Worked Problems in Physics

Nielsen and Chuang Exercise 2.74

Posted in Nielsen/Chuang by rpmuller on June 1, 2010

Exercise 2.74 Suppose a composite of systems {A} and {B} is in the state {\left|a\right>\left|b\right>}, where {\left|a\right>} is a pure state of system {A}, and {\left|b\right>} is a pure state of system {B}. Show that the reduced density operator of system {A} alone is a pure state.

\displaystyle \rho = \left|a\right>\left|b\right>\left<a\right|\left<b\right|,

\displaystyle \mathrm{tr}_B(\rho) = \left|a\right>\left<a\right|\left<b|b\right>= \left|a\right>\left<a\right|,

and is thus a pure state.

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