Maps Preserving Schatten p -Norms of Convex Combinations

Abstract EN

We study maps ϕ of positive operators of the Schatten p -classes ( 1 < p < + ∞ ), which preserve the p -norms of convex combinations, that is, ∥ t ρ + ( 1 - t ) σ ∥ p = ∥ t ϕ ( ρ ) + ( 1 - t ) ϕ ( σ ) ∥ p , ∀ ρ , σ ∈ ? p + ( H ) 1 , t ∈ [ 0,1 ] .

They are exactly those carrying the form ϕ ( ρ ) = U ρ U * for a unitary or antiunitary U .

In the case p = 2 , we have the same conclusion whenever it just holds ∥ ρ + σ ∥ 2 = ∥ ϕ ( ρ ) + ϕ ( σ ) ∥ 2 for all the positive Hilbert-Schmidt class operators ρ , σ of norm 1 .

Some examples are demonstrated.