Linear Algebra and Dirac Notation, Pt. 3

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1 Linear Algebra and Dirac Notation, Pt. 3 PHYS Southern Illinois University February 1, 2017 PHYS Southern Illinois University Linear Algebra and Dirac Notation, Pt. 3 February 1, / 16

2 The tensor product is a construction that combines different vector spaces to form one large vector space. Definition (Tensor Product Space) If H 1 is a Hilbert space with orthonormal basis { a i } d 1 and H 2 is a Hilbert space with orthonormal basis { b i } d 2, then their tensor product space is a vector space H 1 H 2 with orthonormal basis given by { a i b j } d 1,d 2,j=1, called a tensor product basis. Every vector Ψ H 1 H 2 can be written as a linear combination of the basis vectors: d 1 d 2 Ψ = c ij a i b j. j=1 PHYS Southern Illinois University Linear Algebra and Dirac Notation, Pt. 3 February 1, / 16

3 Definition (Tensor Product of Vectors) The tensor product is a map : H 1 H 2 H 1 H 2 that generates the tensor product vector ψ φ H 1 H 2 for ψ H 1 and φ H 2, such that 1 (c ψ ) φ = ψ (c φ ) = c( ψ φ ) ψ H 1, φ H 2, c C, 2 ( ψ + ω ) φ = ψ φ + ω φ ψ, ω H 1, φ H 2, 3 ψ ( φ + ω ) = ψ φ + ψ ω ψ H 1, φ, ω H 2. These three properties are known as bilinearity. Hence, the tensor product is a bilinear map from H 1 H 2 to H 1 H 2. PHYS Southern Illinois University Linear Algebra and Dirac Notation, Pt. 3 February 1, / 16

4 For notation simplicity, the tensor product of two vectors ψ φ is often written as ψ φ, or even just ψφ. Example Suppose ψ = a 0 + b 1 and φ = c 0 + d 1. (a) Express ψ φ in the tensor product basis { 00, 01, 10, 11 }. (b) Express φ ψ in the tensor product basis { 00, 01, 10, 11 }. In this example H 1 = H 2 = C 2. The tensor product space is C 2 C 2, which is also written denoted by C 2 := C 2 C 2. PHYS Southern Illinois University Linear Algebra and Dirac Notation, Pt. 3 February 1, / 16

5 Remark The tensor product space H 1 H 2 is much larger than the set of tensor product vectors! S = { α β : α H 1, β H 2 } H 1 H 2 = span S Not every Ψ H 1 H 2 is the tensor product of two vectors: Ψ α β. As we will see, such vectors Ψ correspond to entangled states in quantum mechanics. PHYS Southern Illinois University Linear Algebra and Dirac Notation, Pt. 3 February 1, / 16

6 Definition (Tensor Product of Operators) If A is an operator on H 1 and B is an operator on H 2, then their tensor product A B is a linear operator on H 1 H 2 with action defined by d 1 d 2 d 2 A B c ij a i b j = c ij A a i B b j. j=1 j=1 If A and C are operators on H 1 and B and D are operators on H 2, then 1 A B + C D is an operator on H 1 H 2 with action (A B + C D) Ψ = A B Ψ + C D Ψ Ψ H1 H 2, 2 (A B)(C D) = AC BD. PHYS Southern Illinois University Linear Algebra and Dirac Notation, Pt. 3 February 1, / 16

7 Recall that any operator A on H 1 and any operator B on H 2 can be expressed as A = d 1 i,j=1 α ij a i a j, B = d 2 i,j=1 β ij b i b j, where { a i } d 1 is a basis for H 1 and { b j } d 2 j=1 is a basis for H 2. Likewise, any operator K on H 1 H 2 can be expressed as K = d 1 d 2 i,k=1 j,l=1 since { a i b j } d 1,d 2 i,j=1 is a basis for H 1 H 2. γ ij,kl a i a k b j b l PHYS Southern Illinois University Linear Algebra and Dirac Notation, Pt. 3 February 1, / 16

8 Example Let { 0, 1 } be the computational basis for C 2. Define the operator σ x to have action σ x 0 = 1 and σ x 1 = 0. What is the action of σ x σ x on an arbitrary Ψ C 2? Example Consider the operator F = defined on C 2. What is its action on an arbitrary Ψ C 2? The operator F is called the SWAP operator. Remark The space of linear operators acting on H 1 H 2 is much larger than the set of tensor product operators! Not every operator on H 1 H 2 is the tensor product of two operators. For example F A B. PHYS Southern Illinois University Linear Algebra and Dirac Notation, Pt. 3 February 1, / 16

9 The matrix representation of tensor products is given in terms of the Kronecker matrix product. a 11 a 12 a 1d1 b 11 b 12 b 1d2 A =. a 21 a 22 a 2d1., B. b 21 b 22 b 2d2 =., a d1 1 a d1 d 1 b d2 1 b d2 d 2 a 11 b 11 a 11 b 12 a 12 b 11 a 12 b 12 a 1d1 b 1d2 A B =. a 11 b 21 a 11 b 22 a 12 b 11 a 12 b 12 a 1d1 b 2d2 a 21 b 11 a 21 b 12 a 22 b 11 a 22 b 12 a 2d1 b 1d2 a d1 1b d2 1 a d1 1b d2 2 a d1 2b d2 1 a d1 2b d2 2 a d1 d 1 b d2 d 2 PHYS Southern Illinois University Linear Algebra and Dirac Notation, Pt. 3 February 1, / 16

