ation is used to recapitulate the basic properties of linear operators and their action on quantum states (here, qubits), which leads to a simple formalization of the concept of quantum measurement. Quantum measurements In Chapter 16, we defined the qubit as a 2D vector noted 1q). In the orthonormal basis of pure states V={|0),|1)}, the qubit has complex coordinates a, B, SO that = As we have seen, higher-dimension qubits can be defined from the extended basis Vn = {|x)}, where /x) is the nth tensor product of the pure state |8i) = |0) or |1), namely: x)=81)082)..8n) (17.1) where the two equivalent notations in the right-hand side can be used for lightening purposes. In this extended space, the qubit |q) can, thus, be expanded into |q)== (17.2) where X represents the complex coordinate of |q) with respect to the n-qubit basis element /x). Regardless of any basis dimension n, the qubit or vector |q) is also referred to as a "ket." Actually, the ket represents the Dirac notation for a quantum state in the space defined by V = {|0), |1)}, which is referred to as Hilbert space. Here, we do not need to elaborate further on the notions of quantum state (as related to the physical world) and of Hilbert space (as defining the continuum of such quantum states). As it turns out, the concept of qubit as a vector, rather than as a quantum state, even as representing an oversimplification, is accurate and wholly sufficient to grasp Dirac formalism. Inner product In a 2D vector space, the scalar product of two vectors x = (x1, x2) and ly=(y1,y2) is defined as the real numberx.y=x1y1+x2y2. In particular, the self-product X . X = x2+x2 = |x|2 corresponds to the vector's length or modulus. In the qubit space V (meaning "defined by any orthogonal basis such as V"), we can introduce the concept of the inner product of two qubits, |q) = a|0) + B|1) and + B'|1) according (17.3) where , B are the complex conjugates of the coordinates a, B, respectively. In particular, we have |q> lq) = which as a real number, represents the modulus of