This final result shows that after the measurement, the state 1/s/ of space V is transformed (17.73) while the ancilla state |k) is transformed into |m). In the rest of this book, we will not have to be concerned about the ancilla space M. Such a space is just used here as a mathematical tool to allow the construction of a quantum measurement operator for joint states. Bearing this in mind, we can now expand the conclusion to measurements of joint states of any dimension from VONOM by means of the n-ancilla space, M i.e., kn) = (17.74) We, thus, know that it is possible to define a measurement operator Eim that can act on the subspace pair /Vi), /ki) and collapse it into the pair Mm/Vi), |m), such that (17.75) We may just forget now about the ancilla bits and consider simply that Eimy1,2,...i..,n (17.76) To provide an illustration, assume that Mm = |m) (m). Letting /Vi) = we obtain Eim/11/1 (17.77) m,Vit1,...,Yn) as obtained within an unobservable phase Finally, consider for simplicity two joint states in 2-state space V X V, namely 10) = 01, 02) and and their superposition 1x) with complex coefficients a, B (satisfying (17.78) Owing to the linearity and distributiveness of the measurement operators, we have E1|X) = = E101,02) + BE1111,122)= a|1,02> + B|1,42> E2|X) = E2(a/0) + B/V) = , O2) + BE2141,42 = a01, + BV1, 2) (17.79) 17.4 Exercises EE2|x) = E2E1|X) = = a|1,2)+ 811,2) = + B) 1, 2), (17.80) which actually illustrates the great simplicity of joint-state measurements and their properties. These can now be applied to the case of single and multiple qubit systems, which are described in Chapter 18. Exercises 17.1 (B): Calculate the moduli and scalar products of the 2-qubits /q =i00)+01)-11), = - |q)=|00)+01)-i10) = - + |11). 17.2 (B): Show that |q) (q is the projector operator over the qubit lq). 17.3 (B): Determine the eigenstates and diagonal representations of the Pauli matrix Y and Hadamard matrix H. 17.4 (T): Show that the transition operator T defined by its matrix element Txy=(x/y) in different orthogon