty principle.

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.

Uncertainty characteristics of generalized quantum measurements

Physical Review A, 2003

The effects of any quantum measurement can be described by a collection of measurement operators ͕M m ͖ acting on the quantum state of the measured system. However, the Hilbert space formalism tends to obscure the relationship between the measurement results and the physical properties of the measured system. In this paper, a characterization of measurement operators in terms of measurement resolution and disturbance is developed. It is then possible to formulate uncertainty relations for the measurement process that are valid for arbitrary input states. The motivation of these concepts is explained from a quantum communication viewpoint. It is shown that the intuitive interpretation of uncertainty as a relation between measurement resolution and disturbance provides a valid description of measurement back action. Possible applications to quantum cryptography, quantum cloning, and teleportation are discussed.

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.

Measurement Uncertainty: Reply to Critics

In a recent publication [PRL 111, 160405 (2013)] we proved a version of Heisenberg's error-disturbance tradeoff. This result was in apparent contradiction to claims by Ozawa of having refuted these ideas of Heisenberg. In a direct reaction [arXiv:1308.3540] Ozawa has called our work groundless, and has claimed to have found both a counterexample and an error in our proof. Here we answer to these allegations. We also comment on the submission [arXiv:1307.3604] by Rozema et al, in which our approach is unfavourably compared to that of Ozawa.

Measurement Uncertainty Relations

2014

Measurement uncertainty relations are quantitative bounds on the errors in an approximate joint measurement of two observables. They can be seen as a generalization of the error/disturbance tradeoff first discussed heuristically by Heisenberg. Here we prove such relations for the case of two canonically conjugate observables like position and momentum, and establish a close connection with the more familiar preparation uncertainty relations constraining the sharpness of the distributions of the two observables in the same state. Both sets of relations are generalized to means of order α rather than the usual quadratic means, and we show that the optimal constants are the same for preparation and for measurement uncertainty. The constants are determined numerically and compared with some bounds in the literature. In both cases the near-saturation of the inequalities entails that the state (resp. observable) is uniformly close to a minimizing one.