We theoretically analyze the polar decomposition for quantum modular values under various pointer states approaches. We consider both the finite-dimensional discrete pointer state and continuous pointer state cases. In contrast… Click to show full abstract
We theoretically analyze the polar decomposition for quantum modular values under various pointer states approaches. We consider both the finite-dimensional discrete pointer state and continuous pointer state cases. In contrast to that, a weak value of an observable is usually divided into its real and imaginary parts; here, we show that separation from the modulus and phase is necessary to a modular value. We show that the modulus of the modular value is related to the pointer post-selection conditional probability, and the phase of the modular value is connected to the summation of a geometric phase and an intrinsic phase. We also discuss a relationship between the modulus and phase, and therein, the derivative of the phase is related to the derivative of the logarithm of the modulus via a Berry-Simon-like connection which is in the form of a weak value. As a consequence, the modulus-phase relation allows us to obtain these polar components whenever the connection is specified. One of the possible appli...
               
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