Update the reference to the book of Csiszar and Korner to include Cambridge reprint. -------------------------------------------------------------------------------- Error in (3.178) -------------------------------------------------------------------------------- After (23.97), before 23.6.2 "We can conclude that the maximally entangled state … achieves the ***quantum*** capacity of the quantum erasure channel because …" -------------------------------------------------------------------------------- The proof of additivity for entanglement-breaking channels is wrong (page 317 --- Theorem 12.3.1). Look at Shor's paper. The initial part of the argument is similar, but then it should go as I(X;B1 B2) = H(B1 B2) - H(B1 B2 | X) The crucial inequality is then H(B1 B2 | X) >= H(B1 | B2 Y X) + H(B2 | X) which is equivalent to strong subadditivity: H(B2 Y | X) + H(B1 B2 | X) >= H(B1 B2 Y | X) + H(B2 | X) Since given Y and X, B1 and B2 are in a product state, the first one is equivalent to H(B1 B2 | X) >= H(B1 | Y X) + H(B2 | X) we then go back to the beginning and use this (along with subadditivity) to get H(B1 B2) - H(B1 B2 | X) <= H(B1) + H(B2) - H(B1 | Y X) - H(B2 | X) I(XY; B1) + I(X; B2) These information quantities are then with respect to correct particular ensembles, so that we find the upper bound chi(N1) + chi(N2) -------------------------------------------------------------------------------- inequality goes the wrong way in Exercise 12.5.4 (Equation 12.163) -------------------------------------------------------------------------------- improve the discussion of Shannon's classical capacity theorem - in particular, the error probability should be Pr { ( Am \intersect A_{m-1}^c \intersect … \intersect A_{1}^c \intersect unconditionally typical set) ^c} where each Ai is the event that the output sequence is not in the conditionally typical set. This is how it should go. ---------------------------------------------------------------- more careful discussion of the CHSH game ( a la Scarani's "device independent outlook on quantum physics") ----------------------------------------------------------------- be more precise in the discussion of the Schmidt decomposition theorem. important to say that the number of Schmidt coefficients is less than the minimum of the local dimensions. This restricts any pure bipartite to be in a subspace of the whole Hilbert space. ----------------------------------------------------------------- include a proof of the Kraus representation theorem (already have this ready---can just insert it in there) ----------------------------------------------------------------- error with joint concavity of fidelity (it is only the root fidelity that obeys this inequality) thank David Reeb for finding this. ----------------------------------------------------------------- include exercise for Araki-Lieb inequality ----------------------------------------------------------------- include diagrams for qudit version of teleportation ----------------------------------------------------------------- prove the inequality ln(x) <= x-1 for 0 <= x <= 1 by Taylor expanding ln(x) = (x-1) - (x-1)^2/2 + (x-1)^3 / 3 - (x-1)^4/4 + … for other values use a derivative argument ----------------------------------------------------------------- We argue in the proof of the following theorem that the coherent information of the joint channel N1 N2 is ***never smaller than*** half the private information of N2 alone. ----------------------------------------------------------------- in the description of a protocol for private classical communication, we don’t need a randomization variable. it can simple be embedded as part of the quantum state... --------------------------------------------------------------------------- add cardinality bounds on the size of classical registers when appropriate put the Fenchel-Eggleston-Caratheodory theorem in an appendix and show how to use the approach. (especially important for triple trade-off) --------------------------------------------------------------------------- don’t call the regions “one-shot” in the trade-off chapter --------------------------------------------------------------------------- In appendix B, in the proof of monotonicity of quantum relative entropy, page 636, equation B.37, it should be A \rho^{-1/2} \rho \rho^{-1/2} instead of (A \rho{-1/2} \tensor I) \rho (\rho^{-1/2} \tensor I) from Ankit Garg and Young Kun Ko --------------------------------------------------------------------------- from Todd Brun I assigned exercise 8.3.1 from your book as a homework problem. But in solving it myself, it seems to me that the two principles you give are not quite enough, by themselves, to prove the converse theorem. You give the two principles: (1) Public communication cannot (by itself) produce private communication or secret key; (2) Secret key bits cannot (by themselves) produce any communication. But I think these are not quite sufficient to prove the converse theorem. Like your proof of the quantum unit resource region, I broke it down into 8 octants. The two principles above are sufficient for 7 of them, but not for the octant (+,-,+), where private channels are used to produce both public channels and secret key bits. To prove optimality in this octant, one needs the additional assumption that the total number of secret key bits and channel uses produced cannot be greater than the number of private channel uses. It doesn’t seem that one can deduce this assumption from the two principles above. (This plays the role of the Holevo bound in the quantum theorem.) Answer: we take a model where private communication can generate public communication. (Alice and Bob can discard information into local trash bins, to which Eve has access.)