No Fine theorem for macrorealism: Limitations of the Leggett-Garg inequality

TL;DR

This paper demonstrates that, unlike Bell inequalities for local realism, no set of inequalities can fully characterize macrorealism, highlighting fundamental differences in their mathematical structures and implications for future tests.

Contribution

It proves that no necessary and sufficient set of inequalities exists for macrorealism, contrasting with Bell inequalities for local realism, and advocates no-signaling in time as a better criterion.

Findings

Bell inequalities are necessary and sufficient for local realism.

No set of inequalities can fully characterize macrorealism.

No-signaling in time is a promising criterion for macrorealism.

Abstract

Tests of local realism and macrorealism have historically been discussed in very similar terms: Leggett-Garg inequalities follow Bell inequalities as necessary conditions for classical behavior. Here, we compare the probability polytopes spanned by all measurable probability distributions for both scenarios and show that their structure differs strongly between spatially and temporally separated measurements. We arrive at the conclusion that, in contrast to tests of local realism where Bell inequalities form a necessary and sufficient set of conditions, no set of inequalities can ever be necessary and sufficient for a macrorealistic description. Fine's famous proof that Bell inequalities are necessary and sufficient for the existence of a local realistic model, therefore cannot be transferred to macrorealism. A recently proposed condition, no-signaling in time, fulfills this criterion, and we show why it is better suited for future experimental tests and theoretical studies of macrorealism. Our work thereby identifies a major difference between the mathematical structures of local realism and macrorealism.