arXiv:quant-ph/0503093

Comparison of quantum and classical probabilities using a generalized density matrix formalism

Jinshan Wu, Shouyong Pei

Published Mar 9, 2005• 0 citations

Abstract

Density Matrix is used to unify the description of classical and quantum objects. The idea is that both states of classical object and states of quantum object can be regarded as vectors in Hilbert space, so that all the states can be represented by density matrices. After constructing the general framework, we apply this to discuss the possibility to map a quantum density matrix onto a classical density matrix, which is also the question explored in Hidden Variable Theory (HVT) of Quantum Mechanics (QM). Bell's Theorem set a boundary between HVT and QM, but the place of the real quantum world depends on experiments. However, here HVT is ruled out just by theoretical consideration. We tackle this question by explicitly constructing the most general HVT, which gives all desired results according QM, and then checking its validity and acceptability. Relation between such HVT and Bell's inequality and Kochen-Specker theorem is also discussed.

Related papers

Hidden Variables in Quantum Mechanics from the Perspective of Boltzmannian Statistical Mechanics

Dustin Lazarovici

This paper examines no-hidden-variables theorems in quantum mechanics from the point of view of statistical mechanics. It presents a general analysis of the measurement process in the Boltzmannian framework that leads to a characterization of (in)compatible measurements and reproduces several features of quantum probabilities often described as “non-classical”. The analysis is applied to versions of the Kochen–Specker and Bell theorems to shed more light on their implications. It is shown how, once the measurement device and the active role of the measurement process are taken into account, contextuality appears as a natural feature of random variables. This corroborates Bell’s criticism that no-go results of the Kochen–Specker type are based on gratuitous assumptions. In contrast, Bell-type theorems are much more profound, but should be understood as nonlocality theorems rather than no-hidden-variables theorems. Finally, the paper addresses misunderstandings and misleading terminology that have confused the debate about hidden variables in quantum mechanics.

Quantum ReportsSep 6, 20241 citation

The Quantum Mechanical Formalism Derived from the Probabilistic Nature of Quantitative Physical Predictions

Hans Rudolf Tschudi

It is a scientific truism nowadays that quantum mechanical predictions are of statistical or rather of probabilistic nature: a quantum theory assigns a probability distribution to each measurable quantity (observable) of a physical system prepared according to a given preparation prescription (state). Such sets of probability distributions represent the most general quantitative predictions of physical theories. The aim of this study is to determine all theoretical frameworks which yield such a kind of output taking the probabilistic nature of quantitative physical predictions as initial hypothesis. Astonishingly, this project succeeds with a minimal number of additional assumptions, which arise quite naturally in the course of the analysis, and which essentially amount to regularity conditions. The result is that there exist only two classes of physical theories providing quantitative probabilistic predictions: classical theories where observables are represented by real functions over a phase space and states are represented by probability measures upon the phase space, and quantum theories where observables are represented by selfadjoint operators acting on a complex Hilbert space and states are represented by state operators. On empirical grounds, classical theories cannot provide the all-encompassing, fundamental frame for the mathematical description of physical systems. Quantum theories remain, and there is no more complete formal frame for quantitative physical theories. As an additional result, it is shown that there exist physical systems which possess a quantum mechanical but no classical description; a prominent example is the spin system of two spin-1/2 particles. The classical hidden variables model, which J. Bell relates to this system in order to prove his theorem, cannot be a true description of it because no such classical model exists, and Bell's theorem loses its foundation.

Mar 5, 20260 citations

Spin-path entanglement in single-neutron interferometer experiments

Yuji Hasegawa, Daniel Erdősi, Dipankar Home +2 more

There are two powerful arguments against the possibility of extending quantum mechanics (QM) into a more fundamental theory yielding a deterministic description of nature. One is the experimental violation of Bell inequalities, which discards local hidden‐variable theories as a possible extension to QM. The other is the Kochen‐Specker (KS) theorem, which stresses the incompatibility of QM with a larger class of hidden‐variable theories, known as noncontextual hidden‐variable theories. We performed experiments with neutron interferometer, which exploits spin‐path entanglements in single neutrons. A Bell‐like state is generated to demonstrate a violation of the Bell‐like inequality and phenomena in accordance with KS theorem: both experiments study quantum contextuality and show clear evidence of the incompatibility of noncontextual hidden variable theories with QM. The value S = 2.202±0.007≰2 is obtained in the new measurement of the Bell‐like inequality, which shows a larger violation than the previous measurement. For the study of KS theorem, the obtained violation 2.291±0.008≰1 clearly shows that quantum mechanical predictions cannot be reproduced by noncontextual hidden variable theories.

AIP conference proceedingsJan 1, 20111 citation

Condition for Any Realistic Theory of Quantum Systems

A. Montina

In quantum physics, the density operator completely describes the state. Instead, in classical physics the mean value of physical quantities is evaluated by means of a probability distribution. We study the possibility to describe pure quantum states and events with classical probability distributions and conditional probabilities and prove that the distributions have to be nonlinear functions of the density operator. Some examples are considered. Finally, we deal with the exponential complexity problem of quantum physics and introduce the concept of classical dimension for a quantum system.

arXiv:quant-ph/0604155Physical Review LettersOct 30, 200641 citations

Density matrices and path integrals

Werner Krauth

Abstract In this chapter, we continue our parallel exploration of statistical and computational physics, but now move to the field of quantum mechanics, where the density matrix generalizes the classical Boltzmann distribution. The density matrix constructs the quantum statistical probabilities from their two origins: the quantum mechanical wave functions and the Boltzmann probabilities of the energy levels. The density matrix can thus be defined in terms of the complete solution of the quantum problem (wave functions, energies), but all this information is available only in simple examples, such as the free particle or the harmonic oscillator. A simple general expression exists for the density matrix only at high temperature.

Sep 14, 20060 citations

On the Bell-Kochen-Specker Paradox

Koji Nagata, Tadao Nakamura

We use the validity of Addition and Multiplication for a hidden variables theory. First, we provide an example that the two operations Addition and Multiplication do not commute with each other as revealed by the analyses that are performed in a finite set of numbers. Our discussion leads to an initial conclusion that Sum rule and Product rule do not commute with each other in a hidden variables theory. If we accept this conclusion, we do not get the Bell- Kochen-Specker paradox. In more detail, quantum mechanics may accept the hidden variables theory. Next, we discuss the validity of operators under an assumption that Sum rule and Product rule commute with each other. In this case, we indeed get the Bell- Kochen-Specker paradox. We got the non-classicality of macroscopic experimental data observed in the Stern-Gerlach experiment and the double-slit experiment. If we detect | ↑> and then we detect | ↓>, the experiments cannot accept the hidden variables theory. We considered whether we can assign the predetermined “hidden ” result to numbers 1 and-1 as in results of measurements with the number of measurements finite (e.g., twice) in the experiments. It turned out that we cannot assign the predetermined hidden result to such results of measurements. The next conclusion indicates interestingly that the Stern-Gerlach experiment cannot accept classical mechanics. The double-slit experiment had led to the same situation, and they were indeed quantum mechanical phenomena.

Sep 1, 20150 citations