QuantumInfo.States.Mixed.Fidelity

For any two quantum states $\rho$ and $\sigma$ represented as density matrices in a $d$-dimensional Hilbert space, the fidelity $F(\rho, \sigma)$ is equal to the trace norm of the product of their square roots: $$F(\rho, \sigma) = \|\sqrt{\sigma} \sqrt{\rho}\|_{\text{tr}}$$ where $\sqrt{\cdot}$ denotes the positive semi-definite square root of a matrix and $\|\cdot\|_{\text{tr}}$ denotes the trace norm (also known as the nuclear norm or Schatten 1-norm), defined as $\|A\|_{\text{tr}} = \text{Tr}(\sqrt{A^\dagger A})$.