# Reference for variational characterization of quantum trace distance in infinite dimensions

Consider two density matrices $$\rho$$ and $$\sigma$$. It is well known that for finite-dimensional systems, the trace distance $$\frac{1}{2}\Vert \rho-\sigma \Vert_1$$ has the variational characterization $$\frac{1}{2}\Vert \rho-\sigma \Vert_1=\max_{P: 0\le 0 \le I}\mathrm{tr}\left[ P(\rho-\sigma)\right]$$ (e.g., see Lemma 9.1.1 in Wilde "Quantum information theory"). Is there a reference for a similar variational characterization for infinite-dimensional Hilbert spaces?

• This is two years late but I'll comment anyway. Would this not be the same definition for the infinite case. I do not see anything intrinsicaly requireing that $\rho$ and $\sigma$ have finite rank. If you now, after two years, have a conclusive answer to your own question I would love to read it. Jun 14 at 2:09

Here is a toy example. Let $$\rho = \sum_{n = 0 }^{\infty}\lambda_{i}|\psi_{i}\rangle \langle\psi_{i}|$$ where $$|\psi_{i}\rangle$$ is some infinite dimensional Hilbert space. Could be the number states for example. Now let $$\sigma = 0$$. Let us compute the simple trace distance $$\| \rho - \sigma\|_{1}$$.
$$\|\rho - \sigma\|_{1} = \|\rho\|_{1} = max_{\substack{P}}Tr\{P\rho\}.$$ We know that $$Tr\{P\rho\}\leq 1$$ for any projector $$P$$, if we use the projector $$P^\prime = \sum_{n}|\psi_{n}\rangle \langle \psi_{n}|$$ notice that we get $$Tr\{P^\prime \rho\} = 1.$$ Hence $$max_{\substack{P}}Tr\{P\rho\} = 1$$ and the maximizing $$P$$ is $$P^\prime$$. As you can see, things work out here and we did not need a new variational definition for the trace distance.