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27 bits, which is very close to the optimal value. This shows the power of data compression codes. The enormous practical importance of data compression in ﬁelds such as telecommunication is self-evident. Data compression allows us to increase the transmission rate or the storage capacity of a computer. To achieve such results, we simply exploit the redundancies that any message contains: for instance, the letters of an (English) text are not equiprobable but appear with diﬀerent frequencies. 9 The von Neumann entropy The quantum analogue of the Shannon entropy is the von Neumann entropy.

48) where Tr2 denotes the partial trace over subsystem 2: Tr2 ρ ≡ 2 α|ρ|α 2 . 49) α We note that it is also possible to deﬁne similarly a reduced density matrix for subsystem 2: ρ2 ≡ Tr1 ρ ≡ 1 i|ρ|i 1 . 50) i The matrix elements of ρ1 in the {|i 1 } basis are given by (ρ1 )ij = 1 i|ρ1 |j 1 = ρiα;jα . 51) α After insertion of this equality into Eq. 47), we obtain A1 = 1 i,j i|ρ1 |j 11 j|A1 |i 1 = 1 i i|ρ1 A1 |i 1 = Tr(ρ1 A1 ). 52) January 25, 2007 11:17 WSPC/Book Trim Size for 9in x 6in qcbook2 Quantum Information Theory 269 Therefore, it is possible to compute the expectation value of an operator acting only on subsystem 1 as if the system were isolated and described by the reduced density matrix ρ1 .

Let us assume that Alice sends a letter (a quantum state) to Bob and that Bob only knows that the letter has been taken from the ensemble {ρi , pi }. Thus, he describes this quantum system by means of the density matrix k p i ρi . 159) i=1 Therefore, k S(ρ) = − Tr ρ log ρ = − λi log λi = H(λ1 , . . 160) i=1 where the λi are the eigenvalues of the density matrix ρ and H(λ1 , . . , λk ) is the Shannon entropy associated with the ensemble {λi }. January 25, 2007 11:17 WSPC/Book Trim Size for 9in x 6in qcbook2 Quantum Information Theory 299 The von Neumann entropy satisﬁes the following properties: 1.