Two Observers

Wenbo Su, Manchao Zhang, Chun-Wang Wu +8 more

The expectation value of the product of two noncommuting observables (POTNO) can be used to test the commutation relation and dig out the correlation between two observables. However, since a POTNO is generally a non-Hermitian operator, its average cannot be measured directly according to standard quantum mechanics. Recently, a theoretical scheme has been proposed to obtain POTNOs, but the observables are limited to projection operators. In this paper, we present a scheme that employs weak measurement to directly measure the expectation value of the POTNO, where the observables are arbitrary operators. Our scheme transforms the measurement of this expectation value into the measurements of several single observable weak values. Experimentally we demonstrate our scheme by measuring the expectation value of the product of two Pauli operators and verify their commutation and anticommutation relations in a trapped-ion system. Moreover, our scheme is applicable to the cases with more than two observables.

Sixia Yu, Nai-Le Liu, Li Li +1 more

We derive a single inequality as the sufficient and necessary condition for two unsharp observables of a two-level system to be jointly measurable in a single apparatus and construct explicitly the joint observables of two jointly measurable observables. By introducing a generalized distinguishability as a measure of the unsharpness of an unsharp measurement, we derive a complementarity inequality, which generalizes Englert's duality inequality, from the condition of joint measurement of two orthogonal unsharp observables.

Massimiliano Sassoli de Bianchi

Benni Reznik, Yakir Aharonov

We explore further the suggestion to describe a pre- and postselected system by a two-state, which is determined by two conditions. Starting with a formal definition of a two-state Hilbert space and basic operations, we systematically recast the basics of quantum mechanics---dynamics, observables, and measurement theory---in terms of two-states as the elementary quantities. We find a simple and suggestive formulation that "unifies" two complementary observables: probabilistic observables and nonprobabilistic "weak" observables. Probabilities are relevant for measurements in the "strong-coupling regime." They are given by the absolute square of a two-amplitude (a projection of a two-state). Nonprobabilistic observables are observed in sufficiently weak measurements and are given by linear combinations of the two-amplitude. As a subclass they include the "weak values" of Hermitian operators. We show that in the intermediate regime, one may observe a mixing of probabilities and weak values. A consequence of the suggested formalism and measurement theory is that the problems of nonlocality and Lorentz noncovariance, of the usual prescription with a "reduction," may be eliminated. We exemplify this point for the Einstein-Podolsky-Rosen experiment and for a system under successive observations.

Leslie Eastman