Incompatible Measurements Research Articles

Stronger error disturbance relations for incompatible quantum measurements

We formulate a new error-disturbance relation, which is free from explicit dependence upon variances in observables. This error-disturbance relation shows improvement over the one provided by the Branciard inequality and the Ozawa inequality for some initial states and for a particular class of joint measurements under consideration. We also prove a modified form of Ozawa's error-disturbance relation. The latter relation provides a tighter bound compared to the Ozawa and the Branciard inequalities for a small number of states.

A Class of Distance-Based Incompatibility Measures for Quantum Measurements

A Class of Distance-Based Incompatibility Measures for Quantum Measurements

A Class of Distance-Based Incompatibility Measures for Quantum Measurements

We discuss a recently proposed class of incompatibility measures for quantum measurements, which is based on quantifying the effect of measurements of one observable on the statistics of the outcome of another. We summarize the properties of this class of measures, and present a tight upper bound for the incompatibility of any set of projective measurements in finite dimensions. We also discuss non-projective measurements, and give a non-trivial upper bound on the mutual incompatibility of a pair of Luders instruments. Using the example of incompatible observables that commute on a subspace, we elucidate how this class of measures goes beyond uncertainty relations in quantifying the mutual incompatibility of quantum measurements.

Information complementarity: A new paradigm for decoding quantum incompatibility

The existence of observables that are incompatible or not jointly measurable is a characteristic feature of quantum mechanics, which lies at the root of a number of nonclassical phenomena, such as uncertainty relations, wave—particle dual behavior, Bell-inequality violation, and contextuality. However, no intuitive criterion is available for determining the compatibility of even two (generalized) observables, despite the overarching importance of this problem and intensive efforts of many researchers. Here we introduce an information theoretic paradigm together with an intuitive geometric picture for decoding incompatible observables, starting from two simple ideas: Every observable can only provide limited information and information is monotonic under data processing. By virtue of quantum estimation theory, we introduce a family of universal criteria for detecting incompatible observables and a natural measure of incompatibility, which are applicable to arbitrary number of arbitrary observables. Based on this framework, we derive a family of universal measurement uncertainty relations, provide a simple information theoretic explanation of quantitative wave—particle duality, and offer new perspectives for understanding Bell nonlocality, contextuality, and quantum precision limit.

Global cardiovascular mortality risk in the adult Polish population: prospective assessment of the cohorts studied in multicentre national WOBASZ and WOBASZ Senior studies.

To develop a global cardiovascular disease (CVD) mortality risk model for the Polish population and to verify these data in the context of the SCORE risk algorithm. We analysed data obtained in two multicentre national population studies, the WOBASZ study which was conducted in 2003-2005 and included 14,769 subjects aged 20-74 years, and the WOBASZ Senior study which was conducted in 2007 and included 1096 subjects above 74 years of age. All these subjects were followed for survival status until 2012 and the cause of death was determined. The mean duration of follow-up was 8.2 years for WOBASZ study participants and about 5 years for WOBASZ Senior study participants. Overall, 1436 subjects died, including 568 due to CVD. For the purpose of our analysis of overall and CVD mortality, 15 established risk factors were selected. Survival was analysed separately in WOBASZ and WOBASZ Senior study participants. Statistical methods included descriptive statistics, Kaplan-Meier curves, Cox proportional hazard models, and the SCORE risk algorithm. Measure of incompatibility of the SCORE risk model to the Polish population was determined as the difference between mortality rates by the SCORE risk quartiles and the Cox approach. During the 8-year follow-up of the WOBASZ study population, mortality due to CVD was 38% among men and 31% among women. The most common causes of CVD mortality were ischaemic heart disease (IHD, 33%) followed by cerebro-vascular disease (17%) in men, and cerebrovascular disease (31%) followed by IHD (23%) in women. We found significant differences between men and women in regard to survival curves for both overall mortality and CVD mortality (p < 0.0001). For overall mortality among men and women, nearly all selected risk factors were shown to be significant in univariate analyses, except for high density lipoprotein cholesterol (HDL-C) level and the total cholesterol/HDL-C ratio in men, and smoking status in women. In multivariate analysis, independent predictors in men included age, glucose level, systolic blood pressure, and smoking status. In women, independent predictors were age, smoking status, and diabetes. During the 5-year follow-up of the WOBASZ Senior study population, mortality due to CVD was 48% among men and 58% among women. The most common cause of CVD mortality in both men and women was IHD (29% and 24%, respectively), followed by cerebrovascular disease (16% and 21%, respectively). We found significant differences between men and women in regard to survival curves for overall mortality (p < 0.0001) but not for CVD mortality (p = 0.0755). Due to the fact that survival curves for CVD mortality did not differ between men and women, we estimated the cut-off age for no survival difference in the WOBASZ study. By selecting the oldest patients and adding them to the WOBASZ Senior cohort, we obtained the cut-off age of 70 years above which the survival curves were not significantly different between men and women. In univariate analyses, independent predictors in men were age and creatinine level. These factors remained significant in multivariate analysis. In women above 74 years of age, independent predictors in univariate analyses included age, HDL-C level, creatinine level, total cholesterol/HDL-C ratio, and smoking status. Age, HDL-C level, creatinine level, and smoking status remained independent predictors of overall mortality in multivariate analysis. For CVD mortality, significant predictors were the same as for overall mortality. In women, significant predictors in uni- and multivariate analyses were age and smoking status. Overall disagreement between CVD mortality rates by the SCORE risk model and the Cox model was 5.7% in men and 2% in women. 1. Long-term follow-up of WOBASZ and WOBASZ Senior study participants allowed assessment of the inde-pendent association of the evaluated cardiovascular risk factors with CVD mortality in the Polish population. 2. Validation of the SCORE risk algorithm to estimate individual global CVD risk in the Polish population showed a high predictive value of this algorithm.

