Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2306.13481 · Journal of Physics A Mathematical and Theoretical

Generalization of Balian–Brezin decomposition for exponentials with linear fermionic part

M. A. Seifi Mirjafarlou, Arash Jafarizadeh, M. A. Rajabpour

Published Apr 12, 2024• 3 citations
arXivHTMLPDF

Abstract

Abstract Fermionic Gaussian states have garnered considerable attention due to their intriguing properties, most notably Wick’s theorem. Expanding upon the work of Balian and Brezin, who generalized properties of fermionic Gaussian operators and states, we further extend their findings to incorporate Gaussian operators with a linear component. Leveraging a technique introduced by Colpa, we streamline the analysis and present a comprehensive extension of the Balian–Brezin decomposition to encompass exponentials involving linear terms. Furthermore, we introduce Gaussian states featuring a linear part and derive corresponding overlap formulas. Additionally, we generalize Wick’s theorem to encompass scenarios involving linear terms, facilitating the expression of generic expectation values in relation to one and two-point correlation functions. We also provide a brief commentary on the applicability of the BB decomposition in addressing the BCH (Zassenhaus) formulas within the so ( N ) Lie algebra.

DOI
10.1088/1751-8121/ad3e44

Related papers

Generalized Wick's theorem for multiquasiparticle overlaps as a limit of Gaudin's theorem

S. Pérez-Martín, Luis Miguel Robledo

By using the extension of the statistical Wick's theorem (Gaudin's theorem) to deal with generalized statistical density operators (those which can be expressed as the product of and operator carrying out a canonical transformations times a density operator) and using the appropriate limits we are able to rederive in a very simple way the standard generalized Wick's theorem for overlaps of mean-field wave functions. Due to the simplicity of the derivation it is now straightforward to consider more involved cases and some of them are discussed. The present derivation also allows us to obtain general and compact formulas for other particular cases of the generalized Wick theorem involving overlaps of multiquasiparticle excitations of product wave functions. The new expressions allow us to reduce the combinatorial complexity of the standard calculation of the above overlaps.

arXiv:0707.3365•Physical Review C•Dec 17, 2007•28 citations

Extension of Wick’s theorem for many-particle matrix elements

K. Tanabe, K. Enami, Naotaka Yoshinaga

Wick's theorem for the expectation value of an operator between many-particle states is generalized to the case of a homogeneous linear transformation operator $\^O$, which leaves invariant the $2N$-dimensional linear space composed of the annihilation and the creation operators for the N fermion or boson single-particle basis. Such a transformation generally belongs to the group $\mathrm{SO}(2N,C)$ for the case of fermion, or the group $\mathrm{Sp}(2N,C)$ for the case of boson. This theorem is applied to derive a reduction theorem for the matrix element of the type $〈\ensuremath{\alpha}|{\ensuremath{\alpha}}_{1}{\ensuremath{\alpha}}_{2}\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}{\ensuremath{\alpha}}_{m}\^O{\ensuremath{\beta}}_{m+1}^{\ifmmode\dagger\else\textdagger\fi{}}{\ensuremath{\beta}}_{m+2}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}{\ensuremath{\beta}}_{2n}^{\ifmmode\dagger\else\textdagger\fi{}}|\ensuremath{\beta}〉,$ whose many-quasiparticle states and vacua appearing in both sides are represented in different quasiparticle pictures as indicated by $\ensuremath{\alpha}$ and $\ensuremath{\beta}.$ The reduction formula is useful in calculating any nuclear transition matrix element for the quantum-number projection calculation associated with the constrained Hartree-Fock-Bogoliubov solutions, and also the generator coordinate method with quantum number projection.

Physical Review C•May 1, 1999•14 citations

Practical formulation of the extended Wick’s theorem and the Onishi formula

Luis Miguel Robledo

The extended Wick's theorem for fermion operators, which is used to compute matrix elements of an arbitrary operator between two different quasiparticle vacuums, is reformulated to deal with quasiparticle vacuums expanded in a finite single particle basis not closed under the canonical transformation relating them. A new expression for the overlap of those quasiparticle vacuums is also given.

