(a) Consider two pure states, ρ1 = |ψ1⟩⟨ψ1| and ρ2 = |ψ2⟩⟨ψ2|. Show by explicit computation that

Step 1: Recall the definition of the trace distance between two density matrices ρ₁ and ρ₂, deno

Question

(a) Consider two pure states, $\rho_1 = |\psi_1\rangle\langle\psi_1|$ and $\rho_2 = |\psi_2\rangle\langle\psi_2|$. Show by explicit computation that $\frac{1}{2}||\rho_1 - \rho_2||_1 = \sqrt{1 - F^2(\rho_1, \rho_2)}$, where $F(\rho_1, \rho_2) = |\langle\psi_1|\psi_2\rangle|$ is the (Uhlmann) fidelity between $\rho_1, \rho_2$. (b) Consider now the case where $\rho_1, \rho_2$ are possibly mixed, and show that $\frac{1}{2}||\rho_1 - \rho_2||_1 \leq \sqrt{1 - F^2(\rho_1, \rho_2)}$. To establish this, you will have to use the fact that trace-norm is contractive under partial trace i.e. if $\rho_1', \rho_2'$ are density matrices over a space $\mathcal{H}_S \otimes \mathcal{H}_E$ larger than the original space $\mathcal{H}_S$ such that $\rho_i = Tr_E(\rho_i')$, then $||\rho_1 - \rho_2||_1 \leq ||\rho_1' - \rho_2'||_1$. Use the definition of $F(\rho_1, \rho_2)$ in terms of purifications of $\rho_1, \rho_2$ together with your result from (a) and contractivity of trace-norm.

          (a) Consider two pure states, $\rho_1 = |\psi_1\rangle\langle\psi_1|$ and $\rho_2 = |\psi_2\rangle\langle\psi_2|$. Show by explicit computation that
$\frac{1}{2}||\rho_1 - \rho_2||_1 = \sqrt{1 - F^2(\rho_1, \rho_2)}$,
where $F(\rho_1, \rho_2) = |\langle\psi_1|\psi_2\rangle|$ is the (Uhlmann) fidelity between $\rho_1, \rho_2$.
(b) Consider now the case where $\rho_1, \rho_2$ are possibly mixed, and show that
$\frac{1}{2}||\rho_1 - \rho_2||_1 \leq \sqrt{1 - F^2(\rho_1, \rho_2)}$.
To establish this, you will have to use the fact that trace-norm is contractive under
partial trace i.e. if $\rho_1', \rho_2'$ are density matrices over a space $\mathcal{H}_S \otimes \mathcal{H}_E$ larger than the
original space $\mathcal{H}_S$ such that $\rho_i = Tr_E(\rho_i')$, then
$||\rho_1 - \rho_2||_1 \leq ||\rho_1' - \rho_2'||_1$.
Use the definition of $F(\rho_1, \rho_2)$ in terms of purifications of $\rho_1, \rho_2$ together with your
result from (a) and contractivity of trace-norm.

(a) Consider two pure states, ρ1 = |ψ1⟩⟨ψ1| and ρ2 = |ψ2⟩⟨ψ2|. Show by explicit computation that
(1)/(2)||ρ1 - ρ2||1 = √(1 - F^2(ρ1, ρ2)),
where F(ρ1, ρ2) = |⟨ψ1|ψ2⟩| is the (Uhlmann) fidelity between ρ1, ρ2.
(b) Consider now the case where ρ1, ρ2 are possibly mixed, and show that
(1)/(2)||ρ1 - ρ2||1 ≤√(1 - F^2(ρ1, ρ2)).
To establish this, you will have to use the fact that trace-norm is contractive under
partial trace i.e. if ρ1', ρ2' are density matrices over a space ℋS ⊗ℋE larger than the
original space ℋS such that = TrE('), then
||ρ1 - ρ2||1 ≤ ||ρ1' - ρ2'||1.
Use the definition of F(ρ1, ρ2) in terms of purifications of ρ1, ρ2 together with your
result from (a) and contractivity of trace-norm.

Added by Jason B.

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