m Eq. (21.39), we find the conditional entropy: S(PA/PB)=S(PAB)-S(PB = 1.251 - 0.650 (21.65) = 0.601 with S(PB/PA) = S(PA/PB). The knowledge of the information in one of the sub- systems, thus, decreases the information (or uncertainty) in the other system by the Quantum information theory amount S(PA) - S(PB/PA) = 0.650 - 0.601 = 0.049 = S(PA;PB), which corresponds to the mutual information in the composite system. Example 21.3: Composite system in pure state As a final example, consider the joint state IMAD) (21.66) This state, which is also noted 1Boo), is one of the four Bell-EPR states (see Eqs. (16.2) and (18.15)). As we have learnt, the Bell-EPR states cannot be generated by the tensor product of two qubits, and they also form an orthonormal basis in the 2-qubit space (referred to as {|Boo>, |Bo1), 1B10), |B11)}). In this basis, (VAB) = 1Boo) is, thus, a pure state and its density operator is, therefore, equal to PAB = (VAB) (VAB). Substitution of the definition in Eq. (21.66) and rearrangement of the different terms into tensor products yields: (11|) (21.67) 0(00+00)(11)+11)(001+11)(11))AB Next, we calculate the reduced density operators PA = trB(PAB) and = trA(PAB), while using the property with the property tr(|i) (jl) = (i|j) = Sij, to obtain: (21.68) (1|A) =1/1/1 = trA(PAB) tr(|0)(1|A) |0) (1|B + tr(|1) (0|A) + tr(|1) (1|A) X |1) (1|B) (21.69) (1|B) where IA, I B are the identity matrices of subsystems A, B. We observe that tr(p2) = tr(p2) = 1/2 < 1, which shows that PA, correspond to mixed states. For the sake of 21.2 Relative, joint, and conditional entropy, and mutual information curiosity, let us now calculate the tensor product PA x PB from Eqs. (21.68) and (21.69). We obtain (21.70) Comparison between Eq. (21.70) and Eq. (21.67) shows that, as expected, PAB # PA x . This provides an indication that the information in the subsystems A, B is correlated or, equivalently, that the subsystem states are entangled. Next, we calculate the quantum information, S(PA), S(PB) associated with the reduced den