conditional entropy can be negative.For example,consider an EPR pair,S(A|B)=0-1 1,but the average definition won't give us this:A and B are in either 00 or 11. On the other hand,the quantum conditional entropy cannot be too negative. IS(A)-S(B)川≤S(A,B)≤S(A)+S(B) Convexity.Suppose Pi's are quantum states and pi's form a distribution. Mutual information. S(A;B)=S(A)+S(B)-S(A,B)=S(A)-S(AIB)=S(B)-S(BA) Classically the mutual information is upper bounded by individual entropy,i.e.I(A;B)< min(H(A),H(B)},but quantum mutual information isn't...Well it still holds---up to a factor of2: 1(A;B)<min(2S(A),2S(B)} Strong subadditivity.S(A,B,C)+S(B)<S(A,B)+S(B,C). Notes The definitions and theorems in this lecture about classical information theory can be found in the standard textbook such as [CT06](Chapter 1).The quantum part can be found in [NC0O](Chapter 11). References [CT06]Thomas Cover,Joy Thomas.Elements of Information Theory,Second Edition, Wiley InterScience,2006.conditional entropy can be negative. For example, consider an EPR pair, S(A|B) = 0 − 1 = 1, but the average definition won't give us this: A and B are in either 00 or 11. On the other hand, the quantum conditional entropy cannot be too negative. |𝑆(𝐴) − 𝑆(𝐵)| ≤ 𝑆(𝐴, 𝐵) ≤ 𝑆(𝐴) + 𝑆(𝐵) Convexity. Suppose 𝜌𝑖 ’s are quantum states and 𝑝𝑖 ’s form a distribution. ∑𝑝𝑖𝑆(𝜌𝑖 ) 𝑖 ≤ 𝑆 (∑𝑝𝑖𝜌𝑖 𝑖 ) ≤ ∑𝑝𝑖𝑆(𝜌𝑖 ) 𝑖 + 𝐻(𝑝) Mutual information. 𝑆(𝐴; 𝐵) = 𝑆(𝐴) + 𝑆(𝐵) − 𝑆(𝐴, 𝐵) = 𝑆(𝐴) − 𝑆(𝐴|𝐵) = 𝑆(𝐵) − 𝑆(𝐵|𝐴) Classically the mutual information is upper bounded by individual entropy, i.e. 𝐼(𝐴; 𝐵) ≤ min{𝐻(𝐴), 𝐻(𝐵)}, but quantum mutual information isn’t… Well it still holds---up to a factor of 2: 𝐼(𝐴; 𝐵) ≤ min{2𝑆(𝐴),2𝑆(𝐵)} Strong subadditivity. 𝑆(𝐴, 𝐵, 𝐶)+ 𝑆(𝐵) ≤ 𝑆(𝐴, 𝐵) + 𝑆(𝐵, 𝐶). Notes The definitions and theorems in this lecture about classical information theory can be found in the standard textbook such as [CT06] (Chapter 1). The quantum part can be found in [NC00] (Chapter 11). References [CT06] Thomas Cover, Joy Thomas. Elements of Information Theory, Second Edition, Wiley InterScience, 2006