Tensor Product of Hilbert Spaces

10 Key excerpts on "Tensor Product of Hilbert Spaces"

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  Matrix Calculus and Kronecker Product

  A Practical Approach to Linear and Multilinear Algebra

  • Willi-Hans Steeb, Yorick Hardy;;;(Authors)
  • 2011(Publication Date)
  • WSPC

    (Publisher)

  Chapter 4 Tensor Product 4.1 Hilbert Spaces In this section we introduce the concept of a Hilbert space (Young [74], Steeb [55], Halmos [25], Berberian [9]). Hilbert spaces play the central rˆole in quantum mechanics. The proofs of the theorems given in this chapter can be found in Prugoveˇ cki [46]. We assume that the reader is familiar with the notation of a linear space. First we introduce the pre-Hilbert space. Definition 4.1. A linear space L is called a pre-Hilbert space if there is defined a numerical function called the scalar product (or inner product ) which assigns to every f, g of “vectors” of L ( f, g ∈ L ) a complex number. The scalar product satisfies the conditions ( a ) ( f, f ) ≥ 0; ( f, f ) = 0 iff f = 0 ( b ) ( f, g ) = ( g, f ) ( c ) ( cf, g ) = c ( f, g ) where c is an arbitrary complex number ( d ) ( f 1 + f 2 , g ) = ( f 1 , g ) + ( f 2 , g ) where ( g, f ) denotes the complex conjugate of ( g, f ). It follows that ( f, g 1 + g 2 ) = ( f, g 1 ) + ( f, g 2 ) and ( f, cg ) = ¯ c ( f, g ) . 255 256 Matrix Calculus and Kronecker Product Definition 4.2. A linear space E is called normed space , if for every f ∈ E there is associated a real number ∥ f ∥ , the norm of the vector f such that ( a ) ∥ f ∥ ≥ 0 , ∥ f ∥ = 0 iff f = 0 ( b ) ∥ cf ∥ = | c |∥ f ∥ where c is an arbitrary complex number ( c ) ∥ f + g ∥ ≤ ∥ f ∥ + ∥ g ∥ . The topology of a normed linear space E is thus defined by the distance d ( f, g ) := ∥ f − g ∥ . If a scalar product is given we can introduce a norm. The norm of f is defined by ∥ f ∥ := √ ( f, f ) . A vector f ∈ L is called normalized if ∥ f ∥ = 1. Let f, g ∈ L . The following identity holds ( parallelogram identity ) ∥ f + g ∥ 2 + ∥ f − g ∥ 2 ≡ 2( ∥ f ∥ 2 + ∥ g ∥ 2 ) . Definition 4.3. Two functions f ∈ L and g ∈ L are called orthogonal if ( f, g ) = 0 .

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  Matrix Calculus, Kronecker Product and Tensor Product

  A Practical Approach to Linear Algebra, Multilinear Algebra and Tensor Calculus with Software Implementations

  • Yorick Hardy, Willi-Hans Steeb(Authors)
  • 2019(Publication Date)
  • WSPC

    (Publisher)

  Chapter 4

  Tensor Product

  4.1Hilbert Spaces

  In this section we introduce the concept of a Hilbert space (Young [74 ], Steeb [55 ], Halmos [27 ], Berberian [9 ], Weidmann [73 ]). Hilbert spaces play the central rôle in quantum mechanics. The proofs of the theorems given in this chapter can be found in Prugovečki [48 ]. We assume that the reader is familiar with the notation of a linear space. First we introduce the pre-Hilbert space.

  Definition 4.1. A linear space L is called a pre-Hilbert space if there is defined a numerical function called the scalar product (or inner product) which assigns to every f, g of “vectors” of L (f, g ∈ L) a complex number. The scalar product , satisfies the conditions

  where denotes the complex conjugate of g, f .

  It follows that f, g1 + g2 = f, g1 + f, g2 and .

  Definition 4.2. A linear space E is called normed space, if for every f ∈ E there is associated a real number ||f||, the norm of the vector f such that

  The topology of a normed linear space E is thus defined by the distance

  If a scalar product is given we can introduce a norm. The norm of f is defined by

  A vector f ∈ L is called normalized if ||f|| = 1.

  Let f, g ∈ L. The following identity holds (parallelogram identity)

  Definition 4.3. Two functions f ∈ L and g ∈ L are called orthogonal if

  Definition 4.4. A sequence {f

  n

  } (n ∈ ) of elements in a normed space E is called a Cauchy sequence if, for every ϵ > 0, there exists a number Mϵ such that ||f

  p

  − f

  q

  || < ϵ for p, q > Mϵ .