10 Example For ψ = a 0 + b 1 and φ = c 0 + d 1, what is the matrix representation of ψ φ? What is the matrix representation of ψ φ? Example What is the matrix representation of σ x σ x? Example What is the matrix representation of F? PHYS Southern Illinois University Linear Algebra and Dirac Notation, Pt. 3 February 1, / 16

11 The Schmidt Decomposition Theorem For any Ψ H 1 H 2, there exists an orthonormal basis { α i } d 1 H 1 and orthonormal basis { β j } d 2 j=1 for H 2 such that for Ψ = r λi α i β i. The numbers λ i > 0 are called the Schmidt coefficients of Ψ and the number r min{d 1, d 2 } is called the Schmidt rank of Ψ. The Schmidt coefficients are unique in the following sense: If Ψ = r λi α i β i = r λ i α i β i for two sets of orthonormal product bases { α i β j } i,j, { α i β j } i,j, and λ i, λ i > 0, then r = r and {λ i } r = {λ i }r. PHYS Southern Illinois University Linear Algebra and Dirac Notation, Pt. 3 February 1, / 16

12 The Schmidt Decomposition Any vector Ψ H 1 H 2 can be written as d 1 d 2 Ψ = c ij a i b j j=1 for any tensor product basis { a i b j } i,j. The Schmidt decomposition says there exists a special tensor product basis { α i β j } i,j such that Ψ = r λi α i β i. Example Give a Schmidt decomposition of the vector Ψ = 1/2[cos θ 00 + cos θ 01 + sin θe iφ 10 sin θe iφ 11 ]. PHYS Southern Illinois University Linear Algebra and Dirac Notation, Pt. 3 February 1, / 16

13 The Schmidt Decomposition - A Deeper Look So far we have only considered linear operators T that map a given vector space H onto itself; T : H H. In general, linear operators map one vector space H 2 onto another H 1 ; T : H 2 H 1. The set of all linear operators mapping H 2 into H 1 forms a vector space itself, denoted by L(H 1, H 2 ). The space L(H 1, H 2 ) can be made into an inner product space (in fact a Hilbert space) by defining the so-called Hilbert-Schmidt inner product: A, B := Tr[A B] for all A, B L(H 1, H 2 ). If { a i } d 1 is an orthonormal basis for H 1 and { b i } d 2 is an orthonormal basis for H 2, then { a i b j } d 1,d 2 i,j=1 is an orthonormal basis for L(H 1, H 2 ). PHYS Southern Illinois University Linear Algebra and Dirac Notation, Pt. 3 February 1, / 16

14 The Schmidt Decomposition - A Deeper Look Recall that one vector space V is isomorphic to another W, denoted by V = W, if there exists an invertible mapping f : V W such that f (aψ + bφ) = af (ψ) + bf (φ) ψ, φ V ; a, b C. Theorem (The First Isomorphism Theorem) L(H 1, H 2 ) = H 1 H 2. The isomorphic mapping f : L(H 1, H 2 ) H 1 H 2 is defined by d 1,d 2 d 1,d 2 f ( a i b j ) = a i b j f c ij a i b j = c ij a i b j i,j=1 for any orthonormal product basis { a i b j } i,j. i,j=1 PHYS Southern Illinois University Linear Algebra and Dirac Notation, Pt. 3 February 1, / 16

15 The Schmidt Decomposition - A Deeper Look Theorem (The Singular Value Decomposition (SVD)) For a given orthonormal basis { a i b j } i,j of L(H 1, H 2 ), any operator Ψ L(H 1, H 2 ) can be decomposed as Ψ = UΛV, where U is a unitary operator on H 1, V is a unitary operator on H 2, and Λ L(H 1, H 2 ) is diagonal in the basis { a i b j } i,j with unique non-negative diagonal elements λ i called the singular values of T. d 1 d 2 U = α i a i, V = β j b j, Λ = j=1 min{d 1,d 2 } λi a i b i. PHYS Southern Illinois University Linear Algebra and Dirac Notation, Pt. 3 February 1, / 16

16 The Schmidt Decomposition - A Deeper Look Ψ = UΛV = min{d 1,d 2 } λi α i β i = Apply isomorphic mapping f : r f (Ψ) = f λi α i β i = (λ i >0) r (λ i >0) r (λ i >0) λi α i β i. λi α i β i. The Schmidt Decomposition of Ψ H 1 H 2 is equivalent to the SVD of its corresponding operator Ψ L(H 1, H 2 ). The Schmidt coefficients of Ψ are the singular values of Ψ, and the Schmidt rank of Ψ is the rank of Ψ. PHYS Southern Illinois University Linear Algebra and Dirac Notation, Pt. 3 February 1, / 16

Principles of Quantum Mechanics Pt. 2

Principles of Quantum Mechanics Pt. 2 PHYS 500 - Southern Illinois University February 9, 2017 PHYS 500 - Southern Illinois University Principles of Quantum Mechanics Pt. 2 February 9, 2017 1 / 13 The