Related papers: Adapted-operator representations: Selective versus…
Operator systems are the unital self-adjoint subspaces of the bounded operators on a Hilbert space. Complex operator systems are an important category containing the C*-algebras and von Neumann algebras, which is increasingly of interest in…
The structure of subspaces of a Hilbert space that are invariant under unitary representations of a discrete group is related to a notion of Hilbert modules endowed with inner products taking values in spaces of unbounded operators. A…
Quantum mechanics requires the operation of quantum computers to be unitary, and thus makes it important to have general techniques for developing fast quantum algorithms for computing unitary transforms. A quantum routine for computing a…
The idea that symmetries simplify or reduce the complexity of a system has been remarkably fruitful in physics, and especially in quantum mechanics. On a mathematical level, symmetry groups single out a certain structure in the Hilbert…
The unitary group acting on the Hilbert space of three quantum bits admits a Lie subgroup, of elements which permute with the symmetric group of permutations. Under the action of such Lie subgroup, the Hilbert space splits into three…
A class of pseudo-hermitian quantum system with an explicit form of the positive-definite metric in the Hilbert space is presented. The general method involves a realization of the basic canonical commutation relations defining the quantum…
We introduce an algorithm for designing Neural Group Actions, collections of deep neural network architectures which model symmetric transformations satisfying the laws of a given finite group. This generalizes involutive neural networks…
The notion of a $U$-statistic for an $n$-tuple of identical quantum systems is introduced in analogy to the classical (commutative) case: given a selfadjoint `kernel' $K$ acting on $(\mathbb{C}^{d})^{\otimes r}$ with $r<n$, we define the…
We propose models of quantum neural networks through Clifford algebras, which are capable of capturing geometric features of systems and to produce entanglement. Due to their representations in terms of Pauli matrices, the Clifford algebras…
Simulating a fermionic system on a quantum computer requires encoding the anti-commuting fermionic variables into the operators acting on the qubit Hilbert space. The most familiar of which, the Jordan-Wigner transformation, encodes…
In this chapter, the Hilbert space framework in the mathematical theory of composite materials is introduced for studying the properties of effective operators. The goal is to introduce some of the key concepts and fundamental theorems in…
We investigate the entanglement properties of pure quantum states describing $n$ qubits. We characterize all multipartite states which can be maximally entangled to local auxiliary systems using controlled operations. A state has this…
Cluster expansions for the exponential of local operators are constructed using tensor networks. In contrast to other approaches, the cluster expansion does not break any spatial or internal symmetries and exhibits a very favourable…
Based on a number of experimentally verified physical observations, it is argued that the standard principles of quantum mechanics should be applied to the Universe as a whole. Thus, a paradigm is proposed in which the entire Universe is…
The pseudo--spectral decomposition of an $N$--particle antisymmetric 1--body positive--semidefinite operator that corresponds to the canonical convex decomposition into the extreme elements of the dual cone of the set of fermion…
In this paper we provide a method for constructing joint distributions for an arbitrary set of observables on finite dimensional Hilbert spaces irrespective of whether the observables commute or not. These distributions have a number of…
In this paper the generalized quantum states, i.e. positive and normalized linear functionals on $C^{*}$-algebras, are studied. Firstly, we study normal states, i.e. states which are represented by density operators, and singular states,…
The necessity of complex numbers in quantum mechanics has long been debated. This paper develops a real Kahler space formulation of quantum mechanics [19], asserting equivalence to the standard complex Hilbert space framework. By mapping…
Proper splittings of operators are commonly used to study the convergence of iterative processes. In order to approximate solutions of operator equations, in this article we deal with proper splittings of closed range bounded linear…
The implementation of a quantum computer requires the realization of a large number of N-qubit unitary operations which represent the possible oracles or which are part of the quantum algorithm. Until now there are no standard ways to…