COMPATIBILITY AND SEPARABILITY FOR CLASSICAL AND QUANTUM ENTANGLEMENT

Abstract

AbstractWe study the concepts of compatibility and separability and their implications for quantum and classicalsystems. These concepts are illustrated on a macroscopic model for the singlet state of a quantum systemof two entangled spin 1/2 with a parameter reflecting indeterminism in the measurement procedure.By varying this parameter we describe situations from quantum, intermediate to classical and studywhich tests are compatible or separated. We prove that for classical deterministic systems the conceptsof separability and compatibility coincide, but for quantum systems and intermediate systems theseconcepts are generally different. 1 Introduction Entanglement is one of the main features of quantum mechanics. The most studied example of quantumentanglement is the one of the two spin 1/2 particles as presented by David Bohm [27]: more specifically inthe situation where the joint quantum entity of the two spin 1/2 particles is in a singlet spin state. JohnBell showed that quantum entanglement can be experimentally tested by means of a set of inequalities,meanwhile called Bell’s inequalities [26]. The ‘violation of Bell’s inequalities’ indicates the presence ofquantum entanglement. In [3] an example of a situation consisting of macroscopic physical entities wasproposed where Bell’s inequalities are violated. Later the example was elaborated to yield a violation withexactly the same value 2√2 as the one that appears in the violation of Bell’s inequalities by the spins in thesinglet spin state [10]. In the macroscopic example the considered entities are two macroscopic particles,and the entanglement is expressed by means of the presence of a rigid rod connecting the two particles.The rod represents the non-local effect that appears with the violation of Bell’s inequalities. In [20] it isshown that not only the singlet spin state, as considered in [3], but all entangled states of coupled spins canbe represented by such a rod-like internal constraint. The quantum probability is obtained in the modelbecause of the introduction of a hidden variable on the measurement apparatus in correspondence with thespin model that was developed in [6, 7].The macroscopic model built in [3] violates Bell’s inequalities in a stronger way than the quantum spinexample, indeed the violation value for the macroscopic model is 4 while for the quantum example it is 2√2.We have proven in [14] that the quantum probability present in the case of the quantum spin example, andalso in the macroscopic example of [3], decreases the violation of Bell’s inequalities. This is the reason thata purely classical non-local situation, like the one considered in [3], violates Bell’s inequalities in a strongerway, with maximum possible violation value 4.We know that two measurements of which each one is performed on one of the two spins of a quantumentangled spin couple are compatible measurements. Indeed, self-adjoint operators H