The commutant of fermionic Gaussian unitaries

• URL: http://arxiv.org/abs/2603.19210v1
• Date: Thu, 19 Mar 2026 17:56:35 GMT
• Title: The commutant of fermionic Gaussian unitaries
• Abstract summary: We characterize the $t$-th order commutants of fermionic Gaussian unitaries and of their particle-preserving subgroup acting on $n$ fermionic modes.<n>These commutants govern Haar averages over the corresponding groups and therefore play a central role in fermionic randomized protocols.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this work, we characterize the $t$-th order commutants of fermionic Gaussian unitaries and of their particle-preserving subgroup acting on $n$ fermionic modes. These commutants govern Haar averages over the corresponding groups and therefore play a central role in fermionic randomized protocols, invariant theory, and resource quantification. Using Howe dualities, we show that the particle-preserving commutant is generated by generalized copy-hopping operators, while that for general Gaussian commutant is generated by generalized quadratic Majorana bilinears together with parity. We then derive closed formulas for the dimensions of both commutants as functions of $t$ and $n$, and develop constructive Gelfand--Tsetlin procedures to obtain explicit orthonormal bases, with detailed low-$t$ examples. Our framework also clarifies the structure of replicated fermionic states and connects naturally to measures of fermionic correlations, generalized Plücker-type constraints, and the stabilizer entropy of fermionic Gaussian states. These results provide a unified algebraic description of higher-order invariants for fermionic Gaussian dynamics.

Related papers

• Computable measures of fermionic non-Gaussianity from the covariance matrix [0.005748392537700255]
  We develop a convex resource theory of fermionic non-Gaussianity and introduce two families of computable measures for pure fermionic states.<n>We analyze both families for stabilizer and translation-invariant states, where they simplify and reveal additional structure.<n>Our work establishes a resource-theoretic framework for computable fermionic non-Gaussianity that unifies notions arising across quantum information, condensed-matter physics, and quantum chemistry.
  arXiv  Detail & Related papers  (2026-07-02T14:38:25Z)
• Generalized symmetries, invariant solutions and conservation laws in the Jaynes-Cummings model [41.99844472131922]
  We investigate the Jaynes--Cummings model (JCM) using Lie symmetry analysis and conservation-law theory.<n>The dynamics is formulated as a system of partial differential equations by projecting the von Neumann equation onto the atomic degrees of freedom.<n>We derive additional conserved currents involving atomic populations, coherence, reduced-state purity, and moments of the field characteristic function.
  arXiv  Detail & Related papers  (2026-06-14T01:42:11Z)
• Constructing Fermionic Dynamics with Closed Moment Hierarchies [0.0]
  We construct a class of completely positive maps and Go-rini--Kossakowski--Sudarshan-Lindblad generators for fermionic systems.<n>We show that for even environment states the linear span of monomials up to any fixed order is invariant, which yields closed equations for low-order moments.
  arXiv  Detail & Related papers  (2026-04-01T20:07:02Z)
• Theory of the Matchgate Commutant [0.0]
  We show that operators coupling different system copies generate the Lie algebra $mathfrakso(k)$, endowing the space of invariants with rich and tractable structure.<n>We also characterize the commutant of the Clifford-matchgate subgroup, showing that restricting to signed permutations of Majorana modes yields a commutant that qualitatively diverges from the matchgate case for $k geq 4$ replicas.
  arXiv  Detail & Related papers  (2026-03-12T19:12:19Z)
• Symmetry-protected topology and deconfined solitons in a multi-link $\mathbb{Z}_2$ gauge theory [45.88028371034407]
  We study a $mathbbZ$ lattice gauge theory defined on a multi-graph with links that can be visualized as great circles of a spherical shell.<n>We show that this leads to state-dependent tunneling amplitudes underlying a phenomenon analogous to the Peierls instability.<n>By performining a detailed analysis based on matrix product states, we prove that charge deconfinement emerges as a consequence of charge-fractionalization.
  arXiv  Detail & Related papers  (2026-03-02T22:59:25Z)
• Representation theory of inhomogeneous Gaussian unitaries [0.0]
  We extend the previous framework to inhomogeneous Gaussian unitaries parameterized by $(M,z,)$.<n>We derive the group multiplication law from the Baker-Campbel-Hausdorff formula.
  arXiv  Detail & Related papers  (2026-02-09T12:55:28Z)
• Measurement-induced transitions for interacting fermions [43.04146484262759]
  We develop a field-theoretical framework that provides a unified approach to observables characterizing entanglement and charge fluctuations.<n>Within this framework, we derive a replicated Keldysh non-linear sigma model (NLSM)<n>By using the renormalization-group approach for the NLSM, we determine the phase diagram and the scaling of physical observables.
  arXiv  Detail & Related papers  (2024-10-09T18:00:08Z)
• Boson-fermion algebraic mapping in second quantization [0.0]
  We present a method to derive the structure at the basis of the mapping of bosonic algebras of creation and operators into fermionic algebras. The structure corresponds to a deformed Grassmann algebra, involving anticommuting Grassmann-type variables.
  arXiv  Detail & Related papers  (2024-09-25T18:37:34Z)
• Quantum Random Walks and Quantum Oscillator in an Infinite-Dimensional Phase Space [45.9982965995401]
  We consider quantum random walks in an infinite-dimensional phase space constructed using Weyl representation of the coordinate and momentum operators. We find conditions for their strong continuity and establish properties of their generators.
  arXiv  Detail & Related papers  (2024-06-15T17:39:32Z)
• Interacting chiral fermions on the lattice with matrix product operator norms [37.69303106863453]
  We develop a Hamiltonian formalism for simulating interacting chiral fermions on the lattice. The fermion doubling problem is circumvented by constructing a Fock space endowed with a semi-definite norm. We demonstrate that the scaling limit of the free model recovers the chiral fermion field.
  arXiv  Detail & Related papers  (2024-05-16T17:46:12Z)
• Conformal field theory from lattice fermions [77.34726150561087]
  We provide a rigorous lattice approximation of conformal field theories given in terms of lattice fermions in 1+1-dimensions. We show how these results lead to explicit error estimates pertaining to the quantum simulation of conformal field theories.
  arXiv  Detail & Related papers  (2021-07-29T08:54:07Z)
• Local optimization on pure Gaussian state manifolds [63.76263875368856]
  We exploit insights into the geometry of bosonic and fermionic Gaussian states to develop an efficient local optimization algorithm. The method is based on notions of descent gradient attuned to the local geometry. We use the presented methods to collect numerical and analytical evidence for the conjecture that Gaussian purifications are sufficient to compute the entanglement of purification of arbitrary mixed Gaussian states.
  arXiv  Detail & Related papers  (2020-09-24T18:00:36Z)
• Fermionic systems for quantum information people [0.0]
  We develop new tools for the treatment of fermionic systems. We consider variants of the notions of fermionic mode correlation and entanglement.
  arXiv  Detail & Related papers  (2020-06-04T18:18:05Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.