Linear Algebra and Dirac Notation, Pt. 1

Linear Algebra and Dirac Notation, Pt. 1 PHYS 500 - Southern Illinois University February 1, 2017 PHYS 500 - Southern Illinois University Linear Algebra and Dirac Notation, Pt. 1 February 1, 2017 1 / 13

Motivation In this lecture we develop the basic tools to study the mathematical theory of finite-dimensional quantum mechanics. We will be primarily focused on how this theory can be applied to quantum information processing. PHYS 500 - Southern Illinois University Linear Algebra and Dirac Notation, Pt. 1 February 1, 2017 2 / 13

Hilbert Space Quantum mechanics describes physical systems in terms of Hilbert Spaces. When we refer to Hilbert space in this course, we mean some d-dimensional complex vector space H with a defined inner product. Definition (Inner Product) A function, : H H C is an inner product on vector space H if for all v, w i H and λ i C: 1 Linearity in the second argument: v, i λ iw i = i λ i v, w i ; 2 Conjugate-commutativity: v, w = w, v ; 3 Non-negativity: v, v 0 with equality iff v = 0. PHYS 500 - Southern Illinois University Linear Algebra and Dirac Notation, Pt. 1 February 1, 2017 3 / 13

Hilbert Space Quantum mechanics describes physical systems in terms of Hilbert Spaces. When we refer to Hilbert space in this course, we mean some d-dimensional complex vector space H with a defined inner product. Definition (Inner Product) A function, : H H C is an inner product on vector space H if for all v, w i H and λ i C: 1 Linearity in the second argument: v, i λ iw i = i λ i v, w i ; 2 Conjugate-commutativity: v, w = w, v ; 3 Non-negativity: v, v 0 with equality iff v = 0. Two vectors v, w H are orthogonal if v, w = 0. A vector v is normalized if v := v, v = 1. PHYS 500 - Southern Illinois University Linear Algebra and Dirac Notation, Pt. 1 February 1, 2017 3 / 13

Dirac Notation - Kets In 1939, P.M. Dirac published a paper that introduced a notation for linear algebra that is commonly referred to today as Dirac Notation. PHYS 500 - Southern Illinois University Linear Algebra and Dirac Notation, Pt. 1 February 1, 2017 4 / 13

Dirac Notation - Kets In 1939, P.M. Dirac published a paper that introduced a notation for linear algebra that is commonly referred to today as Dirac Notation. For a Hilbert space H, every vector ψ H is written as a ket ψ. If ψ, ϕ H, then their inner product is denoted by Notation ψ ϕ := ψ, ϕ. We will be primarily interested in noramlized vectors. We write ψ for an arbitrary ψ H and ψ (without the tilde ) if ψ is normalized and parallel to ψ : ψ = 1 ψ. ψ PHYS 500 - Southern Illinois University Linear Algebra and Dirac Notation, Pt. 1 February 1, 2017 4 / 13

Orthonormal Bases For a d-dimensional Hilbert space H, we can always construct an orthonormal basis { b 1, b 2, b d } with b i H and b i b j = δ ij. PHYS 500 - Southern Illinois University Linear Algebra and Dirac Notation, Pt. 1 February 1, 2017 5 / 13

Orthonormal Bases For a d-dimensional Hilbert space H, we can always construct an orthonormal basis { b 1, b 2, b d } with b i H and b i b j = δ ij. Every element ψ H can be written as a linear combination of basis vectors: d ψ = x i b i. i=1 Normalization requires 1 = ψ ψ = d i=1 x i 2. Every vector space has an infinite number of orthonormal bases. In practice, one identifies some particular basis that is easier to work with and calls it the computational basis. Computational basis vectors are denoted by 0, 1,, d 1. PHYS 500 - Southern Illinois University Linear Algebra and Dirac Notation, Pt. 1 February 1, 2017 5 / 13

