Exercise Let x =

Vector states of a quantum system with n physical states are represented by unique vectors in Cn, the set of n × 1 column vectors1 For example, a photon has vertical or horizontal polarization upon measurement. So, the quantum state of a photon is represented as a unit vector in C2. Furthermore, the theory is better explained using density matrices (positive definite matrices with trace one) to represent quantum states. Thus, we present some basic notation and results on vector and matrices in this chapter. Here are the notation of number systems: N ⊆ Z ⊆ Q ⊆ R ⊆ C. 1 Basics of Vectors The set Cn of n × 1 complex vectors form a vector space. Here is the formal definition. Definition A complex vector space V has two operations: the addition of two vectors x and y in V yielding x+ y, and the scalar multiplication of a complex scalar µ and a vector x ∈ V yielding µx ∈ V such that 1) (x+ y) + z = x+ (y + z) and x+ y = y + x for all x, y, z ∈ V, 2) there is a zero vector 0 ∈ V such that x+ 0 = x for any x ∈ V, 3) for every x ∈ V, there is y such that x+ y = 0, 4) 1x = x for any x ∈ V, 5) for any µ1, µ2 ∈ C and x ∈ V, we have (µ1µ2)x = µ1(µ2x), (µ1 + µ2)x = µ1x+ µ2x, 6) for any µ ∈ C and x, y ∈ V, we have µ(x+ y) = µx+ µy