Perfect correlations between noncommuting observables

Gábor Hofer‐Szabó, Szilárd Szalay

In this paper, we revisit the concept of noncommuting common causes; refute two objections raised against them, the triviality objection and the lack of causal explanatory force; and explore how their existence modifies the EPR argument. More specifically, we show that 1) product states screening off all quantum correlations do not compromise noncommuting common causal explanations; 2) noncommuting common causes can satisfy the law of total probability; 3) perfect correlations can have indeterministic noncommuting common causes; and, as a combination of the above claims, 4) perfect correlations can have noncommuting common causes which are both nontrivial and satisfy the law of total probability.

Giuseppe Nisticò

Somshubhro Bandyopadhyay, Prabha Mandayam

Richard A. Campos

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.

Luis Pedro García-Pintos, Justin Dressel

We analyze the continuous measurement of two noncommuting observables for a qubit, and investigate whether the simultaneously observed noisy signals are consistent with the evolution of an equivalent classical system. Following the approach outlined by Leggett and Garg, we show that the readouts violate macrorealistic inequalities for arbitrarily short temporal correlations. Moreover, the derived inequalities are manifestly violated even in the absence of Hamiltonian evolution, unlike for Leggett-Garg inequalities that use a single continuous measurement. Such a violation should indicate the failure of at least one postulate of macrorealism: either physical quantities do not have well-defined values at all times or the measurement process itself disturbs what is being measured. Nevertheless, for measurements of equal strength we are able to construct a classical stochastic model for a spin that perfectly emulates both the qubit evolution and the observed noisy signals, thus emulating the violations; interestingly, this model also requires an unphysical noise to emulate the readouts, which effectively restricts the ability of an observer to learn information about the spin.

Jeramy Gordon, W. H. Louisell