Measurement uncertainty relations: characterising optimal error bounds for qubits
T Bullock, P Busch
arxiv:1512.00104v3 [quant-ph]

ABSTRACT: In standard formulations of the uncertainty principle, two fundamental features are typically cast as impossibility statements: two noncommuting observables cannot in general both be sharply defined (for the same state), nor can they be measured jointly. The pioneers of quantum mechanics were acutely aware and puzzled by this fact, and it motivated Heisenberg to seek a mitigation, which he formulated in his seminal paper of 1927. He provided intuitive arguments to show that the values of, say, the position and momentum of a particle can at least be unsharply defined, and they can be measured together provided some approximation errors are allowed. Only now, nine decades later, a working theory of approximate joint measurements is taking shape, leading to rigorous and experimentally testable formulations of associated error tradeoff relations. Here we briefly review this new development, explaining the concepts and steps taken in the construction of optimal joint approximations of pairs of incompatible observables. As a case study, we deduce measurement uncertainty relations for qubit observables using two distinct error measures. We provide an operational interpretation of the error bounds and discuss some of the first experimental tests of such relations.

​

Optimal joint measurement of two observables of a qubit

S Yu, CH Oh
arxiv:1402.3785 [quant-ph]

ABSTRACT: Heisenberg's uncertainty relations for measurement quantify how well we can jointly measure two complementary observables and have attracted much experimental and theoretical attention recently. Here we provide an exact tradeoff between the worst-case errors in measuring jointly two observables of a qubit, i.e., all the allowed and forbidden pairs of errors, especially asymmetric ones, are exactly pinpointed. For each pair of optimal errors we provide an optimal joint measurement that is realizable without introducing any ancilla and entanglement. Possible experimental implementations are discussed and Toronto experiment [Rozema et al., Phys. Rev. Lett. 109, 100404 (2012)] can be readily adapted to an optimal joint measurement of two orthogonal observables.

​

Direct tests of measurement uncertainty relations: what it takes

P Busch, N Stevens

Physical Review Letters 114, 070402, 2015
DOI: 10.1103/PhysRevLett.114.070402, arxiv:1407.7752v2 [quant-ph]

ABSTRACT: 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.

​

Focusing in Arthurs-Kelly-type Joint Measurements with Correlated Probes

T Bullock, P Busch

Physical Review Letters 113, 120401, 2014
DOI: 10.1103/PhysRevLett.113.120401, arxiv:1405.5840v2 [quant-ph]

ABSTRACT: Joint approximate measurement schemes of position and momentum provide us with a means of inferring pieces of complementary information if we allow for the irreducible noise required by quantum theory. One such scheme is given by the Arthurs-Kelly model, where information about a system is extracted via indirect probe measurements. So far, only separable uncorrelated probes have been considered. Here, following Di Lorenzo (PRL 110, 120403 (2013)), we extend this model to both entangled and classically correlated probes, achieving full generality. We find the measured observable of the system under consideration to be covariant under phase space translations, and show that correlated probes can produce more precise joint measurement outcomes of position and momentum than the same probes can achieve if applied alone to realize a position or momentum measurement. Contrary to Di Lorenzo's claim, we find that nevertheless there are no violations of Heisenberg's measurement uncertainty relations in these generalized Arthurs-Kelly models.

​

Measurement uncertainty relations

P Busch, P Lahti, RF Werner

Journal of Mathematical Physics 155, 042111, 2014

DOI: 10.1063/1.4871444, arxiv:1312:4392v2 [quant-ph]

ABSTRACT: 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.​

​

Comment on "Experimental Test of Error-Disturbance Uncertainty Relations by Weak Measurement"

P Busch, P Lahti, RF Werner
arxiv:1403.0367 [quant-ph]

ABSTRACT: In this comment on the paper by F. Kaneda, S.-Y. Baek, M. Ozawa and K. Edamatsu [Phys. Rev. Lett. 112, 020402, 2014, arXiv:1308.5868], we point out that the claim of having refuted Heisenberg's error-disturbance relation is unfounded since it is based on the choice of unsuitable and operationally problematical quantifications of measurement error and disturbance. As we have shown elsewhere [PRL 111, 160405, 2013], for appropriate choices of operational measures of error and disturbance, Heisenberg's heuristic relation can be turned into a precise inequality which is a rigorous consequence of quantum mechanics.

​

Comment on "Noise and Disturbance in Quantum Measurements: An Information-Theoretic Approach"

P Busch, P Lahti, RF Werner

arxiv:1403.0368 [quant-ph]

ABSTRACT: In this comment on the work of F. Buscemi, M.J.W. Hall, M. Ozawa and M.M. Wilde [PRL 112, 050401, 2014, arXiv:1310.6603], we point out a misrepresentation of measures of error and disturbance introduced in our recent work [PRL 111, 160405, 2013, arXiv:1306.1565] as being "purely formal, with no operational counterparts". We also exhibit an tension in the authors' message, in that their main result is an error-disturbance relation for state-independent measures, but its importance is declared to be limited to discrete variables. In contrast, we point out the separate roles played by such relations for either state-dependent or state-independent measures of error and disturbance.

