[Paper Review] Quantum Theory as Efficient Representation of Probabilistic Information

Johann Summhammer|ArXiv.org|Jan 25, 2007

Quantum Mechanics and Applications14 references3 citations

TL;DR

This paper proposes that quantum theory emerges as the optimal representation of probabilistic experimental data when information must be stored efficiently with finite bits. By encoding observed relative frequencies as square roots (real or complex), the method minimizes statistical uncertainty and preserves invariance under unitary transformations, revealing quantum theory as a framework that maximizes information fidelity without loss, though slightly better representations may exist for low-trial regimes.

ABSTRACT

Quantum experiments yield random data. We show that the most efficient way to store this empirical information by a finite number of bits is by means of the vector of square roots of observed relative frequencies. This vector has the unique property that its dispersion becomes invariant of the underlying probabilities, and therefore invariant of the physical parameters. This also extends to the complex square roots, and it remains true under a unitary transformation. This reveals quantum theory as a theory for making predictions which are as accurate as the input information, without any statistical loss. Our analysis also suggests that from the point of view of information a slightly more accurate theory than quantum theory should be possible.

Motivation & Objective

• To identify the most efficient way to represent finite probabilistic experimental data using a limited number of bits.
• To determine a representation that preserves information fidelity regardless of underlying probabilities or physical parameters.
• To explain why quantum theory's use of probability amplitudes (complex square roots of frequencies) is uniquely suited for this task.
• To explore whether a more accurate theory than quantum mechanics could exist under finite-trial conditions.

Proposed method

• Mapping observed relative frequencies ν = L/N to the square root of ν, denoted √ν, to minimize statistical dispersion in finite-sample settings.
• Extending the representation to complex square roots by introducing arbitrary phase factors, modeling quantum amplitudes.
• Analyzing the dispersion (statistical uncertainty volume) of the representation vector to assess reliability and invariance under transformations.
• Applying unitary transformations to the complex square root vector and showing that dispersion remains invariant under both input probabilities and transformation parameters.
• Using binomial statistics to compute the probability that S-bit representations of ν or √ν match the true value in the infinite-trial limit.
• Comparing the reliability of different encoding schemes via the probability that S-bit approximations [ν]_S or [√ν]_S are correct, defined by the range of L satisfying |L/N - p| < 1/2^{S+1}.

Experimental results

Research questions

• RQ1 What is the most efficient way to encode finite probabilistic data into a fixed number of bits without statistical loss?
• RQ2 Why does the use of square roots of relative frequencies yield a representation with invariant uncertainty across different probabilities?
• RQ3 How does the complex square root representation relate to the formalism of quantum mechanics, particularly in terms of unitary invariance?
• RQ4 Can a theory more accurate than quantum mechanics exist when only a small number of experimental trials are available?
• RQ5 Under what conditions does the standard quantum amplitude encoding fail to be optimal for predicting combined probabilities?

Key findings

• Encoding relative frequencies as square roots of observed frequencies minimizes statistical dispersion and yields an invariant uncertainty volume across all underlying probabilities.
• The complex square root representation of relative frequencies leads to a dispersion that is invariant not only under changes in the true probability p but also under unitary transformations.
• Unitary evolution preserves the information content of the representation, as the uncertainty volume remains unchanged, explaining the unitary nature of quantum dynamics.
• For N=4000 trials and S=6 bits, the probability of correct S-bit encoding of √ν exceeds 0.95, while that of ν itself drops to 0.68 near p=0.5, demonstrating superior reliability.
• The method reveals that quantum theory is optimal in the infinite-trial limit, but for finite data, especially with p near 0 or 1, better representations may exist.
• The paper suggests that apparent deviations from linear superposition in low-trial scenarios may stem from suboptimal encoding, not from a failure of quantum theory itself.