Derivation of the Born rule based on the minimal set of assumptions

TL;DR

This paper derives the Born rule for quantum measurement probabilities from a minimal set of five assumptions, emphasizing the role of entanglement and environment-assisted invariance, applicable to various measurement devices.

Contribution

It provides a novel derivation of the Born rule using environment-assisted invariance and entanglement, with minimal assumptions, extending applicability to different measurement devices.

Findings

Derivation of the Born rule from five minimal assumptions.

Entanglement and environment-assisted invariance are crucial in the derivation.

Applicable to both ideal and non-ideal measurement devices.

Abstract

The Born rule for probabilities of measurement results is deduced from the set of five assumptions. The assumptions state that: (a) the state vector fully determines the probabilities of all measurement results; (b) between measurements, any quantum system is governed by that part of standard quantum mechanics, which does not refer to measurements; (c) probabilities of measurement results obey the rules of the classical theory of probability; (d) no information transfer is possible without interaction; (e) if two spin-1/2 particles are in the entangled state (sqrt(1-\lambda)|up>|up> + sqrt(\lambda)|down>|down>), and one of spins is measured by the Stern-Gerlach apparatus, then the state of the other spin after this measurement will be either |up> or |down>, corresponding to the measurement result. No one of these assumptions can be omitted. The method of the derivation is based on Zurek's idea of environment-assisted invariance [PRL 90, 120404 (2003)], though taken in rather modified form. Entanglement plays a crucial role in our approach (like in Zurek's one), so the probabilities are first considered in the case when the measured quantum system is entangled with some environment. Probabilities in pure states appear then as a particular case. The method of the present paper can be applied to both ideal and non-ideal measuring devices, irrespective to their constructions and principles of operation.