e" angle 0 and the "longitude" angle Q. The north and south poles correspond to the pure qubit states |0) and |1), respectively. namely 00, 01, 10, and 11, we can call the corresponding elementary two-qubit (or 2-qubit) states |00), |01), |10), and |11), and, similarly to Eq. (16.3), we obtain the most general definition for any 2-qubit state: (16.8) In this definition, the complex amplitudes Aij must satisfy the property (16.9) From the previous description concerning 1-qubits, it is clear that each of the terms |aij|2 in the above summation represents the probability of finding the 2-qubit in the state |ij), or, equivalently, of measuring the classical bit pair ij. We shall note here that the notation |ij) for a pair of qubits |i>, |j), or 2-qubit |i> |j) actually anticipates the concept of tensor state, which is explained in the last section of this chapter. We can even simplify the picture of a 2-qubit by looking at it as a "four-sided die," such as a tetrahedron, hidden in the box. Such a die has four classically measurable outcomes (00, 01, 10, and 11), but in the quantum world there exist an infinity of superpositions of 2-qubit pairs, as I shall now establish. Indeed, the 2-qubit state defined in Eq. (16.8) can also be rewritten in the form (16.10) Quantum bits and quantum gates with = (16.11) 191)1 = The development in Eq. (16.11) illustrates that any 2-qubit state can be conceived as the superposition of two 2-qubits, which we call here |90) (a 2-qubit with the first bit 0), and |q1) (a 2-qubit with the first bit being 1), respectively. According to all expectation, should we have fully assimilated the earlier story of the 1-qubit measurement, if we attempt to measure only the first bit of any 2-qubit state, the latter must collapse into 190) or |91), according to the outcome of the measurement being the 0 or 1 classical bits, with associated probabilities 18012 and 1B12, respectively. The resulting states |90) or |q1), which represent superposition of 2-qubit states (with either 0 or 1 as the first bit