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.
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.
This paper represents one approach to making explicit some of the assumptions and conditions implied in the widespread representation of numbers by composite quantum systems. Any nonempty set and associated operations is a set of natural numbers or a model of arithmetic if the set and operations satisfy the axioms of number theory or arithmetic. This paper is limited to $k\ensuremath{-}\mathrm{ary}$ representations of length L and to the axioms for arithmetic modulo kL. A model of the axioms is described based on an abstract L-fold tensor product Hilbert space Harith. Unitary maps of this space onto a physical parameter based product space Hphy are then described. Each of these maps makes states in Hphy, and the induced operators, a model of the axioms. Consequences of the existence of many of these maps are discussed along with the dependence of Grover's and Shor's algorithms on these maps. The importance of the main physical requirement, that the basic arithmetic operations are efficiently implementable, is discussed. This condition states that there exist physically realizable Hamiltonians that can implement the basic arithmetic operations and that the space-time and thermodynamic resources required are polynomial in L.
Earlier work on modular arithmetic of k-ary representations of length L of the natural numbers in quantum mechanics is extended here to k-ary representations of all natural numbers, and to integers and rational numbers. Since the length L is indeterminate, representations of states and operators using creation and annihilation operators for bosons and fermions are defined. Emphasis is on definitions and properties of operators corresponding to the basic operations whose properties are given by the axioms for each type of number. The importance of the requirement of efficient implementability for physical models of the axioms is emphasized. Based on this, successor operations for each value of j corresponding to addition of k^{j-1} if j>0 and k^{j} if j<0 are defined. It follows from the efficient implementability of these successors, which is the case for all computers, that implementation of the addition and multiplication operators, which are defined in terms of polynomially many iterations of the successors, should be efficient. This is not the case for definitions based on the successor for j=1 only. This is the only successor defined in the usual axioms of arithmetic.
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.
An axiomatic approach to the mathematical formalism of quantum mechanics, based upon a certain concept of conditional probability, has been proposed in two recent papers by the author. It leads to Jordan operator algebras and thus comes rather close to the standard Hilbert space model of quantum mechanics, but still includes the so-called exceptional Jordan algebras, for which a Hilbert space representation does not exist. This approach is now extended by defining a mathematical model of composite systems. Such a model is required for the study of the joint distribution of two quantum observables. A very general type of observables (not only the real-valued observables corresponding to the self-adjoint operators) is considered. The joint distribution is defined, using the concept of conditional probability, and exhibits a certain dependence on the succession of the observations which is different from the classical case and unknown so far in quantum mechanics. Finally, it turns out that, at least in the finite-dimensional case, a really satisfying model of the composite system exists only if each single system is modeled by a complex Jordan matrix algebra (or a direct sum), and the model then becomes the tensor product. This provides some reasoning why the exceptional Jordan algebras can be ruled out, why quantum mechanics needs the complex numbers and the complex Hilbert space, and why the tensor product is the right choice for the model of a composite system.
In this final chapter we discuss composite systems in Quantum Mechanics. First, we lay out the necessary conditions that the state space of a composite system must satisfy by considering the trivial case of two juxtaposed systems. Then we point out the features that the interesting case of a composite system, whose parts interact with each other, must have. We then move to demonstrate how such a system can be naturally described (quantized) by employing tensor product spaces. The definition and the most relevant properties of tensor product spaces are introduced. Then the meaning of “interactions” in quantum mechanics is explored and the most nontrivial aspects of correlation and entanglement are discussed. Finally, in the last section, we introduce systems of identical particles. We describe how to represent the state space of such systems in terms of one-particle states, number operators and occupation number representations. Classification of identical particles into Bosons and Fermions is explained. Finally, introducing symmetric and antisymmetric states, the traditional quantization scheme using the symmetrization postulate is described. The permutation operator is introduced which is the main tool for dealing with identical particle systems in this language. We wind up by indicating how exchange degeneracy is removed which embodies the signature of indistinguishability.