Editing Quantum indeterminacy (section)

== Logical independence and quantum randomness ==

Quantum indeterminacy is often understood as information (or lack of it) whose existence we infer, occurring in individual quantum systems, prior to measurement.  ''Quantum randomness'' is the statistical manifestation of that indeterminacy, witnessable in results of experiments repeated many times. However, the relationship between quantum indeterminacy and randomness is subtle and can be considered differently.<ref>Gregg Jaeger, "Quantum randomness and unpredictability"
Philosophical Transactions of the Royal Society of London A doi/10.1002/prop.201600053 (2016)|Online=http://onlinelibrary.wiley.com/doi/10.1002/prop.201600053/epdf PDF</ref>

In ''classical physics'', experiments of chance, such as coin-tossing and dice-throwing, are deterministic, in the sense that, perfect knowledge of the initial conditions would render outcomes perfectly predictable. The ‘randomness’ stems from ignorance of physical information in the initial toss or throw. In diametrical contrast, in the case of ''quantum physics'', the theorems of Kochen and Specker,<ref>S Kochen and E P Specker, ''The problem of hidden variables in quantum mechanics'', Journal of Mathematics and Mechanics '''17''' (1967), 59–87.</ref> the inequalities of John Bell,<ref>John Bell, ''On the Einstein Podolsky Rosen paradox'', Physics '''1''' (1964), 195–200.</ref> and experimental evidence of [[Alain Aspect]],<ref>Alain Aspect, Jean Dalibard, and Gérard Roger, ''Experimental test of Bell’s inequalities using time-varying analyzers'', ''Physical Revue Letters'' '''49''' (1982), no. 25, 1804–1807.</ref><ref>Alain Aspect, Philippe Grangier, and Gérard Roger, ''Experimental realization of Einstein–Podolsky–Rosen–Bohm gedankenexperiment: A new violation of Bell’s inequalities'', Physical Review Letters '''49''' (1982), no. 2, 91–94.</ref> all indicate that quantum randomness does not stem from any such ''physical information''.

In 2008, Tomasz Paterek et al. provided an explanation in ''mathematical information''. They proved that quantum randomness is, exclusively, the output of measurement experiments whose input settings introduce ''[[Independence (mathematical logic)|logical independence]]'' into quantum systems.<ref>Tomasz Paterek, Johannes Kofler, Robert Prevedel, Peter Klimek, Markus Aspelmeyer, Anton Zeilinger, and Caslav Brukner, "Logical independence and quantum randomness", ''New Journal of Physics'' '''12''' (2010), no. 013019, 1367–2630.</ref><ref>Tomasz Paterek, Johannes Kofler, Robert Prevedel, Peter Klimek, Markus Aspelmeyer, Anton Zeilinger, and Caslav Brukner, "Logical independence and quantum randomness – with experimental data", https://arxiv.org/pdf/0811.4542.pdf (2010).</ref>

Logical independence is a well-known phenomenon in [[Mathematical logic|Mathematical Logic]].  It refers to the null logical connectivity that exists between mathematical propositions (in the same language) that neither prove nor disprove one another.<ref>Edward Russell Stabler, ''An introduction to mathematical thought'', Addison-Wesley Publishing Company Inc., Reading Massachusetts USA, 1948.</ref>

In the work of Paterek et al., the researchers demonstrate a link connecting quantum randomness and ''logical independence'' in a formal system of Boolean propositions. In experiments measuring photon polarisation, Paterek et al. demonstrate statistics correlating predictable outcomes with logically dependent mathematical propositions, and random outcomes with propositions that are logically independent.<ref>Tomasz Paterek, Johannes Kofler, Robert Prevedel, Peter Klimek, Markus Aspelmeyer, Anton Zeilinger, and Caslav Brukner, "Logical independence and quantum randomness", ''New Journal of Physics'' '''12''' (2010), no. 013019, 1367–2630.</ref><ref>Tomasz Paterek, Johannes Kofler, Robert Prevedel, Peter Klimek, Markus Aspelmeyer, Anton Zeilinger, and Caslav Brukner, "Logical independence and quantum randomness – with experimental data", https://arxiv.org/pdf/0811.4542.pdf (2010).</ref>

In 2020, Steve Faulkner reported on work following up on the findings of Tomasz Paterek et al.; showing what logical independence in the Paterek Boolean propositions means, in the domain of Matrix Mechanics proper. He showed how indeterminacy's ''indefiniteness'' arises in evolved density operators representing mixed states, where measurement processes encounter irreversible 'lost history' and ingression of ambiguity.<ref>Steve Faulkner, ''The Underlying Machinery of Quantum Indeterminacy'' (2020). [https://quantum-indeterminacy.science]</ref>