The Extended Bloch Representation of Entanglement and Measurement in Quantum Mechanics

Diederik Aerts,Massimiliano Sassoli De Bianchi,Sandro Sozzo

doi:10.1007/s10773-016-3257-7

Diederik Aerts, Massimiliano Sassoli De Bianchi + Show 1 more

Open Access

https://doi.org/10.1007/s10773-016-3257-7

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Abstract

The quantum formalism can be completed by assuming that a density operator can also represent a pure state. An 'extended Bloch representation' (EBR) then results, in which not only states, but also the measurement-interactions can be represented. The Born rule is obtained as an expression of the subjective lack of knowledge about the measurement-interaction that is each time actualized. Entanglement can also be consistently described in the EBR, as it remains compatible with the principle according to which a composite entity exists only if its components also exist, and therefore are in pure states.

Highlights

There are many different ways to state and understand the famous ‘quantum measurement problem’, associated with as many interpretations of the quantum formalism
A related question is: are the quantum probabilities ontological, or do they appear as the consequence of a condition of lack of knowledge, but lack of knowledge about what?. Another fundamental issue in quantum mechanics (QM), though less publicized than the measurement problem, is understanding what happens to the components of a composite entity when, following their interaction, they become mutually entangled
Two fundamental problems of QM are: (1) finding a coherent and general mechanism explaining the emergence of the quantum probabilities, as described by the ‘Born rule’; and (2) explaining how a composite entity can exist when its components, apparently, may no longer exist, at least according to the physical principle saying that if a physical entity exists necessarily, it must be in a well-defined state [3, 4]

Summary

Introduction

There are many different ways to state and understand the famous ‘quantum measurement problem’, associated with as many interpretations of the quantum formalism (see, e.g., [1]). One of them is that a same operator-state is able to represent an infinity of different statistical mixtures [5, 6] This means that there is a ‘potentiality aspect’ expressed by an operator-state that its standard interpretation as a condition of ‘lack of knowledge about the pure state in which the physical entity is’ is unable to capture. Since “the proof is in the pudding,” our task in this article is to show that, if we assume that density operators represent pure states, and more precisely those operator-states that an entity can enter in during a measurement process, the wavefunction collapse can be described as a physical process which is similar to a ‘weighted symmetry breaking’ and that, quantum probabilities can be understood as epistemic statements expressing a condition of lack of knowledge about which specific measurement-interaction is selected at each run of a measurement.

Measuring on a qubit

Measuring on a N -level system

Two entangled entities

Conclusions