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A. Neumaier

Science Advisor

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- TL;DR Summary
- An illustration is given of the differences in the interpretation of measurement results in the thermal interpretation and in Born's statistical interpretation.

The following two examples illustrate the differences between the thermal interpretation

and Born's statistical interpretation for the interpretation of measurement results.

1. Consider some piece of digital equipment with 3 digit display measuring some physical quantity ##X## using ##N## independent measurements. Suppose the measurement results were 6.57 in 20% of the cases and 6.58 in 80% of the cases. Every engineer or physicist would compute the mean ##\bar X= 6.578## and the standard deviation ##\sigma_X=0.004## and conclude that the true value of the quantity ##X## deviates from ##6.578## by an error of the order of ##0.004N^{-1/2}##.

2. Consider the measurement of a Hermitian quantity ##X\in C^{2\times 2}## of a 2-state quantum system in the pure up state, using ##N## independent measurements, and suppose that we obtain exactly the same results. The thermal interpretation proceeds as before and draws the same conclusion. But Born's statistical interpretation proceeds differently and claims that there is no measurement error. Instead, each measurement result reveals one of the the eigenvalues ##x_1=6.57## or ##x_2=6.58## in an unpredictable fashion with probabilities ##p=0.2## and ##1-p=0.8##, up to statistical errors of order ##O(N^{-1//2})##. For ##X=\pmatrix{6.578 & 0.004 \cr 0.004 & 6.572}##, both interpretations of the results for the 2-state quantum system are consistent with theory. However,

Clearly, the thermal interpretation is much more natural than Born's statistical interpretation, since it needs no other conventions than those that are valid for all cases where multiple measurements of the same quantity produce deviating results.

and Born's statistical interpretation for the interpretation of measurement results.

1. Consider some piece of digital equipment with 3 digit display measuring some physical quantity ##X## using ##N## independent measurements. Suppose the measurement results were 6.57 in 20% of the cases and 6.58 in 80% of the cases. Every engineer or physicist would compute the mean ##\bar X= 6.578## and the standard deviation ##\sigma_X=0.004## and conclude that the true value of the quantity ##X## deviates from ##6.578## by an error of the order of ##0.004N^{-1/2}##.

2. Consider the measurement of a Hermitian quantity ##X\in C^{2\times 2}## of a 2-state quantum system in the pure up state, using ##N## independent measurements, and suppose that we obtain exactly the same results. The thermal interpretation proceeds as before and draws the same conclusion. But Born's statistical interpretation proceeds differently and claims that there is no measurement error. Instead, each measurement result reveals one of the the eigenvalues ##x_1=6.57## or ##x_2=6.58## in an unpredictable fashion with probabilities ##p=0.2## and ##1-p=0.8##, up to statistical errors of order ##O(N^{-1//2})##. For ##X=\pmatrix{6.578 & 0.004 \cr 0.004 & 6.572}##, both interpretations of the results for the 2-state quantum system are consistent with theory. However,

*Born's statistical interpretation deviates radically from engineering practice*, without any apparent necessity.Clearly, the thermal interpretation is much more natural than Born's statistical interpretation, since it needs no other conventions than those that are valid for all cases where multiple measurements of the same quantity produce deviating results.