Symmetries and Geometries of Qubits, and their Uses

The gist The symmetry SU(2) and its geometric Bloch Sphere rendering are familiar for a qubit but extension of symmetries and geometries have been investigated far less for multiple qubits, even just a pair of them, that are central to quantum information.

Introduction to Symmetries

The paper investigates the symmetry aspects of systems in quantum information [1]. It reviews the unitary group SU(2) symmetry of a single two-level system, a “qubit,” and contrasts it with the less utilized SU(4) symmetry for two-qubit systems [1]. The authors bring together both Lie algebraic and geometric perspectives, connecting them to finite projective geometries, combinatorial Steiner designs, and finite quaternionic groups [1]. This synthesis suggests a further development of Felix Klein’s Erlangen Program for symmetries and geometries [2].

Unitary Evolution Operator

The core of the analysis involves the unitary operator U(t) of time evolution for a Hermitian Hamiltonian H(t), which is central to quantum physics and quantum information science [2]. The paper details how this operator can be described algebraically as a product of three exponential factors in Eq. (4), which are related to Pauli spin operators [33]. For the SU(2) case, the description is geometrically rendered as a “spiked unit sphere” with a fiber bundle structure, SU(2)/U(1) × U(1), where S2 is the base manifold and U(1) phase is the fiber [5]. The evolution operator for an arbitrary SU(N) problem can be generalized using a three-term product of block matrices in Eq. (4), which involves solving a matrix Riccati equation for one factor, allowing iteration down to an SU(2) problem [35].

Sub-Group Symmetries and Correspondences

The analysis explores various sub-algebras of the full SU(4) group that arise in physical systems [8]. Key examples include:

1. The trivial example SU(2) × U(1) × SU(2), which involves seven operators and parameters, referred to as the “stabilizer” in quantum error correction [52]. This symmetry leads to a geometrical rendering of two independent spiked Bloch spheres linked by a U(1) fiber element [7].

2. The SO(5) “Desargues sub-group” symmetry, which arises in atomic and molecular four-level systems, is described geometrically by an S4 base manifold and two spiked S2 spheres along with two phases [35]. The evolution operator for this system involves a 5×5 antisymmetric real matrix Fµν(t) forming a 5-dimensional rotation [7].

3. The full SU(4) symmetry, involving all fifteen operators, is shown to complete the geometric picture by adding face centers and body center to the tetrahedron structure [14].

Geometric Algebra and Projective Geometries

The paper establishes deep connections between Lie groups and geometric algebra through several frameworks:

- Geometric Algebra:

The study of tensors, spinors, quaternions, and Clifford algebras has been known as geometric algebra [2]. This framework is linked to the third ladder in Klein’s Erlangen Program [2]. The fifteen generators of the qubit-qubit SU(4) are placed in one-to-one correspondence with finite projective geometries, combinatorial Steiner designs, and finite quaternionic groups [3].

- Finite Projective Geometry:

The fifteen generators of the qubit-qubit SU(4) can be placed in one-to-one correspondence with finite projective geometries, combinatorial Steiner designs, and finite quaternionic groups [3]. Specifically, fifteen points form the important finite projective geometry PG(3,2) [3]. The Fano sub-group symmetry of two-qubits is geometrically represented by the Fano Plane PG(2,2) [10].

- Quaternionic Correspondence:

The Cayley table for the order-16 complex quaternion group is shown in Table V and may be viewed as direct products of C2 with the order-8 groups, whether Q8 or D4 [53]. The binary labelling extends to a 4-binary (txyz) which corresponds to the PG(q, 2) of X-states of q-qubits [16].

Higher Dimensional Spins and Generalization

The symmetry structure permits ready generalization to higher numbers of qubits and qudits [D]. For a system of three qubits, the SU(2) × U(1) × SU(2) × U(1) × SU(2) × U(1) × SU(2) group of 15 generators is the sub-group symmetry for three-qubit X-states [D]. The pure states of q-qubits constitute the finite projective geometry PG(q, 2), and for q = 3, this is geometrically represented by the same 3-simplex tetrahedron used for two qubits [D]. The total number of parameters for a density matrix of q-qubits follows the sequence 3, 15, 63, 255, which corresponds to PG(2q-1,2) [D].

Geometric View and Entanglement

The final section connects the Lie group approach with the geometric algebra approach [IV]. The geometry of entanglement is described by considering a six-dimensional real metric vector space V with a non-degenerate quadratic form Q: V → R [4]. Bilinear combinations of vectors in this space, denoted as Gij, are placed in one-to-one correspondence with the Oi generators introduced earlier [14]. Only the nine Gij with i, j = 1 − 3 that is two-spin operators, as centers support entanglement as per earlier remarks [22]. The geodesic distance with respect to the Mannoury-Fubini-Study metric provides a natural object for quantifying entanglement of the qubits according to the prescription that entanglement is the geodesic distance to the nearest separable state [83].

