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arXiv:quant-ph/0008122 · Physical Review A

Finite-resolution measurement of the nonclassical polarization statistics of entangled photon pairs

Holger Friedrich Hofmann

Published Mar 16, 2001• 4 citations
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Abstract

By limiting the resolution of quantum measurements, measurement-induced changes of the quantum state can be reduced, permitting subsequent measurements of variables that do not commute with the initially measured property. It is then possible to determine correlations between noncommuting variables experimentally. The application of this method to the polarization statistics of entangled photon pairs reveals that negative conditional probabilities between nonorthogonal polarization components are responsible for the violation of Bell's inequalities. Such negative probabilities can also be observed in finite-resolution measurements of the polarization of a single photon. The violation of Bell's inequalities therefore originates from local properties of the quantum statistics of single-photon polarization.

DOI
10.1103/physreva.63.042106

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By limiting the resolution of quantum measurements, measurement-induced changes of the quantum state can be reduced, permitting subsequent measurements of variables that do not commute with the initially measured property. It is then possible to determine correlations between noncommuting variables experimentally. The application of this method to the polarization statistics of entangled photon pairs reveals that negative conditional probabilities between nonorthogonal polarization components are responsible for the violation of Bell's inequalities. Such negative probabilities can also be observed in finite-resolution measurements of the polarization of a single photon. The violation of Bell's inequalities therefore originates from local properties of the quantum statistics of single-photon polarization.
By using finite resolution measurements it is possible to simultaneously obtain noisy information on two non-commuting polarization components of a single photon. This method can be applied to a pair of entangled photons with polarization statistics that violate Bell's inequalities. The theoretically predicted results show that the non-classical nature of entanglement arises from negative joint probabilities for the non-commuting polarization components. These negative probabilities allow a "disentanglement" of the statistics, providing new insights into the non-classical properties of quantum information.
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.
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.
Bipartite quantum entangled systems can exhibit measurement correlations that violate Bell inequalities, revealing the profoundly counter-intuitive nature of the physical universe. These correlations reflect the impossibility of constructing a joint probability distribution for all values of all the different properties observed in Bell inequality tests. Physically, the impossibility of measuring such a distribution experimentally, as a set of relative frequencies, is due to the quantum back-action of projective measurements. Weakly coupling to a quantum probe, however, produces minimal back-action, and so enables a weak measurement of the projector of one observable, followed by a projective measurement of a noncommuting observable. By this technique it is possible to empirically measure weak-valued probabilities for all of the values of the observables relevant to a Bell test. The marginals of this joint distribution, which we experimentally determine, reproduces all of the observable quantum statistics including a violation of the Bell inequality, which we independently measure. This is possible because our distribution, like the weak values for projectors on which it is built, is not constrained to the interval [0,1][0,1][0,1]. It was first pointed out by Feynman that, for explaining singlet-state correlations within ``a [local] hidden variable view of nature \dots{} everything works fine if we permit negative probabilities.'' However, there are infinitely many such theories. Our method, involving ``weak-valued probabilities,'' singles out a unique set of probabilities, and moreover does so empirically.
Bipartite quantum entangled systems can exhibit measurement correlations that violate Bell inequalities, revealing the profoundly counter-intuitive nature of the physical universe. These correlations reflect the impossibility of constructing joint probability distribution for all values of all the different properties observed in Bell inequality tests. Physically, the impossibility of measuring such distribution experimentally, as set of relative frequencies, is due to the quantum back-action of projective measurements. Weakly coupling to quantum probe, however, produces minimal back-action, and so enables weak measurement of the projector of one observable, followed by projective measurement of non-commuting observable. By this technique it is possible to empirically measure probabilities for all of the values of the observables relevant to Bell test. The marginals of this joint distribution, which we experimentally determine, reproduces all of the observable quantum statistics including violation of the Bell inequality, which we independently measure. This is possible because our distribution, like the weak values for projectors on which it is built, is not constrained to the interval [0, 1]. It was first pointed out by Feynman that, for explaining singlet-state correlations within a [local] hidden variable view of nature ... everything works fine if we permit negative probabilities. However, there are infinitely many such theories. Our method, involving weak-valued singles out unique set of probabilities, and moreover does so empirically.