I'm curious as to whether a statement of the following form can be proven: $$ D(\rho_{AB} || \tau_{AB}) \leq D(\rho_{A}|| \tau_{A}) + |B| $$ Where $D(\cdot || \cdot )$ is the standard quantum relative entropy, and $\rho, \tau$ are both density matrices. Is there any literature on the subject? Do counter examples exist?
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No. Let $$\begin{align*} \rho_{AB} &= (1-\varepsilon) |\phi^+\rangle \langle \phi^+| + \varepsilon |\psi^-\rangle \langle \psi^-| \\ \tau_{AB} &= \varepsilon |\phi^+\rangle \langle \phi^+| + (1-\varepsilon) |\psi^-\rangle \langle \psi^-| \end{align*} $$ Then $D(\rho_{AB}\| \tau_{AB}) = (1-2\varepsilon) \log \frac{1-\varepsilon}{\varepsilon}$, which goes to infinity as $\varepsilon \to 0$. Your hopeful upper bound, on the other hand, is just $|B| = 2$, since $$ \rho_A = \tau_A = \frac12 I, $$ and therefore $D(\rho_A \| \tau_A) = 0$.
Mateus Araújo
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