Conclusion

The paper demonstrates how Lie groups, geometric algebra, and finite projective geometries provide a unified framework for understanding quantum correlations like entanglement [1]. The correspondence between continuous Lie groups and discrete finite groups offers further insight into quantum information problems [3].

reference:2103.14105page2,extension of symmetries and geometries have been investigated far less for multiple qubits, even just a pair of them, that are central to quantum information.

reference:2103.14105page3,This review brings together both the Lie algebraic and group representation perspective of quantum physics and the geometric algebraic one, along with connections to complex quaternions.

reference:2103.14105page6,The fifteen generators of the qubit-qubit SU(4) symmetry can be placed in one-to-one correspondence with finite projective geometries, combinatorial Steiner designs, and finite quaternionic groups.

reference:2103.14105page7,The same number of generators in re-arranged combinations describe SO(6), the group of six-dimensional rotations, and its closely allied non-compact groups SO(4,2) of the hydrogen atom in non-relativistic quantum mechanics [2].

reference:2103.14105page8,This Review is broadly in two parts. Section II consists the first, dealing with the time evolution operator of multiple qubits in quantum information, derived on the basis of a Lie algebra of generators that close under commutation.

reference:2103.14105page9,The logic is that such local measurements on A or B alone cannot change quantum correlations but can provide all classical correlations.

reference:2103.14105page27,The geodesic distance with respect to this metric provides a natural object for quantifying entanglement of the qubits according to the prescription that entanglement is the geodesic distance to the nearest separable state [83, 84].

reference:2103.14105page26,The total number of parameters for a density matrix of q-qubits is the sequence 3, 15, 63, 255, which is PG(2q-1,2) [71].

reference:2103.14105page24,The total number of parameters for a density matrix of q-qubits is the sequence 3, 15, 63, 255, which is PG(2q-1,2) [71].

reference:2103.14105page27,The geometric literature has also extended beyond two-qubits to q-qubits.

reference:2103.14105page26,The roles of W(5, 2) for three qubits and W(7, 2) for four has also been discussed recently [92].

reference:2103.14105page8,The full SU(4) amounts to a simple modification of the previously considered Spin(5) by adding a term proportional to the unit operator to the diagonal blocks and making the four F5µ complex, with F6µ absorbed as their imaginary parts [35].

reference:2103.14105page9,The full SU(4) amounts to a simple modification of the previously considered Spin(5) by adding a term proportional to the unit operator to the diagonal blocks and making the four F5µ complex, with F6µ absorbed as their imaginary parts [35].

reference:2103.

1. Increased Correlation Analysis for Multi-Qubit Systems (SU(4)): The system can analyze quantum correlations such as entanglement [1], quantum discord [5, 6] by exploiting the fifteen generators of the continuous SU(4) Lie group for two-qubits and its connections to finite projective geometries, combinatorial Steiner designs, and finite quaternionic groups.

2. Enhanced Time Evolution Operator Construction: The system can construct the unitary time evolution operator U(t) for any SU(N) problem by using the general form derived from the step-up/down combinations σ± ≡ σx ± iσy and its generalization to block matrices, which is key for handling complex Hamiltonians in multi-qubit systems.

3. Symmetry Identification via Subgroup Analysis: The system can efficiently identify relevant physical subsystems by utilizing the commutator table of SU(4) operators (Table I) to locate sub-groups like SU(2) × U(1) × SU(2) or SO(5), which correspond to specific physical scenarios such as stabilizer symmetries in quantum error correction.

4. Geometric State Representation: The system can characterize the physical states of multiple qubits by mapping them onto geometric objects, specifically relating the X-states of two qubits to the Fano triangle (Fig. 5) or its higher-dimensional generalizations, which provides a geometric rendering of their properties.

5. Higher Dimensional Spin Handling (Qudits): The system can extend its analysis to higher dimensional systems by using the generalized evolution operator derived from Eq. (11) and Eq. (12), which leads to a linear Bloch-like equation in six dimensions, allowing for the description of state spaces as a Grassmannian manifold [35].

The symmetry SU(2) and its geometric Bloch Sphere rendering are familiar for a qubit (spin-1/2) but extension of symmetries and geometries have been investigated far less for multiple qubits, even just a pair of them, that are central to quantum information. In the last two decades, two different approaches with independent starting points and motivations have come together for this purpose. One was to develop the unitary time evolution of two or more qubits for studying quantum correlations, exploiting the relevant Lie algebras and especially sub-algebras of the Hamiltonians involved, and arriving at connections to finite projective geometries and combinatorial designs. Independently, geometers studying projective ring lines and associated finite geometries have come to parallel conclusions. This review brings together both the Lie algebraic and group representation perspective of quantum physics and the geometric algebraic one, along with connections to complex quaternions. Together, all 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 may provide further insight into problems in quantum information. Extensions are considered for multiple qubits and higher spin or higher dimensional qudits.