# The quantum key establishment protocol

Consider the quantum key establishment protocol. Assume that bit values, emitter basis, and receiver lter orientation are selected independently and uniformly at random (i.e., 50% chance for each of the bit values/ bases/orientations).

1)What is the joint entropy of the emitter bit value and receiver bit value?

My assumption is: H(X=0,Y=0) = 3/4 H(X=0,Y=1) = 1/4 H(X=1,Y=0) = 1/4 H(X=1,Y=1) = 3/4

H(Y=0,X=0) = 3/4 H(Y=0,X=0) = 1/4 H(Y=0,X=0) = 1/4 H(Y=0,X=0) = 3/4

I could not go further here.

2)What is the mutual information between the emitter bit value and the receiver bit value?

In my opinion, the filters can measure the polarasation of a photon if the filter and the photon do not agree the outcome is uniformly random. So, with respect to receiver bit value and emitter bit value we can find the sifted key.

Note:It is not a homework question.

• Whether or not it's homework, it would help if you could provide some context about where you came across this problem, how far you are in your studies (from the phrasing and the image, I assume this is an educational exercise of some kind), and which specific part of the problem you're stuck on. – Ilmari Karonen Aug 18 '18 at 23:59
• @IlmariKaronen I have edited. – Ashley Watson Aug 19 '18 at 7:05
• The notation in your edit looks odd to me. Are you sure you're not confusing entropy with probability, or something? You might want to review the definitions of joint entropy and mutual information and see if that clarifies the problem at all. – Ilmari Karonen Aug 19 '18 at 9:25

Your "assumption" are conditional probabilities, and they should be calculated, not "assumed". For example, let Alice sends bit $0$; what is the probability that Bob receives bit $0$? Bob has 2 equally likely choices: either his basis coincides with the Alice's basis or not; in the 1st case Bob receives $0$ bit with probability $1$, in the 2nd case Bob receives $0$ bit with probability $1/2$. So the conditional probability that Bob receives $0$ bit given Alice sent $0$ bit is $$P_{X,Y}(0,0|X=0) = 1/2\cdot 1 + 1/2\cdot 1/2 = 3/4$$
Since Alice sends $0$ and $1$ bits equally likely, $$P_{X,Y}(0,0) = 1/2\cdot 3/4 = 3/8$$ and this I believe is one of the 4 probabilities that should be inserted into the Joint Entropy Formula