The relation between the scaling of Husimi functions and the linear phase insensitive amplification of the corresponding quantum states and its implications
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Using the description of the linear phase insensitive amplification of a quantum state available in the literature we prove that to every state rho(t), produced from the initial state rho(0) by such amplification, there corresponds a phase space distribution lambda(2)D(0)(lambda(q), lambda p), where D-0(q, p) is the Husimi function of the initial state. The scaling parameter lambda satisfies the relation 0 LT lambda LT 1 and decreases with time while, as we have shown earlier, the scaled function is again a Husimi distribution. We prove, using these facts, that if the definition of the phase insensitive amplification is physically correct then the distribution of the phase of a state is unambiguously given as the corresponding marginal distribution obtained from its Husimi distribution represented in polar coordinates.