4
votes
0answers
94 views

A tight lower bound for the entropy of the XOR of two random variables

Let $U$ be the uniform random variable over $n$-bit binary strings, and let $X$ be another random variable that is dependent on $U$ and ranges over $n$-bit binary strings. Assuming $I(X;U) \le ...
0
votes
0answers
53 views

Von Neumann Entropy Inequality

Suppose $\rho_1$ and $\rho_2$ are density matrices (and thus Hermitian, positive semi-definite matrices) and $\hat{w}$ is the solution of the following optimization problem $$ \hat{w} = ...
0
votes
1answer
75 views

Entropy Inequality

I have very hard time to prove the following inequality or to show a contradiction. $H(X_1,X_2,X_3) + H(X_1,X_2,X_4)+ H(X_1,X_3,X_4) + H(X_2,X_3,X_4) \leq 3(H(X_1,X_2) + H(X_3,X_4))$ The problem ...
1
vote
0answers
139 views

Inequality involving KL divergence

Following is a part of an answer which was not resolved when I tried to answer a question in mathoverflow. I thought it would be nice to discuss that here. Let $P$ and $Q$ be two distinct ...
22
votes
2answers
1k views

An information theory inequality which relates to Shannon Entropy

For $a_1,...,a_n,b_1,...,b_n>0,\quad$ define $a:=\sum a_i,\ b:=\sum b_i,\ s:=\sum \sqrt{a_ib_i}$. Is the following inequality true?: $${\frac{\Bigl(\prod a_i^{a_i}\Bigr)^\frac1a}a \cdot ...