On Heisenberg's inequality and Bell's inequality

Xiaoling Yin

We investigate the violation of the Bell inequality for thermal states of qubits in the two-qubit and three-qubit Heisenberg models with Dzyaloshinski-Moriya(DM) interaction.The results indicate that the Bell inequality can be violated in the case of two-qubit Heisenberg model but not violated in the case of three-qubit Heisenberg model at any DM interaction and temperature.

Gao Mingzhe

Michael Cowling, John Price

M. R. Clover

By implicitly assuming that all possible Bell-measurements occur simultaneously, all proofs of Bell's Theorem violate Heisenberg's Uncertainty Principle. This assumption is made in the original form of Bell's inequality, in Wigner's probability inequalities, and in the ``nonlocality without inequalities'' arguments. The introduction of time into derivations of these variants of Bell's theorem results in extra terms related to the time order of the measurements used in constructing correlation coefficients. Since the same locality assumptions are made in the Heisenberg-compliant derivations of this paper, only time-independent classical local hidden variable theories are forbidden by violations of the original Bell inequalities; time-dependent quantum local hidden variable theories can satisfy this new bound and agree with experiment. We further point out that factorizable wavefunctions have been used to describe some EPR experiments and can be used to describe others. These will generate local de Broglie-Bohm trajectories in the description of the data. This second, independent, line of argument also shows that violation of Bell's inequality is only evidence that Heisenberg's Uncertainty Principle cannot be ignored.

Ulrich Faigle

Bell's inequality plays an important role with respect to the Einsteinian question about the physical reality of quantum theory. While Bell's inequality is usually viewed within the geometric framework of a Hilbert space quantum model, the present note extends the theory of Heisenberg measurements to quantum systems with representations in general orthogonal geometric spaces and, in particular, the Minkowski spaces of relativity theory. A Feynmanian numerical example exhibits two measurements that admit a joint probabilistic interpretation in Minkowski space while they are not jointly observable in Hilbert space. The analysis shows that probabilistic interpretations of quantum measurements may depend not only on the measuring instruments and the system states but also on the geometric space in which the measurements are conducted. In particular, an explicit numerical example is given of a Heisenberg measurement with a complete set of common observables that violates Bell's inequality in Minkowski space but, mutatatis mutandis, satisfies it in Hilbert space.

Pierre A. Millette