​

Measurement Uncertainty: Reply to Critics

P Busch, P Lahti, RF Werner

arxiv:1402.3102 [quant-ph]

ABSTRACT: 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.

​

Heisenberg uncertainty for qubit measurements

P Busch, P Lahti, RF Werner

Physical Review A 89, 012129, 2014

DOI: 10.1103/PhysRevA.89.012129, arxiv:1311.0837v3 [quant-ph]

ABSTRACT: 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.

​

Proof of Heisenberg's error-disturbance relation

P Busch, P Lahti, RF Werner

Physical Review Letters 111, 160405, 2013
DOI: 10.1103/PhysRevLett.111.160405, arxiv:1306.1565v2 [quant-ph]

ABSTRACT: While the slogan "no measurement without disturbance" has established itself under the name Heisenberg effect in the consciousness of the scientifically interested public, a precise statement of this fundamental feature of the quantum world has remained elusive, and serious attempts at rigorous formulations of it as a consequence of quantum theory have led to seemingly conflicting preliminary results. Here we show that despite recent claims to the contrary [Rozema et al, Phys. Rev. Lett. 109, 100404 (2012)], Heisenberg-type inequalities can be proven that describe a trade-off between the precision of a position measurement and the necessary resulting disturbance of momentum (and vice versa). More generally, these inequalities are instances of an uncertainty relation for the imprecisions of any joint measurement of position and momentum. Measures of error and disturbance are here defined as figures of merit characteristic of measuring devices. As such they are state independent, each giving worst-case estimates across all states, in contrast to previous work that is concerned with the relationship between error and disturbance in an individual state.

​

Coexistence of qubit effects
P Busch, H-J Schmidt

Quantum Information Processing 9, 143, 2010

DOI: 10.1007/s11128-009-0109-x,  arxiv:0802.4167v3 [quant-ph]

ABSTRACT: Two quantum events, represented by positive operators (effects), are coexistent if they can occur as possible outcomes in a single measurement scheme. Equivalently, the corresponding effects are coexistent if and only if they are contained in the ranges of a single (joint) observable. Here we give several equivalent characterizations of coexistent pairs of qubit effects. We also establish the equivalence between our results and those obtained independently by other authors. Our approach makes explicit use of the Minkowski space geometry inherent in the four-dimensional real vector space of selfadjoint operators in a two-dimensional complex Hilbert space.

​

Approximate joint measurements of qubit observables

P Busch, T Heinosaari

Quantum Information and Computation 8, 797, 2008

arxiv:0706.1415v2 [quant-ph]

ABSTRACT: Joint measurements of qubit observables have recently been studied in conjunction with quantum information processing tasks such as cloning. Considerations of such joint measurements have until now been restricted to a certain class of observables that can be characterized by a form of covariance. Here we investigate conditions for the joint measurability of arbitrary pairs of qubit observables. For pairs of noncommuting sharp qubit observables, a notion of approximate joint measurement is introduced. Optimal approximate joint measurements are shown to lie in the class of covariant joint measurements. The marginal observables found to be optimal approximators are generally not among the coarse-grainings of the observables to be approximated. This yields scope for the improvement of existing joint measurement schemes. Both the quality of the approximations and the intrinsic unsharpness of the approximators are shown to be subject to Heisenberg-type uncertainty relations.

​

Universal joint-measurement uncertainty relation for error bars

P Busch, DB Pearson

Journal of Mathematical Physics 48, 082103, 2007

DOI: 10.1063/1.2759831, arxiv:math-ph/0612074v2

ABSTRACT: We formulate and prove a new, universally valid uncertainty relation for the necessary errors bar widths in any approximate joint measurement of position and momentum.

​

Noise and disturbance in quantum mechanics

P Busch, T Heinonen (now Heinosaari), P Lahti

Physics Letters A 320, 261, 2004

DOI: 10.1016/j.physleta.2003.11.036, arXiv:quant-ph/0312006v1

ABSTRACT: The operational meaning of some measures of noise and disturbance in measurements is analyzed and their limitations are pointed out. The cases of minimal noise and least disturbance are characterized.

​

The uncertainty relation for joint measurement of position and momentum​

RF Werner

Quantum Information and Computation 4, 546, 2004

arXiv:quant-ph/0405184v1

ABSTRACT: We prove an uncertainty relation, which imposes a bound on any joint measurement of position and momentum. It is of the form (ΔP)(ΔQ)≥Cℏ, where the `uncertainties' quantify the difference between the marginals of the joint measurement and the corresponding ideal observable. Applied to an approximate position measurement followed by a momentum measurement, the uncertainties become the precision ΔQ of the position measurement, and the perturbation ΔP of the conjugate variable introduced by such a measurement. We also determine the best constant C, which is attained for a unique phase space covariant measurement.

​

​

​