Heisenberg's Relation Research Articles

Heisenberg's error-disturbance relations: A joint measurement-based experimental test

The Heisenberg's error-disturbance relation is a cornerstone of quantum physics. It was recently shown to be not universally valid and two different approaches to reformulate it were proposed.The first one focuses on how error and disturbance of two observables, A and B, depend on a particular quantum state. The second one asks how a joint measurement of A and B affects their eigenstates. Previous experiments focused on the first approach. Here, we focus on the second one. Firstly, we propose and implement an extendible method for quantum walk-based joint measurements of noisy Pauli operators to test the error-disturbance relation for qubits. Then, we formulate and experimentally test a new universally valid relation for the three mutually unbiased observables. We therefore establish a fundamentally new method of testing error-disturbance relations.

On the invariant properties of quantum uncertainty relations with respect to parameters of virtual photons responsible for interaction processes in quantum particle systems

Uncertainty relations (UR) are considered based on analysis of the peculiarities of virtual photons involved in electromagnetic inter-particle interaction processes in the frame of quantum field theory. Based on the virtual photon approach, particular properties of UR are discussed with respect to both species of the uncertainty relations dependent or independent on the velocity of particle moving in the coordinate–time space. In the frame of this approach, the underlying physical reason for a universal property of Heisenberg's UR is established, which consists in their invariance on the frequency of virtual photons. In line with the virtual photon approach, UR properties are discussed for the case of bound electron states, as well as peculiarities of Heisenberg's relations at inter-particle distances smaller than Compton length resulting from the decrease of virtual photon's contribution into the interaction processes at short distances.

Experimental Test of Residual Error-Disturbance Uncertainty Relations for Mixed Spin-½ States.

The indeterminacy inherent in quantum measurements is an outstanding character of quantum theory, which manifests itself typically in the uncertainty principle. In the last decade, several universally valid forms of error-disturbance uncertainty relations were derived for completely general quantum measurements for arbitrary states. Subsequently, Branciard established a form that is optimal for spin measurements for some pure states. However, the bound in his inequality is not stringent for mixed states. One of the present authors recently derived a new bound tight in the corresponding mixed state case. Here, a neutron-optical experiment is carried out to investigate this new relation: it is tested whether error and disturbance of quantum measurements disappear or persist in mixing up the measured ensemble. The attainability of the new bound is experimentally observed, falsifying the tightness of Branciard's bound for mixed spin states.

Bose Description of the Deformed Parabose Oscillator with Z p -Graded Reflection Symmetry

In this paper Bose description of the deformed parabose oscillator with Z p -graded reflection symmetry is discussed. New lowering and raising operators are obtained from p Bose operators and their adjoint ones. These new step operators are shown to be satisfy the Heisenberg relation of Wigner algebra type.

Direct tests of measurement uncertainty relations: what it takes.

The uncertainty principle being a cornerstone of quantum mechanics, it is surprising that, in nearly 90 years, there have been no direct tests of measurement uncertainty relations. This lacuna was due to the absence of two essential ingredients: appropriate measures of measurement error (and disturbance) and precise formulations of such relations that are universally valid and directly testable. We formulate two distinct forms of direct tests, based on different measures of error. We present a prototype protocol for a direct test of measurement uncertainty relations in terms of value deviation errors (hitherto considered nonfeasible), highlighting the lack of universality of these relations. This shows that the formulation of universal, directly testable measurement uncertainty relations for state-dependent error measures remains an important open problem. Recent experiments that were claimed to constitute invalidations of Heisenberg's error-disturbance relation, are shown to conform with the spirit of Heisenberg's principle if interpreted as direct tests of measurement uncertainty relations for error measures that quantify distances between observables.

Heisenberg's uncertainty relation: Violation and reformulation

The uncertainty relation formulated by Heisenberg in 1927 describes a trade-off between the error of a measurement of one observable and the disturbance caused on another complementary observable so that their product should be no less than a limit set by Planck's constant. In 1980, Braginsky, Vorontsov, and Thorne claimed that this relation leads to a sensitivity limit for gravitational wave detectors. However, in 1988 a model of position measurement was constructed that breaks both this limit and Heisenberg's relation. Here, we discuss the problems as to how we reformulate Heisenberg's relation to be universally valid and how we experimentally quantify the error and the disturbance to refute the old relation and to confirm the new relation.

Certainty in Heisenberg's uncertainty principle: Revisiting definitions for estimation errors and disturbance

We revisit the definitions of error and disturbance recently used in error-disturbance inequalities derived by Ozawa and others by expressing them in the reduced system space. The interpretation of the definitions as mean-squared deviations relies on an implicit assumption that is generally incompatible with the Bell-Kochen-Specker-Spekkens contextuality theorems, and which results in averaging the deviations over a non-positive-semidefinite joint quasiprobability distribution. For unbiased measurements, the error admits a concrete interpretation as the dispersion in the estimation of the mean induced by the measurement ambiguity. We demonstrate how to directly measure not only this dispersion but also every observable moment with the same experimental data, and thus demonstrate that perfect distributional estimations can have nonzero error according to this measure. We conclude that the inequalities using these definitions do not capture the spirit of Heisenberg's eponymous inequality, but do indicate a qualitatively different relationship between dispersion and disturbance that is appropriate for ensembles being probed by all outcomes of an apparatus. To reconnect with the discussion of Heisenberg, we suggest alternative definitions of error and disturbance that are intrinsic to a single apparatus outcome. These definitions naturally involve the retrodictive and interdictive states for that outcome, and produce complementarity and error-disturbance inequalities that have the same form as the traditional Heisenberg relation.

