eloped in this section has shown that given a message of length n, and any E > 0 such that E=S-R (with R <1), there exists a typical subspace S2, spanned by a subset of corresponding E-typical codewords, where message compression is possible with high fidelity F. Consistently with the formal description, the example also showed that (a) for messages of increasing length, the fidelity F can be made arbitrarily high, and (b) the dimension of the typical subspace dim(S) asymptotically converges towards en(Sts). As the parameter E can be chosen to be arbitrarily small, the limiting value for the subspace dimension is ens, which corresponds to the best achievable compression factor R 2 S. Exercises 22.1 (T): Given two tensor states |abc), |def), where |a), ,b),c),d),e),\f)are qubits, provide a general formula for the space-overlap factor |(abc| def)|2 and apply this formula to all the nine possible cases: where a is a parameter. (Clue: refer to the qubit tensor-product definition in Eq. (16.55) as applied to the present case.) 22.2 (B): What is the best achievable compression factor for a quantum message of arbitrary long length, whose qubits are generated by the density operator p=1(62) 22.3 (M): A qubit source is characterized by the density operator p=1(62) Assuming a quantum message length of n=5 qubits, determine the E-typical subspace and corresponding quantum codewords for which E=0.3. Quantum channel noise and channel capacity This chapter introduces the notion of noisy quantum channels, and the different types of "quantum noise" that affect qubit messages passed through such channels. The main types of noisy channel reviewed here are the depolarizing, bit-flip, phase-flip, and bit-phase-flip channels. Then the quantum channel capacity X is defined through the Holevo-Schumacher-Westmoreland (HSW) theorem. Such a theorem can conceptu- ally be viewed as the elegant quantum counterpart of Shannon's (noisy) channel coding theorem, which was described in Chapter 13. Here, I shall not venture into the complex proof 