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CRC Press eBooks

Quantum measurements

Ole Keller

Published May 13, 2014• 0 citations
Publisher

Abstract

Starting from two Hilbert spaces H1 and H2 of dimension N1 and N2, respectively, one can construct a larger Hilbert space H = H1 ⊗H2, (32.1) a sort of union of the two. The space H is called the tensor product of H1 and H2. State vectors and operators of these spaces will be denoted by an index, (1) or (2). Associated with a pair of state vectors |φ(1)〉 and |χ(2)〉 belongs a state vector |φ(1)〉 ⊗ |χ(2)〉 ≡ |φ(1)〉|χ(2)〉 ≡ |φ(1)χ(2)〉 (32.2) in the combined Hilbert space. Let us choose basis sets {|un(1)〉} and {|uk(2)〉} for H1 and H2. In terms of these one can express the arbitrary vectors |φ(1)〉 and |χ(2)〉 as follows: |φ(1)〉 = ∑ n an|un(1)〉, (32.3) |χ(2)〉 = ∑ k bk|vk(2)〉. (32.4) In the Hilbert space H one may use a basis {|un(1)⊗ |vk(2)〉〉} for expansion of the tensor product in Eq. (32.2). Thus, |φ(1)〉 ⊗ |χ(2)〉 = ∑ n,k anbk|un(1)〉 ⊗ |vk(2)〉. (32.5) The components of a tensor product state vector in H hence are products, anbk, of components of the two state vectors in the Hilbert spaces H1 and H2. The most general state vector in H, |ψ〉, can in the basis {|un(1)〉 ⊗ |vk(2)〉} be expanded as follows: |ψ〉 = ∑ n,k cnk|un(1)〉 ⊗ |vk(2)〉, (32.6) where the double index nk on the expansion coefficient cnk specifies the vector axes in H. Although an arbitrary vector |ψ〉 in H cannot be written as a single vector product of two vectors in H1 and H2 [Eq. (32.5)], |ψ〉 can always be decomposed into a linear combination of tensor product vectors [Eq. (32.6)]. This result relates to the fact that the elements of an arbitrary matrix {cnk} cannot in general be given as the elements {anbk} of a dyadic product of two vectors with components {an} and {bk}. The scalar product in H of two arbitrary basis vectors, |un(1)〉 ⊗ |vk(2)〉 and |un′(1)〉 ⊗ |vk′(2)〉, is given by 〈un′(1)vk′(2)|un(1)vk(2)〉 = 〈un′(1)|un(1)〉〈vk′ (2)|vk(2)〉 = δnn′δkk′ . (32.7) of The last member of Eq. (32.7) is obtained provided each of the bases {|un(1)〉} and |vk(2)〉 is orthonormal in its respective Hilbert space. If this is the case, the basis in H, i.e., {|un〉⊗ |vk(2)〉}, is orthonormal, too.