Heisenberg uncertainty for qubit measurements

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 new 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 as 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.

The Heisenberg Relation - Mathematical Formulations

We study some of the possibilities for formulating the Heisenberg relation of quantum mechanics in mathematical terms. In particular, we examine the framework discussed by Murray and von Neumann, the family (algebra) of operators affiliated with a finite factor (of infinite linear dimension).

HEISENBERG UNCERTAINTY RELATION REVISITED — UNIVERSALITY OF ROBERTSON'S RELATION

It is shown that all the known uncertainty relations are the secondary consequences of Robertson's relation. The basic idea is to use the Heisenberg picture so that the time development of quantum mechanical operators incorporate the effects of the measurement interaction. A suitable use of triangle inequalities then gives rise to various forms of uncertainty relations. The assumptions of unbiased measurement and unbiased disturbance are important to simplify the resulting uncertainty relations and to give the familiar uncertainty relations, such as a naive Heisenberg error-disturbance relation. These simplified uncertainty relations are however, valid only conditionally. Quite independently of uncertainty relations, it is shown that the notion of precise measurement is incompatible with the assumptions of unbiased measurement and unbiased disturbance. We can thus naturally understand the failure of the naive Heisenberg's error-disturbance relation, as was demonstrated by the recent spin measurement by Erhart et al.

The velocity operator in quantum mechanics in noncommutative space

We tested the consequences of noncommutative (NC from now on) coordinates xk, k = 1, 2, 3 in the framework of quantum mechanics. We restricted ourselves to 3D rotationally invariant NC configuration spaces with dynamics specified by the Hamiltonian \documentclass[12pt]{minimal}\begin{document}$\hat{H} = \hat{H}_0 + \hat{U}$\end{document}Ĥ=Ĥ0+Û, where \documentclass[12pt]{minimal}\begin{document}$\hat{H}_0$\end{document}Ĥ0 is an analogue of kinetic energy and \documentclass[12pt]{minimal}\begin{document}$\hat{U} = \hat{U}(\hat{r})$\end{document}Û=Û(r̂) denotes an arbitrary rotationally invariant potential. We introduced the velocity operator by \documentclass[12pt]{minimal}\begin{document}$\hat{V}_k = - i [\hat{X}_k, \hat{H}]$\end{document}V̂k=−i[X̂k,Ĥ] (\documentclass[12pt]{minimal}\begin{document}$\hat{X}_k$\end{document}X̂k being the position operator), which is a NC generalization of the usual gradient operator (multiplied by −i). We found that the NC velocity operators possess various general, independent of potential, properties: (1) uncertainty relations \documentclass[12pt]{minimal}\begin{document}$[\hat{V}_i,\hat{X}_j]$\end{document}[V̂i,X̂j] indicate an existence of a natural kinetic energy cut-off, (2) commutation relations \documentclass[12pt]{minimal}\begin{document}$[\hat{V}_i,\hat{V}_j] = 0$\end{document}[V̂i,V̂j]=0, which is non-trivial in the NC case, (3) relation between \documentclass[12pt]{minimal}\begin{document}$\hat{V}^2$\end{document}V̂2 and \documentclass[12pt]{minimal}\begin{document}$\hat{H}_0$\end{document}Ĥ0 that indicates the existence of maximal velocity and confirms the kinetic energy cut-off, (4) all these results sum up in canonical (general, not depending on a particular form of the central potential) commutation relations of Euclidean group E(4) = SO(4)▷T(4), (5) Heisenberg equation for the velocity operator, relating acceleration \documentclass[12pt]{minimal}\begin{document}$\dot{\hat{V}}_k = -i[\hat{V}_k, \hat{H}]$\end{document}V̂̇k=−i[V̂k,Ĥ] to derivatives of the potential.

On independence of some pseudocharacters on braid groups

It is proved that the pseudocharacter defined on the braid group by the signature of braid closures is linearly independent of all pseudocharacters obtained from the twist number via the Malyutin operators, provided that the number of strands is greater than 4. This pseudocharacter is shown to have a nontrivial kernel part. It is observed that the operators $I$ and $R$ defined by Malyutin on the space of pseudocharacters satisfy the Heisenberg relation, and that some of Malyutin’s results are standard consequences of this fact.

