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Local optimization on pure Gaussian state manifolds

by Bennet Windt, Alexander Jahn, Jens Eisert, Lucas Hackl

Submission summary

As Contributors: Lucas Hackl
Arxiv Link: (pdf)
Date accepted: 2021-03-02
Date submitted: 2021-01-23 14:48
Submitted by: Hackl, Lucas
Submitted to: SciPost Physics
Academic field: Physics
  • High-Energy Physics - Theory
  • Quantum Physics
  • Statistical and Soft Matter Physics
Approach: Theoretical


We exploit insights into the geometry of bosonic and fermionic Gaussian states to develop an efficient local optimization algorithm to extremize arbitrary functions on these families of states. The method is based on notions of gradient descent attuned to the local geometry which also allows for the implementation of local constraints. The natural group action of the symplectic and orthogonal group enables us to compute the geometric gradient efficiently. While our parametrization of states is based on covariance matrices and linear complex structures, we provide compact formulas to easily convert from and to other parametrization of Gaussian states, such as wave functions for pure Gaussian states, quasiprobability distributions and Bogoliubov transformations. We review applications ranging from approximating ground states to computing circuit complexity and the entanglement of purification that have both been employed in the context of holography. Finally, 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.

Published as SciPost Phys. 10, 066 (2021)

Author comments upon resubmission

We implemented the changes requested by the referees.

List of changes

A detailed list of changes is provided in our reply to the referees' lists of requested changes.

Reports on this Submission

Anonymous Report 2 on 2021-2-26 Invited Report

  • Cite as: Anonymous, Report on arXiv:2009.11884v3, delivered 2021-02-26, doi: 10.21468/SciPost.Report.2615


The authors have adequately addressed all concerns and questions I had. Optimization over Gaussian states is important in many areas of physics and the method proposed and explained in this work, which exploits the mathematical structure of this set, is likely to provide important advances in efficiency and stability and has a broad range of applicability. The authors have made available a well-documented Mathematica-package, implementing their approach.
In addition, the manuscript provides a valuable summary of properties and representations of (pure) Gaussian states and carefully shows how to treat bosonic and fermionic states in a single formalism and it my well become a useful reference for these matters.
In summary, I consider this a high-quality contribution and gladly recommend publication.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Report 1 by Michael Kaicher on 2021-2-26 Invited Report

  • Cite as: Michael Kaicher, Report on arXiv:2009.11884v3, delivered 2021-02-26, doi: 10.21468/SciPost.Report.2613


When discussing fermionic and bosonic states, one typically does so distinctly, see e.g. [1]. This paper shows how both types of Gaussian states may be talked about in a unified way by means of Kaehler structures. A strength of this paper is that is summarizes various parametrizations of Gaussian states and shows how to convert between them and doing so in a unified way for bosonic and fermionic Gaussian states.

It tackles one of the most important problems in the study of many-body quantum systems, namely extremizing an arbitrary function (such as the energy expectation value) within the family of Gaussian states using an efficient algorithm based on gradient descent.

It gives a very thorough introduction and supports its claims (a conjecture concerning the entanglement of purification) using both analytical and numerical arguments.

It includes toy examples (for bosonic and fermionic systems, respectively) in order to make the introduction part more accessible to the reader.

It provides a Mathematica package ("GaussianOptimization.m") as well as a example notebook that includes the functions used in the numerical example section, and can be accessed by the reader.

It very clearly highlights whenever a certain representation is used.

[1] Shi, T., Demler, E., & Cirac, J. I. (2018). Variational study of fermionic and bosonic systems with non-Gaussian states: Theory and applications. Annals of Physics, 390, 245-302


The weaknesses were corrected in the resubmitted version.


In this paper, a unified approach to the study of Gaussian states for bosonic and fermionic systems is given.

After providing a review of Gaussian states and introducing a unified notation for both bosonic and fermionic systems in Chapter II, the paper gives a comprehensive summary of the various representations of Gaussian states and how to switch among them in Chapter III, summarized in Table II.

In Chapter IV, the paper shows how to extremize an arbitrary function (such as the energy expectation value) on the family of fermionic and bosonic Gaussian states based on a gradient descent approach in a way which does not require having to evaluate the inverse metric in each iteration (this is a typically quite computationally costly step, see e.g. [1]).

In Chapter V, the authors provide three prominent examples of analytical functions which one wishes to find the global/local extremum, namely the energy expectation value (whose respective Gaussian state gives an approximate ground state to a given problem Hamiltonian), the Gaussian EoP and Complexity of Purification (CoP), which is a correlation measure in composite many-body systems.

This paper provides numerical and analytical support for two conjectures in Chapter VI, namely the Gaussian optimality (conjecture 1)-, and minimum purification conjecture (conjecture 2 - both defined on page 26 of the arXiv manuscript), which (if both hold true) state that Gaussian purifications "are sufficient to compute the Entanglement of Purification (EoP) of arbitrary mixed Gaussian states". It studies two example quantum mechanical systems in order to check if these two conjectures hold: The Klein-Gordon scalar field for bosons and the transverse field Ising model for fermions.

The results for conjecture 1 are summarized in Table V and they show the numerically computed values of the EoP for non-Gaussian- and Gaussian states for the fermionic transverse field Ising model. As for all studied system sizes, the EoP values are smaller for Gaussian states than for non-Gaussian states, using Eq.168 it follows that the conjecture holds that optimal purification of a mixed Gaussian state is Gaussian.

Evidence that conjecture 2 holds for the studied bosonic and fermionic systems is given in Table VI. While it is argued why the number of degrees of freedom of the purifying systems $A′$
and $B′$ must be identical to the number of degrees of freedom in $A$ and $B$ in the main text of VI.C (in other words, why $N_A+N_B=N_{A′}+N_{B′}$), Table VI validates conjecture 2 by showing that the minimal EoP is obtained for $N_A=N_{A′}$ and $N_B=N_{B′}$.

The results of the paper are discussed and summarizes in Chapter VII.

Requested changes


  • validity: high
  • significance: high
  • originality: good
  • clarity: high
  • formatting: perfect
  • grammar: perfect

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