rs 12a), 12b) are (22.5) and in the eigenstate basis {12a), 12b)}, the density operator p has the diagonal form: (22.6) Finally, the VN entropy associated with any message symbol, or per-qubit entropy is S(p) = - log Ab = f [cos2(r/8)] =0.6008. 22.1 Quantum data compression and fidelity Let us now calculate the overlap between the eigenstates 12a), 12b) and the quantum symbols |a), |b). Using Eqs. (22.3) and (22.5), we obtain: (22.7) The overlap between the subspaces defined or "spanned" by either 12a) or 12b) and the full space defined or spanned by |a), |by is, thus, observed to be significantly greater in the first case (i.e., 85% VS. 15%, or da VS. Ab). We may now introduce the concept of fidelity, defined by F (v/p/) (22.8) where 1/5/ is an arbitrary measurement state. Here, since Pa = pb = 1/2, the fidelity reduces to F = From Eq. (22.7), it is clear that the fidelity is a maximum for the measurement choice /V/> = 12a), regardless of whether |a), |b) was sent by the originator, which gives F = 0.853. Hence, {14}} = {12a)} corresponds to a one-dimensional "likely subspace" with which any message symbol is most strongly overlapping. Let us now extend the above notion of "likely subspace" to the case of a message of three qubits long. The eight equally-likely message outcomes |M) are |M) =|aaa), |aab), |aba), |abb), |baa), |bab), |bba), |bbb). (22.9) Let 1/>/ = 12,2j2k) be an arbitrary measurement state with Ai, Aj, Ak = Na, Ab. Using the property = |(2,2,2k/xyz)| together with the result in Eq. (22.7), we obtain the space overlaps applicable to any message |M>: (22.10) (22.11) (22.12) (22.13) It is seen that the subspace defined by So = {12ahaha), 12ahahb), 12ahbha), 12bhahal) is the most likely, while its orthogonal subspace complement, S2 = {12a2bdb), 12bhadb), 12bhbda), 1232b2b)}, is the least likely. Such a statement means that mak- ing any measurement of the message using a state 1/>/ from the eigenstate basis A = {12a), 12b)}03 is more likely to project the state in the subspace S than in the subspace s