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Completely positive quantum operations are frequently discussed in thecontexts of statistical mechanics and quantum information. They are customarilygiven by maps forming positive operator-values measures. To intuitivelyunderstand physical meanings of such abstract operations, the method ofphase-space representations is examined. This method enables one to grasp theoperations in terms of the classical statistical notions. As an example ofphysical importance, here, the phase-space representation of the completelypositive quantum operation arising from the single-mode subdynamics of thetwo-mode squeezed vacuum state, which maps from the vacuum state at vanishingtemperature to mixed states with perfect decoherence including the thermalstate, is studied. It is found in the $P$ representation that remarkably thisoperation is invertible, implying that coherence lost by the quantum operationcan be recovered.

Phase-space representation of completely positive quantum operation: Two-mode squeezed vacuum and thermal states