Error-disturbance Uncertainty Relation Research Articles

Error-disturbance uncertainty relations in Faraday measurements

We examine error-disturbance relations in the quantum measurement of spin systems using an atom-light interface scheme. We model a single spin-1/2 system that interacts with a polarized light meter via a Faraday interaction. We formulate the error and disturbance of the model and examine the uncertainty relations. We find that, for the coherent light meter in pure polarization, both the error and disturbance behave like the cyclic oscillations due to the Faraday rotation in both the light and spin polarizations. We also examine a class of polarization-squeezed light meter, where we apply the phase-space approximation and characterize the role of squeezing. We derive the error-disturbance relations for these cases and find that the Heisenberg-Arthurs-Kelly uncertainty is violated while the tight Branciard-Ozawa uncertainty always holds. We note that, in the limit of weak interaction strength, the error and disturbance come to obey the unbiasedness condition and hence the Heisenberg-Arthurs-Kelly relation holds. The work contributes to our understanding of the quantum measurement of spin systems under the atom-light interface framework and may hold potential applications in quantum metrology, quantum state estimation, and control.

Experimental test of error-disturbance uncertainty relation with continuous variables

The uncertainty relation is one of the fundamental principles in quantum mechanics and plays an important role in quantum information science. We experimentally test the error-disturbance uncertainty relation (EDR) with continuous variables for Gaussian states. Two incompatible continuous-variable observables, amplitude and phase quadratures of an optical mode, are measured simultaneously using a heterodyne measurement system. The EDR values with continuous variables for coherent, squeezed, and thermal states are verified experimentally. Our experimental results demonstrate that Heisenberg’s EDR with continuous variables is violated, while Ozawa’s and Branciard’s EDRs with continuous variables are validated.

Experimental test of error-tradeoff uncertainty relation using a continuous-variable entangled state

Heisenberg’s original uncertainty relation is related to measurement effect, which is different from the preparation uncertainty relation. However, it has been shown that Heisenberg’s error disturbance uncertainty relation is not valid in some cases. We experimentally test the error-tradeoff uncertainty relation by using a continuous-variable Gaussian Einstein–Podolsky–Rosen (EPR)-entangled state. Based on the quantum correlation between the two entangled optical beams, the errors on amplitude and phase quadratures of one EPR optical beam coming from joint measurement are estimated, respectively, which are used to verify the error–tradeoff relation. Especially, the error–tradeoff relation for error-free measurement of one observable is verified in our experiment. We also verify the error–tradeoff relations for nonzero errors and mixed state by introducing loss on one EPR beam. Our experimental results demonstrate that Heisenberg’s error–tradeoff relation is violated in some cases for a continuous-variable system, while the Ozawa’s and Branciard’s relations are valid.

Heisenberg's error-disturbance uncertainty relation: Experimental study of competing approaches

Over the past few years, Heisenberg's error-disturbance uncertainty relation has experienced increased attention since several experimental publications verified the theoretical findings of Ozawa predicting the violation and thus necessary reformulation of Heisenberg's relation. However, soon after their appearance, an alternative theory was presented by Busch and co-workers, which proclaimed the validity of Heisenberg's relation and thus gave rise to heated debates. Here, we present an experimental comparison of the competing approaches by applying them to the same neutron optical measurement apparatus. Empirical results for the different definitions of error and disturbance are presented for special input states and configurations of the apparatus to illustrate the opposing approaches. The inequalities restricting errors and disturbances are particularly emphasized. Despite the strong controversy, in the case of projectively measured qubit observables, both approaches lead to equal outcomes.

Uncertainty relations: An operational approach to the error-disturbance tradeoff

The notions of error and disturbance appearing in quantum uncertainty relations are often quantified by the discrepancy of a physical quantity from its ideal value. However, these real and ideal values are not the outcomes of simultaneous measurements, and comparing the values of unmeasured observables is not necessarily meaningful according to quantum theory. To overcome these conceptual difficulties, we take a different approach and define error and disturbance in an operational manner. In particular, we formulate both in terms of the probability that one can successfully distinguish the actual measurement device from the relevant hypothetical ideal by any experimental test whatsoever. This definition itself does not rely on the formalism of quantum theory, avoiding many of the conceptual difficulties of usual definitions. We then derive new Heisenberg-type uncertainty relations for both joint measurability and the error-disturbance tradeoff for arbitrary observables of finite-dimensional systems, as well as for the case of position and momentum. Our relations may be directly applied in information processing settings, for example to infer that devices which can faithfully transmit information regarding one observable do not leak any information about conjugate observables to the environment. We also show that Englert's wave-particle duality relation [PRL 77, 2154 (1996)] can be viewed as an error-disturbance uncertainty relation.

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.

Error-disturbance uncertainty relations studied in neutron optics

Heisenberg's uncertainty principle is probably the most famous statement of quantum physics and its essential aspects are well described by a formulations in terms of standard deviations. However, a naive Heisenberg-type error-disturbance relation is not valid. An alternative universally valid relation was derived by Ozawa in 2003. Though universally valid Ozawa's relation is not optimal. Recently, Branciard has derived a tight error-disturbance uncertainty relation (EDUR), describing the optimal trade-off between error and disturbance. Here, we report a neutron-optical experiment that records the error of a spin-component measurement, as well as the disturbance caused on another spin-component to test EDURs. We demonstrate that Heisenberg's original EDUR is violated, and the Ozawa's and Branciard's EDURs are valid in a wide range of experimental parameters, applying a new measurement procedure referred to as two-state method.