Physical Review C•Dec 1, 1994•48 citations

Matrix Elements of Fermionic Gaussian Operators in Arbitrary Pauli Bases: A Pfaffian Formula

M. A. Rajabpour, M. A. Seifi Mirjafarlou, Reyhaneh Khasseh

Fermionic Gaussian operators are foundational tools in quantum many-body theory, numerical simulation of fermionic dynamics, and fermionic linear optics. While their structure is fully determined by two-point correlations, evaluating their matrix elements in arbitrary local spin bases remains a nontrivial task, especially in applications involving quantum measurements, tomography, and basis-rotated simulations. In this work, we derive a fully explicit and general Pfaffian formula for the matrix elements of fermionic Gaussian operators between arbitrary Pauli product states. Our approach introduces a pair of sign-encoding matrices whose classification leads to a Lie algebra isomorphic to $\mathfrak{so}(2L)$. This algebraic structure not only guarantees consistency of the Pfaffian signs but also reveals deep connections to Clifford algebras. The resulting framework enables scalable computations across diverse fields -- from quantum tomography and entanglement dynamics to algebraic structure in fermionic circuits and matchgate computation. Beyond its practical utility, our construction sheds light on the internal symmetries of Gaussian operators and offers a new lens through which to explore their role in quantum information and computational models.

arXiv:2506.02809•Jun 3, 2025•0 citations

General Wick's theorem for bosonic and fermionic operators

Luca Ferialdi, Lajos Diósi

Wick's theorem provides a connection between time ordered products of bosonic or fermionic fields, and their normal ordered counterparts. We consider a generic pair of operator orderings and we prove, by induction, the theorem that relates them. We name this the general Wick's theorem (GWT) because it carries Wick's theorem as special instance, when one applies the GWT to time and normal orderings. We establish the GWT both for bosonic and fermionic operators, i.e., operators that satisfy c-number commutation and anticommutation relations respectively. We remarkably show that the GWT is the same, independent of the type of operator involved. By means of a few examples, we show how the GWT helps treat demanding problems by reducing the amount of calculations required.

arXiv:2110.02920•Physical Review A•Nov 15, 2021•9 citations

Exponential de Finetti Theorems for Fermionic Gaussian States

Jędrzej Burkat, Michał Studziński, Sergii Strelchuk

We prove an exponential variant of the Gaussian de Finetti theorem: the subsystems of permutation-invariant, free-fermionic Gaussian states are well approximated by convex combinations of almost-i.i.d. states that are Gaussian on subsets of their parts. Our result provides an error bound between the original state and its approximants that decays exponentially in the number of unconstrained parts, becoming super-exponential when the subsystem under consideration is small. The dependence of our bound on the local Hilbert space dimension is polylogarithmic, giving an exponential improvement over the standard de Finetti theorem of [Nat. Phys. 3, 645-649]. Previous works considered Gaussian-symmetric states, which are supported on the trivial irrep of the tensor matchgate representation. We extend these results to the broader class of Gaussian-invariant states, which contains, for example, i.i.d. copies of single-replica mixed Gaussian states. We show that Gaussian-invariant states are precisely the partial traces of Gaussian-symmetric states on locally enlarged replicas and always admit a purification into a larger Gaussian-symmetric state. As an application, we derive explicit convergence bounds for a hierarchy of convex relaxations of polynomial optimisation over fermionic Gaussian states. For variational energy minimisation, the hierarchy gives lower bounds converging to the global Gaussian minimum. Combined with upper bounds from Gaussian trial states, these bounds quantify how far a variational calculation can be from the best energy attainable within the Gaussian family.

