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arXiv:2103.14105 · Symmetry

Symmetries and Geometries of Qubits, and Their Uses

A. R. P. Rau

Published Sep 18, 2021• 10 citations

Abstract

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.

DOI: 10.3390/sym13091732

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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.

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A two-sphere (``Bloch'' or ``Poincare'') is familiar for describing the dynamics of a spin-

1∕2

particle or light polarization. Analogous objects are derived for unitary groups larger than SU(2) through an iterative procedure that constructs evolution operators for higher-dimensional

\mathrm{SU}(N)

in terms of lower-dimensional ones. We focus, in particular, on the SU(4) of two qubits which describes all possible logic gates in quantum computation and entangled states in quantum-information sciences. For a general Hamiltonian of SU(4) with 15 parameters, and for Hamiltonians of its various subgroups so that fewer parameters suffice, we derive Bloch-like rotation of unit vectors analogous to the one familiar for a single spin in a magnetic field. The unitary evolution of a quantal spin pair is thereby expressed as rotations of real, many-dimensional vectors. Correspondingly, the manifolds involved are Bloch two-spheres along with higher dimensional manifolds such as a four-sphere for the SO(5) subgroup and an eight-dimensional Grassmannian manifold for the general SU(4). The latter may also be viewed as two, mutually orthogonal, real six-dimensional unit vectors moving on a five-sphere with an additional phase constraint. This geometrical picture for two spins provides the extension and generalization of the Bloch sphere that has proved invaluable for the understanding of the dynamics of a single spin.

FINDING: Spin-1/2 qubit symmetries unify Lorentz/Poincaré group structure with Bloch sphere geometry via SU(2) representations, while dissipative dynamics extend the Bloch ball to gl(3,R)⊕R³ semigroup actions. | MATH: SU(2) ≅ Spin(3) double-covers SO(3); Bloch sphere S² ≅ SU(2)/U(1); Poincaré sphere = same SU(2) parameterization for polarization; dissipative semigroup: gl(3,R) ⊕ R³ acting on Bloch vector r ∈ ℝ³; fermionic time-reversal symmetry ⇒ trivial Bloch bundle (Z₂ index triviality); Wilson lines: holonomy ∮ A · dk over Brillouin zone torus. | CONNECTION: SU(2) binary polyhedral groups (binary tetrahedral 2T, octahedral 2O, icosahedral 2I) are finite subgroups of SU(2) — their orders (24, 48, 120) relate to crystallographic root systems (A₃, B₃, H₃). The icosahedral group 2I has golden ratio φ = (1+√5)/2 ≈ 1.618 embedded in its character table and quaternion generators; the Bloch sphere's equatorial projection yields stereographic ratios; dissipation's gl(3,R) contains SO(3) rota Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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.

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)\otimes 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.