If I have two logical states of a quantum code that are close in trace distance i.e. $\vert 0\rangle_L$ and $U_L\vert 0\rangle_L$ where $\|U_L - I\|_{\diamond} \leq \delta$, what is an upper bound (if any) on how close can the corresponding physical states be?
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$\begingroup$ What exactly do you mean with physical state in this context? Don't you mean that the operators $U_L$ and $\mathbb I$ are $\delta$-close and aren't you interested in the distance between $\rho = |0\rangle_L \langle 0|_L$ and $\sigma = U_L|0\rangle_L \langle 0|_L U^\dagger_L$? $\endgroup$– qubitzerCommented May 31 at 8:30
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