Notes on the firewall paradox, complexity, and quantum theory

TL;DR

This paper explores the implications of the firewall paradox and quantum complexity on the foundations of mathematics and quantum theory, linking thermodynamic decoding problems to set theory and computational complexity.

Contribution

It offers a novel perspective by connecting the firewall paradox to foundational issues in mathematics and quantum theory, emphasizing the indistinguishability of computational and proof theoretic complexity.

Findings

Highlights the connection between quantum complexity and set theory foundations.

Suggests the firewall paradox has implications for axiomatic mathematics.

Reevaluates the role of randomness in quantum theory foundations.

Abstract

We investigate what it means to apply the solution, proposed to the firewall paradox by Harlow and Hayden, to the famous quantum paradoxes of Sch\"odinger's Cat and Wigner's Friend if ones views these as posing a thermodynamic decoding problem (as does Hawking radiation in the firewall paradox). The implications might point to a relevance of the firewall paradox for the axiomatic and set theoretic foundations underlying mathematics. We reconsider in this context the results of Benioff on the foundational challenges posed by the randomness postulate of quantum theory. A central point in our discussion is that one can mathematically not naturally distinguish between computational complexity (as central to the approach of Harlow and Hayden and further developed by Susskind) and proof theoretic complexity (since they represent the same concept on a Turing machine), with the latter being related to a finite bound on Kolmogorov entropy (due to Chaitin incompleteness).