Inner product with maximally entangled state

[rmp251]

Consider the maximum inner product of a density operator with a maximally entangled state. (So, given a density operator, we're maximizing over all maximally entangled states.)

I'm pretty sure the minimum value (a lower bound on the maximum) for this is 1/n², using the density operator 1/n² identity. How can I prove that is the minimum?

Thanks!
Reuben

 

[tom.stoer]

can you please write down the density operator you are talking about; and what you mean by 'maximum inner product of a density operator with a maximally entangled state'.

 

[rmp251]

Sorry I realize that was a little incomplete. Let \mathcal{X} and \mathcal{Y} be complex Euclidean spaces with dim(\mathcal{X})=dim(\mathcal{Y})=n.

Define

<br /> M(\rho) = \max\left\{&lt;u u^{\ast},\rho&gt;\,:\,u\in\mathcal{X}\otimes\mathcal{Y}\;\text{is maximally entangled}<br /> \right\}<br />

for any density operator \rho on \mathcal{X}\otimes\mathcal{Y}.

For \rho=\frac{1}{n^2}\text{I} we get M=\frac{1}{n^2}

How can I show that that is the minimum value of M over all possible density operators \rho?