DOI
10.1201/b16917-44

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Starting from two Hilbert spaces H1 and H2 of dimension N1 and N2, respectively, one can construct a larger Hilbert space H = H1 ⊗H2, (32.1) a sort of union of the two. The space H is called the tensor product of H1 and H2. State vectors and operators of these spaces will be denoted by an index, (1) or (2). Associated with a pair of state vectors |φ(1)〉 and |χ(2)〉 belongs a state vector |φ(1)〉 ⊗ |χ(2)〉 ≡ |φ(1)〉|χ(2)〉 ≡ |φ(1)χ(2)〉 (32.2) in the combined Hilbert space. Let us choose basis sets {|un(1)〉} and {|uk(2)〉} for H1 and H2. In terms of these one can express the arbitrary vectors |φ(1)〉 and |χ(2)〉 as follows: |φ(1)〉 = ∑ n an|un(1)〉, (32.3) |χ(2)〉 = ∑ k bk|vk(2)〉. (32.4) In the Hilbert space H one may use a basis {|un(1)⊗ |vk(2)〉〉} for expansion of the tensor product in Eq. (32.2). Thus, |φ(1)〉 ⊗ |χ(2)〉 = ∑ n,k anbk|un(1)〉 ⊗ |vk(2)〉. (32.5) The components of a tensor product state vector in H hence are products, anbk, of components of the two state vectors in the Hilbert spaces H1 and H2. The most general state vector in H, |ψ〉, can in the basis {|un(1)〉 ⊗ |vk(2)〉} be expanded as follows: |ψ〉 = ∑ n,k cnk|un(1)〉 ⊗ |vk(2)〉, (32.6) where the double index nk on the expansion coefficient cnk specifies the vector axes in H. Although an arbitrary vector |ψ〉 in H cannot be written as a single vector product of two vectors in H1 and H2 [Eq. (32.5)], |ψ〉 can always be decomposed into a linear combination of tensor product vectors [Eq. (32.6)]. This result relates to the fact that the elements of an arbitrary matrix {cnk} cannot in general be given as the elements {anbk} of a dyadic product of two vectors with components {an} and {bk}. The scalar product in H of two arbitrary basis vectors, |un(1)〉 ⊗ |vk(2)〉 and |un′(1)〉 ⊗ |vk′(2)〉, is given by 〈un′(1)vk′(2)|un(1)vk(2)〉 = 〈un′(1)|un(1)〉〈vk′ (2)|vk(2)〉 = δnn′δkk′ . (32.7) of The last member of Eq. (32.7) is obtained provided each of the bases {|un(1)〉} and |vk(2)〉 is orthonormal in its respective Hilbert space. If this is the case, the basis in H, i.e., {|un〉⊗ |vk(2)〉}, is orthonormal, too.
There is a notational discrepancy in how mathematicians and physicists treat Hilbert spaces. In physics, one often works with finite-dimensional Hilbert spaces treated as ℂ n , and hence wants to write the Hermitian inner product as (x, y) = x ∗ y. This definition dictates the use of a Hermitian inner product that is antilinear in the first argument, the convention followed, for example, in Reed & Simon [246]. This is also the choice adopted in Dirac’s bracket notation, whose usage in quantum mechanics is very widespread. Our notation is chosen to extend-as closely as possible-the traditional notation of standard finite-dimensional matrix algebra to arbitrary complex inner product spaces and associated linear operators. In matrix algebra, column vectors and the corresponding matrices with one column are identical objects, row vectors are the linear functionals, and the adjoint is the conjugate transpose. For example, ℍ = ℍ × = ℂ n is the space of column vectors of size n, the dual space ℍ ∗ is the space of row vectors of size n, and the operator product ϕ ∗ ψ of a row vector ϕ ∗ and a column vector ψ is the standard Hermitian inner product of the column vectors ϕ and ψ. We use Greek lower case letters to write vectors, thus emphasizing their intended use as quantum state vectors in quantum mechanics. On the other hand, mathematicians working on reproducing kernel Hilbert spaces use an inner product (x, y) antilinear in the second argument, related to the physicist’s inner product (x, y) by (x, y) =( ). This is the convention followed, for example, in Rudin [252]. Although the two ways of defining the inner product lead to fully equivalent theories, all details look a bit different, a fact that has to be taken into account when reading the literature on the subject. For example, in the description based on the physical tradition, it is preferable to work with the antidual space in place of the dual space used in the mathematical tradition. In functional analysis, linear operators in Hilbert spaces are usually considered each with their own domain. But many computations in quantum mechanics require the consideration of algebras of operators with a common domain. The latter is a Euclidean space, a dense subspace of a Hilbert space. This space and its antidual play in many respects a more basic role in quantum physics than the Hilbert space itself. Therefore, and to avoid possible confusion caused by the different traditions, we give in the present chapter a self-contained introduction to Euclidean spaces and their associated spaces. All proofs are carried out in detail.
