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

Wigner Function and Entanglement Entropy for Bosons from Non-Equilibrium Field Theory

Ahana Chakraborty, Rajdeep Sensarma

Published Oct 24, 2018• 1 citation
OpenAlex

Abstract

We propose a new method of calculating entanglement entropy of a many-body interacting Bosonic system (open or closed) in a field theoretic approach without replica methods. The Wigner function and Renyi entropy of a Bosonic system undergoing arbitrary non-equilibrium dynamics can be obtained from its Wigner characteristic function, which we identify with the Schwinger Keldysh partition function in presence of quantum sources turned on at the time of measurement. For non-interacting many body systems, starting from arbitrary density matrices, we provide exact analytic formulae for Wigner function and entanglement entropy in terms of the single particle Green's functions. For interacting systems, we relate the Wigner characteristic to the connected multi-particle correlators of the system. We use this formalism to study the evolution of an open quantum system from a Fock state with negative Wigner function and zero entropy, to a thermal state with positive Wigner function and finite entropy. The evolution of the Renyi entropy is non-monotonic in time for both Markovian and non-Markovian dynamics. The entropy is also found to be anti-correlated with negativity of the Wigner function of a $2$ -mode open quantum system.

Related papers

Nonequilibrium Dynamics of Renyi Entropy for Bosonic Many-Particle Systems

Ahana Chakraborty, Rajdeep Sensarma

We propose a new field theoretic method for calculating Renyi entropy of a subsystem of many interacting bosons without using replica methods. This method is applicable to dynamics of both open and closed quantum systems starting from arbitrary initial conditions. Our method identifies the Wigner characteristic of a reduced density matrix with the partition function of the whole system with a set of linear sources turned on only in the subsystem, and uses this to calculate the subsystem's Renyi entropy. We use this method to study the evolution of Renyi entropy in a noninteracting open quantum system starting from an initial Fock state. We find a relation between the initial state and final density matrix which determines whether the entropy shows nonmonotonic behavior in time. For non-Markovian dynamics, we show that the entropy approaches its steady-state value as a power law with exponents governed by nonanalyticities of the bath. We illustrate that this field-theoretic method can be used to study large bosonic open quantum systems.

arXiv:1810.10545•Physical Review Letters•Nov 12, 2021•8 citations

Entanglement entropy of fermions from Wigner functions: Excited states and open quantum systems

Saranyo Moitra, Rajdeep Sensarma

We formulate a new ``Wigner characteristics''-based method to calculate entanglement entropies of subsystems of Fermions using Keldysh field theory. This bypasses the requirements of working with complicated manifolds for calculating R\'enyi entropies for many-body systems. We provide an exact analytic formula for R\'enyi and von Neumann entanglement entropies of noninteracting open quantum systems, which are initialized in arbitrary Fock states. We use this formalism to look at entanglement entropies of momentum Fock states of one-dimensional Fermions. We show that the entanglement entropy of a Fock state can scale either logarithmically or linearly with subsystem size, depending on whether the number of discontinuities in the momentum distribution is smaller or larger than the subsystem size. We also use this formalism to describe entanglement dynamics of an open quantum system starting with a single domain wall at the center of the system. Using entanglement entropy and mutual information, we understand the dynamics in terms of coherent motion of the domain wall wavefronts, creation and annihilation of domain walls, and incoherent exchange of particles with the bath.

arXiv:2006.16271•Physical review. B./Physical review. B•Nov 25, 2020•12 citations

Entanglement entropy from correlation functions of scalar fields in and out of equilibrium

