Projection operators are central to the algebraic formulation of quantum theory because both wavefunction and hermitian operators(observables) have spectral decomposition in terms of the spectral projections. Projection operators are hermitian operators which are idempotents also. We call them quantum idempotents. They are also important for the conceptual understanding of quantum theory because projection operators also represent observation process on quantum system. In this paper we explore the algebra of quantum idempotents and show that they generate Iterant algebra (defined in the paper), Lie algebra, Grassmann algebra and Clifford algebra which is very interesting because these later algebras were introduced for the geometry of spaces and hence are called geometric algebras. Thus the projection operator representation gives a new meaning to these geometric algebras in that they are also underlying algebras of quantum processes and also they bring geometry closer to the quantum theory. It should be noted that projection operators not only make lattices of quantum logic but they also span projective geometry. We will give iterant representations of framed braid group algebras, parafermion algebras and the $su(3)$ algebra of quarks. These representations are very striking because iterant algebra encodes the spatial and temporal aspects of recursive processes. In that regard our representation of these algebras for physics opens up entirely new perspectives of looking at fermions,spins and parafermions(anyons).
Projection operators are central to the algebraic formulation of quantum theory because both wavefunction and hermitian operators(observables) have spectral decomposition in terms of the spectral projections. Projection operators are hermitian operators which are idempotents also. We call them quantum idempotents. They are also important for the conceptual understanding of quantum theory because projection operators also represent observation process on quantum system. In this paper we explore the algebra of quantum idempotents and show that they generate Iterant algebra (defined in the paper), Lie algebra, Grassmann algebra and Clifford algebra which is very interesting because these later algebras were introduced for the geometry of spaces and hence are called geometric algebras. Thus the projection operator representation gives a new meaning to these geometric algebras in that they are also underlying algebras of quantum processes and also they bring geometry closer to the quantum theory. It should be noted that projection operators not only make lattices of quantum logic but they also span projective geometry. We will give iterant representations of framed braid group algebras, parafermion algebras and the su(3) algebra of quarks. These representations are very striking because iterant algebra encodes the spatial and temporal aspects of recursive processes. In that regard our representation of these algebras for physics opens up entirely new perspectives of looking at fermions,spins and parafermions(anyons).
We review our approach to quantum mechanics adding also some new interesting results. We start by giving proof of two important theorems on the existence of the and Clifford algebras. This last algebra gives proof of the von Neumann basic postulates on the quantum measurement explaining thus in an algebraic manner the wave function collapse postulated in standard quantum theory. In this manner we reach the objective to expose a self-consistent version of quantum mechanics. We give proof of the quantum like Heisenberg uncertainty relations, the phenomenon of quantum Mach Zender interference as well as quantum collapse in some cases of physical interest We also discuss the problem of time evolution of quantum systems as well as the changes in space location. We also give demonstration of the Kocken-Specher theorem, and also we give an algebraic formulation and explanation of the EPR . By using the same approach we also derive Bell inequalities. Our formulation is strongly based on the use of idempotents that are contained in Clifford algebra. Their counterpart in quantum mechanics is represented by the projection operators that are interpreted as logical statements, following the basic von Neumann results. Using the Clifford algebra we are able to invert such result. According to the results previously obtained by Orlov in 1994, we are able to give proof that quantum mechanics derives from logic. We show that indeterminism and quantum interference have their origin in the logic.
In this paper we reexamine the definition of parafermions and parabosons by means of Green's triple relations, and extend these relations by including a parity operator P which is also determined by means of triple relations. As a consequence, we are dealing with new algebraic structures. It is shown that the algebra underlying a set of n parafermions together with P is the orthogonal Lie algebra so(2n+2). The Fock spaces correspond to particular irreducible representations of so(2n+2), and the action of P in these spaces leads to interesting observations. Next, we show that the algebra underlying a set of n parabosons together with P is the orthosymplectic Lie superalgebra osp(2∣2n). In this case, the Fock spaces correspond to certain irreducible infinite-dimensional representations of osp(2∣2n). Both for parafermions and parabosons the spectrum of P is closely related to the so-called order of statistics p, introduced by Green.
This paper presents a unified algebraic framework for n-qubit quantum computation rooted in the complex Clifford algebra Cl(2n,C). We leverage the fundamental isomorphism Cl(2n,C)≅M(2n,C) to directly identify the algebra of all n-qubit quantum operators (including states, gates, and observables) with elements of Cl(2n,C). This operator-centric perspective treats quantum states, whether pure or mixed, as density multivectors ρ∈Cl(2n,C) satisfying standard physical constraints (ρ†=ρ, 2n⟨ρ⟩0=1, positivity), and unitary gates U as multivectors obeying U†U=I. Quantum evolution ρ′=UρU† and measurements p(m)=2n⟨Pmρ⟩0 are thus described by intrinsic Clifford algebraic operations. This contrasts with, yet encompasses, specific realizations like spinor representations based on Witt ideals (S=Cl(2n,C)∏jfjfj†), which are shown to be particular constructions within this universal algebra. By analyzing Deutsch's algorithm through these different lenses, we demonstrate the generality and unifying power of the operator-centric multivector approach. This formulation not only provides a comprehensive algebraic language that naturally handles mixed states and general quantum processes but also opens avenues for applying the rich geometric and structural tools of Clifford algebras—such as grade decomposition and invariant theory—to gain deeper insights into quantum information, entanglement, and dynamics.
