# 1-norm and symmetry

Define the fidelity function for positive operators by $$F(\rho, \sigma) = \lVert \sqrt{\rho}\sqrt{\sigma}\rVert_1$$. Here, $$\lVert\cdot\rVert_1$$ is the Schatten 1-norm and defined as $$\lVert A\rVert_1 = \operatorname{Tr}(\sqrt{A^{\dagger}A})$$.

I'm having some trouble showing that $$F$$ is symmetric in its arguments. Physics textbooks do it through the idea of purifications but I thought there should be a mathematical argument.

How does one see that $$F(A,B) = F(B,A)$$ given that $$F(\rho, \sigma) = \operatorname{Tr}(\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}})$$?

My original idea to prove it was to try and use cyclicity of trace and assume that $$\sqrt{AB} = \sqrt{A}\sqrt{B}$$, but it seems this is invalid even for positive operators? Can someone also comment on this seemingly simple statement being false?