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Physica Scripta

Tensor-product representation of qubits and tensor realization of one-qubit operators

Петер Адам, V. A. Andreev, József Janszky, Margarita A. Man’ko, Vladimir I. Man’ko

Published Mar 1, 2013• 1 citation
Publisher

Abstract

We consider one-particle spin states. We show that all the information contained in the state can be written in an equivalent form using three completely decohered mixed states. The density matrix of such a system has the form of tensor product of three diagonal matrices. The unitary operators defined in the space of one-particle spin states are represented by some transformation of tensor products of diagonal matrices. We determine the explicit form of the unitary transformations acting on the representing three-qubit system and corresponding basic operators acting in the space of the initial qubit states.

DOI
10.1088/0031-8949/2013/t153/014001

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We consider one-particle spin states. We show that all the information contained in the state can be written in an equivalent form using three completely decohered mixed states. The density matrix of such a system has the form of tensor product of three diagonal matrices. The unitary operators defined in the space of one-particle spin states are represented by some transformation of tensor products of diagonal matrices. We determine the explicit form of the unitary transformations acting on the representing three-qubit system and corresponding basic operators acting in the space of the initial qubit states.
A method for producing tensor-product representation of qubits via physical processes is developed. The tensor-product representation of a qubit consists of three completely decohered qubits and contains all information on the original system. The advantage of this representation consists in the fact that the representing system can be easily stabilized against further external influences. The problem of retransforming the information preserved in the representation into the state of the initial qubit is also analyzed.
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.
Pure states of three-qubit quantum systems are represented by eight-dimensional complex vectors, whereas their unitary transformations are described by 8×88 \times 88×8 complex matrices. Motivated by this correspondence, we extend our algebraic model for two-qubit systems to eight-dimensional spaces endowed with commutative and associative multiplication. To each eight-dimensional number we assign a matrix and prove that the resulting correspondence is a homomorphic bijection. We derive conditions under which the associated matrix is unitary and hence represents a three-qubit quantum gate. This leads to a specific algebraic notion of a unitary state: a normalized state whose multiplication matrix is unitary. We construct 16,384 basis spaces; within each fixed space, multiplication generates a commuting family of quantum gates. For any two unitary states in the class considered, we give an explicit transition-gate construction within one of these algebraic families. For real components, the unitary states in a representative basis space form a union of thirty-two four-dimensional manifolds in R8\mathbb{R}^{8}R8. The construction therefore yields new structured classes of three-qubit unitary operators and invariant state sets, while its extension to larger multiqubit systems remains open.
The representation of numbers by tensor product states of composite quantum systems is examined. Consideration is limitedto k — ary representations of length L and arithmetic mod K L An abstract representation on an L fold tensor product Hilbert space H arith of number states and operators for the basic arithmetic operation is described. Unitary maps onto a physical parameter based tensor product space H phy are defined and the relations between these two spaces and the dependence of algorithm dynamics on the unitary maps is discussed. The important condition of efficient implementation by physically realizable Hamiltonians of the basic arithmetic operations is also discussed.