Tensor product operators on finite dimensional Hilbert spaces are studied. The focus is on bilinear tensor product operators. A tensor product operator on a pair of Hilbert spaces is a maximally general bilinear operator into a target Hilbert space. By 'maximally general' is meant every bilinear operator from the same pair of spaces to any Hilbert space factors into the composition of the tensor product operator with a uniquely determined linear mapping on the target space. There are multiple distinct tensor product operators of the same type; there is no "the" tensor product. Distinctly different tensor product operators can be associated with different parts of a multipartite system without difficulty. Separability of states, and locality of operators and observables is tensor product operator dependent. The same state in the target state space can be inseparable with respect to one tensor product operator but separable with respect to another, and no tensor product operator is distinguished relative to the others; the unitary operator used to construct a Bell state from a pair of |0>'s being highly tensor product operator-dependent is a prime example. The relationship between two tensor product operators of the same type is given by composition with a unitary operator. There is an equivalence between change of tensor product operator and change of basis in the target space. Among the gains from change of tensor product operator is the localization of some nonlocal operators as well as separability of inseparable states. Examples are given.
Tensor product operators on finite dimensional Hilbert spaces are studied. The focus is on bilinear tensor product operators. A tensor product operator on a pair of Hilbert spaces is a maximally general bilinear operator into a target Hilbert space. By 'maximally general' is meant every bilinear operator from the same pair of spaces to any Hilbert space factors into the composition of the tensor product operator with a uniquely determined linear mapping on the target space. There are multiple distinct tensor product operators of the same type; there is no "the" tensor product. Distinctly different tensor product operators can be associated with different parts of a multipartite system without difficulty. Separability of states, and locality of operators and observables is tensor product operator dependent. The same state in the target state space can be inseparable with respect to one tensor product operator but separable with respect to another, and no tensor product operator is distinguished relative to the others; the unitary operator used to construct a Bell state from a pair of |0>'s being highly tensor product operator-dependent is a prime example. The relationship between two tensor product operators of the same type is given by composition with a unitary operator. There is an equivalence between change of tensor product operator and change of basis in the target space. Among the gains from change of tensor product operator is the localization of some nonlocal operators as well as separability of inseparable states. Examples are given.
Quantum computing is fundamentally based on the mathematical structure of complex Hilbert spaces, linear operators, tensor products, and unitary transformations. Among these concepts, tensor products provide the formal mechanism for constructing the state space of composite quantum systems, whereas linear and unitary operators describe quantum transformations, measurements, and computational gates. This article examines the mathematical relationship between linear operators and tensor products and analyzes their role in the formation, manipulation, and characterization of quantum entanglement. Particular attention is given to two-qubit systems, separable and entangled states, Schmidt decomposition, controlled operations, and the role of non-factorizable operators in quantum circuits. The analysis demonstrates that tensor products perform a dual function: they provide the mathematical representation of composite systems and enable the construction of local operations acting on individual subsystems. At the same time, genuinely entangling transformations, such as the controlled-NOT operation, cannot in general be represented as a single tensor product of independent local operators. Recent research further demonstrates that tensor-product and operator-Schmidt decompositions can be used to characterize operator entanglement, reduce circuit complexity, and construct efficient representations of quantum processes. The results establish that linear algebra and tensor-product structures are not merely formal tools but constitute the mathematical foundation connecting quantum entanglement with quantum computation.
The situation of two independent observers conducting measurements on a joint quantum system is usually modelled using a Hilbert space of tensor product form, each factor associated to one observer. Correspondingly, the operators describing the observables are then acting non-trivially only on one of the tensor factors. However, the same situation can also be modelled by just using one joint Hilbert space, and requiring that all operators associated to different observers commute, i.e. are jointly measurable without causing disturbance. The problem of Tsirelson is now to decide the question whether all quantum correlation functions between two independent observers derived from commuting observables can also be expressed using observables defined on a Hilbert space of tensor product form. Tsirelson showed already that the distinction is irrelevant in the case that the ambient Hilbert space is of finite dimension. We show here that the problem is equivalent to the question whether all quantum correlation functions can be approximated by correlation function derived from finite-dimensional systems. We also discuss some physical examples which fulfill this requirement.
Relations between states and maps, which are known for quantum systems in finite-dimensional Hilbert spaces, are formulated rigorously in geometrical terms with no use of coordinate (matrix) interpretation. In a tensor product realization they are represented simply by a permutation of factors. This leads to natural generalizations for infinite-dimensional Hilbert spaces and a simple proof of a generalized Choi Theorem. The natural framework is based on spaces of Hilbert-Schmidt operators [Formula: see text] and the corresponding tensor products [Formula: see text] of Hilbert spaces. It is proved that the corresponding isomorphisms cannot be naturally extended to compact (or bounded) operators, nor reduced to the trace-class operators. On the other hand, it is proven that there is a natural continuous map [Formula: see text] from trace-class operators on [Formula: see text] (with the nuclear norm) into compact operators mapping the space of all bounded operators on [Formula: see text] into trace class operators on [Formula: see text] (with the operator-norm). Also in the infinite-dimensional context, the Schmidt measure of entanglement and multipartite generalizations of state-maps relations are considered in the paper.
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.
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.
Separable operators and PPT operators on a finite dimensional tensor product space are introduced.Then,general forms of separable self-adjoint operators,separable projective operators and separable positive operators are obtained.The relationship between separable operators and PPT operators are proved,and some examples which are PPT operators but not separable operators are given.Lastly,separable maps and PPT maps are introduced and discussed,and it is proved that every separable map maps a PPT operator as a PPT operator.