o the corresponding probabilities being = = = 0.942 = 1 - 8 and p(s) = + 0.0031 = 0.058 = 8, respec- tively. We may then define E as the projector on the likely four-dimensional subspace Quantum data compression S2, according to = Idahaha) (2ahahbl (22.14) + (hadbhal + 12bhaha) (Abhadal, or as expressed in the eigenstate basis A: (22.15) In the same eigenstate basis, the message's density operator = pon takes the diagonal form: (22.16) Based on the definitions in Eqs. (22.15) and (22.16), we clearly have (22.17) We shall now refer to So as the typical subspace, which is reminiscent of the typical set of classical information theory described in Chapter 13. Likewise, any message of the type 1/typ) = Idahaha), 12ahadb), 12ahbha), 12bhaha) represents the quantum analog of the typical sequences described in that chapter. We shall now refer to them as typical states. At this stage, it is only an intuitive notion that as the message length n is increased, the probability associated with the typical subspace So asymptotically increases and that associated with S2 asymptotically decreases, meaning that the uncertainty 8 may become arbitrarily small, and from Eq. (22.17), we may be able to achieve the condition tr(pME) = p(S2) 2 1. These observations will be discussed again. We are now going to exploit the possibility of partitioning the space into typical and atypical subspaces in order to achieve quantum compression in the message. Assume, for instance, that we want to reduce the 3-qubit message to a 2-qubit, compressed, message. The following describes how the originator must proceed. First, the originator should apply a unitary transformation U, such that the four typical states (indifferently called 22.1 Quantum data compression and fidelity 1/typ)) are transformed into states of the form /xy0) = |xy) 0010) and the other four (atypical) states are transformed into states of the form /xy1) = |xy) (||). This is just a specific way of re-encoding the original message 1/s/. Call the result of this re-encoding. With the k