Material homogeneity and strain compatibility in thin elastic shells

We discuss several issues regarding material homogeneity and strain compatibility for materially uniform thin elastic shells from the viewpoint of a three-dimensional theory, with small thickness, as well as a two-dimensional Cosserat theory. A relationship between inhomogeneity and incompatibility measures under the two descriptions is developed. More specifically, we obtain explicit forms of intrinsic dislocation density tensors characterizing the inhomogeneity of a dislocated Cosserat shell. We also formulate a system of governing equations for the residual stress field emerging out of strain incompatibilities which in turn are related to inhomogeneities. The equations are simplified for several cases under the Kirchhoff–Love assumption.

Noise robustness of the incompatibility of quantum measurements

The existence of incompatible measurements is a fundamental phenomenon having no explanation in classical physics. Intuitively, one considers given measurements to be incompatible within a framework of a physical theory, if their simultaneous implementation on a single physical device is prohibited by the theory itself. In the mathematical language of quantum theory, measurements are described by POVMs (positive operator valued measures), and given POVMs are by definition incompatible if they cannot be obtained via coarse-graining from a single common POVM; this notion generalizes noncommutativity of projective measurements. In quantum theory, incompatibility can be regarded as a resource necessary for manifesting phenomena such as Clauser-Horne-Shimony-Holt (CHSH) Bell inequality violations or Einstein-Podolsky-Rosen (EPR) steering which do not have classical explanation. We define operational ways of quantifying this resource via the amount of added classical noise needed to render the measurements compatible, i.e., useless as a resource. In analogy to entanglement measures, we generalize this idea by introducing the concept of incompatibility measure, which is monotone in local operations. In this paper, we restrict our consideration to binary measurements, which are already sufficient to explicitly demonstrate nontrivial features of the theory. In particular, we construct a family of incompatibility monotones operationally quantifying violations of certain scaled versions of the CHSH Bell inequality, prove that they can be computed via a semidefinite program, and show how the noise-based quantities arise as special cases. We also determine maximal violations of the new inequalities, demonstrating how Tsirelson's bound appears as a special case. The resource aspect is further motivated by simple quantum protocols where our incompatibility monotones appear as relevant figures of merit.

Surmounting intrinsic quantum-measurement uncertainties in Gaussian-state tomography with quadrature squeezing.

We reveal that quadrature squeezing can result in significantly better quantum-estimation performance with quantum heterodyne detection (of H. P. Yuen and J. H. Shapiro) as compared to quantum homodyne detection for Gaussian states, which touches an important aspect in the foundational understanding of these two schemes. Taking single-mode Gaussian states as examples, we show analytically that the competition between the errors incurred during tomogram processing in homodyne detection and the Arthurs-Kelly uncertainties arising from simultaneous incompatible quadrature measurements in heterodyne detection can often lead to the latter giving more accurate estimates. This observation is also partly a manifestation of a fundamental relationship between the respective data uncertainties for the two schemes. In this sense, quadrature squeezing can be used to overcome intrinsic quantum-measurement uncertainties in heterodyne detection.

Entropic Uncertainty Relation and Information Exclusion Relation for multiple measurements in the presence of quantum memory.

