 both 0 and 1, or, more accurately, a linear superposition of states 0 and 1. Unlike classical bits, which correspond to only two possible points on a line (x = 0 or X = 1), qubits can correspond to an infinity of points in a 2D space. We shall see later how such points can be represented visually. As we have seen from Chapter 15, the classical or von Neumann computer retrieves information by reading the bit contents in its inner memory. Any such reading operation is, thus, analogous to a physical measurement of the memory contents. Likewise, the information carried by qubits can be measured. The key difference between the classical bit and the qubit is that, as we have seen, the latter is a superposition of the states |0) and |1) in a 2D space. It is not possible to physically measure the two complex amplitudes a, B, which define the qubit state. This amounts to saying that it is not possible to "measure" a qubit state. Rather, any physical measurement of a qubit yields a classical bit, which must be either 0 or 1. Such a measurement is not deterministic, but associated with certain probabilities. The probability that the qubit measure yields the bit 0 is proportional to /a/2, and that of yielding the bit 1 is proportional to 1812. Since there exist only two measurement possibilities, we must have (16.4) Since is the length or magnitude of the qubit vector, this result expresses the property that the qubit is a unitary vector. Thus a qubit with a = 0 or B = 0 exists in the pure state |0) or |1). The corresponding measurements yield the exact result 0 with probability |a2 or 1, with probability IB21=1, respectively. Another case of interest is where the measurement outcome has equal chances of being 0 or 1. In general, there is no way to tell the measurement outcome, even if the qubit state is known, because the measurement is probabilistic. Given the property in Eq. (16.4) concerning the qubit amplitudes, the most general definition of the qubit is, 16.1 Quantum bits therefore, (16.5) where a', B' are any comple