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arXiv:1807.05672 · Physical Review A

Experimental evaluation of the nonclassical relation between measurement errors using entangled photon pairs as a probe

M. Iinuma, Masayuki Nakano, Holger Friedrich Hofmann, Yutaro Suzuki

Published Dec 7, 2018• 5 citations
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Abstract

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.

DOI
10.1103/physreva.98.062109

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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.
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.
Joint measurements of noncommuting observables are characterized by unavoidable measurement uncertainties that can be described in terms of the error statistics for input states with well-defined values for the target observables. However, a complete characterization of measurement errors must include the correlations between the errors of the two observables. Here, we show that these correlations appear in the experimentally observable measurement statistics obtained by performing the joint measurement on maximally entangled pairs. For two-level systems, the results indicate that quantum theory requires imaginary correlations between the measurement errors of $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{X}$ and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Y}$ since these correlations are represented by the operator product $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{X}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Y}=i\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Z}$ in the measurement operators. Our analysis thus reveals a directly observable consequence of noncommutativity in the statistics of quantum measurements.
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.
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 quantum mechanics, joint measurements of noncommuting observables are only possible if a minimal unavoidable measurement uncertainty is accepted. On the other hand, correlations between noncommuting observables can exceed classical limits, as demonstrated by the violation of Bell's inequalities. Here, the relation between the uncertainty-limited statistics of joint measurements and the limits on expectation values of possible input states is analyzed. It is shown that the experimentally observable statistics of joint measurements explain the uncertainty limits of local states, but result in less restrictive bounds when applied to identify the limits of nonlocal correlations between two separate quantum systems. A tight upper bound is obtained for the four correlations that appear in the violation of Bell's inequalities and the statistics of pure states saturating the bound is characterized. The results indicate that the limitations of quantum nonlocality are a necessary consequence of the local features of joint measurements, suggesting the possibility that quantum nonlocality could be explained in terms of the local characteristics of quantum statistics.
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.