Necessity of sine-cosine joint measurement

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Abstract
The quantum measurement of trigonometric variables is revisted. We show that the probability distributions of the sine and cosine operators of Susskind and Glogower [Physics 1, 49 (1964)] suffer unphysical features for nonclassical states. We suggest that any measurement of a trigonometric variable needs necessarily a joint measurement of the two cosine-sine phase quadratures. In this way unphysical quantum statistics are avoided, and no violation of the trigonometric calculus occurs for expected values. We show that this trigonometric measurement can be defined in general terms in the framework of quantum estimation theory.