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A relation between values of a unitarily invariant norm of Hermitian operatorbefore and after action of completely positive map is studied. If the norm isjointly defined on both the input and output Hilbert spaces, one defines ashrinking factor under the restriction of given map to Hermitian operators. Asit is shown, for any unitarily invariant norm this shrinking factor is notlarger than the maximum of two values for the spectral norm and the trace norm.

A Shrinking Factor for Unitarily Invariant Norms under a Completely Positive Map