We consider algebras underlying Hilbert spaces used by quantum information algorithms. We show how one can arrive at equations on such algebras which define n-dimensional Hilbert space subspaces which in turn can simulate quantum systems on a quantum system. In doing so we use MMP diagrams and linear algorithms. MMP diagrams are tractable since an n-block of an MMP diagram has n elements while an n-block of a standard lattice diagram has 2^n elements. An immediate test for such an approach is a generation of minimal and arbitrary Kochen-Specker vectors and we present a minimal state-independent Kochen-Specker set of seven vectors from a Hilbert space with more than four dimensions.
We consider algebras underlying Hilbert spaces used by quantum information algorithms. We show how one can arrive at equations on such algebras which define n-dimensional Hilbert space subspaces which in turn can simulate quantum systems on a quantum system. In doing so we use MMP diagrams and linear algorithms. MMP diagrams are tractable since an n-block of an MMP diagram has n elements while an n-block of a standard lattice diagram has 2^n elements. An immediate test for such an approach is a generation of minimal and arbitrary Kochen-Specker vectors and we present a minimal state-independent Kochen-Specker set of seven vectors from a Hilbert space with more than four dimensions.
We use classes of Hilbert lattice equations for an alternative representation of Hilbert lattices and Hilbert spaces of arbitrary quantum systems that might enable a direct introduction of the states of the systems into quantum computers. More specifically, we look for a way to feed a quantum computer with algebraic equations of n-th order underlying an infinite dimensional Hilbert space description of quantum systems. A number of new results on states defined on Hilbert lattices are presented and discussed and a number of recently obtained results in the field of Hilbert space equations are reviewed.
We give a constructive and exhaustive definition of Kochen–Specker (KS) vectors in a Hilbert space of any dimension as well as of all the remaining vectors of the space. KS vectors are elements of any set of orthonormal states, i.e., vectors in an n -dimensional Hilbert space, , to which it is impossible to assign 1s and 0s in such a way that no two mutually orthogonal vectors from the set are both assigned 1 and that not all mutually orthogonal vectors are assigned 0. Our constructive definition of such KS vectors is based on algorithms that generate MMP diagrams corresponding to blocks of orthogonal vectors in , on algorithms that single out those diagrams on which algebraic 0 – 1 states cannot be defined, and on algorithms that solve nonlinear equations describing the orthogonalities of the vectors by means of statistically polynomially complex interval analysis and self-teaching programs. The algorithms are limited neither by the number of dimensions nor by the number of vectors. To demonstrate the power of the algorithms, all four-dimensional KS vector systems containing up to 24 vectors were generated and described, all three-dimensional vector systems containing up to 30 vectors were scanned, and several general properties of KS vectors were found.
Abstract. We give a constructive and exhaustive definition of Kochen-Specker (KS) vectors in a Hilbert space of any dimension as well as of all the remaining vectors of the space. KS vectors are elements of any set of orthonormal states, i.e., vectors in n-dim Hilbert space, H n, n ≥ 3 to which it is impossible to assign 1s and 0s in such a way that no two mutually orthogonal vectors from the set are both assigned 1 and that not all mutually orthogonal vectors are assigned 0. Our constructive definition of such KS vectors is based on algorithms that generate MMP diagrams corresponding to blocks of orthogonal vectors in R n, on algorithms that single out those diagrams on which algebraic 0-1 states cannot be defined, and on algorithms that solve nonlinear equations describing the orthogonalities of the vectors by means of statistically polynomially complex interval analysis and self-teaching programs. The algorithms are limited neither by the number of dimensions nor by the number of vectors. To demonstrate the power of the algorithms, all 4-dim KS vector systems containing up to 24 vectors were generated and described, all 3-dim vector systems containing up to 30 vectors were scanned, and several general properties of KS vectors were found.
Quantum information theory has become a central area in the two fields of quantum mechanics and information science. The focus of this work is mathematical modelling of quantum information with the use of operator algebras, in a way that accentuates the modelling, and acting or manipulation of observables and quantum states. The fundamental aim is to utilise C*-algebras, von Neumann algebras, and other frameworks in operator algebra to formally describe the behaviour of quantum systems, understand the nature of quantum entanglement, and support quantum error correction. Operator algebra usage provides a firm mathematical foundation that guarantees accuracy and contributes to the incorporation of topological mechanisms and algebraic rules necessary for explaining noncommutative concepts in quantum mechanics. In this paper, we will present some original notions of operator algebra and explain their usefulness when it comes to modeling the key dynamics, features of quantum states, measurement, and evolution. We will use a method that combines algebraic techniques to chart qubit interactions, entropic inequalities, and characterize transformations of level in unitary evolution. Through the use of Hilbert spaces and bounded linear operators, we build models in multi-qubit systems and see what the specifications have to say about quantum computing and quantum communication protocols. The paper provide a comprehensive discussion, including algebraic formulations, simulation outputs, and a critical comparison with traditional Hilbert space methods. It is quite notable that the operator algebras can say more about the non-local correlations and decoherence processes. There are extensive literature reviews, useful modelling tools, and implementation outcomes in the paper when symbolic computation tools have been utilised. This combination gives both theoretical strength and modelling instruments to develop quantum technologies in the future.
Classical computing architectures encode information into discrete binary digits residing in isolated regions of phase space. Quantum computing replaces this paradigm by leveraging complex state vectors defined within continuous, multi-dimensional Hilbert spaces. This paper provides a unified mathematical framework examining finite-dimensional Hilbert spaces in quantum information processing. It traces the structural scaling of quantum state spaces from single-qubit systems up through composite tensor product structures and multi-qubit registers. By detailing computational basis representations, state transformations, probability amplitudes, and multipartite entanglement classes, such as Bell, GHZ, and W states, this paper demonstrates how exponential dimensional scaling enables quantum parallelism while presenting computational barriers to classical simulation.
Algorithms for finding arbitrary sets of Kochen‐Specker (KS) qubits as well as all the remaining vectors in a space of an arbitrary dimension are presented. The algorithms are based on linear MMP diagrams which generate orthogonalities of KS qubits, on an algebraic definition of states on the diagrams, and on nonlinear equations corresponding to MMP diagrams whose solutions are either KS qubits or the remaining vectors of a chosen space depending on whether the diagrams allow 0–1 states or not. The complexity of the algorithms is polynomial. New results obtained with the help of the algorithms are presented.