By Marinescu D., Marinescu G.

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Pn ). Since V is positive then pi ≥ 0, hence they have unambiguously defined positive square roots. Define: Q= √ √ √ √ V to be represented by diag ( p1 , p2 , . . , pn ) . The modulus of any operator V is defined as √ | V |= V† V ≥ 0, √ it is always positive. One can also consider V† V ≥ 0 but, unless V is normal, that is not the same as (VV† ) ≥ 0. The operator Q is invertible if: | Q || Q |−1 = I If an operator Q is invertible then it can be written as a product of a positive-definite operator, V, and a unitary operator, U: Q = VU, with V =| Q |= QQ† and U = | Q |−1 Q.

J , . . | k . . | (n−1) } ∈ Hn . A Hermitian operator A = A† applied to state | ψ = k αk | k , with αk = k | ψ , produces a new state | ϕ : |ϕ =A|ψ or ϕ |= ψ | A† = ψ | A. It follows that k k j|A|k k|ψ . j | A | k αk = αk | k = j|ϕ = j|A|ψ = j|A k The expressions j | ϕ give the amount that each basis state | j contributes to the state | ϕ ; this amount is given in terms of a linear superposition of k | ψ , the probability amplitudes in each basis state of the original state, | ψ . The numbers j | A | k tell how much of each amplitude k | ψ goes to the sum for each | j ; these coefficients are the components Ajk of the matrix A = [Ajk ] associated with the linear operator A Ajk = j | A | k .

A1 , A2 ] = A1 A2 − A2 A1 = 0 P5. The eigenvectors of a projector operator, Pψ =| ψ ψ |, where ψ | ψ = 1, are either perpendicular to or collinear to the vector | ψ and their eigenvalues are 0 and 1, respectively. Proof: Assume | ϕ is an eigenvector of Pψ corresponding to eigenvalue λ: Pψ | ϕ = λ | ϕ or, |ψ ψ|ϕ =λ|ϕ . The inner product ψ | ϕ is a number γ γ|ψ =λ|ϕ This implies that λ = 0 if γ = 0, when | ψ and | ϕ are perpendicular λ = 1 if γ = 1, when | ψ and | ϕ are parallel and normalized. or, An operator A is unitary if An A†n = A†n An = In .