  Definition 4.5. A normed space E is said to be complete if every Cauchy sequence of elements in E converges to an element in E.

  Example 4.1. Let be the rational numbers. Since the sum and product of two rational numbers are again rational numbers we obviously have a pre-Hilbert space with the scalar product q1 , q2 ≔ q1 q2 . However, the pre-Hilbert space is not complete. Consider the sequence

  with n = 1, 2, . . . . The sequence fn

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  Quantum Mechanics

  • Eugene Stefanovich(Author)
  • 2018(Publication Date)
  • De Gruyter

    (Publisher)

  This means that if ? 1 ∈ ℒ 1 and ? 2 ∈ ℒ 2 are atoms, then the meet of their images ? 1 ( ? 1 )∧? 2 ( ? 2 ) is also an atomic proposition in ℒ 1 + 2 . The next theorem [54, 2] allows us to translate the above properties from the lan-guage of quantum logic to the more familiar language of Hilbert spaces. Theorem 6.2 (Matolcsi) . Suppose that H 1 , H 2 and H 1 + 2 are three complex Hilbert spaces corresponding to the propositional lattices ℒ 1 , ℒ 2 and ℒ 1 + 2 from Postulate 6.1 . Suppose, also, that ? 1 and ? 2 are two mappings whose existence is required by the Postulate. Then the Hilbert space H 1 + 2 of the composite system 1 + 2 is equal to one of the following four possible tensor products 1 : H 1 + 2 = H 1 ⊗ H 2 , H 1 + 2 = H ∗ 1 ⊗ H 2 , or H 1 + 2 = H 1 ⊗ H ∗ 2 or H 1 + 2 = H ∗ 1 ⊗ H ∗ 2 . The proof of this theorem is beyond the scope of our book. So, we have four ways to connect two one-particle Hilbert spaces into one two-particle space. In quantum mechanics, only the first possibility is used: H 1 + 2 = H 1 ⊗ 1 A definition of the tensor product of two Hilbert spaces can be found in Appendix F.4. The asterisk marks dual Hilbert spaces, introduced in Appendix F.3. 6.1 Hilbert space of multiparticle system | 125 H 2 . 2 In particular, this means that if particle 1 is in the state | 1 ⟩ ∈ H 1 and particle 2 is in the state | 2 ⟩ ∈ H 2 , then the state of the composite system is described by the vector | 1 ⟩ ⊗ | 2 ⟩ ∈ H 1 ⊗ H 2 . 6.1.2 Particle observables in multiparticle systems The functions ? 1 and ? 2 from Postulate 6.1 map propositions (projections) from Hilbert spaces H 1 and H 2 of individual particles into the Hilbert space H 1 + 2 = H 1 ⊗ H 2 of the composite system. Therefore, they also map observables of particles from H 1 and H 2 to H 1 + 2 . For example, consider an observable of particle 1, which is repre-sented in the Hilbert space H 1 by a Hermitian operator with spectral resolution (1.9), F 1 = ∑ f fP 1 f .

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  Entangled Systems

  New Directions in Quantum Physics

  • Jürgen Audretsch(Author)
  • 2008(Publication Date)
  • Wiley-VCH

    (Publisher)

  of the type: “If Alice does this to subsystem S A , then Bob will measure that on subsystem S B ”. Existence We will once again assume, in agreement with the standard interpretation from Sect. 1.2, that such subsystems are not just abstract auxiliary constructions like the quantum systems in the minimal interpretation, but rather that they exist in reality. With this, we do not mean to imply that a state can be ascribed to an individual subsystem which is independent of the state of the other subsystem. In entangled systems, precisely this independence does not exist. This is the cause of many startling quantum-physical effects. It is furthermore not meant by our assumption of existence in reality that similar elementary particles of the same type, such as two photons, have individual identities and are therefore distinguishable. The assumption that the photons exist cannot lead us to such conclusions. The possibility of separate manipulations, and not the individuality of quantum objects, defines the subsystem (compare Sect. 7.9). 7.2 The Product Hilbert Space We first wish to supply the mathematical formalism which we need to formulate the physics of composite systems. We require for this purpose the product Hilbert space . 7.2.1 Vectors The tensor product H AB of two Hilbert spaces H A and H B , whose dimensions need not be the same, H AB = H A ⊗ H B (7.1) 7.2 The Product Hilbert Space 117 is itself a Hilbert space. We call H A and H B the factor spaces . For each pair of vectors | ϕ A ∈ H A and | χ B ∈ H B , there is a product vector in H AB , which can be written in different ways | ϕ A ⊗ | χ B =: | ϕ A | χ B =: | ϕ A , χ B =: | ϕ, χ . (7.2) It is linear in each argument with respect to multiplication by complex numbers. With λ, µ ∈ C | ϕ A ⊗ ( λ | χ B 1 + µ | χ B 2 ) = λ | ϕ A ⊗ | χ B 1 + µ | ϕ A ⊗ | χ B 2 , (7.3) and ( λ | ϕ A 1 + µ | ϕ A 2 ) ⊗ | χ B = λ | ϕ A 1 ⊗ | χ B + µ | ϕ A 2 ⊗ | χ B .

