ann entropy. 21.5 (M): Given the density matrices PA, PB and identity matrices IA, I B of two subsys- tems A, B with dimensions n X n and pxp, respectively, and the density matrix PAB of the joint system with dimension n X p, show the following relations: (1) PA D P B = (PA (3) = using for (3) the property valid for any observable Ux in a composite system XY tr[pxy(UxQIy)]=tr(pxUx) with a similar relation applying to any observable Uy. (Clue: start from the operator tensor-product definition in Eq. (16.54), and assume, for simplicity and without loss of generality, that n = 2, p = 3) 21.6 (B): Show that the 2-qubit state Quantum information theory (a) Is not a pure state; (b) Cannot be decomposed as a tensor product (VA) x 1/BB of some subsystem states (VA),IVB). 21.7 (M): Determine the von Neumann entropy of the composite system having density operator p(000 111011 Quantum data compression This chapter describes the principle of compression in quantum communication chan- nels. The underlying concept is that it is possible to convey "faithfully" a quantum message with a large number of qubits, while transmitting a compressed version of this message with a reduced number of qubits through the channel. Beyond the mere notion of fidelity, which characterizes the quality of quantum message transmission, the description brings the new concept of typicality in the space defined by all possi- ble "quantum codewords." The theorem of Schumacher's quantum compression states that for a qubit source with von Neumann entropy S, the message compression factor R has S + E for the lower bound, where E is any nonnegative parameter that can be made arbitrarily small for sufficiently long messages (hence, R 2 S is the best possible compression factor). An original graphical and numerical illustration of the effect of Schumacher's quantum compression and the evolution of the typical quantum-codeword subspace with increasing message length is provided. Quantum data compression and fidelity In this chapter, we have reached the stage w