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?

  • $\begingroup$ 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. $\endgroup$
    – Hldngpk
    Jun 14, 2022 at 2:09

1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.