The uncertainty principle is an important tenet and active field of research in quantum physics. Information-theoretic uncertainty relations, formulated using entropies, provide one approach to quantifying the extent to which two non-commuting observables can be jointly measured. Recent theoretical analysis predicts that general quantum measurements are necessary to saturate some such uncertainty relations and thereby overcome certain limitations of projective measurements. Here, we experimentally test a tight information-theoretic measurement uncertainty relation with neutron spin- 1 / 2 qubits. The noise associated to the measurement of an observable is defined via conditional Shannon entropies and a tradeoff relation between the noises for two arbitrary spin observables is demonstrated. The optimal bound of this tradeoff is experimentally obtained for various non-commuting spin observables. For some of these observables this lower bound can be reached with projective measurements, but we observe that, in other cases, the tradeoff is only saturated by general quantum measurements (i.e. positive-operator valued measures) as predicted theoretically. These results showcase experimentally the advantage obtainable by general quantum measurements over projective measurements when probing certain uncertainty relations.
The uncertainty principle is an important tenet and active field of research in quantum physics. Information-theoretic uncertainty relations, formulated using entropies, provide one approach to quantifying the extent to which two non-commuting observables can be jointly measured. Recent theoretical analysis predicts that general quantum measurements are necessary to saturate some such uncertainty relations and thereby overcome certain limitations of projective measurements. Here, we experimentally test a tight information-theoretic measurement uncertainty relation with neutron spin- 1 / 2 qubits. The noise associated to the measurement of an observable is defined via conditional Shannon entropies and a tradeoff relation between the noises for two arbitrary spin observables is demonstrated. The optimal bound of this tradeoff is experimentally obtained for various non-commuting spin observables. For some of these observables this lower bound can be reached with projective measurements, but we observe that, in other cases, the tradeoff is only saturated by general quantum measurements (i.e. positive-operator valued measures) as predicted theoretically. These results showcase experimentally the advantage obtainable by general quantum measurements over projective measurements when probing certain uncertainty relations.
Information-theoretic uncertainty relations formulate the joint immeasurability of two noncommuting observables in terms of information entropies.The tradeoff of the accuracy in the outcome of two successive measurements manifests in entropic noise-disturbance uncertainty relations.Recent theoretical analysis predicts that projective measurements are not optimal, with respect to the noise-disturbance tradeoffs.Therefore, the results in our previous paper [Phys. Rev. Lett. 115, 030401 (2015)] are outperformed by general quantum measurements.Here, we experimentally test a tight information-theoretic measurement uncertainty relation for three-outcome positive-operator-valued measures, using neutron spin-1 2 qubits.The obtained results violate the lower bound for projective measurements as theoretically predicted.
The emergence of the uncertainty principle has celebrated its 90th anniversary recently. For this occasion, the latest experimental results of uncertainty relations quantified in terms of Shannon entropies are presented, concentrating only on outcomes in neutron optics. The focus is on the type of measurement uncertainties that describe the inability to obtain the respective individual results from joint measurement statistics. For this purpose, the neutron spin of two non-commuting directions is analyzed. Two sub-categories of measurement uncertainty relations are considered: noise–noise and noise–disturbance uncertainty relations. In the first case, it will be shown that the lowest boundary can be obtained and the uncertainty relations be saturated by implementing a simple positive operator-valued measure (POVM). For the second category, an analysis for projective measurements is made and error correction procedures are presented.
Information-theoretic definitions for noise and disturbance in quantum measurements were given in [Phys. Rev. Lett. 112, 050401 (2014)] and a state-independent noise-disturbance uncertainty relation was obtained. Here, we derive a tight noise-disturbance uncertainty relation for complementary qubit observables and carry out an experimental test. Successive projective measurements on the neutron's spin-1/2 system, together with a correction procedure which reduces the disturbance, are performed. Our experimental results saturate the tight noise-disturbance uncertainty relation for qubits when an optimal correction procedure is applied.
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.
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.
Information-theoretic definitions for the noise associated with a quantum measurement and the corresponding disturbance to the state of the system have recently been introduced [F. Buscemi et al., Phys. Rev. Lett. 112, 050401 (2014)]. These definitions are invariant under relabeling of measurement outcomes, and lend themselves readily to the formulation of state-independent uncertainty relations both for the joint estimate of observables (noise-noise relations) and the noise-disturbance tradeoff. Here we derive such relations for incompatible qubit observables, which we prove to be tight in the case of joint estimates, and present progress towards fully characterizing the noise-disturbance tradeoff. In doing so, we show that the set of obtainable noise-noise values for such observables is convex, whereas the conjectured form for the set of obtainable noise-disturbance values is not. Furthermore, projective measurements are not optimal with respect to the joint-measurement noise or noise-disturbance tradeoffs. Interestingly, it seems that four-outcome measurements are needed in the former case, whereas three-outcome measurements are optimal in the latter.