Entropy Inequalities

Hey I am reading this paper Entropy Inequalities by Araki and Lieb.

I am trying to prove the following lemma: $$S^1+S^2\leq S^{12}+S^{23}$$ using the following lemmas:

1. $S^{123}+S^{2}\leq S^{12}+S^{23}$

2. For a pure density matrix $\rho^{12}$: $Tr^1\left[f(\rho^1)\right]=Tr^2\left[f(\rho^2)\right]$.

and:

1. There exists a pure density matrix $\rho^{12}$ and a $\rho^{1}$ such that $\rho^1=Tr^2\left[\rho^{12}\right]$.

So I tried the following:

Using the second lemma on the first one we can write:

$S^{123}+S^{1}\leq S^{12}+S^{13}$

Now subtracting the RHS of the new inequality from the LHS of the 1st lemma and the LHS of the new inequality from the RHS of the 1st lemma yields:

$S^{123}+S^2-S^{12}-S^{13}=S^{12}+S^{13}-S^{123}-S^2$

or:

$S^1+S^2 \leq 2(S^{12}-S^{123})+S^{13}+S^{23}$

Now we can use the theorem 1 from their paper $S^{123}\leq S^{12}+S^{23}$

which when plugging in yields:

$S^1+S^2\leq S^{23}-S^{13}+2\,\eta$

But this is not the inequality I wanted to prove. I get a different one.

• Do you mean you want to show $S^1+S^2\leq S^{13}+S^{23}$? Your inequality is false if 1 is empty, and $\rho^{23}$ pure. – Holographer Feb 6 '15 at 15:48
• Yes I know that my inequality has to be wrong. But I can't see where I am going wrong and I can't see how else I can use those lemmas to prove the inequality that Lieb proves. – onephys Feb 7 '15 at 8:53