The symmetry SU(2) and its geometric Bloch Sphere rendering have been successfully applied to the study of a single qubit (spin-1/2); however, the extension of such symmetries and geometries to multiple qubits—even just two—has been investigated far less, despite the centrality of such systems for quantum information processes. In the last two decades, two different approaches, with independent starting points and motivations, have been combined for this purpose. One approach has been to develop the unitary time evolution of two or more qubits in order to study quantum correlations; by exploiting the relevant Lie algebras and, especially, sub-algebras of the Hamiltonians involved, researchers have arrived at connections to finite projective geometries and combinatorial designs. Independently, geometers, by studying projective ring lines and associated finite geometries, have come to parallel conclusions. This review brings together the Lie-algebraic/group-representation perspective of quantum physics and the geometric–algebraic one, as well as their connections to complex quaternions. Altogether, this may be seen as further development of Felix Klein’s Erlangen Program for symmetries and geometries. In particular, the fifteen generators of the continuous SU(4) Lie group for two qubits can be placed in one-to-one correspondence with finite projective geometries, combinatorial Steiner designs, and finite quaternionic groups. The very different perspectives that we consider may provide further insight into quantum information problems. Extensions are considered for multiple qubits, as well as higher-spin or higher-dimensional qudits.
Standard quantum information is formulated over complex Hilbert spaces, where the pure state space of a single qubit is geometrically encoded by the first Hopf fibration S3→S2. Beyond this familiar setting, the normed division algebras C, H and O provide a hierarchy of increasingly rich algebraic and topological structures that has motivated several extensions of quantum theory and quantum computation. This review synthesizes the literature connecting quaternionic and octonionic frameworks, Clifford algebras, spinors, projective spaces and Hopf fibrations. We emphasize a central conceptual point that is often blurred in the literature: the second and third Hopf fibrations play two distinct roles, namely, (i) as kinematical descriptions of hypercomplex single-particle state spaces such as the quaternionic projective line HP1≃S4, and (ii) as entanglement-sensitive descriptions of multi-qubit complex systems, especially two- and three-qubit Hilbert spaces. On the algebraic side, Clifford and geometric algebras provide a natural language for rotations, spinors, and gate synthesis, while quaternionic Hilbert modules furnish a mathematically consistent extension of standard qubit kinematics and dynamics. By contrast, octonionic models face major obstructions due to non-associativity, which affects inner products, tensor products, spectral theory and circuit composition. We therefore distinguish carefully between robust results, partial constructions and speculative directions. The outcome is a unified geometric review of hypercomplex quantum information, together with a map of open problems at the interface of topology, noncommutative algebra, and quantum computation.
In this thesis, the complex Clifford algebra and its representation are introduced as a foundation of our geometrical description of quantum states. An inner product and a metric are defined on the matrix representation in a natural way, so that it has the structure of a Hilbert space and a Riemannian manifold. In particular, the set of D x D pure density operators which de-scribe n-qubit pure states, where D = 2\\(^n\\) and n is a natural number, is characterised as the set of D x D positive operators whose associated quadratic forms satisfy the Fierz identities and the normalisation condition. Then this set is, in fact, nothing but the complex projective space ‚Äövëvá‚Äövëv¥\\(^{D-1}\\). Further-more, the fibre bundle S\\(^{2D-1}\\) ‚Äövúv¿ ‚Äövëvá‚Äövëv¥\\(^{D-1}\\) is derived from the construction of the space of D x D pure density operators, and its bundle projection provides the natural correspondence between the two formulations of quantum mechanics, namely, the state vector formulation and the density operator formulation. The single qubit state case and the two-qubit pure state case are ex-plored intensively as examples. For the single qubit case, both pure and mixed states are discussed explicitly in terms of the Clifford algebra description along with the `mixedness' of a single qubit state, and the Riemannian structure of the space of single qubit pure density operators is examined. For the two-qubit case, the Clifford algebra description is discussed explicitly in relation to the Fierz identities, and, by employing the reduced density operators, it is shown that the concurrence of a two-qubit pure density operator coincides with the 'mixedness' of the reduced density operators. In addition, from this viewpoint, the \EPR paradox\" is examined as a more geometrically precise illustrated example."