Entropy inequality of joint von Neumann entropy given some marginal inequalities Assume that I have a pure, multipartite state $\omega_{ABCD}$ and a unitary $U(\omega_{ABCD})=\tau_{ABCD}$. The effect of $U$ on $\omega$ results in
$$
S(\omega_B)&lt;S(\tau_B) 
$$ 
and
$$
S(\omega_A)&gt;S(\tau_A).
$$
$S$ is the von Neumann entropy and the states $\omega_A,\tau_A$ etc are the marginal density matrices (for instance, $\omega_A$= $Tr_{BCD}{\omega_{ABCD}}$). This effect of $U$ is an assumption, I don't say it always holds.

QUESTION:
Comparing the joint entropies $S(\omega_{AB})$ and $S(\tau_{AB})$, is this true
$$
S(\tau_{AB})-S(\omega_{AB})\leq S(\tau_{B})-S(\omega_{B})
$$
and if yes, why?

EDIT: Another assumption is that $S(\omega_{AB})\geq S(\omega_B)$.
Why should this be true?

Let the dimensions of $A, B, C, D$ all be the same. 

Let $\omega_{AB}$ and $\omega_{CD}$ both be pure states, with 
$$S(\omega_A) = S(\omega_B) = S(\omega_C) = S(\omega_D).$$

Now, let $U$ perform a cyclic permutation on $A, B, C, D$. 

We have $S(\tau_{AB}) = 2 S(\tau_B)$, and  $S(\omega_{AB}) = 0$, while
$S(\tau_B) = S(\omega_B)$. 

Now all you have to do is show that you can perturb this construction slightly to get the desired inequalities in your hypothesis.