Error-disturbance uncertainty relations in neutron spin measurements

Heisenberg’s uncertainty principle in a formulation of uncertainties, intrinsic to any quantum system, is rigorously proven and demonstrated in various quantum systems. Nevertheless, Heisenberg’s original formulation of the uncertainty principle was given in terms of a reciprocal relation between the error of a position measurement and the thereby induced disturbance on a subsequent momentum measurement. However, a naive generalization of a Heisenberg-type error-disturbance relation for arbitrary observables is not valid. An alternative universally valid relation was derived by Ozawa in 2003. Though universally valid, Ozawa’s relation is not optimal. Recently, Branciard has derived a tight error-disturbance uncertainty relation (EDUR), describing the optimal trade-off between error and disturbance under certain conditions. Here, we report a neutron-optical experiment that records the error of a spin-component measurement, as well as the disturbance caused on another spin-component to test EDURs. We demonstrate that Heisenberg’s original EDUR is violated, and Ozawa’s and Branciard’s EDURs are valid in a wide range of experimental parameters, as well as the tightness of Branciard’s relation.

Uncertainty Relation and Quantum Cheshire-Cat: Studied with Neutron Polarimeter and Interferometer

Fundamental phenomena in quantum mechanics are investigated by the use of matter-wave optics: in the studies neutron polarimeter and interferometer are exploited. Successive measurements of 1/2-spin of the neutron are carried out to test the error-disturbance uncertainty relation. The experimental results confirm the violation of Heisenberg's original reciprocal relation for measurement error and disturbance, and the validity of the reformulated generally valid relation. In addition, as an example of a counterfactual phenomenon of quantum mechanics, interferometric experiment is performed to observe the so-called quantum Cheshire-Cat: a particle and its magnetic moment travel through the interferometer along different beam paths. The results of our experiment suggest that, with suitable pre- and post-selections, neutrons travel along one of the arms of the interferometer, while their spin is located in the other arm.

Investigations of fundamental phenomena in quantum mechanics with neutrons

Neutron interferometer and polarimeter are used for the experimental investigations of quantum mechanical phenomena. Interferometry exhibits clear evidence of quantum-contextuality and polarimetry demonstrates conflicts of a contextual model of quantum mechanics á la Leggett. In these experiments, entanglements are achieved between degrees of freedom in a single-particle: spin, path and energy degrees of freedom are manipulated coherently and entangled. Both experiments manifest the fact that quantum contextuality is valid for phenomena with matter waves with high precision. In addition, another experiment is described which deals with error-disturbance uncertainty relation: we have experimentally tested error-disturbance uncertainty relations, one is derived by Heisenberg and the other by Ozawa. Experimental results confirm the fact that the Heisenberg's uncertainty relation is often violated and that the new relation by Ozawa is always larger than the limit. At last, as an example of a counterfactual phenomenon of quantum mechanics, observation of so-called quantum Cheshire Cat is carried out by using neutron interferometer. Experimental results suggest that pre- and post-selected neutrons travel through one of the arms of the interferometer while their magnetic moment is located in the other arm.

Experimental Test of Error-Disturbance Uncertainty Relations by Weak Measurement

We experimentally test the error-disturbance uncertainty relation (EDR) in generalized, strength-variable measurement of a single photon polarization qubit, making use of weak measurement that keeps the initial signal state practically unchanged. We demonstrate that the Heisenberg EDR is violated, yet the Ozawa and Branciard EDRs are valid throughout the range of our measurement strength.

Error-Disturbance Uncertainty Relations in Neutron-Spin Measurements

In his seminal paper, which was published in 1927, Heisenberg originally introduced a relation between the precision of a measurement and the disturbance it induces onto another measurement. Here, we report a neutron-optical experiment that records the error of a spin-component measurement as well as the disturbance caused on a measurement of another spin-component to test error-disturbance uncertainty relations (EDRs). We demonstrate that Heisenberg’s original EDR is violated and the Ozawa and Branciard EDRs are valid in a wide range of experimental parameters.

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.

Universally valid Heisenberg uncertainty relation

A universally valid Heisenberg uncertainty relation is proposed by combining the universally valid error-disturbance uncertainty relation of Ozawa with the relation of Robertson. This form of the uncertainty relation, which is defined with the same mathematical rigor as the relations of Kennard and Robertson, incorporates both of the intrinsic quantum fluctuations and measurement effects.

Experimental demonstration of a universally valid error–disturbance uncertainty relation in spin measurements

The uncertainty principle generally prohibits determination of certain pairs of quantum mechanical observables with arbitrary precision and forms the basis of indeterminacy in quantum mechanics. It was Heisenberg who used the famous gamma-ray microscope thought experiment to illustrate this indeterminacy. A lower bound was set for the product of the measurement error of an observable and the disturbance caused by the measurement. Later on, the uncertainty relation was reformulated in terms of standard deviations, which focuses solely on indeterminacy of predictions and neglects unavoidable recoil in measuring devices. A correct formulation of the error-disturbance relation, taking recoil into account, is essential for a deeper understanding of the uncertainty principle. However, the validity of Heisenberg's original error-disturbance uncertainty relation is justifed only under limited circumstances. Another error-disturbance relation, derived by rigorous and general theoretical treatments of quantum measurements, is supposed to be universally valid. Here, we report a neutron optical experiment that records the error of a spin-component measurement as well as the disturbance caused on another spin-component measurement. The results confirm that both error and disturbance completely obey the new, more general relation but violate the old one in a wide range of an experimental parameter.