quant-pharXiv:2607.25779•Jul 29, 2026•0 citations
Abstract Fermionic Gaussian states have garnered considerable attention due to their intriguing properties, most notably Wick’s theorem. Expanding upon the work of Balian and Brezin, who generalized properties of fermionic Gaussian operators and states, we further extend their findings to incorporate Gaussian operators with a linear component. Leveraging a technique introduced by Colpa, we streamline the analysis and present a comprehensive extension of the Balian–Brezin decomposition to encompass exponentials involving linear terms. Furthermore, we introduce Gaussian states featuring a linear part and derive corresponding overlap formulas. Additionally, we generalize Wick’s theorem to encompass scenarios involving linear terms, facilitating the expression of generic expectation values in relation to one and two-point correlation functions. We also provide a brief commentary on the applicability of the BB decomposition in addressing the BCH (Zassenhaus) formulas within the so ( N ) Lie algebra.
By using the extension of the statistical Wick's theorem (Gaudin's theorem) to deal with generalized statistical density operators (those which can be expressed as the product of and operator carrying out a canonical transformations times a density operator) and using the appropriate limits we are able to rederive in a very simple way the standard generalized Wick's theorem for overlaps of mean-field wave functions. Due to the simplicity of the derivation it is now straightforward to consider more involved cases and some of them are discussed. The present derivation also allows us to obtain general and compact formulas for other particular cases of the generalized Wick theorem involving overlaps of multiquasiparticle excitations of product wave functions. The new expressions allow us to reduce the combinatorial complexity of the standard calculation of the above overlaps.
Wick's theorem for the expectation value of an operator between many-particle states is generalized to the case of a homogeneous linear transformation operator Oˆ\^OOˆ, which leaves invariant the 2N2N2N-dimensional linear space composed of the annihilation and the creation operators for the N fermion or boson single-particle basis. Such a transformation generally belongs to the group SO(2N,C)\mathrm{SO}(2N,C)SO(2N,C) for the case of fermion, or the group Sp(2N,C)\mathrm{Sp}(2N,C)Sp(2N,C) for the case of boson. This theorem is applied to derive a reduction theorem for the matrix element of the type $〈\ensuremath{\alpha}|{\ensuremath{\alpha}}_{1}{\ensuremath{\alpha}}_{2}\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}{\ensuremath{\alpha}}_{m}\^O{\ensuremath{\beta}}_{m+1}^{\ifmmode\dagger\else\textdagger\fi{}}{\ensuremath{\beta}}_{m+2}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}{\ensuremath{\beta}}_{2n}^{\ifmmode\dagger\else\textdagger\fi{}}|\ensuremath{\beta}〉,$ whose many-quasiparticle states and vacua appearing in both sides are represented in different quasiparticle pictures as indicated by $\ensuremath{\alpha}$ and $\ensuremath{\beta}.$ The reduction formula is useful in calculating any nuclear transition matrix element for the quantum-number projection calculation associated with the constrained Hartree-Fock-Bogoliubov solutions, and also the generator coordinate method with quantum number projection.
Fermionic Gaussian operators are foundational tools in quantum many-body theory, numerical simulation of fermionic dynamics, and fermionic linear optics. While their structure is fully determined by two-point correlations, evaluating their matrix elements in arbitrary local spin bases remains a nontrivial task, especially in applications involving quantum measurements, tomography, and basis-rotated simulations. In this work, we derive a fully explicit and general Pfaffian formula for the matrix elements of fermionic Gaussian operators between arbitrary Pauli product states. Our approach introduces a pair of sign-encoding matrices whose classification leads to a Lie algebra isomorphic to so(2L)\mathfrak{so}(2L)so(2L). This algebraic structure not only guarantees consistency of the Pfaffian signs but also reveals deep connections to Clifford algebras. The resulting framework enables scalable computations across diverse fields -- from quantum tomography and entanglement dynamics to algebraic structure in fermionic circuits and matchgate computation. Beyond its practical utility, our construction sheds light on the internal symmetries of Gaussian operators and offers a new lens through which to explore their role in quantum information and computational models.
Wick's theorem provides a connection between time ordered products of bosonic or fermionic fields, and their normal ordered counterparts. We consider a generic pair of operator orderings and we prove, by induction, the theorem that relates them. We name this the general Wick's theorem (GWT) because it carries Wick's theorem as special instance, when one applies the GWT to time and normal orderings. We establish the GWT both for bosonic and fermionic operators, i.e., operators that satisfy c-number commutation and anticommutation relations respectively. We remarkably show that the GWT is the same, independent of the type of operator involved. By means of a few examples, we show how the GWT helps treat demanding problems by reducing the amount of calculations required.
We prove an exponential variant of the Gaussian de Finetti theorem: the subsystems of permutation-invariant, free-fermionic Gaussian states are well approximated by convex combinations of almost-i.i.d. states that are Gaussian on subsets of their parts. Our result provides an error bound between the original state and its approximants that decays exponentially in the number of unconstrained parts, becoming super-exponential when the subsystem under consideration is small. The dependence of our bound on the local Hilbert space dimension is polylogarithmic, giving an exponential improvement over the standard de Finetti theorem of [Nat. Phys. 3, 645-649]. Previous works considered Gaussian-symmetric states, which are supported on the trivial irrep of the tensor matchgate representation. We extend these results to the broader class of Gaussian-invariant states, which contains, for example, i.i.d. copies of single-replica mixed Gaussian states. We show that Gaussian-invariant states are precisely the partial traces of Gaussian-symmetric states on locally enlarged replicas and always admit a purification into a larger Gaussian-symmetric state. As an application, we derive explicit convergence bounds for a hierarchy of convex relaxations of polynomial optimisation over fermionic Gaussian states. For variational energy minimisation, the hierarchy gives lower bounds converging to the global Gaussian minimum. Combined with upper bounds from Gaussian trial states, these bounds quantify how far a variational calculation can be from the best energy attainable within the Gaussian family.