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  Hilbert Space Methods in Quantum Mechanics

  • Werner O. Amrein(Author)
  • 2009(Publication Date)
  • EPFL PRESS

    (Publisher)

  CHAPTER 1 Hilbert Spaces Hilbert space sets the stage for standard quantum theory: the pure states of a physical system are identified with the unit rays of a Hilbert space H and observables with self- adjoint operators acting in H. In this initial chapter we present the essential concepts and prove the basic results concerning separable Hilbert spaces (Sections 1.1 - 1.3). In Section 1.4 we then introduce L 2 spaces, which are of special importance for quantum mechanics. This requires some familiarity with measure theory, and we include a short description of the necessary concepts from this theory. 1.1 Definition and elementary properties 1.1.1. Throughout this text a Hilbert space means a complex linear vector space, equipped with a Hermitian scalar product, which is complete and admits a countable basis. More precisely a (separable) Hilbert space H is defined by the four postulates (H1) - (H4) stated below: (H1) H is a linear vector space over the field C of complex numbers: With each couple {f,g} of elements of H there is associated another element of H, denoted f + g, and with each couple {α,f }, α ∈ C, f ∈ H, there is associated an ele- ment αf of H, and these associations have the following properties (where f,g,h ∈ H and α,β ∈ C): f + g = g + f f + (g + h) = (f + g) + h (1.1) α(f + g) = αf + αg (α + β)f = αf + βf (1.2) α(βf ) = (αβ)f 1 f = f. (1.3) Furthermore there exists a unique element 0 ∈ H (called the zero vector) such that 1 0 + f = f , 0 f = 0 ∀f ∈ H. (1.4) 1 Here 0 denotes the complex number α = 0. 2 HILBERT SPACES (H2) H is equipped with a strictly positive scalar product 2 : With each couple {f,g} of elements of H there is associated a complex number (f,g), and this association has the following properties 3 : (g,f ) = (f,g) ∀f,g ∈ H (1.5) (f,g + αh) = (f,g) + α(f,h) ∀ α ∈ C, ∀f,g,h ∈ H (1.6) (f,f ) > 0 except for f = 0. (1.7) One then defines bardblf bardbl := [(f,f )] 1/2 . (1.8) (H3) H is complete: Each Cauchy sequence in H has a limit in H.

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  An Introduction to the Theory of Reproducing Kernel Hilbert Spaces

  • Vern I. Paulsen, Mrinal Raghupathi(Authors)
  • 2016(Publication Date)
  • Cambridge University Press

    (Publisher)

  We now have enough tools at our disposal to describe the RKHS that arises from the product of kernels. Recall that if H i , i = 1, 2 are Hilbert spaces, then we can form their tensor product, H 1 ⊗ H 2 , which is a new Hilbert space. If ·, · i , i = 1, 2, denotes the respective inner products on the spaces, then to form this new space we first endow the algebraic tensor product with the inner product obtained by setting  f ⊗ g, h ⊗ k  =  f , h  1 g, k  2 and extending linearly and then completing the algebraic tensor product in the induced norm. One of the key facts about this completed tensor product is that it contains the algebraic tensor product faithfully; that is, the inner product satisfies u , u  > 0 for any u  = 0 in the algebraic tensor product. Now if H 1 and H 2 are RKHSs on sets X and S, respectively, then it is natural to want to identify an element of the algebraic tensor product, u = ∑ n i =0 h i ⊗ f i with the function, ˆ u (x , s ) = ∑ n i =0 h i (x ) f i (s ). The following theorem shows that not only is this identification well-defined, but that it also extends to the completed tensor product. 5.5 Products of kernels and tensor products of spaces 73 Theorem 5.11. Let H 1 and H 2 be RKHSs on sets X and S, with reproducing kernels K 1 and K 2 . Then K given by K ((x , s ), ( y , t )) = K 1 (x , y ) K 2 (s , t ) is a kernel function on X × S and the map u → ˆ u extends to a well-defined, linear isometry from H 1 ⊗ H 2 onto the reproducing kernel Hilbert space H( K ). Proof . Set k 1 y (x ) = K 1 (x , y ) and k 2 t (s ) = K 2 (s , t ). Note that if u = ∑ n i =1 h i ⊗ f i , then u , k 1 y ⊗k 2 t  H 1 ⊗H 2 = ∑ n i =1 h i , k 1 y  H 1  f i , k 2 t  H 2 = ˆ u ( y , t ). Thus, we may extend the mapping u → ˆ u from the algebraic tensor product to the completed tensor product as follows. Given u ∈ H 1 ⊗ H 2 , define a function on X × S by setting ˆ u ( y , t ) = u , k 1 y ⊗ k 2 t  H⊗H 2 .

