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Uncertainty relations are one of the fundamental principles in physics. Itbegan as a fundamental limitation in quantum mechanics, and today the word {\ituncertainty relation} is a generic term for various trade-off relations innature. In this letter, we improve the Kennard-Robertson uncertainty relation,and clarify how much coherence we need to implement quantum measurement underconservation laws. Our approach systematically improves and reproduces theprevious various refinements of the Kennard-Robertson inequality. As a directconsequence of our inequalities, we improve a well-known limitation of quantummeasurements, the Wigner-Araki-Yanase-Ozawa theorem. This improvement gives anasymptotic equality for the necessary and sufficient amount of coherence toimplement a quantum measurement with the desired accuracy under conservationlaws.

Coherence-variance uncertainty relation and coherence cost for quantum measurement under conservation laws