The Heisenberg uncertainty principle shows that no one can specify the values of the non-commuting canonically conjugated variables simultaneously. However, the uncertainty relation is usually applied to two incompatible measurements. We present tighter bounds on both entropic uncertainty relation and information exclusion relation for multiple measurements in the presence of quantum memory. As applications, three incompatible measurements on Werner state and Horodecki’s bound entangled state are investigated in details.

Measurement incompatibility and channel steering

Incompatible measurements in quantum theory always lead to Einstein-Podolsky-Rosen (EPR)-Schr\"{o}dinger steering. Channel steering which is a generalized notion of EPR-Schr\"{o}dinger steering, has been introduced recently. Here we establish a connection between lack of joint measureability and channel steering.

A method of determining attribute weights in evidential reasoning approach based on incompatibility among attributes

Decisions on attribute weights are important problems in multiple attribute decision making. Many methods have been proposed to create attribute weights which are used to aggregate attributes in a simple additive weighting way. In this paper, a method of deriving attribute weights from incompatibility among attributes and possible constraints on the weights is developed based on the evidential reasoning approach in which attribute aggregation is nonlinear rather than linear. The incompatibility is a flexible combination of deviation incompatibility and decision incompatibility with a relaxation coefficient. The deviation incompatibility measures differences between assessments of alternatives on each attribute and the decision incompatibility quantifies differences between assessments of alternatives on one attribute and the aggregated assessments of the alternatives. For a specific alternative, two pairs of optimization problems with a constraint on the difference between potential weights and the combination of deviation incompatibility and decision incompatibility are designed to generate the favorable intervals of attribute weights and those of utilities of assessment grades. A problem of car performance assessment is investigated to demonstrate the applicability of the proposed method. The method is validated by comparison with other methods of producing attribute weights using the problem.

Entropic uncertainty relations for multiple measurements

We present the entropic uncertainty relations for multiple measurement settings in quantum mechanics. Those uncertainty relations are obtained for both cases with and without the presence of quantum memory. They take concise forms which can be proven in a unified method and easy to calculate. Our results recover the well known entropic uncertainty relations for two observables, which show the uncertainties about the outcomes of two incompatible measurements. Those uncertainty relations are applicable in both foundations of quantum theory and the security of many quantum cryptographic protocols.

Combining PM2.5 Component Data from Multiple Sources: Data Consistency and Characteristics Relevant to Epidemiological Analyses of Predicted Long-Term Exposures.

BackgroundRegulatory monitoring data have been the exposure data resource most commonly applied to studies of the association between long-term PM2.5 components and health. However, data collected for regulatory purposes may not be compatible with epidemiological studies.ObjectivesWe studied three important features of the PM2.5 component monitoring data to determine whether it would be appropriate to combine all available data from multiple sources for developing spatiotemporal prediction models in the National Particle Component and Toxicity (NPACT) study.MethodsThe NPACT monitoring data were collected in an extensive monitoring campaign targeting cohort participant residences. The regulatory monitoring data were obtained from the Chemical Speciation Network (CSN) and the Interagency Monitoring of Protected Visual Environments (IMPROVE). We performed exploratory analyses to examine features that could affect our approach to combining data: comprehensiveness of spatial coverage, comparability of analysis methods, and consistency in sampling protocols. In addition, we considered the viability of developing spatiotemporal prediction models given a) all available data, b) NPACT data only, and c) NPACT data with temporal trends estimated from other pollutants.ResultsThe number of CSN/IMPROVE monitors was limited in all study areas. The different laboratory analysis methods and sampling protocols resulted in incompatible measurements between networks. Given these features we determined that it was preferable to develop our spatiotemporal models using only the NPACT data and under simplifying assumptions.ConclusionsInvestigators conducting epidemiological studies of long-term PM2.5 components need to be mindful of the features of the monitoring data and incorporate this understanding into the design of their monitoring campaigns and the development of their exposure prediction models.CitationKim SY, Sheppard L, Larson TV, Kaufman JD, Vedal S. 2015. Combining PM2.5 component data from multiple sources: data consistency and characteristics relevant to epidemiological analyses of predicted long-term exposures. Environ Health Perspect 123:651–658; http://dx.doi.org/10.1289/ehp.1307744

Joint measurability, Einstein-Podolsky-Rosen steering, and Bell nonlocality.

