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.
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.
Up to now we have limited ourselves to states of a single particle. In the present chapter we shall introduce the description of two-particle states. Once this case is understood, it will be easy to generalize to any number of particles. States of two (or more) particles lead to very rich configurations called entangled states. A remarkable feature is that two entangled quantum particles, even at arbitrarily large spatial separations, continue to form a single entity and no classical probabilistic model is able to reproduce the correlation between particles. In the first section we shall present the essential mathematical formalism, that of the tensor product. This will permit us in Section 6.2 to describe quantum mixtures using the state operator formalism. Section 6.3 is devoted to the study of important physical consequences like the Bell inequalities and interference experiments involving entangled states, which will lead us to a deeper understanding of quantum physics. Finally, in the last section we shall briefly review applications to measurement theory and quantum information theory. The latter is undergoing rapid development at present and has applications to quantum computing, cryptography, and teleportation. The tensor product of two vector spaces Definition and properties of the tensor product We wish to construct the space of states of two physical systems which we assume initially to be completely independent. Let ℌ N 1 and ℌ M 2 be the spaces of states of the two systems, of dimension N and M , respectively.
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.
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.
Hilbert spaces provide the fundamental mathematical framework for describing quantum mechanical systems. Their structure, characterized by an inner product and completeness, allows for the representation of quantum states as vectors and physical observables as self-adjoint operators. Key quantum phenomena such as superposition and entanglement find natural expression within this formalism. Superposition, where a quantum system can exist in multiple states simultaneously, is represented by linear combinations of basis vectors in the Hilbert space. Entanglement, a non-classical correlation between quantum systems, is described by non-separable state vectors in a tensor product of Hilbert spaces. These concepts are pivotal in quantum computing, where the unit of information, the qubit, is a two-level quantum system whose state is a vector in a two-dimensional complex Hilbert space (ℂ²). Quantum gates, which perform operations on qubits, are represented by unitary operators acting on these state vectors. The power of quantum computation, particularly in algorithms like Shor's or Grover's, stems from the ability to exploit superposition and entanglement, processes intrinsically described within the Hilbert space framework. Thus, a thorough understanding of Hilbert spaces is indispensable for grasping the principles of quantum mechanics and for advancing the field of quantum information and computation [1, 7]. The transition from classical bits to quantum qubits, and from classical logic gates to quantum unitary operations, is entirely predicated on the mathematical properties endowed by Hilbert spaces.
Hilbert spaces provide the fundamental mathematical framework for describing quantum mechanical systems. Their structure, characterized by an inner product and completeness, allows for the representation of quantum states as vectors and physical observables as self-adjoint operators. Key quantum phenomena such as superposition and entanglement find natural expression within this formalism. Superposition, where a quantum system can exist in multiple states simultaneously, is represented by linear combinations of basis vectors in the Hilbert space. Entanglement, a non-classical correlation between quantum systems, is described by non-separable state vectors in a tensor product of Hilbert spaces. These concepts are pivotal in quantum computing, where the unit of information, the qubit, is a two-level quantum system whose state is a vector in a two-dimensional complex Hilbert space (ℂ²). Quantum gates, which perform operations on qubits, are represented by unitary operators acting on these state vectors. The power of quantum computation, particularly in algorithms like Shor's or Grover's, stems from the ability to exploit superposition and entanglement, processes intrinsically described within the Hilbert space framework. Thus, a thorough understanding of Hilbert spaces is indispensable for grasping the principles of quantum mechanics and for advancing the field of quantum information and computation [1, 7]. The transition from classical bits to quantum qubits, and from classical logic gates to quantum unitary operations, is entirely predicated on the mathematical properties endowed by Hilbert spaces.
Even the simplest mathematical concepts can have surprisingly deep struc-ture and hidden connections to other topics. In this article we explore rotations of two-dimensional objects. After viewing rotations first concretely as small matrices, then more abstractly as members of a "group", we consider how rotations transform simple functions (x, y, x -y , etc.) and their products. Transforming products of three or more functions is best captured by introducing multi-dimensional generalizations of vectors and matrices known as tensors and chaining these tensors together. The resulting object known as a tensor train (or matrix product state) is one of the most powerful computational tools for applications such as modeling fluid flows, understanding properties of metals and crystals, predicting financial markets, and simulating quantum computers.