Orthonormal Bases Proposition The inner product between any two vectors ψ, φ H is given by ψ φ = d i=1 y i x i, where φ = d i=1 x i b i, ψ = d i=1 y i b i, and { b 1, b 2,, b d } is any orthonormal basis for H. PHYS 500 - Southern Illinois University Linear Algebra and Dirac Notation, Pt. 1 February 1, 2017 6 / 13

Orthonormal Bases Proposition The inner product between any two vectors ψ, φ H is given by ψ φ = d i=1 y i x i, where φ = d i=1 x i b i, ψ = d i=1 y i b i, and { b 1, b 2,, b d } is any orthonormal basis for H. Proof PHYS 500 - Southern Illinois University Linear Algebra and Dirac Notation, Pt. 1 February 1, 2017 6 / 13

Dual Space and Bras Definition (Dual Space) For a Hilbert space H, its dual space H is the vector space consisting of all linear functions acting on H. PHYS 500 - Southern Illinois University Linear Algebra and Dirac Notation, Pt. 1 February 1, 2017 7 / 13

Dual Space and Bras Definition (Dual Space) For a Hilbert space H, its dual space H is the vector space consisting of all linear functions acting on H. For every f H, there exists a unique η H such that f ( ψ ) = η ψ ψ H. The function f is called the dual vector of η and it is sometimes denoted by the bra η := f. PHYS 500 - Southern Illinois University Linear Algebra and Dirac Notation, Pt. 1 February 1, 2017 7 / 13

Dual Space and Bras Definition (Dual Space) For a Hilbert space H, its dual space H is the vector space consisting of all linear functions acting on H. For every f H, there exists a unique η H such that f ( ψ ) = η ψ ψ H. The function f is called the dual vector of η and it is sometimes denoted by the bra η := f. If { b 1, b 2,, b d } is an arbitrary orthonormal basis and ψ = d i=1 x i b i, then the bra of ψ is given by ψ = d i=1 x i b i. PHYS 500 - Southern Illinois University Linear Algebra and Dirac Notation, Pt. 1 February 1, 2017 7 / 13

Dual Space and Bras Remark Think of the inner product between two vectors ψ and φ as multiplication of a bra onto a ket. PHYS 500 - Southern Illinois University Linear Algebra and Dirac Notation, Pt. 1 February 1, 2017 8 / 13

Dual Space and Bras Remark Think of the inner product between two vectors ψ and φ as multiplication of a bra onto a ket. For φ = d i=1 x i b i and ψ = d i=1 y i b i, their inner product is ψ φ = = = ( d i=1 d i=1 j=1 d i=1 ) yi b i d y i x i. d x j b j j=1 y i x j ( b i )( b j ) = d d i=1 j=1 y i x j b i b j PHYS 500 - Southern Illinois University Linear Algebra and Dirac Notation, Pt. 1 February 1, 2017 8 / 13

Dual Space and Bras In the basis { b i }, kets are represented by column vectors and bras are represented by row vectors as: x 1 φ =. x 2. x d ψ. = ( y 1 y 2 y d ). PHYS 500 - Southern Illinois University Linear Algebra and Dirac Notation, Pt. 1 February 1, 2017 9 / 13

Dual Space and Bras In the basis { b i }, kets are represented by column vectors and bras are represented by row vectors as: x 1 φ =. x 2. x d ψ. = ( y 1 y 2 y d ). Their inner product is given by standard matrix multiplication: x 1 ψ φ = ( y1 y2 yd ) x 2. = x d d i=1 y i x i. PHYS 500 - Southern Illinois University Linear Algebra and Dirac Notation, Pt. 1 February 1, 2017 9 / 13

The Outer Product and Linear Operators Definition (Linear Operator) A linear operator on H is a linear transformation T : H H. PHYS 500 - Southern Illinois University Linear Algebra and Dirac Notation, Pt. 1 February 1, 2017 10 / 13

