Summary
Aquantum, or operator-valued, wavelet is defined for a general density operator\(\hat \rho \), in a basis generated by a general observable\(\hat \theta \) by defining an operator-valued dilation. The scale changing part of the dilation is shown to correspond to the Yuen squeeze operator. The wavelet gives a family of operator-valued coefficients which represent a given density operator in the eigenbasis of\(\hat \theta \), possibly a complete set of commuting observables. The wavelet is given in both the Heisenberg and Schrödinger pictures. Then aninverse problem is formulated which allows an unknown density operator to be calculated in terms of the family of all wavelet operators. It is interesting that a limiting process is required to obtain a unique inverse, when one exists. Then the Heisenberg-picture dilation is applied to two known examples: the unitary process of phase sensitive amplification and the irreversible process of number amplification.
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D’Ariano, G.M., DeFacio, B. A quantum wavelet for quantum optics. Il Nuovo Cimento B 108, 753–763 (1993). https://doi.org/10.1007/BF02741873
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DOI: https://doi.org/10.1007/BF02741873