The quantum theory of a finite quantum system with L degrees of freedom is usually set up by associating it with a Hilbert space of dimension d ( L ) and identifying observables and states in the matrix algebra . For the case d = 2 m , m integer, this algebra can be identified with the Clifford algebra . The case of d = 2 m dimensions is simply realized by a system with m dichotomic degrees of freedom, an m -qubit system for instance. The physically relevant new point is the appearance of a new (symmetry-?)group SO (2 m ). A possible interpretation of the space in which this group operates is proposed. It is shown that the eigenvalues of m -qubit-type states only depend on SO (2 m )-invariants. We use this fact to determine state parameter domains (generalized Bloch spheres) for states classified as SO (2 m )-tensors. The classification of states and interactions of components of a physical m -qubit system as k -tensors and pseudotensors (0 ⩽ k ⩽ m ) leads to rules similar to those found in elementary quantum mechanics. The question of electromagnetic interactions is shortly broached. We sketch, pars pro toto, a graphical interpretation of tensor contractions appearing in perturbative expansions.
Certain families of quantum mechanical models can be described and solved efficiently on a classical computer, including qubit or qudit Clifford circuits and stabilizer codes, free-boson or free-fermion models, and certain rotor and GKP codes. We show that all of these families can be described as instances of the same algebraic structure, namely quadratic functions over abelian groups, or more generally over (super) Hopf algebras. Different kinds of degrees of freedom correspond to different "elementary" abelian groups or Hopf algebras:
Z2 for qubits,
Zd for qudits,
R for continuous variables, both
Z and
R/Z for rotors, and a super Hopf algebra
F for fermionic modes. Objects such as states, operators, superoperators, or projection-operator valued measures, etc, are tensors. For the solvable models above, these tensors are quadratic tensors based on quadratic functions. Quadratic tensors with
n degrees of freedom are fully specified by only
O(n2) coefficients. Tensor networks of quadratic tensors can be contracted efficiently on the level of these coefficients, using an operation reminiscent of the Schur complement. Our formalism naturally includes models with mixed degrees of freedom, such as qudits of different dimensions. We also use quadratic functions to define generalized stabilizer codes and Clifford gates for arbitrary abelian groups. Finally, we give a generalization from quadratic (or 2nd order) to
ith order tensors, which are specified by
O(ni) coefficients but cannot be contracted efficiently in general.
In the study of d-dimensional quantum channels
(dgeq2), an assumption which is not very restrictive, and which has a natural physical interpretation, is that the corresponding Kraus operators form a representation of a Lie algebra. Physically, this is a symmetry algebra for the interaction Hamiltonian. This paper begins a systematic study of channels defined by representations; the famous Werner-Holevo channel is one element of this infinite class. We show that the channel derived from the defining representation of SU(n) is a depolarizing channel for all
n, but for most other representations this is not the case. Since the Bloch sphere is not appropriate here, we develop technology which is a generalization of Bloch's technique. Our method works by representing the density matrix as a polynomial in symmetrized products of Lie algebra generators, with coefficients that are symmetric tensors. Using these tensor methods we prove eleven theorems, derive many explicit formulas and show other interesting properties of quantum channels in various dimensions, with various Lie symmetry algebras. We also derive numerical estimates on the size of a generalized ``Bloch sphere'' for certain channels. There remain many open questions which are indicated at various points through the paper.
A generalization is presented of the kinematical algebra so(5), shown previously to be relevant for the description of the internal dynamics (Zitterbewegung) of Dirac’s electron. The algebra so(n+2) is proposed for the case of a compact quantum system with n degrees of freedom. Associated wave equations follow from boosting these compact quantum systems. There exists a contraction to the kinematical algebra of a system with n degrees of freedom of the usual type, by which the commutation relations between n coordinate operators Qi and corresponding momentum operators Pi, occurring within the so(n+2) algebra, go over into the usual canonical commutation relations. The so(n+2) algebra is contrasted with the sl(l,n) superalgebra introduced recently by Palev in a similar context: because so(n+2) has spinor representations, its use allows the possibility of interpreting the half-integral spin in terms of the angular momentum of internal finite quantum systems. Connection is made with the ideas of Weyl on the possible use in quantum mechanics of ray representation of finite Abelian groups, and so also with other recent works on finite quantum systems. Possible directions of future research are indicated.
Lie groups, and therefore Lie algebras, are fundamental structures in quantum physics that determine the space of possible trajectories of evolving systems. However, classification and characterization methods for these structures are often impractical for larger systems. In this work, we provide a comprehensive classification of Lie algebras generated by an arbitrary set of Pauli operators, from which an efficient method to characterize them follows. By mapping the problem to a graph setting, we identify a reduced set of equivalence classes: the free-fermionic Lie algebra, the set of all anti-symmetric Paulis on n qubits, the Lie algebra of symplectic Paulis on n qubits, and the space of all Pauli operators on n qubits, as well as controlled versions thereof. Moreover, out of these, we distinguish 6 Clifford inequivalent cases and find a simple set of canonical operators for each, which allow us to give a physical interpretation of the dynamics of each class. Our findings reveal a no-go result for the existence of small Lie algebras beyond the free-fermionic case in the Pauli setting and offer efficiently computable criteria for universality and extendibility of gate sets. These results bear significant impact in ideas in a number of fields like quantum control, quantum machine learning, or classical simulation of quantum circuits.
In the study of d-dimensional quantum channels (d⩾2), an assumption which includes many interesting examples, and which has a natural physical interpretation, is that the corresponding Kraus operators form a representation of a Lie algebra. Physically, this is a symmetry algebra for the interaction Hamiltonian. This paper begins a systematic study of channels defined by representations; the famous Werner-Holevo channel is one element of this infinite class. We show that the channel derived from the defining representation of su(n) is a depolarizing channel for all n, but for most other representations this is not the case. Since the standard Bloch sphere only exists for the qubit representation of su(2), we develop a consistent generalization of Bloch’s technique. By representing the density matrix as a polynomial in Lie algebra generators, we determine a class of positive semidefinite matrices which represent quantum states for various channels defined by finite-dimensional representations of semisimple Lie algebras. We also give a general method for finding positive semidefinite matrices using Lie algebraic trace identities. This includes an analysis of channels based on the exceptional Lie algebra g2 and the Clifford algebra.