# Limit of a particular trace norm.

I have the following problem.

Let $$\mathbf{\hat{\rho}}(t)$$ and $$\mathbf{\hat{\sigma}}(t)$$ be two trace class positive operators acting on a Hilbert space of infinite dimension for all $$t > 0$$. More precisely assume that $$\mathbf{\hat{\rho}}(t):= \int p_{i}(x)e^{-ixt\mathbf{\hat{B}}}\big|\psi\big\rangle \big\langle \psi\big|e^{ixt\mathbf{\hat{B}}}dx$$ $$\mathbf{\hat{\sigma}}(t):= \int p_{j}(x)e^{-ixt\mathbf{\hat{B}}}\big|\psi\big\rangle \big\langle \psi\big|e^{ixt\mathbf{\hat{B}}}dx$$

where $$\mathbf{\hat{B}}$$ is a self-adjoint operator with purely absolutely continuous spectrum and $$\big|\psi\big\rangle$$ is any vector in the Hilbert space in question, and $$p_{i}$$ and $$p_{j}$$ are probability distributions with compact support which is nonoverlapping. I am trying to prove that $$\lim_{t\rightarrow \infty}\big\|\sqrt{\mathbf{\hat{\rho}}(t)}\sqrt{\mathbf{\hat{\sigma}}(t)}\big\|_{1}= 0 \;\; (quantum\; fidelity)$$

However, this has proven to be quite a challenge since there are no good upper bounds for the quantum fidelity in the general case were both of the operators in question are not pure. I have tried using the following celebrated bound.

$$\big\|\sqrt{\mathbf{\hat{\rho}}(t)}\sqrt{\mathbf{\hat{\sigma}}(t)}\big\|_{1}\leq\sqrt{1-\big\|\mathbf{\hat{\rho}}(t)-\mathbf{\hat{\sigma}}(t)\big\|_{1}^{2}}$$ but this just replaces a very difficult problem with one of equal complexity.

For the simpler version of this problem where

$$\mathbf{\hat{\rho}}(t):=e^{-ix_{i}t\mathbf{\hat{B}}}\big|\psi\big\rangle \big\langle \psi\big|e^{ix_{i}t\mathbf{\hat{B}}}$$

and

$$\mathbf{\hat{\sigma}}(t):=e^{-ix_{j}t\mathbf{\hat{B}}}\big|\psi\big\rangle \big\langle \psi\big|e^{ix_{j}t\mathbf{\hat{B}}}$$

with $$x_{i}\neq x_{j}$$ and all of the other assumptions preserved I can easily show the analogous hypothesis.

Here

$$\lim_{t\rightarrow \infty}\big\|\sqrt{\mathbf{\hat{\rho}}(t)}\sqrt{\mathbf{\hat{\sigma}}(t)}\big\|_{1} = \big|\langle \psi\big|e^{-t(x_{i}-x_{j})\mathbf{\hat{B}}}\big|\psi\big\rangle\big| = \int e^{-t(x_{i}-x_{j})\lambda}d\mu_{\psi}(\lambda)$$ where $$d\mu_{\psi}(\lambda)$$ is the absolutely continnuous spectral measure afforded by $$\big|\psi\rangle$$. Owing to the Riemann Lebegues lemma indeed $$\lim_{t\rightarrow \infty}\int e^{-t(x_{i}-x_{j})\lambda}d\mu_{\psi}(\lambda) = 0$$. Due to this result, I am led to believe that the more general case where $$\mathbf{\hat{\rho}}(t)$$ and $$\mathbf{\hat{\sigma}}(t)$$ are uncountable mixtures as presented above, we should have the same sort of behavior as $$t\rightarrow \infty$$. However, the quantum fidelity is unwieldy. Any help tackling this problem would be greatly appreciated.

• This also holds when only one of the density operators is pure by a similar argument (combined with the dominated convergence theorem). But the general case does indeed seem harder. Sep 6, 2023 at 10:34
• @LL3.14 How so? Regard the indices of the xs. One is $x_{i}$ whilst the other is $x_{j}$. Sep 6, 2023 at 19:55
• @MaoWao You are correct, I have that case under control as well. Computing Fidelitites is simple when at least one of the terms in the fidelity is a pure state (i.e. a projector). This case, with both states not being pure, seems insurmountable. Sep 6, 2023 at 19:57
• No, I am saying there us two times "$\lim ...$" Sep 6, 2023 at 20:55
• I fixed it. Thank you @LL3.14 Sep 6, 2023 at 20:57