2006/11/21 by Fedor Herbut · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Algorithm #Connection (principal bundle) #Eigenvalues and eigenvectors #Geometry #Mathematical physics #Mathematics #Observable #Operator (biology) #Philosophy #Physics #Quantum #Quantum Information and Cryptography #Quantum Mechanics and Applications #Quantum mechanics #Quantum state #Spectroscopy and Quantum Chemical Studies #State (computer science) #Statistical physics #TRACE (psycholinguistics) #Theoretical physics #quant-ph
paper · pdf · doi:10.1088/1751-8113/40/22/013
published as Journal of Physics A: Math. and Theor. 40, 5949 (2007) · 25 pages, Latex2e and Revtex4
arxiv created 2006/11/21 · openalex publication_date 2007/05/14 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
Zurek's derivation of the Born rule from envariance (environment-assisted invariance) is retraced in an attempt to present a more detailed derivation, which is extended to encompass the trace-rule form of the quantum probability law in its full generality. The investigation begins by a review presentation of Schmidt decomposition, and a detailed and complete theory of twin unitaries, which are the other face of envariance, and which stand in close connection with twin observables. The trace rule tr( E ρ), with E being an event (projector) and ρ a quantum state (density operator), is derived in five stages. The first three achieve the same as Zurek's derivation of Born's rule, but in a different way, using the full power of twin unitaries. Stage 4 extends the Zurek-like derivation to states (ray projectors) that are not eigenstates of the density operator. Finally, stage 5 utilizes the assumption of continuity of the probability law in the density operators to extend the trace rule to isolated (not correlated) systems.