Worked Problems in Physics

Nielsen and Chuang Exercise 2.31

Posted in Nielsen/Chuang by rpmuller on March 6, 2010

Exercise 2.31 Show that {A\otimes B} is positive if {A,B} are positive.

\displaystyle  A\otimes B = (U\Lambda U^\dagger)\otimes(U'\Lambda'U'^\dagger) = (U\otimes U')(\Lambda\otimes\Lambda')(U^\dagger\otimes U'^\dagger)

The final decomposition represents an eigendecomposition in the tensor space. {U\otimes U'} is unitary (ex 2.29), and {U^\dagger\otimes U'^\dagger} is its adjoint. {\Lambda\otimes\Lambda'} is a diagonal matrix with nonnegative values. Thus {A\otimes B} is positive.

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