Complex Hilbert spaces play an important role in the description of quantum systems. In fact, with every quantum system there is associated an infinite dimensional or a finite dimensional separable complex Hilbert space ℍ that consists of the states of the quantum system. In physics terminology, the Hilbert space ℍ is usually referred to as the space of (pure) states. Throughout this monograph, the mathematical description of a quantum system shall be based on a certain complex (separable) Hilbert space ℍ, and therefore the quantum system will simply be denoted by ℍ. The quantum system ℍ is said to be a finite-dimensional system if ℍ is a finite-dimensional complex Hilbert space. Otherwise, the quantum system ℍ is said to be an infinite-dimensional system. The Hilbert space ℍ representing a composite quantum system that consists of n subsystems is a tensor product of the Hilbert spaces of n component systems described by ℍ 1 , … ℍ n , i.e., ℍ = ℍ 1 ⊗ … ⊗ ℍ n . In the theory of Hilbert spaces, a tensor product of Hilbert spaces is a way to create a new Hilbert space out of 2 (or more) Hilbert spaces. In the following, we illustrate the concept and construction of tensor product ℍ ⊗ 핂 of only 2 Hilbert spaces ℍ and 핂. The concepts and constructions can be easily extended to more than 2 Hilbert spaces such as ℍ 1 ⊗ ℍ 2 ⊗ … ⊗ ℍ n . Recall that the tensor product of ℍ with 핂 denoted by ℍ ⊗ 핂 is a new Hilbert space that consists of elements ϕ ⊗ φ ( ϕ ∈ ℍ and φ ∈ 핂) and is equipped with the Hilbertian inner product 〈·, ·〉 ℍ⊗핂 :(ℍ ⊗ 핂) × (ℍ ⊗ 핂) → C defined by 〈 ϕ 1 ⊗ φ 1 , ϕ 2 ⊗ φ 2 〉 ℍ⊗핂 = 〈 ϕ 1 , ϕ 2 〉 ℍ 〈 φ 1 , φ 2 〉 핂 , ϕ i ∈ ℍ and φ i ∈ 핂 for i = 1, 2.
From here, we take up the task of reformulating quantum mechanics in the language of inner product space. We start by introducing vector space through the familiar example of a set of finite dimensional, complex, column matrices under addition and multiplication by scalars. The inner product is then introduced as a dot product generalized to complex vectors leading to the idea of an inner product space . This is adequate to show how the concept of a quantum state can be described by a normalized vector in an inner product space so that the state space can be identified with the inner product space. Amplitudes are associated with inner products , and observables with orthonormal bases in this space. Hence the laws of quantum mechanics are reformulated. We then point out that the now reformulated laws are independent of the space of description and the reader is urged to contemplate an universality which would reflect the independence with respect to the representation space . This creates the stage for the introduction of the abstract inner product space and the formulation of quantum mechanics in this setting. The chapter concludes with this description. The transformed laws are now called postulatesof quantum mechanics .
The study of quantum systems involving two or more qubits necessitates the development of mathematical tools which capture the interactions between the qubits. This leads to the construction of objects called tensors which are elements of a vector space called a tensor product space. This material is essential since tensors are required for defining the means of data representation in quantum computation. The tensor product is the result of combining two vectors in a particular way designed to capture the interaction between the various parts of the individual vectors. The results of finding tensor products reside in a vector space called the tensor product space, the elements of which are called tensors. The formal construction of a tensor product space is somewhat abstract and can be difficult for the student to understand.
Linear algebra is the natural mathematical language of quantum mechanics. For this reason, the present chapter starts with a review of Hilbert spaces for vectors and operators. We introduce vector and Hilbert spaces along with inner products and metrics. The Dirac notation is developed for the Euclidean vector spaces as a starting point for the concepts of complete orthonormal sets of vectors, closure, dual vector spaces, and adjoint operators. The Dirac delta function in various forms and the principal part is introduced in Appendix 5 as an essential tool. The chapter turns to the main use of Dirac notation for function spaces; the concepts of norm, inner product, and closure are discussed. Fourier, Cosine, and Sine series are discussed as examples of expansions in complete orthonormal sets of functions. Although Hilbert spaces are interesting mathematical objects with important physical applications, the study of linear algebra remains incomplete without a study of linear operators (i.e., linear transformations). In fact, the set of linear transformations itself forms a vector space and therefore has a basis set. The basis set for the operator is linked with the basis sets for the spaces that it operates between. The linear operator can be discussed as an abstract operator or through an isomorphism as a matrix or as a generalized expansion in operator space. A Hermitian (a.k.a., self-adjoint) operator produces a basis set within a Hilbert space. The basis set comes from the eigenvector equation for the particular operator. The fact that a Hermitian operator produces a complete set (of orthornomal vectors) has special importance for quantum mechanics. Observables such as energy or momentum correspond to Hermitian operators. Complete sets make it possible to represent every possible result of a measurement of the observable by an object (vector) in the theory. The Hermitian operators have real eigenvalues which represent the results of the measurement.