Mrinal Kanti Sarkar, Saranyo Moitra, Rajdeep Sensarma

We show that odd order Rényi entropies $S^{(2q+1)}$ of a system of interacting scalar fields can be calculated as the free energy of $2q+1$ replicas of the system with additional quadratic inter-replica couplings in the subsystem at the time of measurement of the entropy. These couplings replace boundary field matching conditions. This formalism works both in and out of thermal equilibrium, for closed as well as open quantum systems, and provides a general dictionary between measurable correlation functions and entanglement entropy. $S^{(2q+1)}$ can be analytically continued to calculate the von Neumann entropy $S^{\mathrm{vN}}$. We provide an exact formula relating $S^{(2q+1)}$ and $S^{\mathrm{vN}}$ with correlation functions in a non-interacting theory. For interacting theories, we provide rules for constructing all possible Feynman diagrams for $S^{(2q+1)}$. We show that the boundary matching conditions cannot be completely eliminated while calculating Rényi entropies of even order due to presence of zero modes in replica space.

arXiv:2510.15035•Physical Review A•Jun 3, 2026•0 citations

Nonequilibrium field theory for dynamics starting from arbitrary athermal initial conditions

Ahana Chakraborty, Pranay Gorantla, Rajdeep Sensarma

Schwinger Keldysh field theory is a widely used paradigm to study nonequilibrium dynamics of quantum many-body systems starting from a thermal state. We extend this formalism to describe nonequilibrium dynamics of quantum systems starting from arbitrary initial many-body density matrices. We show how this can be done for both bosons and fermions, and for both closed and open quantum systems, using additional sources coupled to bilinears of the fields at the initial time, calculating Green's functions in a theory with these sources, and then taking appropriate set of derivatives of these Green's functions with respect to initial sources to obtain physical observables. The set of derivatives depends on the initial density matrix. The physical correlators in a dynamics with arbitrary initial conditions do not satisfy Wick's theorem, even for noninteracting systems. However, our formalism constructs intermediate ``$n\ensuremath{-}\mathrm{particle}$ Green's functions'' that obey Wick's theorem and provide a prescription to obtain physical correlation functions from them. This allows us to obtain analytic answers for all physical many-body correlation functions of a noninteracting system even when it is initialized to an arbitrary density matrix. We use these exact expressions to obtain an estimate of the violation of Wick's theorem and relate it to presence of connected multiparticle initial correlations in the system. We illustrate this new formalism by calculating density and current profiles in many-body fermionic and bosonic open quantum systems initialized to nontrivial density matrices. We have also shown how this formalism can be extended to interacting many-body systems.

arXiv:1810.08692•Physical review. B./Physical review. B•Feb 19, 2019•24 citations

Linear entropy in quantum phase space

L. Rosales-Zárate, P. D. Drummond

We calculate the quantum Renyi entropy in a phase-space representation for either fermions or bosons. This can also be used to calculate purity and fidelity, or the entanglement between two systems. We show that it is possible to calculate the entropy from sampled phase-space distributions in normally ordered representations, although this is not possible for all quantum states. We give an example of the use of this method in an exactly soluble thermal case. The quantum entropy cannot be calculated at all using sampling methods in classical symmetric (Wigner) or antinormally ordered (Husimi) phase spaces, due to inner-product divergences. The preferred method is to use generalized Gaussian phase-space methods, which utilize a distribution over stochastic Green's functions. We illustrate this approach by calculating the reduced entropy and entanglement of bosonic or fermionic modes coupled to a time-evolving, non-Markovian reservoir.