We investigate the relation between the incompatibility of quantum measurements and quantum nonlocality. We show that a set of measurements is not jointly measurable (i.e., incompatible) if and only if it can be used for demonstrating Einstein-Podolsky-Rosen steering, a form of quantum nonlocality. Moreover, we discuss the connection between Bell nonlocality and joint measurability, and give evidence that both notions are inequivalent. Specifically, we exhibit a set of incompatible quantum measurements and show that it does not violate a large class of Bell inequalities. This suggests the existence of incompatible quantum measurements which are Bell local, similarly to certain entangled states which admit a local hidden variable model.

DECIPHERING THE ATMOSPHERIC COMPOSITION OF WASP-12b: A COMPREHENSIVE ANALYSIS OF ITS DAYSIDE EMISSION

WASP-12b was the first planet reported to have a carbon-to-oxygen ratio (C/O) greater than one in its dayside atmosphere. However, recent work to further characterize its atmosphere and confirm its composition has led to incompatible measurements and divergent conclusions. Additionally, the recent discovery of stellar binary companions ~1" from WASP-12 further complicates the analyses and subsequent interpretations. We present a uniform analysis of all available Hubble and Spitzer Space Telescope secondary-eclipse data, including previously-unpublished Spitzer measurements at 3.6 and 4.5 microns. The primary controversy in the literature has centered on the value and interpretation of the eclipse depth at 4.5 microns. Our new measurements and analyses confirm the shallow eclipse depth in this channel, as first reported by Campo and collaborators and used by Madhusudhan and collaborators to infer a carbon-rich composition. To explain WASP-12b's observed dayside emission spectrum, we implemented several recent retrieval approaches. We find that when we exclude absorption due to C2H2 and HCN, which are not universally considered in the literature, our models require implausibly large atmospheric CO2 abundances, regardless of the C/O. By including C2H2 and HCN in our models, we find that a physically-plausible carbon-rich solution achieves the best fit to the available photometric and spectroscopic data. In comparison, the best-fit oxygen-rich models have abundances that are inconsistent with the chemical equilibrium expectations for hydrogen-dominated atmospheres and are 670 times less probable. Our best-fit solution is also 7.3*10^{6} times more probable than an isothermal blackbody model.

Entropic uncertainty from effective anticommutators

We investigate entropic uncertainty relations for two or more binary measurements, for example, spin-$\frac{1}{2}$ or polarization measurements. We argue that the effective anticommutators of these measurements, i.e., the anticommutators evaluated on the state prior to measuring, are an expedient measure of measurement incompatibility. Based on the knowledge of pairwise effective anticommutators we derive a class of entropic uncertainty relations in terms of conditional R\'enyi entropies. Our uncertainty relations are formulated in terms of effective measures of incompatibility, which can be certified in a device-independent fashion. Consequently, we discuss potential applications of our findings to device-independent quantum cryptography. Moreover, to investigate the tightness of our analysis we consider the simplest (and very well studied) scenario of two measurements on a qubit. We find that our results outperform the celebrated bound due to Maassen and Uffink [Phys. Rev. Lett. 60, 1103 (1988)] and provide an analytical expression for the minimum uncertainty which also outperforms some recent bounds based on majorization.

Measures of disturbance and incompatibility for quantum measurements

We propose a class of incompatibility measures for quantum observables based on quantifying the effect of a measurement of one observable on the statistics of the outcomes of another. Specifically, for a pair of observables $A$ and $B$ with purely discrete spectra, we compare the following two probability distributions: one resulting from a measurement of $A$ followed by a measurement of $B$ on a given state, and the other obtained from a measurement of $B$ alone on the same state. We show that maximizing the distance between these two distributions over all states yields a valid measure of the incompatibility of observables $A$ and $B$, which is zero if and only if they commute and is strictly greater than zero (and less than or equal to one) otherwise. For finite dimensional systems, we obtain a tight upper bound on the incompatibility of any pair of observables and show that the bound is attained when the observables are totally non-degenerate and associated with mutually unbiased bases. In the process, we also establish an important connection between the incompatibility of a pair of observables and the maximal disturbances due to their measurements. Finally, we indicate how these measures of incompatibility and disturbance can be extended to the more general class of non-projective measurements. In particular, we obtain a non-trivial upper bound on the incompatibility of one L\"uders instrument with another.

Improved entropic uncertainty relations and information exclusion relations

The uncertainty principle can be expressed in entropic terms, also taking into account the role of entanglement in reducing uncertainty. The information exclusion principle bounds instead the correlations that can exist between the outcomes of incompatible measurements on one physical system, and a second reference system. We provide a more stringent formulation of both the uncertainty principle and the information exclusion principle, with direct applications for, e.g., the security analysis of quantum key distribution, entanglement estimation, and quantum communication. We also highlight a fundamental distinction between the complementarity of observables in terms of uncertainty and in terms of information.

