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Basic Notation and Background Basic Notation and Background Chi-Kwong Li Department of Mathematics, The College of William and Mary, Williamsburg, Virginia, USA; Department of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi, P.R. of China. Chi-Kwong Li Basic Notation and Background In physics, we use the bra and ket vector notation (Dirac notation). Let x1 t . n |xi = (x1, . , xn) = . ∈ C . xn Then n∗ hx| = (¯x1,..., x¯n) ∈ C is the dual vector. Hilbert spaces The mathematical platform of quantum mechanics/computing is Hilbert space (complete inner product space) V . n We mainly focus on finite dimensional complex inner product space C , the set of n × 1 column vectors. n∗ n Let C be the dual vector space of C consisting of 1 × n row vectors Chi-Kwong Li Basic Notation and Background Hilbert spaces The mathematical platform of quantum mechanics/computing is Hilbert space (complete inner product space) V . n We mainly focus on finite dimensional complex inner product space C , the set of n × 1 column vectors. n∗ n Let C be the dual vector space of C consisting of 1 × n row vectors In physics, we use the bra and ket vector notation (Dirac notation). Let x1 t . n |xi = (x1, . , xn) = . ∈ C . xn Then n∗ hx| = (¯x1,..., x¯n) ∈ C is the dual vector. Chi-Kwong Li Basic Notation and Background The norm(length) of |xi is ( n )1/2 1/2 t 1/2 X kxk = hx|xi = {(¯x1,..., x¯n)(x1, . , xn) } = |xj | . j=1 (a) For |xi ∈ Cn, we can construct its transpose |xit, the conjugate |xi, the conjugate transpose |xi† (instead of |xi∗). n Pn ∗ (b) The inner product of |xi, |yi ∈ C is hx|yi = j=1 xj yj. The vectors are orthogonal if their inner product is zero. (c) Given a set of vectors S in Cn, we can determine whether it is a linearly independent set, a generating set, an orthonormal set, a basis, or an orthonormal basis. Chi-Kwong Li Basic Notation and Background Matrices Linear maps (transformations/functions) on finite dimensional vector spaces n n can be identified with matrices, namely, A : C → C so that |xi 7→ A|xi. Operations and Properties Let Mn be the set (vector space/algebra) of n × n matrices. (a) One can perform A + B, AB and µA for A, B ∈ Mn and µ ∈ C. (b) One can compute the eigenvalues and eigenvectors of A ∈ Mn. Chi-Kwong Li Basic Notation and Background n Let {|e1i,..., |eni} be the standard basis for C . Then X Aij = hei|A|ej i and A = Aij |eiihej |. i,j Pn The trace of A is defined by Tr(A) = j=1 Ajj . Exercise 1) If A is m × n and B is n × m, then Tr(AB) = Tr(BA). n 2) If R is an n × n matrix, and |ψi ∈ C , then hψ|R|ψi = Tr(R|ψihψ|). Chi-Kwong Li Basic Notation and Background Gram-Schmidt process and orthogonal projectors n If S is linearly independent set in C , one can apply the Gram-Schmidt process to S to get an orthonormal set. If |eki is a unit vector, then the projection of a vector |vi in the direction of |eki is |vi − Pk|vi, where P = |ekihek| is the projection operator. The vector |vi − Pk|vi is orthogonal to |eki. If {|e1i,..., |eni} is an orthonormal basis and Pk = |ekihek| for k = 1, . , n, then 2 Pn (i) Pk = Pk, (ii) Pj Pk = 0 for j 6= k, (iii) k=1 Pk = In. Chi-Kwong Li Basic Notation and Background More notation, definitions and examples t Let A ∈ Mn. One can compute its transpose A , the conjugate A and the conjugate transpose A†. The matrix is Hermitian if A = A†; it is skew-Hermitian if A = −A†; it is normal if AA† = A†A; it is unitary if A† = A−1. If A is real and At = A−1, the A is a real orthogonal matrix. In quantum information science, the following Pauli matrices are useful: 0 1 0 −i 1 0 σ = I , σ = , σ = , σ = . 0 2 x 1 0 y i 0 z 0 −1 Exercises 1) The Pauli matrices are trace zero Hermitian unitary matrices. 2) Let {i, j, k} = {x, y, z}, then σiσj = iγij σk = −σj σi, and [σi, σj ] = 2iγij , where γij = 1 for (i, j) = (x, y), (y, z), (z, x). Chi-Kwong Li Basic Notation and Background If AA† = A†A, then (U †AU)(U †A†U) = (U †A†U)(U †AU). Comparing the (1, 1), (2, 2),..., (n, n) entries on both sides, we see that U †AU is in diagonal form. Some useful facts Theorem (Schur Triangularization Lemma) Every matrix in Mn is unitarily similar to a matrix in upper or lower triangular form. Sketch of proof. Solve det(λI − A) = 0 to get A|xi = λ|xi with hx|xi = 1. † λ A12 Construct U1 with |xi as the first column. Then U AU = . 