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Conference on Lasers and Electro-Optics Pacific Rim (CLEO-PR)

Correlation between initial and final result in a sequential quantum measurement

M. Iinuma, Yutaro Suzuki, Taiki Nii, Ryuji Kinoshita, Holger Friedrich Hofmann

Published Aug 1, 2015• 0 citations
Publisher

Abstract

Measurement error can be evaluated using the concepts introduced by Ozawa in 2003. Importantly, this evaluation takes into account the non-classical effects of quantum correlations. We investigate this effect by using a sequence of non-commuting measurements of photon polarization with a variable measurement resolution for the intermediate measurement. It is shown that quantum correlations between the initial and the final measurement outcomes result in a significant reduction of the measurement errors for the target observable, even though the final measurement is not sensitive to the polarization component of interest.

DOI
10.1109/cleopr.2015.7375883

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Measurement error can be evaluated using the concepts introduced by Ozawa in 2003. Importantly, this evaluation takes into account the non-classical effects of quantum correlations. We investigate this effect by using a sequence of non-commuting measurements of photon polarization with a variable measurement resolution for the intermediate measurement. It is shown that quantum correlations between the initial and the final measurement outcomes result in a significant reduction of the measurement errors for the target observable, even though the final measurement is not sensitive to the polarization component of interest.
In general, it is difficult to evaluate measurement errors when the initial and final conditions of the measurement make it impossible to identify the correct value of the target observable. Ozawa proposed a solution based on the operator algebra of observables which has recently been used in experiments investigating the error-disturbance trade-off of quantum measurements. Importantly, this solution makes surprisingly detailed statements about the relations between measurement outcomes and the unknown target observable. In the present paper, we investigate this relation by performing a sequence of two measurements on the polarization of a photon, so that the first measurement commutes with the target observable and the second measurement is sensitive to a complementary observable. While the initial measurement can be evaluated using classical statistics, the second measurement introduces the effects of quantum correlations between the noncommuting physical properties. By varying the resolution of the initial measurement, we can change the relative contribution of the nonclassical correlations and identify their role in the evaluation of the quantum measurement. It is shown that the most striking deviation from classical expectations is obtained at the transition between weak and strong measurements, where the competition between different statistical effects results in measurement values well outside the range of possible eigenvalues.
In order to evaluate a quantum correlation between two measurement errors in a joint measurement, we performed correlation measurements of entangled photon pairs with two identified joint measurements, where two different non-commuting polarizations are jointly measured for each photon. The joint measurements were realized by filtering the polarization oriented the direction, which is halfway in two observables. The experimentally observed correlations corresponded to the ideal correlations expected for the entangled state, except for the correlation between the products of the two non-commuting polarizations. This result exceeds the limits that apply to classical error statistics and suggests that the correlation between the measurement errors in each local measurement is imaginary.
A sequential measurement of two non-commuting quantum observables results in a joint probability distribution for all output combinations that can be explained in terms of an initial joint quasi-probability of the non-commuting observables, modified by the resolution errors and back-action of the initial measurement. Here, we show that the error statistics of a sequential measurement of photon polarization performed at different measurement strengths can be described consistently by an imaginary correlation between the statistics of resolution and back-action. The experimental setup was designed to realize variable strength measurements with well-controlled imaginary correlation between the statistical errors caused by the initial measurement of diagonal polarizations, followed by a precise measurement of the horizontal/vertical polarization. We perform the experimental characterization of an elliptically polarized input state and show that the same complex joint probability distribution is obtained at any measurement strength.
We have experimentally evaluated the nonclassical relation between measurement errors in a joint measurement of two noncommuting polarizations by using entangled photon pairs as a probe. The joint measurement was realized by filtering polarization directions that are sensitive to both of the two target polarizations, and the same filtering procedures were applied independently to the two photons of an entangled pair. Since the statistically independent measurement errors of two identical measurements performed on entangled pairs will reduce an overall visibility of the correlation by the square of the local visibilities, the squared visibilities of the local measurements can be obtained directly from the measurement data. We apply this method to determine the visibility of the correlation between the products of the two noncommuting polarizations, a characteristic of the measurement errors that cannot be obtained locally since there is no self-adjoint operator that describes this product of measurement outcomes as a single-photon observable. The experimental results clearly show that the square of the product visibility is negative, indicating the nonclassical nature of the statistical relation between the measurement errors of noncommuting polarizations.
The origin of non-classical correlations is difficult to identify since the uncertainty principle requires that information obtained about one observable invariably results in the disturbance of any other non-commuting observable. Here, this problem is addressed by investigating the uncertainty trade-off between measurement errors and disturbance for measurement interactions controlled by the state of a single qubit, where the measurement is described by a quantum coherent superposition of a fully projective measurement and the identity operation. It is shown that the measurement statistics obtained from a quantum controlled measurement of followed by a projective measurement of can be explained in terms of a simple combination of resolution and back-action errors acting on an intrinsic joint probability of the non-commuting observables defined by the input state of the system. These intrinsic joint probabilities are consistent with the complex-valued joint probabilities recently observed in weak measurements of quantum systems and provide direct evidence of non-commutativity in the form of imaginary correlations between the non-commuting operators. In quantum controlled measurements, these imaginary correlations can be converted into well-defined contributions to the real measurement statistics, allowing a direct experimental observation of the less intuitive aspects of quantum theory.
Abstract Bell’s inequalities are defined by sums of correlations involving non-commuting observables in each of the two systems. Violations of Bell’s inequalities are only possible because the precision of any joint measurement of these observables will be limited by quantum mechanical uncertainty relations. In this paper we explore the relation between the local measurement uncertainties and the magnitude of the correlations by preparing polarization entangled photon pairs and performing joint measurements of non-commuting polarization components at different uncertainty trade-offs. The change in measurement visibility reveals the existence of a non-trivial balance between the measurement uncertainties where the probabilities of a specific pair of measurement outcomes approaches zero because of the particular combination of enhancement and suppression of the experimentally observed correlations. The occurrence of these high-contrast results shows that the quantum correlations between the photons are close to their maximal value, confirming that the Cirel’son bound of Bell’s inequality violations is defined by the minimal uncertainties that limit the precision of joint measurements.