other dimension the 1 information component. We shall now formalize the 2D vector representation of the qubit. Given a 2D vector space with basis , , one can then define the qubit state, q, under the linear combination (16.1) where complex numbers, which represent the vector's coordinates in the 2D space. Assuming that (, v) is an orthonormal basis, and using the column To recall essential basics, complex numbers Z are defined as ib, where (a, b) are real numbers and i the "pure imaginary" basis having the property i2 = -1. The real numbers a and b are called the real and the imaginary parts of z, respectively. The length or modulus of Z is defined as b2. Complex numbers can be equivalently written as Z = |z|eio where O = is the argument of z, and where ei = cos O + i sin O is the imaginary-exponential function. The complex-conjugate of z, indifferently called z* or , is defined as z=a = |z|e-ia A key property is z*z = zz = |z|2 = a2 + b2. To recall from vector-space algebra, an orthonormal basis is any set of n vectors 1, ...., un, which satisfy for all i, j = 1, the scalar-product condition = Sij, where Sij is the Kronecker symbol (Sij = 1 for and Sij = 0 for i # j). For i = j, the condition gives 2 = | 1, or Quantum bits and quantum gates representation for vectors, we have (16.2) As detailed further on in this chapter, in QIT a vector or qubit state q is noted instead as /q), where the arrow |) is called a ket (as in the second half of the word bracket). Consistently, and by analogy with classical bits, the two elementary states u and v are noted |0) and |1), respectively. The qubit can, thus, be defined as the vector linear combination: (16.3) This shows that a qubit can be conceived as a superposition of the two elementary orthogonal states |0) and |1). In QIT jargon, the complete set of states |0), |1) is referred to as the computational basis. The two complex coordinates a, B, are also called the qubit amplitudes. It is seen from the above that, contrary to the classical information bits, the qubit can be