NO OTHERNESS

Abstract

Quantum mechanics predicts that entangled particles will exhibit correlations that seem to transcend the limits of locality, which implies faster-than-light communication. Further, quantum contextuality indicates that the results of measurements are not objective but can depend on the context of the measurements. This paper employs decoherence and information theory to resolve the paradox of quantum nonlocality and the phenomenon of quantum contextuality, by demonstrating that observed correlations and contextual dependencies arise from pre-existing conditions rather than from nonlocal interactions or hidden variables.

Introduction

Nonlocality and contextuality have long been issues in quantum mechanics, but decoherence theory and information theory provide the tools for addressing those problems. By showing that entanglement and measurement lead to classical correlations without nonlocal interactions or predetermined hidden values, we can attribute those phenomena to interactions with the environment and the nature of information gain.

Mathematical Formulations

1. Decoherence

Decoherence explains how quantum systems transition to classical behavior through interactions with their environment. The mathematical description of decoherence is captured by the master equation for the density matrix (\(\rho(t)\)) of a system interacting with its environment:

\[\frac{d\rho(t)}{dt} = -\frac{i}{\hbar} [H, \rho(t)] + \mathcal{D}[\rho(t)]\]

where \(\mathcal{D}[\rho(t)]\) represents the decoherence term that accounts for environmental interactions. 

Decoherence makes quantum superposition states “collapse” into classical states.

2. Information Theory

In information theory, the entropy (\(S\)) of a system can be quantified using von Neumann entropy:

\[S(\rho) = -\text{Tr}(\rho \log \rho)\]

The reduction in entropy following a measurement shows the information gain and the observed correlations between entangled particles. Entanglement implies pre-existing correlations that reflect shared quantum states, which encompasses both quantum nonlocality and contextuality.

Decoherence and Measurement

1. Decoherence

When a quantum system becomes entangled with its environment, coherence is lost, and the system appears to collapse into a definite classical state. This clarifies why definite classical outcomes are observed in quantum experiments.

2. Measurement and Effective Collapse

Measurement is framed as an interaction with the environment, resulting in decoherence that manifests as an apparent collapse of the wave function into classical outcomes. This process does not require faster-than-light communication but rather involves the system’s entanglement with its environment.

Quantum Contextuality and Measurement

1. The Kochen-Specker Theorem

The Kochen-Specker theorem shows that it is impossible to assign definite values to all measurement outcomes in a way consistent with quantum mechanics and classical realism. Measurement outcomes are context-dependent and cannot be explained by predetermined values.

2. Decoherence and Contextuality

Decoherence plays a significant role in how measurement outcomes are influenced by the specific context. The loss of coherence due to environmental interactions results in classical-like outcomes that vary with measurement context. This refines our understanding of contextuality and its relation to measurement processes.

Information Theory and Contextuality

1. Information Gain 

The entropy of a quantum system is influenced by the measurement context. The von Neumann entropy quantifies uncertainty, with measurement processes reducing this uncertainty to yield context-dependent information about the system’s state.

2. Quantum Information and Contextuality

Quantum information theory reveals that measurement outcomes are influenced by the context in which they occur. The entanglement with the environment and the information gained during measurement demonstrate that outcomes are not predetermined but emerge through environmental interactions influenced by the measurement setup.

Empirical Validation

Our resolutions of nonlocality and contextuality are supported by empirical observations of decoherence in quantum systems and the successful application of information theory to quantum measurement. Studies such as those by Joos et al. (2003) and Zurek (2003) demonstrate how decoherence affects entangled states and measurement outcomes. Further, numerous quantum information experiments validate the predictions regarding entropy and measurement.

Implications

Decoherence and information theory comprehensively address both quantum nonlocality and contextuality. The correlations observed in entangled particles arise from pre-existing conditions without the need for nonlocal communication. Decoherence explains the transition from quantum superpositions to classical outcomes, while information theory illuminates the roles of entropy and context in measurement.

Conclusion

Decoherence and information theory effectively resolve the issues of quantum nonlocality and contextuality, by demonstrating that the correlations and contextual dependencies in quantum systems arise from environmental interactions and pre-existing conditions. This reinforces the foundational principles of quantum mechanics, affirming its completeness and augmenting our knowledge of entanglement, measurement, and information. 

References

– Aspect, A., Dalibard, J., & Roger, G. (1982). Experimental Test of Bell’s Inequalities Using Time‐Averaged Correlations. Physical Review Letters, 49(25), 1804-1807.
– Kochen, S., & Specker, E. P. (1967). The Problem of Hidden Variables in Quantum Mechanics. Journal of Mathematics and Mechanics, 17, 59-87.
– Landauer, R. (1961). Irreversibility and Heat Generation in the Computing Process. IBM Journal of Research and Development, 5(3), 183-191.
– Zurek, W. H. (2003). Decoherence, einselection, and the existential interpretation (the rough guide). Philosophical Transactions of the Royal Society A, 361(1808), 117-136.
– Joos, E., Zeh, D. J., Kiefer, C., Giulini, D., Kupsch, J., & Stamatescu, I.-O. (2003). Decoherence and the Appearance of a Classical World in Quantum Theory. Springer.
– Zurek, W. H. (2003). Decoherence and the Transition from Quantum to Classical. Physics Today, 44(10), 36-44.
– Nielsen, M. A., & Chuang, I. L. (2010). Quantum Computation and Quantum Information. Cambridge University Press.