The Outer Product and Linear Operators Definition (Linear Operator) A linear operator on H is a linear transformation T : H H. Definition (Outer Product) For two vectors ψ and φ their outer product is the linear operator denoted by ψ φ and has the action ψ φ : η ψ φ η = φ η ψ. PHYS 500 - Southern Illinois University Linear Algebra and Dirac Notation, Pt. 1 February 1, 2017 10 / 13

The Outer Product and Linear Operators Definition (Linear Operator) A linear operator on H is a linear transformation T : H H. Definition (Outer Product) For two vectors ψ and φ their outer product is the linear operator denoted by ψ φ and has the action ψ φ : η ψ φ η = φ η ψ. Remark Think of the outer product between two vectors ψ and φ as multiplication of a ket onto a bra. PHYS 500 - Southern Illinois University Linear Algebra and Dirac Notation, Pt. 1 February 1, 2017 10 / 13

The Outer Product and Linear Operators An arbitrary linear operator on H can be written as T = i,j t ij b i b j, where { b 1, b 2,, b d } is an arbitrary orthonormal basis. The numbers t ij are called the components of T with respect to the basis {b i }. The matrix representation of T in this basis is given by matrix T b : t 11 t 12 t 1d T =. t 21 t 22 t 2d T b =., where t ij = b i T b j. t d1 t d2 t dd PHYS 500 - Southern Illinois University Linear Algebra and Dirac Notation, Pt. 1 February 1, 2017 11 / 13

The Outer Product and Linear Operators An arbitrary linear operator on H can be written as T = i,j t ij b i b j, where { b 1, b 2,, b d } is an arbitrary orthonormal basis. The numbers t ij are called the components of T with respect to the basis {b i }. The matrix representation of T in this basis is given by matrix T b : t 11 t 12 t 1d T =. t 21 t 22 t 2d T b =., where t ij = b i T b j. t d1 t d2 t dd The components are also denoted by [[T b ]] ij := t ij = b i T b j. PHYS 500 - Southern Illinois University Linear Algebra and Dirac Notation, Pt. 1 February 1, 2017 11 / 13

The Outer Product and Linear Operators Example Consider the operator T = 1 4 0 0 + 3 4 1 1. What is the matrix representation of T in (a) The computation basis { 0, 1 }, (b) The Hadamard basis { +, }, where + = 1/2( 0 + 1 ) and = 1/2( 0 1 )? (c) What is the action of T on the vector ψ = cos θ + + sin θ? PHYS 500 - Southern Illinois University Linear Algebra and Dirac Notation, Pt. 1 February 1, 2017 12 / 13

The Adjoint Operator Definition (Adjoint) For a linear operator T on H, the adjoint of T, denoted by T, is the unique operator satisfying ψ T φ = φ T ψ ψ, φ H. PHYS 500 - Southern Illinois University Linear Algebra and Dirac Notation, Pt. 1 February 1, 2017 13 / 13

The Adjoint Operator Definition (Adjoint) For a linear operator T on H, the adjoint of T, denoted by T, is the unique operator satisfying ψ T φ = φ T ψ ψ, φ H. For an orthonormal basis { b i } d i=1, the matrix components of T are related to the matrix components of T by [[T b ]] ij = b i T b j = b j T b i = [[T b ]] ji. PHYS 500 - Southern Illinois University Linear Algebra and Dirac Notation, Pt. 1 February 1, 2017 13 / 13

The Adjoint Operator Definition (Adjoint) For a linear operator T on H, the adjoint of T, denoted by T, is the unique operator satisfying ψ T φ = φ T ψ ψ, φ H. For an orthonormal basis { b i } d i=1, the matrix components of T are related to the matrix components of T by [[T b ]] ij = b i T b j = b j T b i = [[T b ]] ji. The matrix representation for T is obtained by taking the conjugate transpose of the matrix representation for T. PHYS 500 - Southern Illinois University Linear Algebra and Dirac Notation, Pt. 1 February 1, 2017 13 / 13