Quantum mechanics for a four‐state‐system is derived from classical statistics. Entanglement, interference, the difference between identical fermions or bosons and the unitary time evolution find an interpretation within a classical statistical ensemble. Quantum systems are subsystems of larger classical statistical systems, which include the environment or the vacuum. They are characterized by incomplete statistics in the sense that the classical correlation function cannot be used for sequences of measurements in the subsystem.Quantum mechanics for a four‐state‐system is derived from classical statistics. Entanglement, interference, the difference between identical fermions or bosons and the unitary time evolution find an interpretation within a classical statistical ensemble. Quantum systems are subsystems of larger classical statistical systems, which include the environment or the vacuum. They are characterized by incomplete statistics in the sense that the classical correlation function cannot be used for sequences of measurements in the subsystem.

arXiv:0809.2671AIP conference proceedingsJan 1, 201010 citations

A classical field theory comeback? The classical field viewpoint on triparticle entanglement

Andrei Yu. Khrennikov

We present the fundamentals of a prequantum model with hidden variables of classical field type—prequantum classical statistical field theory (PCSFT). In some sense, this is the comeback of classical field theory, but with an extension to massive particles. All quantum averages (including correlations of entangled systems) can be represented as classical field averages and correlations. In the literature, entanglement of biparticle systems (in particular, biphotons) has been represented in the PCSFT framework. In this paper, we consider triparticle entanglement. It is well known that in conventional quantum mechanics, the cases of biparticle and triparticle entanglement are quite different. Roughly speaking, triparticle entanglement is not reducible to biparticle entanglement. The PCSFT approach to entanglement (as a correlation of classical signals) reflects this quantum situation. The PCSFT representations of correlations in biparticle and triparticle systems differ essentially. In particular, from the mathematical viewpoint the latter is fundamentally more complicated than the former.We present the fundamentals of a prequantum model with hidden variables of classical field type—prequantum classical statistical field theory (PCSFT). In some sense, this is the comeback of classical field theory, but with an extension to massive particles. All quantum averages (including correlations of entangled systems) can be represented as classical field averages and correlations. In the literature, entanglement of biparticle systems (in particular, biphotons) has been represented in the PCSFT framework. In this paper, we consider triparticle entanglement. It is well known that in conventional quantum mechanics, the cases of biparticle and triparticle entanglement are quite different. Roughly speaking, triparticle entanglement is not reducible to biparticle entanglement. The PCSFT approach to entanglement (as a correlation of classical signals) reflects this quantum situation. The PCSFT representations of correlations in biparticle and triparticle systems differ essentially. In particular, from the mathematical viewpoint the latter is fundamentally more complicated than the former.

Physica ScriptaFeb 1, 20116 citations

Probability and Entanglement in Quantum Mechanics

James M. Espinosa

Bulletin of the American Physical SocietyMar 23, 20070 citations

Necessity of entanglement for the typicality argument in statistical mechanics

Pedro Silva Correia, Gabriel Dias Carvalho, Thiago Rodrigues de Oliveira

Typicality arguments replace the postulated mixed state ensembles of statistical mechanics with pure states sampled uniformly at random, explaining why most microstates of large systems exhibit thermal behavior. This paradigm has been revived in quantum contexts, where entanglement is deemed essential, but no clear quantitative link between entanglement structure and typicality has been established. Here, we study pure quantum states with controlled multipartite entanglement and show that when entanglement grows with system size $N$ , fluctuations in macroscopic observables decay exponentially with $N$ , whereas if entanglement remains finite, one recovers only the classical $1/\sqrt{N}$ suppression. Our work thus provides a quantitative connection between entanglement structure and the emergence of typicality, demonstrating that entanglement is crucial for thermalization in small quantum systems but unnecessary to justify equilibrium in macroscopic ensembles. This unifies classical and quantum foundations of statistical mechanics by pinpointing exactly when and why entanglement matters.Typicality arguments replace the postulated mixed state ensembles of statistical mechanics with pure states sampled uniformly at random, explaining why most microstates of large systems exhibit thermal behavior. This paradigm has been revived in quantum contexts, where entanglement is deemed essential, but no clear quantitative link between entanglement structure and typicality has been established. Here, we study pure quantum states with controlled multipartite entanglement and show that when entanglement grows with system size $N$, fluctuations in macroscopic observables decay exponentially with $N$, whereas if entanglement remains finite, one recovers only the classical $1/\sqrt{N}$ suppression. Our work thus provides a quantitative connection between entanglement structure and the emergence of typicality, demonstrating that entanglement is crucial for thermalization in small quantum systems but unnecessary to justify equilibrium in macroscopic ensembles. This unifies classical and quantum foundations of statistical mechanics by pinpointing exactly when and why entanglement matters.

arXiv:2504.21090Physical Review AMay 21, 20260 citations

N

, fluctuations in macroscopic observables decay exponentially with

N

, whereas if entanglement remains finite, one recovers only the classical

1/\sqrt{N}

suppression. Our work thus provides a quantitative connection between entanglement structure and the emergence of typicality, demonstrating that entanglement is crucial for thermalization in small quantum systems but unnecessary to justify equilibrium in macroscopic ensembles. This unifies classical and quantum foundations of statistical mechanics by pinpointing exactly when and why entanglement matters.