The use of meta-analytic software to derive hypotheses for EMDR

In a comment directed at the fallacies of meta-analyses, Kazrin,Durac, and Agteros (1979) imagined a process which they termed ameta-meta-analysis. Stating that “the strength of the meta-meta-analysis, like the unfortunate person stranded on a desert island,is that it will look at anything” (p. 397), they whimsically arguedhow this process, directed at psychotherapy effectiveness, utilised850,000 therapists and 2,000,000 clients. Such tongue in cheekwarnings regarding the misuse of meta-analyses while evaluatingpsychotherapy effectiveness should be taken seriously. Indeed,looking at the inclusion and exclusion criteria within meta-analyses, Barlow (2010) warns us of cherry-picking studies statingthat “it is easy tosit backand ‘pick-off’ any newstudyand concludethat it’snot‘perfect’” (p. 16). For this reason, Barlow argues, weneed to be very cautious when similar meta-analyses includedifferent studies and come to different conclusions.Lee and Cuijpers (2013), in their meta-analysis of ‘trials’ lookingpurely at the utility of eye movements in EMDR, concatenate eyemovements, negative imagery, autobiographical memories con-taining distress, and randomised experimental study or treatmentwith patients or students to place their inclusion criteria. Havingset-up the meta-analysis exclusion criteria they accordinglyexclude large numbers of studies which tested this exact phe-nomenon for various reasons of technicality or preference (e.g.,Devilly & Brown, 2011; Devilly & Spence, 1999; Jensen, 1994;Merckelbach, Hogervorst, Kampman, & de Jongh, 1994; Pitmanet al., 1996; Rothbaum, Astin, & Marsteller, 2005; Taylor et al.,2003; to name just a few) or cherry-picking (or combining)incompatible outcome measures when the studies have beenincluded (e.g., Dunn, Schwartz, Hatfield & Weigele, 1996 present awealth of physiological data rather unsympathetic to EMDR, whichwas not factored into the analysis). The authors then wish toinclude two of the studies twice each in the same meta-analysis(incidentally one of these twice counted studies was conductedby the first author of the meta-analysis). One of the main reasonsforameta-analysisistoreducecommonmethodvariancebiasfromindividual research groups. Whether this actually increases or de-creases the derived effect size is irrelevant in comparison to theprinciple of introducing error from the same study twice. Besidesnegating this benefit, the current approach also requires moreclarificationonhowonesetofconfidenceintervalsaroundaneffectsize mean was computed for each individual study from multipleoutcome measures (their Figure 2 and 3). We assume that the in-terval around this average was computed based upon total samplesize rather than the number of measures combined or a multipli-cation of number of measures by sample size or some other esti-mate of error.Lee’s twice counted study utilised students, as opposed to pa-tients, as did some other studies included in the meta-analysis andalso, by their own admission, deviated extensively from the EMDRprotocol. We are unclear why one would accept this study andsimilar studies, yet not include large trials which compared expo-sure with and without eye movements (i.e., PE vs EMDR). If indeedthe argument is that it is the eye movements which make thedifference, then excluding them appears quite inconsistent.Lee & Cuijpers obtain an effect size which they claim is signifi-cant. The fact that an effect size of 0.27 is significant in this meta-analysis (when controlling for two extreme outliers) from 12studies(or13ifweincludeLee’sstudytwice)wouldsuggestthatwehave complete agreement amongst these studies. However, with 5out of the 12 studies finding effect sizes in exactly the opposite di-rection or finding a zero sum result, and with us being aware ofexcluded studies also finding opposite results, we remain uncon-vincedbythismeta-analysis.Excludingstudieswhichcontrolledforplacebo effects (such as tapping and clicking) and not includingstudies unless it focussed on a negative autobiographical memory(i.e., excluding studies which concentrated on trauma words, anxi-ety, etc), and at the same time excluding any comparison groups

Operational measure of incompatibility of noncommuting observables

Uncertainty relations are often considered to be a measure of incompatibility of noncommuting observables. However, such a consideration is not valid in general, motivating the need for an alternate measure that applies to any set of noncommuting observables. We present an operational approach to quantifying incompatibility without invoking uncertainty relations. Our measure aims to capture the incompatibility of noncommuting observables as manifest in the non-orthogonality of their eigenstates. We prove that this measure has all the desired properties: it is zero when the observables commute, strictly greater than zero when they do not, and is maximum when they are mutually unbiased. We also obtain tight upper bounds on this measure for any N noncommuting observables and compute it exactly when the observables are mutually unbiased.