Abstract
We may try to discuss gently the fact of “symmetric measurements outcomes” that is free from the order of measurements themselves, which fact might be extended to considering naturally the uncertainty principle. For two symmetric measurement outcomes, sometimes, the two measured observables are commutative. In this specific and symmetric example, we introduce a supposition that the operation Addition is equivalent to the operation Multiplication, and then, we may be apt to have an example of an inconsistency, probably due to the nature of Matrix theory based on non-commutativeness. We show here the inconsistency in an arbitrary dimensional unitary space when measuring commuting observables/an observable. We would say that the trial above might be categorized into an inconsistency example in the effect of the uncertainty principle, if we are forgiven for describing the above.
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We cannot introduce a supposition that the sum rule is equivalent to the product rule if \(A_1\) and \(A_2\) are not commutative. Therefore, \([A_1,A_2]= 0\) is an important sufficient condition for our interesting objective
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Acknowledgements
We thank Soliman Abdalla, Jaewook Ahn, Josep Batle, Mark Behzad Doost, Ahmed Farouk, Han Geurdes, Preston Guynn, Shahrokh Heidari, Wenliang Jin, Hamed Daei Kasmaei, Janusz Milek, Mosayeb Naseri, Santanu Kumar Patro, Germano Resconi, and Renata Wong for their valuable support.
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Nagata, K., Diep, D.N. & Nakamura, T. Two symmetric measurements may cause an unforeseen effect. Quantum Inf Process 22, 94 (2023). https://doi.org/10.1007/s11128-023-03841-5
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DOI: https://doi.org/10.1007/s11128-023-03841-5