The key dynamic properties of fermionic systems, like controllability, reachability, and simulability, are investigated in a general Lie-theoretical frame for quantum systems theory.It just requires knowing drift and control Hamiltonians of an experimental set-up.Then one can easily determine all the states that can be reached from any given initial state.Likewise all the quantum operations that can be simulated with a given set-up can be identified.Observing the parity superselection rule, we treat the fully controllable and quasifree cases of fermions, as well as various translation-invariant and particle-number conserving cases.We determine the respective dynamic system Lie algebras to express reachable sets of pure (and mixed) states by explicit orbit manifolds.
The key dynamic properties of fermionic systems, like controllability, reachability, and simulability, are investigated in a general Lie-theoretical frame for quantum systems theory.It just requires knowing drift and control Hamiltonians of an experimental set-up.Then one can easily determine all the states that can be reached from any given initial state.Likewise all the quantum operations that can be simulated with a given set-up can be identified.Observing the parity superselection rule, we treat the fully controllable and quasifree cases of fermions, as well as various translation-invariant and particle-number conserving cases.We determine the respective dynamic system Lie algebras to express reachable sets of pure (and mixed) states by explicit orbit manifolds.
Understanding the dynamical properties of quantum systems is an essential task in quantum computing, quantum control, and many-body physics. Tools such as representation theory and Lie theory provide crucial information on reachability and computational power. However, this information can be difficult to access exactly or compute efficiently for arbitrary generating sets. Here we focus on the setting of Pauli strings, which satisfy numerous exceptional properties that simplify the problem. We find deep connections between Pauli Lie algebras and certain subgroups of the Clifford group generated by transvections, through the symplectic properties of the Pauli strings. This allows us to give an invariant-based perspective on these objects and their reachability, in the language of Pauli orbits, symmetries, and invariant subspaces. The invariant-based approach provides efficient algorithms for identifying Lie algebras and orbits, as well as a simple framework for analyzing structured Pauli generating sets. We also show in an elementary way that Clifford subgroups generated by transvections provide 3-designs for the corresponding Pauli Lie groups. We illustrate the framework through structured examples from variational quantum algorithms, restricted quantum computation, many-body systems, and random circuits.
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.
The ability to navigate and control the state space of a quantum system is fundamental to quantum computing, quantum information processing, and advanced spectroscopy. This paper explores a rigorous mathematical framework for quantifying state space accessibility by leveraging the theory of operator commutation relations, particularly within the context of Lie algebras. We demonstrate that the span of iterated commutators of system Hamiltonians and control operators forms a dynamical Lie algebra, whose dimension and structure directly dictate the reachable states and unitary transformations. For closed quantum systems, the Lie algebra rank condition provides a powerful criterion for complete controllability, asserting that if the dynamical Lie algebra spans the entire space of anti-Hermitian operators, then any state can be reached from any other, and any unitary operation can be implemented. For open quantum systems, while environmental interactions introduce decoherence and dissipation, the Lie algebraic approach can still offer insights into the attainable subspaces and the limits of control, often involving Lie semigroups and Lie wedges. We present methods to analyze the accessibility of both pure and mixed states, illustrating how the algebraic properties encoded in commutation relations provide a precise measure of the manipulability of quantum systems. The implications extend to the design of robust quantum control protocols, optimized state preparation, and the development of error-resilient quantum technologies.
Determining the unitary dynamics accessible from finite Hamiltonian resources is a central problem in Hamiltonian engineering and quantum control. Dynamical Lie algebras (DLAs) connect available control Hamiltonians with the reachable dynamics, but their use as a design tool for modifying Hamiltonian generator sets remains less developed. In this work, we develop a finite-dimensional DLA framework for three generator-set operations: composition, invariance, and reduction. For composition, we construct direct sums of component DLAs using spectral projectors on an auxiliary register. For invariance, we analyze when modifications of Pauli-string generating sets preserve the generated Lie algebra, and introduce algebraic diagnostics for added generators. For reduction, we consider compact reductive DLAs and show how projection onto selected simple ideals gives reduced generating sets whose Lie closures are the corresponding ideal sums. We illustrate these results with finite-dimensional examples and numerical checks, including direct-sum dimension addition, central-spin invariance diagnostics, and DLA-based ansatz reduction for block-local Hamiltonians. The results show how DLA structure can be used to diagnose controllability and guide Hamiltonian generator design under explicit algebraic assumptions.
The characterization of entanglement is a fundamental issue for Quantum Information Theory. But the definition of entanglement depends on the notion of locality, and thus on the tensor product structure of the state space of the composite system. This notion is affected by the presence of superselection rules that restrict the accessible Hilbert space to a direct sum of subspaces. Indistinguishability of particles imposes one such restriction, namely to totally symmetric or totally antisymmetric states. The entanglement can in this case be defined with respect to partitions of modes in the second quantization formalism. For fermionic systems the Fock space of m modes is isomorphic to the space of m qubits, but the action of creation and annihilation operators is not local, due to their anticommutation. Conservation of the parity of fermion number imposes another relevant superselection rule. It requires that local physical observables commute with the local parity operator. Taking into account the considerations above, it is possible to define the set of separable states or equivalently the concept of entanglement for fermionic systems in a number of ways. Here we analyze systematically these possibilities and the relation among the various sets of separable states. We also discuss the behavior of the different classes when taking several copies of the state, as well as the characterization of the sets in terms of the usual criteria regarding the tensor product.
Abstract The operator algebra of fermionic modes is isomorphic to that of qubits, the difference between them is twofold: the embedding of subalgebras corresponding to mode subsets and multiqubit subsystems on the one hand, and the parity superselection in the fermionic case on the other. We discuss these two fundamental differences extensively, and illustrate these through the Jordan–Wigner representation in a coherent, self-contained, pedagogical way, from the point of view of quantum information theory. Our perspective leads us to develop useful new tools for the treatment of fermionic systems, such as the fermionic (quasi-)tensor product, fermionic canonical embedding, fermionic partial trace, fermionic products of maps and fermionic embeddings of maps. We formulate these by direct, easily applicable formulas, without mode permutations, for arbitrary partitionings of the modes. It is also shown that fermionic reduced states can be calculated by the fermionic partial trace, containing the proper phase factors. We also consider variants of the notions of fermionic mode correlation and entanglement, which can be endowed with the usual, local operation based motivation, if the parity superselection rule is imposed. We also elucidate some other fundamental points, related to joint map extensions, which make the parity superselection inevitable in the description of fermionic systems.