Introduction to Quantum Physics and Information Processing

Characterization of Quantum Information 221

1. Relative entropy is non-negative:

H (p(x) k q(x)) ≥ 0, (11.20)

with equality iff p(x) = q(x).

2. Mutual information is non-negative:

I(X; Y ) ≥ 0 (11.21)

with equality iff X and Y are independent.

3. Conditioning reduces entropy:

H(X|Y ) ≤ H(X), (11.22)

with equality iff X and Y are independent.

4. Subadditivity:

H(X, Y ) ≤ H(X) + H(Y ), (11.23)

with equality iff X and Y are independent.

Example 11.1.3. The ancient Indian game of dice used two cuboids marked 1

to 4 on the long faces. When the pair is rolled, the results are two independent

sets of events; let’s call them X and Y . Suppose a trickster uses loaded