0 A22 † By induction, U2 A22U2 is in upper triangular form. 1 0 † Let U = U1 . Then U AU is in triangular form. 0 U2 Chi-Kwong Li Basic Notation and Background Some useful facts Theorem (Schur Triangularization Lemma) Every matrix in Mn is unitarily similar to a matrix in upper or lower triangular form. Sketch of proof. Solve det(λI − A) = 0 to get A|xi = λ|xi with hx|xi = 1. † λ A12 Construct U1 with |xi as the first column. Then U AU = . 0 A22 † By induction, U2 A22U2 is in upper triangular form. 1 0 † Let U = U1 . Then U AU is in triangular form. 0 U2 If AA† = A†A, then (U †AU)(U †A†U) = (U †A†U)(U †AU). Comparing the (1, 1), (2, 2),..., (n, n) entries on both sides, we see that U †AU is in diagonal form. Chi-Kwong Li Basic Notation and Background Theorem (Spectral Theorem) If A ∈ Mn is normal, then there is a unitary U † such that UAU = diag (λ1, . , λn); hence if U has columns |u1i,..., |uni then A = λ1|u1ihu1| + ··· + λn|unihun|. (a) For any positive integer m, m m m A = λ1 |u1ihu1| + ··· + λn |unihun|. The formula holds for negative integers m as well if A is invertible. (b) If f(z) is an analytic function, then n X f(A) = f(λj )|uj ihuj |. j=1 z Pn λj (c) In particular, if f(z) = e , then f(A) = j=1 e |uj ihuj |. Chi-Kwong Li Basic Notation and Background Theorem (Singular Value Decomposition) For every m × n matrix A, there † are unitary U ∈ Mm and V ∈ Mn so that U AV = D such that the (j, j) 2 2 entries of D is sj for 1 ≤ j ≤ min{m, n}, where s1 ≥ s2 ≥ · · · are the eigenvalues of A†A. Sketch of Proof. Construct unitary V so that V †A†AV is in diagonal form with diagonal 2 2 entries s1, . , sn. Note that the columns of AV are orthogonal vectors with lengths s1, . , sn. Let the first k = min{m, n} columns of AV be s1|u1i, . , sk|uki, and let U ∈ Mm be unitary so that the first k columns are |u1i,..., |uki. Then U †AV has the asserted form. Chi-Kwong Li Basic Notation and Background Definition Given two real vectors x = (x1, . , xn) and y = (y1, . , yn), we Pn Pn say that x is majorized by y if j=1 xj = j=1 yj and the sum of the k largest entries of x is not larger than that of y for k = 1, . , n − 1. Theorem The vector of diagonal entries (d1, . , dn) of a Hermitian matrix in Mn is majorized by the vector of its eigenvalues (λ1, . , λn). Theorem Every unitary matrix U ∈ Mn is a product of no more than n(n − 1)/2 tridiagonal unitary matrices, each of them differs from In by a 2 × 2 principal submatrix. Sketch of proof on white board! Remark Using the Gray code labeling of 2m × 2m, all the tridiagonal unitary matrices involve a change of two basic vectors with binary labels differ in one position. Chi-Kwong Li Basic Notation and Background Theorem Every unitary matrix U ∈ Mn is a product of no more than n(n − 1)/2 tridiagonal unitary matrices, each of them differs from In by a 2 × 2 principal submatrix. Sketch of proof on white board! Remark Using the Gray code labeling of 2m × 2m, all the tridiagonal unitary matrices involve a change of two basic vectors with binary labels differ in one position. Definition Given two real vectors x = (x1, . , xn) and y = (y1, . , yn), we Pn Pn say that x is majorized by y if j=1 xj = j=1 yj and the sum of the k largest entries of x is not larger than that of y for k = 1, . , n − 1. Theorem The vector of diagonal entries (d1, . , dn) of a Hermitian matrix in Mn is majorized by the vector of its eigenvalues (λ1, . , λn). Chi-Kwong Li Basic Notation and Background Tensor products Let A = (aij ) and B be two rectangular matrices or vectors. Then their tensor product (Kronecker product) is the matrix A ⊗ B = (aij B). The following equalities hold: (A⊗B)(C⊗D) = (AC)⊗(BD),A⊗(B+C) = A⊗B+A⊗C, (A⊗B)† = A†⊗B†. Note that A, B, C can be |ψ1i, |ψ2i, |ψ3i vectors. If A ∈ Mm and B ∈ Mn are invertible, then (A ⊗ B)−1 = A−1 ⊗ B−1. P Every T on Mmn can be written as T = j=1 cj Aj ⊗ Bj with cj ∈ C so that m n for any vectors |ui ∈ C , |vi ∈ C , ! X X T |ui|vi = cj Aj ⊗ Bj |ui ⊗ |vi = cj Aj |ui ⊗ Bj |vi.
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