Ex10 17

$\def\abs#1{|#1|}\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\def\tr{\mathord{\mbox{tr}}}\mathbf{Exercise\ 10.17}$

Maximally entangled bipartite states.

Let $\ket\psi$ be a state of the form

(1)
\begin{align} \ket\psi = \frac{1}{\sqrt{m}} \sum_{i=1}^m \ket{\phi_i^A}\otimes\ket{\phi_i^B} \end{align}

where the $\{\ket{\phi_i^A} \}$ and $\{\ket{\phi_i^B} \}$ are orthonormal sets. Show that the vector $\lambda^\psi$ is majorized by $\lambda^\phi$ for all states $\ket\phi \in A\otimes B$.