2019/05/31 by Yurii V. Brezhnev
Computer Science · Mathematics · Physics and Astronomy · #Axiom #Geometry #Hilbert space #Mathematics #Open quantum system #POVM #Physics #Pure mathematics #Quantum #Quantum Information and Cryptography #Quantum Mechanics and Applications #Quantum dynamics #Quantum mechanics #Quantum operation #Quantum probability #Quantum process #Quantum state #Quantum statistical mechanics #State vector #Statistical Mechanics and Entropy #quant-ph
paper · pdf · doi:10.1098/rspa.2020.0282
published as Proc. Royal Soc. A (2020) v.476(2244), 20200282 (22pp) · 10 pages, no figures; 'Discussion'-section much improved (+1 page)
arxiv created 2019/10/10 · openalex publication_date 2020/12/01 · openalex created_date 2020/12/21 · arxiv updated 2020/12/24 · openalex updated_date 2026/08/05
We deduce the Born rule from a purely statistical take on quantum theory within minimalistic math-setup. No use is required of quantum postulates. One exploits only rudimentary quantum mathematics—a linear, not Hilbert’, vector space—and empirical notion of the Statistical Length of a state. Its statistical nature comes from the lab micro-events (detector-clicks) being formalized into the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> </mml:math> -coefficients of quantum superpositions. We also comment that not only has the use not been made of quantum axioms (scalar-product, operators, interpretations , etc.), but that the involving thereof would be, in a sense, inconsistent when deriving the rule. In point of fact, the quadratic character of the statistical length, and even not (the ‘physics’ of) Born’s formula, represents a first step in constructing the mathematical structure we name the Hilbert space of quantum states.