arXiv:1106.3372•Physical Review A•Oct 17, 2011•12 citations

Theory of post-selected entanglement transitions in monitored bosons

Ilia Komissarov, Emanuele G. Dalla Torre, Ahana Chakraborty

Entanglement phase transitions driven by quantum measurements have emerged as a central paradigm in open quantum many-body physics. Such phase transitions are well established for systems with finite local Hilbert-space dimensions, such as qubits and fermions, while their realization in bosonic systems with unbounded local occupation numbers remains poorly understood. Even in the absence of interactions, number states of bosons are intrinsically non-Gaussian, preventing the use of standard correlation-matrix approaches. To address this problem, we develop a replica-free Keldysh field-theoretic framework that expresses the Renyi entropy of bosonic systems initialized in on-site Fock states in terms of permanents of matrices constructed from single-particle Green's functions. Applying this framework to a continuously monitored one-dimensional cross-stitch lattice conditioned on the no-click trajectory, we uncover a transition from volume-law to logarithmic entanglement scaling. We show that the transition is controlled by a restructuring of the non-Hermitian spectrum that changes the number of long-lived modes from extensive to finite. In the strongly monitored regime, bosons dynamically condense into a microscopic number of slowest-decaying modes, producing logarithmic entanglement scaling, whereas an extensive manifold of long-lived modes at weak monitoring gives rise to volume-law entanglement. Our results establish a distinct mechanism for measurement-induced entanglement transitions in free bosonic systems and provide a computationally efficient diagnostic of the measurement-induced bosonic condensation.