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  Quantum Mechanics in Hilbert Space

  • Eduard Prugovecki(Author)
  • 2003(Publication Date)
  • Academic Press

    (Publisher)

  C H A P T E R I Basic Ideas of Hilbert Space Theory The central object of study in this chapter is the infinite-dimensional Hilbert space. The main goal is to give a rigorous analysis of the problem of expanding a vector in a Hilbert space in terms of an orthogonal basis containing a countable infinity of vectors. We first review in $1 a few key theorems on vector spaces in general, and in $2 we investigate the basic properties of vector spaces on which an inner product is defined. In order to define convergence in an inner-product space, we introduce in 93 the concept of metric. In $4 we give the basic concepts and theorems on separable Hilbert spaces, con- centrating especially on properties of orthonormal bases. We conclude the chapter by illustrating some of the physical applications of these mathe- matical results with the initial-value problem in wave mechanics. 1, Vector Spaces 1.1. VECTOR SPACES OVER FIELDS OF SCALARS A mathematical space is in general a set endowed with some given structure. Such a structure can be given, for instance, by means of certain operations which are defined on the elements of that set. These operations are then required to obey certain general rules, which are called the postulates or the axioms of the mathematical space. Definition 1.1. Any set 9'- on which the operations of vector addition and multiplication by a scalar are defined is said to be a vector 11 12 I. Basic Ideas of Hilbert Space Theory space (or linear space, or linear manifold). The operation of vector addition is a mapping, (f, g) - f + g, ( f , g ) E v x v> f + g E v , of Y x Y into Y, while the operation of multiplication by a scalar a f h m a field+ 9 is a mapping ( a , f ) -+ 4, (%f) E x vs af E v, of 9 x Y into Y. These two vector operations are required to satisfy the following axioms for any f, g, h E Y and any scalars a, b E 9: (1) f + g = g + f (commutativity of vector addition). (2) (f + g) + h = f + (g + h) (associativity of vector addition).

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  Picturing Quantum Processes

  A First Course in Quantum Theory and Diagrammatic Reasoning

  • Bob Coecke, Aleks Kissinger(Authors)
  • 2017(Publication Date)
  • Cambridge University Press

    (Publisher)

  The full definition of a Hilbert space goes like this. Definition 5.118 A (possibly infinite-dimensional) Hilbert space is a complex inner product space that is additionally Cauchy complete. That is, for any (Cauchy) convergent sequence (v i ) i of vectors in H, there exists v ∈ H such that v i − v → 0. This means we can take limits of sequences, so we can hit Hilbert spaces (and hence quantum mechanics) with a whole big bag of tools from functional analysis. For one thing, having limits around allows one to define infinite sums: ∞  i=0 ψ i := lim n→∞ n  i=0 ψ i There are two examples of Hilbert spaces that have historically played a major role in quantum mechanics as (equivalent) presentations of the quantum mechanical state space. Example 5.119 L 2 is the set of all functions ψ : R n → C whose ‘squared-integrals’ are finite:  ψ(x)ψ(x)dx < ∞ This forms a Hilbert space by letting: (λ 1 ψ + λ 2 φ)(x) := λ 1 ψ(x) + λ 2 φ(x) and ψ |φ :=  ψ(x)φ(x)dx Example 5.120  2 is the set of all (countably infinite) sequences of complex numbers, whose ‘squared sums’ are finite: ∞  i=0 a i a i < ∞ 5.6 Advanced Material 239 This forms a Hilbert space by letting: λ 1 (a i ) i + λ 2 (b i ) i := (λ 1 a i + λ 2 b i ) i and (a i ) i |(b i ) i  := ∞  i=0 a i b i Both L 2 and  2 allow one to express the state of a quantum particle and compute its position or momentum. What is surprising is not only that these are both the same kind of mathematical object (an infinite-dimensional Hilbert space), but that they are in fact isomorphic Hilbert spaces (see Section 5.7 for more details). So, this brings us back to Q3 from Section 2.3: Why don’t infinite dimensional Hilbert spaces play a role in this book? Or in the majority of quantum computing, for that matter? The short answer is that many things just don’t seem to work as well. Suppose, for example, that we take the caps and cups that we know (and love) by now, and try to run them in infinite dimensions.

