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American Journal of Physics

Correlation coefficient for incompatible observables of the quantum harmonic oscillator

Richard A. Campos

Published Aug 1, 1998• 14 citations
Publisher

Abstract

We explore, using elementary examples of harmonic-oscillator superposition states, the quantum-mechanical measures of covariance and correlation coefficient for two noncommuting observables. A Schrödinger cat state which provides perfect correlation or anticorrelation of the position and momentum operators is presented. We employ quantum and classical phase-space distributions as graphical illustrations of correlation coefficients for the two theories.

DOI
10.1119/1.18937

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We explore, using elementary examples of harmonic-oscillator superposition states, the quantum-mechanical measures of covariance and correlation coefficient for two noncommuting observables. A Schrödinger cat state which provides perfect correlation or anticorrelation of the position and momentum operators is presented. We employ quantum and classical phase-space distributions as graphical illustrations of correlation coefficients for the two theories.
Schrodinger's famous cat has long been misunderstood. According to quantum theory and experiments with entangled systems, an entangled state such as the Schrodinger's cat state is neither a superposition of states of either subsystem nor a superposition of compound states of the composite system, but rather a nonlocal superposition of correlations between pairs of states of the two subsystems. The entangled post-measurement state that results from an ideal measurement is not paradoxical, but is merely a coherent superposition of two statistical correlations at "zero phase angle," i.e. at 100% positive correlation. Thus the state of the radioactive nucleus and Schrodinger's cat is as follows: an undecayed nucleus is 100% positively correlated with an alive cat, and (i.e. superposed with) a decayed nucleus is 100% positively correlated with a dead cat. The superposition consists merely in the fact that both correlations are simultaneously true. Despite many published statements to the contrary, this superposition is not paradoxical. It is in fact what one expects intuitively.
The two-mode even and odd coherent Schrödinger cat states are introduced for the generalized stationary and non-stationary harmonic oscillator. The expected values of positions and momenta and their dispersion matrices are calculated for the Schrödinger cat states. The quadrature squeezing and correlation phenomenon are studied and the Wigner and Husimi functions are constructed explicitly for these states. The photon statistics of the two-mode even and odd coherent states of the generalized oscillator is described in terms of the photon distribution function, which is expressed in terms of Hermite polynomials. The photon number, means and dispersions show oscillations which are characteristics of squeezing and correlation phenomenon.
The entangled state that results when a detector measures a superposed quantum system has spawned decades of concern about the problem of definite outcomes or "Schrodinger's cat." This state seems to describe a detector in an indefinite or "smeared" situation of indicating two macroscopic configurations simultaneously. This would be paradoxical. Since all entangled states are known to have nonlocal properties, and since measurements have obvious nonlocal characteristics, it's natural to turn to nonlocality experiments for insight into this question. Unlike the measurement situation where the phase is fixed at zero for perfect correlations, nonlocality experiments cover the full range of superposition phases and can thus show precisely what entangled states superpose. For two-state systems, these experiments reveal that the measurement state is not a superposition of two macroscopically different detector states but instead a superposition of two coherent correlations between distinct detector states and corresponding system states. In the measurement situation (i.e. at zero phase), and assuming the Schrodinger's cat scenario, the entangled state can be read as follows: An undecayed nucleus is perfectly correlated with an alive cat, AND a decayed nucleus is perfectly correlated with a dead cat, where "AND" indicates the superposition. This is not paradoxical.
Two different approaches to a characterization of the degree of (in)compatibility of quantum observables are investigated. First, recent examples of the (partial) commutativity of spectral measures of incompatible observables are proved to be generic. The analysis is extended to the case of compatible or incompatible unsharp, or stochastic observables, leading to a general criterion for commutativity of position and momentum effects. Further, a recently proposed information theoretic quantification of the (in)compatibility of noncommuting observables is generalized, and the relation between ‘‘maximal information,’’ ‘‘minimal uncertainty,’’ partial commutativity, and strict correlation is further clarified. Both approaches are illustrated in a number of examples.