tual physical measurements, for instance, in quantum cryptography (see Chapter 25). Quantum measurements on joint states Chapter 16, we introduced the concept of tensor states, which correspond to systems being described by more than one quantum state, e.g., /V1) and /V2). Here, we shall call such systems composite systems. As we have seen, the corresponding state can be written in the tensor form 1/4/> = 1/1/1 x 1422), or 1/4/ = 1/1/1/2/2), or, for short 1/>/ = 141,422, and even, sometimes, 1/4/ = 1/14/22, e.g., the notations 1/>/ = i) X |j) = |i>|j) = j) = |ij) are all equivalent. These tensor states are also referred to as joint states. Regardless of the notation conciseness for joint states, however, it is important to Quantum measurements keep in mind the underlying tensor product concept, which is also associated with that of tensor operators. Thus, the tensor operator U = A O B acts on the joint state /V/ = according to the rules U/V) = AV1) x B/V2) and (V|U+ : = (, hence, the imperative need of keeping the single quantum state notation in the right order sequence inside the ket 1.), and choose the appropriate notation for the tensor products when operators are being used. We are now interested in defining a generic measurement operator for joint states. The following may look complicated at first but, as we shall see, the result makes complete sense in view of what we have learnt from the previous section. Assume first a single-state space V, as defined by the orthonormal basis ={|xi}} (for simplicity we identify here the "space" and one corresponding "basis" with the same letter V). Assume next that {M}} represents a valid set of measurement operators for V (meaning that M+Mm are positive and verify the completeness relation). For the set {Mm} there is a space M of orthonormal states {|m)}, corresponding to each of the different possibilities of post-measurement states resulting from the action of Mm. I shall refer to M as an ancilla space. Let |m) be a fixed state of M, to be used as an arbitrary refere