quant-pharXiv:2609.19251•Sep 18, 2026•0 citations
We propose a new method of calculating entanglement entropy of a many-body interacting Bosonic system (open or closed) in a field theoretic approach without replica methods. The Wigner function and Renyi entropy of a Bosonic system undergoing arbitrary non-equilibrium dynamics can be obtained from its Wigner characteristic function, which we identify with the Schwinger Keldysh partition function in presence of quantum sources turned on at the time of measurement. For non-interacting many body systems, starting from arbitrary density matrices, we provide exact analytic formulae for Wigner function and entanglement entropy in terms of the single particle Green's functions. For interacting systems, we relate the Wigner characteristic to the connected multi-particle correlators of the system. We use this formalism to study the evolution of an open quantum system from a Fock state with negative Wigner function and zero entropy, to a thermal state with positive Wigner function and finite entropy. The evolution of the Renyi entropy is non-monotonic in time for both Markovian and non-Markovian dynamics. The entropy is also found to be anti-correlated with negativity of the Wigner function of a 222 -mode open quantum system.
We propose a new field theoretic method for calculating Renyi entropy of a subsystem of many interacting bosons without using replica methods. This method is applicable to dynamics of both open and closed quantum systems starting from arbitrary initial conditions. Our method identifies the Wigner characteristic of a reduced density matrix with the partition function of the whole system with a set of linear sources turned on only in the subsystem, and uses this to calculate the subsystem's Renyi entropy. We use this method to study the evolution of Renyi entropy in a noninteracting open quantum system starting from an initial Fock state. We find a relation between the initial state and final density matrix which determines whether the entropy shows nonmonotonic behavior in time. For non-Markovian dynamics, we show that the entropy approaches its steady-state value as a power law with exponents governed by nonanalyticities of the bath. We illustrate that this field-theoretic method can be used to study large bosonic open quantum systems.
We formulate a new ``Wigner characteristics''-based method to calculate entanglement entropies of subsystems of Fermions using Keldysh field theory. This bypasses the requirements of working with complicated manifolds for calculating R\'enyi entropies for many-body systems. We provide an exact analytic formula for R\'enyi and von Neumann entanglement entropies of noninteracting open quantum systems, which are initialized in arbitrary Fock states. We use this formalism to look at entanglement entropies of momentum Fock states of one-dimensional Fermions. We show that the entanglement entropy of a Fock state can scale either logarithmically or linearly with subsystem size, depending on whether the number of discontinuities in the momentum distribution is smaller or larger than the subsystem size. We also use this formalism to describe entanglement dynamics of an open quantum system starting with a single domain wall at the center of the system. Using entanglement entropy and mutual information, we understand the dynamics in terms of coherent motion of the domain wall wavefronts, creation and annihilation of domain walls, and incoherent exchange of particles with the bath.
We show that odd order Rényi entropies S(2q+1)S^{(2q+1)}S(2q+1) of a system of interacting scalar fields can be calculated as the free energy of 2q+12q+12q+1 replicas of the system with additional quadratic inter-replica couplings in the subsystem at the time of measurement of the entropy. These couplings replace boundary field matching conditions. This formalism works both in and out of thermal equilibrium, for closed as well as open quantum systems, and provides a general dictionary between measurable correlation functions and entanglement entropy. S(2q+1)S^{(2q+1)}S(2q+1) can be analytically continued to calculate the von Neumann entropy SvNS^{\mathrm{vN}}SvN. We provide an exact formula relating S(2q+1)S^{(2q+1)}S(2q+1) and SvNS^{\mathrm{vN}}SvN with correlation functions in a non-interacting theory. For interacting theories, we provide rules for constructing all possible Feynman diagrams for S(2q+1)S^{(2q+1)}S(2q+1). We show that the boundary matching conditions cannot be completely eliminated while calculating Rényi entropies of even order due to presence of zero modes in replica space.
Schwinger Keldysh field theory is a widely used paradigm to study nonequilibrium dynamics of quantum many-body systems starting from a thermal state. We extend this formalism to describe nonequilibrium dynamics of quantum systems starting from arbitrary initial many-body density matrices. We show how this can be done for both bosons and fermions, and for both closed and open quantum systems, using additional sources coupled to bilinears of the fields at the initial time, calculating Green's functions in a theory with these sources, and then taking appropriate set of derivatives of these Green's functions with respect to initial sources to obtain physical observables. The set of derivatives depends on the initial density matrix. The physical correlators in a dynamics with arbitrary initial conditions do not satisfy Wick's theorem, even for noninteracting systems. However, our formalism constructs intermediate ``$n\ensuremath{-}\mathrm{particle}$ Green's functions'' that obey Wick's theorem and provide a prescription to obtain physical correlation functions from them. This allows us to obtain analytic answers for all physical many-body correlation functions of a noninteracting system even when it is initialized to an arbitrary density matrix. We use these exact expressions to obtain an estimate of the violation of Wick's theorem and relate it to presence of connected multiparticle initial correlations in the system. We illustrate this new formalism by calculating density and current profiles in many-body fermionic and bosonic open quantum systems initialized to nontrivial density matrices. We have also shown how this formalism can be extended to interacting many-body systems.
We calculate the quantum Renyi entropy in a phase-space representation for either fermions or bosons. This can also be used to calculate purity and fidelity, or the entanglement between two systems. We show that it is possible to calculate the entropy from sampled phase-space distributions in normally ordered representations, although this is not possible for all quantum states. We give an example of the use of this method in an exactly soluble thermal case. The quantum entropy cannot be calculated at all using sampling methods in classical symmetric (Wigner) or antinormally ordered (Husimi) phase spaces, due to inner-product divergences. The preferred method is to use generalized Gaussian phase-space methods, which utilize a distribution over stochastic Green's functions. We illustrate this approach by calculating the reduced entropy and entanglement of bosonic or fermionic modes coupled to a time-evolving, non-Markovian reservoir.
Entanglement phase transitions driven by quantum measurements have emerged as a central paradigm in open quantum many-body physics. Such phase transitions are well established for systems with finite local Hilbert-space dimensions, such as qubits and fermions, while their realization in bosonic systems with unbounded local occupation numbers remains poorly understood. Even in the absence of interactions, number states of bosons are intrinsically non-Gaussian, preventing the use of standard correlation-matrix approaches. To address this problem, we develop a replica-free Keldysh field-theoretic framework that expresses the Renyi entropy of bosonic systems initialized in on-site Fock states in terms of permanents of matrices constructed from single-particle Green's functions. Applying this framework to a continuously monitored one-dimensional cross-stitch lattice conditioned on the no-click trajectory, we uncover a transition from volume-law to logarithmic entanglement scaling. We show that the transition is controlled by a restructuring of the non-Hermitian spectrum that changes the number of long-lived modes from extensive to finite. In the strongly monitored regime, bosons dynamically condense into a microscopic number of slowest-decaying modes, producing logarithmic entanglement scaling, whereas an extensive manifold of long-lived modes at weak monitoring gives rise to volume-law entanglement. Our results establish a distinct mechanism for measurement-induced entanglement transitions in free bosonic systems and provide a computationally efficient diagnostic of the measurement-induced bosonic condensation.