Jacobi polynomials and generalized Clifford algebra‐valued Appell sequences

Appell sequences in Clifford analysis are defined as polynomial families on which the Heisenberg algebra acts through a raising and a lowering operator satisfying the canonical Heisenberg relation. Recently, these sequences have gained new interest, as they are connected to the topic of special functions (such as harmonic or monogenic Gegenbauer polynomials) and branching rules for certain irreducible representations of the spin group. In this paper, we will explain how Jacobi polynomials appear quite naturally in the setting of Appell sequences related to certain branching problems. Copyright © 2013 John Wiley & Sons, Ltd.

Violation of Heisenberg's error-disturbance uncertainty relation in neutron-spin measurements

In its original formulation, Heisenberg's uncertainty principle dealt with the relationship between the error of a quantum measurement and the thereby induced disturbance on the measured object. Meanwhile, Heisenberg's heuristic arguments have turned out to be correct only for special cases. A new universally valid relation was derived by Ozawa in 2003. Here, we demonstrate that Ozawa's predictions hold for projective neutron-spin measurements. The experimental inaccessibility of error and disturbance claimed elsewhere has been overcome using a tomographic method. By a systematic variation of experimental parameters in the entire configuration space, the physical behavior of error and disturbance for projective spin-1/2 measurements is illustrated comprehensively. The violation of Heisenberg's original relation, as well as, the validity of Ozawa's relation become manifest. In addition, our results conclude that the widespread assumption of a reciprocal relation between error and disturbance is not valid in general.

Experimental violation and reformulation of the Heisenberg's error-disturbance uncertainty relation

The uncertainty principle formulated by Heisenberg in 1927 describes a trade-off between the error of a measurement of one observable and the disturbance caused on another complementary observable such that their product should be no less than the limit set by Planck's constant. However, Ozawa in 1988 showed a model of position measurement that breaks Heisenberg's relation and in 2003 revealed an alternative relation for error and disturbance to be proven universally valid. Here, we report an experimental test of Ozawa's relation for a single-photon polarization qubit, exploiting a more general class of quantum measurements than the class of projective measurements. The test is carried out by linear optical devices and realizes an indirect measurement model that breaks Heisenberg's relation throughout the range of our experimental parameter and yet validates Ozawa's relation.

Aspects of universally valid Heisenberg uncertainty relation

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 original experiment, fail in the present context of spin-measurements, and the cause of their failure is identified with the assumptions of unbiased measurement and disturbance. It is also shown that all the universally valid uncertainty relations are derived from Robertson's relation and thus the essence of the uncertainty relation is exhausted by Robertson's relation as is widely accepted.

TRANSFORMATION LAWS OF THE COMPONENTS OF CLASSICAL AND QUANTUM FIELDS AND HEISENBERG RELATIONS

This paper recalls and shows the origin of the transformation laws of the components of classical and quantum fields. They are considered from the "standard" and fiber bundle point of view. The results are applied to the derivation of the Heisenberg relations in quite general setting, in particular, in the fiber bundle approach. All conclusions are illustrated in a case of transformations induced by the Poincaré group.

Heisenberg uncertainty relations for photons

The idea to base the uncertainty relation for photons on the electromagnetic energy distribution in space enabled us to derive a sharp inequality that expresses the uncertainty relation [Phys. Rev. Lett. {\bf 108}, 140401 (2012)]. An alternative version of the uncertainty relation derived in this paper is closer in spirit to the original Heisenberg relation because it employs the analog of the position operator for the photon---the center of energy operator. The noncommutativity of the components of the center of energy operator results in the increase of the bound $3/2\,\hbar$ in the standard Heisenberg uncertainty relation in three dimensions. This difference diminishes with the increase of the photon energy. In the limiting case of infinite momentum frame, the lower bound in the Heisenberg uncertainty relations for photons is the same as in nonrelativistic quantum mechanics.

Uncertainty Relation for Photons

The uncertainty relation for the photons in three dimensions that overcomes the difficulties caused by the nonexistence of the photon position operator is derived in quantum electrodynamics. The photon energy density plays the role of the probability density in configuration space. It is shown that the measure of the spatial extension based on the energy distribution in space leads to an inequality that is a natural counterpart of the standard Heisenberg relation. The equation satisfied by the photon wave function in momentum space which saturates the uncertainty relations has the form of the Schrödinger equation in coordinate space in the presence of electric and magnetic charges.

Planar quantum squeezing and atom interferometry

We obtain a lower bound on the sum of two orthogonal spin component variances in a plane. This gives a novel planar uncertainty relation which holds even when the Heisenberg relation is not useful. We investigate the asymptotic, large $J$ limit, and derive the properties of the planar quantum squeezed states that saturate this uncertainty relation. These states extend the concept of spin squeezing to any two conjugate spin directions. We show that planar quantum squeezing can be achieved experimentally as the ground state of a Bose-Einstein condensate in two coupled potential wells with a critical attractive interaction. These states reduce interferometric phase noise at all phase angles simultaneously. This is useful for one-shot interferometric phase-measurements where the measured phase is completely unknown. Our results can also be used to derive entanglement criteria for multiple spins $J$ at separated sites, with applications in quantum information.