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Physical Review A

Nonclassicality in the statistics of noncommuting observables: Nonclassical states are more compatible than classical states

Alfredo Luis

Published Jul 8, 2011• 4 citations
Publisher

Abstract

We study nonclassicality in the product of the probabilities of noncommuting observables. We show that within the quantum theory, nonclassical states can provide larger probability product than classical states, so that nonclassical states approach the nonfluctuating states of the classical theory more closely than classical states. This is particularized to relevant complementary observables such as conjugate quadratures, phase and number, quadrature and number, and orthogonal angular momentum components.

DOI
10.1103/physreva.84.012106

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We study nonclassicality in the product of the probabilities of noncommuting observables. We show that within the quantum theory, nonclassical states can provide larger probability product than classical states, so that nonclassical states approach the nonfluctuating states of the classical theory more closely than classical states. This is particularized to relevant complementary observables such as conjugate quadratures, phase and number, quadrature and number, and orthogonal angular momentum components.
According to Born's rule quantum probabilities are given by the overlap between the system state and measurement states in a quite symmetrical way. This means that both contribute to any observed nonclassical effect that is usually attributed just to the observed light state. This is relevant since typical measurement are highly nonclassical by themselves, such as number states and quadrature eigenstates. We show that nonclassical effects only arise provided that the measurement is itself nonclassical. Otherwise there is a classical-like model accounting for the observed statistics.
Nonclassicality is an interesting property of light having applications in many different contexts of quantum optics, quantum information and computation. Nonclassical states produce substantial amount of reduced noise in optical communications. Furthermore, they often behave as sources of entangled quantum states, which are the most elementary requirement for quantum teleportation. We study various nonclassical properties of coherent states and Schrödinger cat states in a setting of noncommutative space resulting from the generalized uncertainty relation, first, in a complete analytical fashion and, later, by computing their entanglement entropies, which in turn provide supporting arguments behind our analytical results. By using standard theoretical frameworks, they are shown to produce considerably improved squeezing and nonclassicality and, hence, significantly higher amount of entanglement in comparison to the usual quantum mechanical models. Both the nonclassicality and the entanglement can be enhanced further by increasing the noncommutativity of the underlying space. In addition, we find as a by-product some rare explicit minimum uncertainty quadrature and number squeezed states, i.e., ideal squeezed states.
An observable criterion is derived that allows one to distinguish nonclassical states of the harmonic oscillator from those having a classical counterpart. A quantum state is shown to have no classical counterpart if and only if the characteristic functions of the quadrature distributions or the s-parametrized phase-space distributions exhibit a slower decay than for the ground state of the oscillator. This renders it possible to experimentally check the failure of the P function to be a probability measure.
Any nonclassical effect noticed in measured statistics is usually attributed just to the light state being observed. According to Born's rule, quantum probabilities are given by the overlap between the system state and measurement states in a quite symmetrical way. We demonstrate that the nonclassicality of the measurement is a necessary condition to obtain nonclassical statistics. To this end, we present a detector characterization for two-observable joint detection processes based on detector tomography. This is particularized to the most common signatures of nonclassical light.
We derive upper bounds on the statistics of phase and phase difference that are satisfied by all classical states. They are obtained by finding the maximum projection of classical states on phase states. For a single-mode phase, meaningful bounds are obtained conditioned to a fixed mean number of photons. We also derive classical bounds for the projection on phase-coherent states, discussing their relation with phase-state bounds within the context of analytic representations. We find states with nonclassical phase properties disclosed by the violation of these classical bounds. These are quadrature and SU(222) squeezed states and phase-coherent states.