Derivative of fidelity with respect to time

Question

Consider the quantum fidelity between two states defined as $$ F(\rho(t),\sigma(t)):=\text{Tr}\left(\sqrt{\sqrt{\rho(t)}\sigma(t)\sqrt{\rho(t)}}\right)^2 $$ Does $dF/dt$ have a closed form equation?

Answer 1

It's not going to have a simple closed-form expression, but I'll give you some tools that you can use to build up an expression for the derivative yourself.

First consider matrix functions of the form $\sqrt{A(t)}$, where $A(t)$ is a continuously differentiable positive definite $n\times n$ matrix. (Things can get a little hairy if $A$ is not positive definite.) We wish to compute the derivative $ \left.\frac{d}{dt} \sqrt{A(t)}\right|_{t=0}. $ We may suppose without loss of generality that $A(0)$ is diagonal, $A(0)=\operatorname{diag}(\alpha_1,\dots,\alpha_n)$. Let $D$ be the $n\times n$ matrix whose entries are given by $$ D_{i,j} = \left\{ \begin{array}{ll} \frac{\sqrt{\alpha_i}-\sqrt{\alpha_j}}{\alpha_i-\alpha_j} &\quad \text{if }\alpha_i\neq\alpha_j\\ \frac{1}{2}\frac{1}{\sqrt{\alpha_i}}&\quad \text{if }\alpha_i=\alpha_j. \end{array}\right. $$ The desired derivative is $$ \left.\frac{d}{dt} \sqrt{A(t)}\right|_{t=0} = D\circ A'(0), $$ where $\circ$ denotes the entrywise product of matrices.

You can find the derivation of this in many textbooks on matrix analysis. I'll list a few here:

• Theorem 6.6.30 in $[1]$
• Theorem 5.9 in $[2]$
• Section V.3 in $[3]$

$[1]$ Roger A. Horn, Charles R. Johnson. Topics in matrix analysis (1991).

$[2]$ Fumio Hiai, Dénes Petz. Introduction to matrix analysis and applications (2014).

$[3]$ Rajendra Bhatia. Matrix Analysis (1997).