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Very briefly, gate fidelity refers to a way to compare how "close" two gates, or more generally operations, are to each other. As discussed e.g. in (Magesan et al. 2012), if one wants to compare the action of an operation $\mathcal E$ and a gate $\mathcal U$ on a given state $\rho$, one can define their "gate fidelity" as the quantum ...


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Fidelity is a distance measure between quantum states. The gate fidelity uses fidelity to decide how noisy a quantum gate is. Take two copies of a state, apply your implementation of a gate on one copy and apply the ideal gate on another copy (this can be done on paper, not in a lab) and compute the fidelity between the two outputs. This is the gate fidelity....


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As @rnva points out these are not the same quantities. To give some clarity as to why they are both referred to as $D_{\min}$ it is best to look at the as limiting cases of $\alpha$-R'enyi divergences. First, we have the sandwiched divergences which for $\alpha \in (0, 1) \cup (1, \infty)$ are defined as $$ \widetilde{D}_{\alpha}(\rho\|\sigma) = \frac{1}{\...


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No, there's not a lot you can say. Consider these two cases, both with $\epsilon=0$. First, the obvious one, $\rho=\sigma=|\psi\rangle\langle\psi|\otimes |\psi\rangle\langle\psi|$. Clearly $\rho_r-\sigma_r=0$. Second, let $|\psi^\perp\rangle$ be orthogonal to $|\psi\rangle$. You can have $$\rho=(|\psi\rangle\langle\psi|\otimes |\psi^\perp\rangle\langle\psi^\...


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