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Physica Scripta

Local realism does not impose the commutative algebraic structure into all quantum observables

Koji Nagata, Tadao Nakamura

Published Jun 14, 2013• 1 citation
Publisher

Abstract

Malley and Fine (Phys. Rev. A 2006 73 066102) discussed that all quantum observables would commute simultaneously if we accept a realistic theory of the Bell type (a local realistic theory) for quantum events, provided that all quantum events, including every quantum state and every observable (including every projector), are reproduced by a realistic theory of the Bell type for quantum events. We point out that their claim leads us to a contradiction. We study the relation between a local realistic theory and commutativity of quantum observables in a finite-dimensional space. We show that a realistic theory of the Bell type for quantum events does not impose the commutative algebraic structure into the set of all quantum observables if all quantum events are reproduced by a local realistic theory. We discuss that a violation of Bell locality is derived within a realistic theory of the Kochen?Specker type within quantum events.

DOI
10.1088/0031-8949/88/01/015007

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Malley and Fine (Phys. Rev. A 2006 73 066102) discussed that all quantum observables would commute simultaneously if we accept a realistic theory of the Bell type (a local realistic theory) for quantum events, provided that all quantum events, including every quantum state and every observable (including every projector), are reproduced by a realistic theory of the Bell type for quantum events. We point out that their claim leads us to a contradiction. We study the relation between a local realistic theory and commutativity of quantum observables in a finite-dimensional space. We show that a realistic theory of the Bell type for quantum events does not impose the commutative algebraic structure into the set of all quantum observables if all quantum events are reproduced by a local realistic theory. We discuss that a violation of Bell locality is derived within a realistic theory of the Kochen?Specker type within quantum events.
Recently, [{arXiv:0810.3134}] is accepted and published. We would like to study the relation between a local realistic theory and commutativity of observables in a finite-dimensional space. We would like to conjecture that a realistic theory of the Bell type (a local realistic theory) for events would not imply the commutative algebraic structure into the set of all observables if all experimental events would be reproduced by a local realistic theory. We would like to suggest that a violation of Bell locality would be derived within a realistic theory of the Kochen-Specker type within quantum events.
Malley discussed [Phys. Rev. A 69, 022118 (2004)] that all quantum observables in a hidden-variable model for quantum events must commute simultaneously. In this comment, we discuss that Malley's theorem is indeed valid for the hidden-variable theoretical assumptions, which were introduced by Kochen and Specker. However, we give an example that the local hidden-variable (LHV) model for quantum events preserves noncommutativity of quantum observables. It turns out that Malley's theorem is not related to the LHV model for quantum events, in general.
Under a standard set of assumptions for a hidden-variable model for quantum events we show that all observables must commute simultaneously. This seems to be an ultimate statement about the inapplicability of the usual hidden-variable model for quantum events. And, despite Bell's complaint that a key condition of von Neumann's was quite unrealistic, we show that these conditions, under which von Neumann produced the first no-go proof, are entirely equivalent to those introduced by Bell and Kochen and Specker. As these conditions are also equivalent to those under which the Bell-Clauster-Horne inequalities are derived, we see that the experimental violations of the inequalities demonstrate only that quantum observables do not commute.
Bell’s theorem is a foundational result in quantum theory, demonstrating that no local hidden-variable model satisfying certain assumptions can reproduce quantum correlations. While mathematically correct within its formal framework, the theorem implicitly relies on a specific representational choice: measurement outcomes are modeled as scalar, commutative variables (±1). This assumption, though natural for flat and static configurations, is not mandated by physical necessity. In this work, we examine the consequences of relaxing this representational constraint when measurement outcomes arise as projections of states with nontrivial topology. We show that when physical correlations are naturally organized as elements of an S³-valued structure — such as those associated with spin and rotational degrees of freedom — the induced measurement algebra becomes noncommutative. Under these conditions, the key algebraic factorizations employed in Bell-type derivations no longer apply. We construct an explicit local, realistic model in which observable correlations reproduce the quantum expectation E(a, b) = − cos θ and violate the CHSH bound S ≤ 2, without invoking non-local interactions or superluminal influences. The effective S³×I structure arises not as a fundamental substrate, but as the natural geometric form taken by the actualization of physical processes within an underlying relational framework. Importantly, this structure was not introduced to address quantum entanglement specifically; it emerged independently in the study of cosmological expansion, emergent gravity, and mass generation. Its relevance to Bell-type correlations follows as a consequence of geometry, not as an ad hoc modification. These results suggest that quantum entanglement may reflect topological features of physical realization rather than intrinsic non-locality. Possible experimental implications and connections to geometric and substrate-based approaches to fundamental physics
Using the Kochen–Specker contradiction, I prove that ‘local realistic’ theories predict nontrivial violations of the quantum mechanical EPR-type perfect anticorrelations. The proof invokes the same stochastic local realism conditions used in Bell arguments. For a class of theories called ‘orthodox spin theories’, the perfect anticorrelations used in the proof emerge from rotational symmetry. Therefore, an orthodox spin theorist must abandon either the spirit of relativity, as encoded by local realism, or the letter of relativity, which demands rotational invariance.
One of the essential features of quantum theory is the noncommutativity of observables. Moreover, under the non-contextual realistic interpretations of quantum theory, all quantum observables must commute {[J. D. Malley, Phys. Rev. A {\\bf 69}, 022118 (2004)]}. Experimental realizations of violations of Bell inequality say the noncommutativity of observables in an entangled state. Here we show that commutativity can be tested experimentally using uncorrelated states, with no dependence on the algebraic structure of the observables under study. The test is derived from the bounds of an inequality valid for all commuting observables. For nnn spin-1/2 systems we show that an nnn-partite uncorrelated state violates the inequality by an amount that grows exponentially with nnn.