nce, which we might index |m) =||0). We can now work with the joint states 14>10) = 1/4, 0) from the extended space V O M of joint states /v/|m) 14, m). Next, define the tensor operator U whose action on 14, 0) results in: (17.66) We then have for any two states 1/s/, 14'): (17.67) The above result shows that the transformation U preserves the inner product of all states 14, 0) as a unitary operator for the subspace V x {|0}}. Actually, we can make U a unitary operator for the entire space V X M, which applies to any state 14, m),6 Indeed, define U' as the sum of two projectors on the two orthogonal spaces V x {|0}} and V x {|m 0)} for any state 1/s/ E V, as follows: U' = UV, 0) (v), 0| + 14, m> (V), m. It is clear that since U is unitary, U' is unitary. We also have U'ly, 0) = U'ly, 0) and, for k # 0, U'V k) = UV, 0) (V, k| + m/V, k) = USOk + 14, m)8mk 14, k). 17.3 Quantum measurements on joint states according to the alternative definition: Uk/V, (17.68) As an exercise, it is possible to check that (V', KUUV,K = which shows that the new definition of U (i.e., Uk) in Eq. (17.68) is indeed unitary over V x M. Having found the above unitary operator U that conserves the inner product of all joint states 14, in V X M, we can now introduce the joint-state measurement (JSM) operator Em Em=U+PmU, (17.69) Pm=IyQm)m, (17.70) where Iv is the identity operator on V and |m) (m/ the projector over |m) E M.7 Next, we shall calculate the bracket function p(m) = (V), kEm/, k), which, using Eq. (17.66) with |0) = |k), is: p(m)=(y,kU+PmUv,k = (2mm) (17.71) We immediately recognize from the result that p(m) is the measurement probability associated with the operator Mm on the state (v) E V, as defined in Eq. (17.46). After Eq. (17.50), and writing the JSM operator Em = (U+P +)PmU, we obtain the joint post-measurement state as the result of the transformation: (17.72) 7 The full definition of the JSM operator actually being Em=(PmU)+PmU=U+PPmU=U+PmU, where we used the projector and Hermitian properties of Pm. Quantum measurements 