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  Quantum Mechanics in Hilbert Space

  • Eduard Prugovecki(Author)
  • 1982(Publication Date)
  • Academic Press

    (Publisher)

  C H A P T E R I Basic Ideas of Hilbert Space Theory The central object of study in this chapter is the infinite-dimensional Hilbert space. The main goal is to give a rigorous analysis of the problem of expanding a vector in a Hilbert space in terms of an orthogonal basis containing a countable infinity of vectors. We first review in $1 a few key theorems on vector spaces in general, and in 52 we investigate the basic properties of vector spaces on which an inner product is defined. In order to define convergence in an inner-product space, we introduce in $3 the concept of metric. In $4 we give the basic concepts and theorems on separable Hilbert spaces, con- centrating especially on properties of orthonormal bases. We conclude the chapter by illustrating some of the physical applications of these mathe- matical results with the initial-value problem in wave mechanics. 1, Vector Spaces 1 . 1 . VECTOR SPACES OVER FIELDS OF SCALARS A mathematical space is in general a set endowed with some given structure. Such a structure can be given, for instance, by means of certain operations which are defined on the elements of that set. These operations are then required to obey certain general rules, which are called the postulates or the axioms of the mathematical space. Definition 1.1. Any set V on which the operations of vector addition and multiplication by a scalar are defined is said to be a vector 11 12 I. Basic Ideas of Hilbert Space Theory space (or linear space, or linear manifold). The operation of vector addition is a mapping,* (f,g)+f+g, ( f , d E Y x Y, f + g e Y , of Y x Y into Y, while the operation of multiplication by a scalar a from a field+ F is a mapping (a,f)* 4, (%f) E F x v, a f e v, of F x Y into V. These two vector operations are required to satisfy the following axioms for anyf, g, h E V‘ and any scalars a, b E F : (1) f + g = g +f(commutativity of vector addition). (2) (f + g) + h = f + (g + h) (associativity of vector addition).

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  Negative Quantum Channels

  • James M. McCracken(Author)
  • 2022(Publication Date)
  • Springer

    (Publisher)

  is notation will be important in later discussions when it is applied to both the reduced system and the bath Hilbert spaces. e state of the composite system is defined in a tensor product Hilbert space H S ˝ H B where H S is the Hilbert space of the reduced system and H B is the Hilbert space of the bath. e reduced system will have some state S 2 S .H S / ; the bath will have some state B 2 S .H B / ; and the composite system will have some state SB 2 S .H SB / : e relationship of the composite system to its component states will be important and is referred to as the system-bath correlation. A product state is defined by the following relationship: SB D S ˝ B ; where S 2 S .H S / and B 2 S .H B /. If the state is not a product state, it might still be a separable state, i.e., it might have the form SB D X ij p ij S i ˝ B j ; 1.4. MATHEMATICAL STRUCTURE OF OPEN SYSTEMS 7 where S i 2 S .H S /, B j 2 S .H B /, p ij 0 8i; j and P ij p ij D 1. e set of separable states of the composite system will be denoted G SB . Notice if SB is a product state then SB 2 G SB , but the converse is not necessarily true. e composite state is entangled if and only if it is not separable. 1.4.2 PARTIAL TRACE e relationship between the states of the subsystems and the composite system can be defined by looking at expectation values. e partial trace will be defined as the operation that ensures consistent measurement statistics between an observable and its trivial extension to a higher di- mensional space. e trivial extension of an operator is an operator that acts on any extended space as the identity operator. Let A 2 B.H X / be an observable with the trivial extension Q A D A ˝ I 2 B.H XY / D B.H X ˝ H Y / where I is the identity operator. In this example, A acts on space X , so it’s trivial extension, Q A, acts on the extended space, Y , as identity. If fjx i ig is a basis of H X and f ˇ ˇ y j ˛ g is a basis of H Y , than fjx i i ˝ ˇ ˇ y j ˛ ˇ ˇ x i y j ˛ g is a basis of H XY .

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