Reference for variational characterization of quantum trace distance in infinite dimensions

Question

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 $$\frac{1}{2}\Vert \rho-\sigma \Vert_1=\max_{P: 0\le P \le I}\mathrm{tr}\left[ P(\rho-\sigma)\right]$$ (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?

Answer 1

For infinite-dimensional systems this result holds without change as is, e.g., developed in Chapter 3.4.3 of "The Mathematical Language of Quantum Theory" (2012) by Heinosaari & Ziman

Answer 2

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.