Abstract
The usual no-cloning theorem implies that two quantum states are identical or orthogonal if we allow a cloning to be on the two quantum states. Here, we investigate a relation between the no-cloning theorem and the projective measurement theory that the results of measurements are either + 1 or − 1. We introduce the Kochen-Specker (KS) theorem with the projective measurement theory. We result in the fact that the two quantum states under consideration cannot be orthogonal if we avoid the KS contradiction. Thus the no-cloning theorem implies that the two quantum states under consideration are identical in that case. It turns out that the KS theorem with the projective measurement theory says a new version of the no-cloning theorem. Next, we investigate a relation between the no-cloning theorem and the measurement theory based on the truth values that the results of measurements are either + 1 or 0. We return to the usual no-cloning theorem that the two quantum states are identical or orthogonal in the case.
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Notes
In probability theory, the law of large numbers is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed. The strong law of large numbers states that the sample average converges almost surely to the expected value.
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Nagata, K., Nakamura, T., Farouk, A. et al. No-Cloning Theorem, Kochen-Specker Theorem, and Quantum Measurement Theories. Int J Theor Phys 58, 1845–1853 (2019). https://doi.org/10.1007/s10773-019-04078-8
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DOI: https://doi.org/10.1007/s10773-019-04078-8