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I'm trying to prove that $-\log{d} \leq H(A|B) \leq \log{d}$ for von Neumann entropy.

Now, for this to make sense I should give some definitions. System $A$ lives in Hilbert space $\mathcal{H}_A$, system $B$ in $\mathcal{H}_B$, and these are finite dimensional systems. Moreover, system A has dimension $d$. $H(A|B)$ is the conditional entropy, which I here take to be $H(A|B) = H(AB) - H(B)$, where $H(A) = -Tr(\rho_A \log{\rho_A})$, $\rho_A$ being the density operator of system $A$, so that $H(AB)$ is the density operator of the joint system.

Now, I can prove the upperbound quite easily:

$H(A|B) = H(AB) - H(B) \leq H(A) + H(B) - H(B) \leq \log{d}$, where both inequalities follow from the positivity of the relative entropy $H(A||B) = Tr(\rho_A\log{\rho_A}-\rho_A\log{\rho_B})$ quite easily. If needed, I can write those two line proofs in an update.

However, for the lowerbound I'm quite lost. I'm pretty sure I need to make some clever choice of $A$ and $B$ for the relative entropy again and abuse its positivity, but I can't figure out what to do. Could anyone give me a hint?

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  • $\begingroup$ What is $d$ here? $\endgroup$
    – Danu
    Jan 10 '15 at 15:13
  • $\begingroup$ The dimension of Hilbert space $A$, so the number of elements in its basis. $\endgroup$
    – user129412
    Jan 10 '15 at 15:28
  • $\begingroup$ Oh, don't know why I missed that initially. Thanks. $\endgroup$
    – Danu
    Jan 10 '15 at 16:13
  • $\begingroup$ One can show this using strong subadditivity of the von Neumann entropy using a purification, but this seems like a bit of overkill. $\endgroup$ Jan 10 '15 at 16:14
  • $\begingroup$ As in, $H(AB|C) \leq H(A|C) + H(B|C)$, I suppose? I can look at that for a bit. What purification would you use? $\endgroup$
    – user129412
    Jan 10 '15 at 16:35
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The lower bound $-\log\,d\le H(A|B)$ follows from strong subadditivity, $$ H(ABC)+H(B)\le H(AB)+H(BC)\ . $$ To this end, choose a purification $\lvert\psi_{ABC}\rangle$ of $\rho_{AB}$. Then, $H(ABC)=0$, and $H(BC)=H(A)$, and thus, we have $H(B)\le H(AB) + H(A)$ (this is also known as the Araki-Lieb inequality), which implies $$ -\log\,d\le -H(A)\le H(AB)-H(B)=H(A|B)\ . $$

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  • $\begingroup$ Thanks for the answer. I see indeed that it follows from strong subadditivity, however I don't see exactly how this is the Araki Lieb inequality. I tried doing it from that myself, and failed. I thought the Araki Lieb was the fact that $|H(A) - H(B)| \leq H(AB) \leq H(A) + H(B)$. How does one rewrite that to $H(B) \leq H(AB) + H(A)$? Maybe I'm just making a simple mistake, but the way I see it it states that $H(AB) - H(A) \leq H(B)$, which is not easily rewritten to the form you state as far as I can see. In fact, taking H(AB) to the other side and multiplying by -1 seems to give the opposite. $\endgroup$
    – user129412
    Jan 11 '15 at 13:09
  • $\begingroup$ Disregard that last sentence starting with in fact, I was too late to edit that out. However, the rest is still a question! $\endgroup$
    – user129412
    Jan 11 '15 at 13:16
  • $\begingroup$ $|H(A)-H(B)|\le H(AB)$ implies $H(B)-H(A)\le H(AB)$, which is equivalent to $H(B)\le H(AB)+H(A)$. $\endgroup$ Jan 11 '15 at 13:26
  • $\begingroup$ BTW, the Araki-Lieb inequality can be proven without using strong subadditivity (indeed, the Araki-Lieb paper predates the proof of strong subadditivity), so one indeed prove this without strong subadditivity. $\endgroup$ Jan 11 '15 at 15:36
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    $\begingroup$ The triangle equality follows from subadditivity by making use of duality as follows: Let R be a third purifying reference system, so that $|\psi\rangle_{RAB}$ is a purification of $\rho_{AB}$. Then $H(B) = H(RA) \leq H(A) + H(R) = H(A) + H(AB)$. This implies $H(AB) \geq H(B) - H(A)$, and you get the absolute values by doing the proof with A and B reversed. $\endgroup$
    – user129412
    Jan 12 '15 at 19:39

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