and to the measurement value Am. In the case where the measurement fails, we may write |q') = for the input state annihilation. 18.2 Measuring n-qubits q')=12m) Figure 18.2 Quantum-circuit representation of measurement apparatus to measure qubits with operator Mm, yielding measurement value Am. Measuring n-qubits In this section, I describe quantum measurements of higher-order or n-qubits, which have fundamental implications in both QIT and quantum computing (QC). This is a turning point, where we shall begin to grasp the strange beauty (and the oddity!) of the quantum properties of qubits. As we have seen in Chapter 17, any base-state measurement Eim on an n-state /V/ from VOn results in the collapsing effect Eim/V Y = (18.8) In the case of an n-qubit 1q) = /X1,X2,...,Xn), with xi=0,1,we have: (18.9) With this measurement tool in hand, we can now perform measurements on any qubit superpositions or mixtures in the subspace VO2 (for instance), such as: (18.10) with u, V being complex amplitudes satisfying = 1, SO that |r) has a modulus of one. For each qubit i in the subspace V, there are two possible measurement operators, namely Ei and Ei2. To illustrate the action of these operators, consider the following example (using the compact notation /xi,xj) = |xixj)): (18.11) If we want to measure a classical bit 0 in the first qubit position (i = 1), we use E10 and obtain (18.12) =-(100) + 101)) Qubit measurements, coding, and teleportation Thus, the probability of measuring 0 in the first qubit is: = 3/is (18.13) It is now a straightforward exercise to evaluate mentally the other probabilities: (18.14) Next, we shall look at a peculiar family of some 2-qubits, which are of utmost interest for qubit measurements and for the "quantum manipulation" of information. In Chapter 15, we identified four state superpositions, referred to as Einstein-Podolsky-Roser (EPR) or Bell states (or pairs), and defined: 1800)= 1001)= (18.15) (1910)= These EPR-Bell states may seem innocuous at first glance, but they actually reserve a cer