A quantum spin-1/2, and its associated su(2) algebra of Pauli spin matrices are familiarly linked to Clifford algebra and quaternions. Somewhat more loosely, we develop connections between the su(4) algebra of two spins and of its sub-algebras, which are important throughout the field of quantum information, with octonions, the projective plane of seven elements, and entities in projective geometry and design theory called partial Steiner systems of 10 and 15 elements. Although not the exact correspondence between a single spin and quaternions, the close analogy between these geometrical objects and the 15 operators of a two-spin system may be useful to both projective geometry and quantum information.
A quantum spin-1/2, and its associated su(2) algebra of Pauli spin matrices are familiarly linked to Clifford algebra and quaternions. Somewhat more loosely, we develop connections between the su(4) algebra of two spins and of its sub-algebras, which are important throughout the field of quantum information, with octonions, the projective plane of seven elements, and entities in projective geometry and design theory called partial Steiner systems of 10 and 15 elements. Although not the exact correspondence between a single spin and quaternions, the close analogy between these geometrical objects and the 15 operators of a two-spin system may be useful to both projective geometry and quantum information.
The link between a quantum spin 1/2 and its associated su(2) algebra of Pauli spin matrices with Clifford algebra and quaternions is well known. A pair of spins or qubits, which are important throughout the field of quantum information for describing logic gates and entangled states, has similarly an su(4) algebra. We develop connections between this algebra and its subalgebras with the projective plane of seven elements (also related to octonions) and other entities in projective geometry and design theory.
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 paper the geometry of two and three-qubit states under local unitary groups is discussed. We first review the one qubit geometry and its relation with Riemannian sphere under the action of group SU(2). We show that the quaternionic stereographic projection intertwines between local unitary group SU(2)⊗SU(2) and quaternionic Möbius transformation. The invariant term appearing in this operation is related to concurrence measure. Yet, there exists the same intertwining stereographic projection for much more global group Sp(2), generalizing the familiar Bloch sphere in 2-level systems. Subsequently, we introduce octonionic stereographic projection and octonionic conformal map (or octonionic Möbius maps) for three-qubit states and find evidence that they may have invariant terms under local unitary operations which shows that both maps are entanglement sensitive.
This paper develops a geometric model for coupled two-state quantum systems (qubits), which is formulated using geometric (aka Clifford) algebra. It begins by showing how Euclidean spinors can be interpreted as entities in the geometric algebra of a Euclidean vector space. This algebra is then lifted to Minkowski space-time and its associated geometric algebra, and the insights this provides into how density operators and entanglement behave under Lorentz transformations are discussed. The direct sum of multiple copies of space-time induces a tensor product structure on the associated algebra, in which a suitable quotient is isomorphic to the matrix algebra conventionally used in multi-qubit quantum mechanics. Finally, the utility of geometric algebra in understanding both unitary and nonunitary quantum operations is demonstrated on several examples of interest in quantum information processing.
Clifford algebras are used for definition of spinors. Because of using spin-1/2 systems as an adequate model of quantum bit, a relation of the algebras with quantum information science has physical reasons. But there are simple mathematical properties of the algebras those also justifies such applications. First, any complex Clifford algebra with 2n generators, Cl(2n,C), has representation as algebra of all 2^n x 2^n complex matrices and so includes unitary matrix of any quantum n-gate. An arbitrary element of whole algebra corresponds to general form of linear complex transformation. The last property is also useful because linear operators are not necessary should be unitary if they used for description of restriction of some unitary operator to subspace. The second advantage is simple algebraic structure of Cl(2n) that can be expressed via tenzor product of standard "building units" and similar with behavior of composite quantum systems. The compact notation with 2n generators also can be used in software for modeling of simple quantum circuits by modern conventional computers.