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How Real is Incomputability in Physics?

José Manuel Agüero Trejo, Cristian S. Calude, Michael J. Dinneen, Arkady Fedorov, Anatoly Kulikov, Rohit Navarathna, Karl Svozil

TL;DR

Theoretically, it is proved that every infinite sequence generated by these quantum protocols is strongly incomputable -- no algorithm computing any bit of such a sequence can be proved correct.

Abstract

A physical system is determined by a finite set of initial conditions and "laws" represented by equations. The system is computable if we can solve the equations in all instances using a "finite body of mathematical knowledge". In this case, if the laws of the system can be coded into a computer program, then given the initial conditions of the system, one can compute the system's evolution. Are there incomputable physical systems? This question has been theoretically studied in the last 30-40 years. In this paper, we experimentally show for the first time the strong incomputability of a quantum experiment, namely the outputs of a quantum random number generator. Moreover, the experimental results are robust and statistically significant.

How Real is Incomputability in Physics?

TL;DR

Theoretically, it is proved that every infinite sequence generated by these quantum protocols is strongly incomputable -- no algorithm computing any bit of such a sequence can be proved correct.

Abstract

A physical system is determined by a finite set of initial conditions and "laws" represented by equations. The system is computable if we can solve the equations in all instances using a "finite body of mathematical knowledge". In this case, if the laws of the system can be coded into a computer program, then given the initial conditions of the system, one can compute the system's evolution. Are there incomputable physical systems? This question has been theoretically studied in the last 30-40 years. In this paper, we experimentally show for the first time the strong incomputability of a quantum experiment, namely the outputs of a quantum random number generator. Moreover, the experimental results are robust and statistically significant.
Paper Structure (12 sections, 4 theorems, 9 equations, 3 figures, 2 tables, 2 algorithms)

This paper contains 12 sections, 4 theorems, 9 equations, 3 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. 3D-QRNG -- Theory
  3. Notation and definitions
  4. The quantum protocol
  5. Localised Kochen-Specker Theorem
  6. 3D-QRNG -- Physical Realisation
  7. Data Generation
  8. Testing Incomputability
  9. Why do we need testing?
  10. Theory
  11. Experimental analysis
  12. Conclusions

Key Result

Theorem 1

Assume a quantum system prepared in the state $\ket{\psi}$ in a dimension $n\ge 3$ Hilbert space ${\mathbf C}^n$, and let $\ket{\phi}$ be any quantum state such that $0<\abs{\bra{\psi}\ket{\phi}}<1$. If the following three conditions are satisfied: i) admissibility, ii) non-contextuality and iii) ei

Figures (3)

  • Figure 1: QRNG setup proposed in abbott2012strongrandomness; the values $\frac{1}{2},\frac{1}{2}$ (in blue) correspond to the outcome probabilities
  • Figure 2: Blueprint for a new QRNG; the values $\frac{1}{4}, \frac{1}{2},\frac{1}{4}$ (in blue) correspond to the outcome probabilities of setups prepared in the state $\ket{\psi} = \ket{\pm 1}$
  • Figure 3: Fourth Chaitin-Schwartz-Solovay-Strassen test: distribution of the average Z-liar counts for all odd composite numbers less than 50

Theorems & Definitions (4)

  • Theorem 1: Localised Kochen-Specker Theorem Abbott:2010uqacs-2015-info6040773PhysRevLett.119.240501aguero_trejo_new_2021
  • Theorem 2: abbott2012strongrandomnessaguero_trejo_new_2021
  • Theorem 3: aguero_trejo_new_2021
  • Theorem 4: aguero_trejo_new_2021