#  Quantum hydrodynamics from local thermal pure states

Shoichiro Tsutsui<EMAIL_ADDRESS>Theoretical Research Division,
Nishina Center for Accelerator-Based Science, RIKEN, Wako 351-0198, Japan
Masaru Hongo<EMAIL_ADDRESS>Department of Physics, University of
Illinois, Chicago, Illinois 60607, USA  RIKEN iTHEMS, RIKEN, Wako 351-0198,
Japan  Shintaro Sato<EMAIL_ADDRESS>Department of Physics,
Waseda University, Shinjuku, Tokyo 169-8555, Japan  Takahiro Sagawa
<EMAIL_ADDRESS>Department of Applied Physics and Quantum-Phase
Electronics Center (QPEC), The University of Tokyo, 7-3-1 Hongo, Byunkyo-ku,
Tokyo 113-8656, Japan

###### Abstract

We provide a pure state formulation for hydrodynamic dynamics of isolated
quantum many-body systems. A pure state describing quantum systems in local
thermal equilibrium is constructed, which we call a local thermal pure quantum
($\ell\mathrm{TPQ}$) state. We show that the thermodynamic functional and the
expectation values of local operators (including a real-time correlation
function) calculated from the $\ell\mathrm{TPQ}$ state converge to those from
a local Gibbs ensemble in the large fluid-cell limit. As a numerical
demonstration, we investigate a one-dimensional spin chain and observe the
hydrodynamic relaxation obeying the Fourier’s law. We further prove the second
law of thermodynamics and the quantum fluctuation theorem, which are also
validated numerically. The $\ell\mathrm{TPQ}$ formulation gives a useful
theoretical basis to describe the emergent hydrodynamic behavior of quantum
many-body systems furnished with a numerical efficiency, being applicable to
both the non-relativistic and relativistic regimes.

††preprint: RIKEN-QHP-497

Introduction.— Nonequilibrium dynamics of isolated quantum many-body systems
has been one of the central issues appearing at the crossroad of statistical
physics and quantum physics. In particular, hydrodynamics Landau and Lifshitz
(1987); Forster (1975); Chaikin and Lubensky (2000) is an emergent macroscopic
theory to investigate such dynamics under local equilibrium conditions. Recent
experimental technologies have promoted further developments of hydrodynamics
theory for isolated quantum systems, ranging from ultracold atoms Schäfer and
Teaney (2009); Cao  _et al._ (2011a, b); Adams  _et al._ (2012); Schaefer
(2014); Elliott  _et al._ (2014); Joseph  _et al._ (2015); Baird  _et al._
(2019); Patel  _et al._ (2020) to quark-gluon plasma Romatschke and Romatschke
(2007); Teaney (2010); Song  _et al._ (2011); Niemi  _et al._ (2011); Heinz
and Snellings (2013); Gale  _et al._ (2013); Jeon and Heinz (2015); Derradi de
Souza  _et al._ (2016); Busza  _et al._ (2018); Romatschke and Romatschke
(2019). Furthermore, quantum or classical integrable systems, where
conventional hydrodynamics is believed to break down, is now becoming a hot
field on the basis of generalized hydrodynamics Castro-Alvaredo  _et al._
(2016); Bertini  _et al._ (2016); Doyon and Yoshimura (2017); De Nardis  _et
al._ (2018); Ruggiero  _et al._ (2020).

A recently-developed microscopic derivation of hydrodynamics is based on an
extended notion of the statistical ensembles, called the local Gibbs (LG)
ensemble both in classical Sasa (2014); Mabillard and Gaspard (2020, 2021) and
quantum systems Hayata  _et al._ (2015); Hongo (2017, 2019); Hongo and Hattori
(2021). An advantage of the local ensemble formulation lies in the fact that
it does not require the quasi-particle description and thus is applicable to
strongly-coupled systems (see also another approach Bhattacharyya  _et al._
(2008); Rangamani (2009); Hartnoll (2009); Hubeny  _et al._ (2012) based on
the holographic principle and the pioneering works Nakajima (1957); Mori
(1958); McLennan (1960); Luttinger (1964); Mori (1965); Kawasaki and Gunton
(1973); Zubarev (1974) along the line of the linear response theory). However,
under unitary evolutions of isolated quantum systems, a pure state never falls
into any mixed state such as the Gibbs ensemble. Thus, it is still an open
problem how macroscopic hydrodynamics emerges under unitary evolutions of pure
states as observed in ultracold atoms Trotzky  _et al._ (2012); Parsons  _et
al._ (2015); Kaufman  _et al._ (2016); Gross and Bloch (2017); Ohl de Mello
_et al._ (2019); Ebadi  _et al._ (2020).

In this Letter, we provide a pure state-based derivation of the hydrodynamic
equation for isolated quantum many-body systems. We introduce a special class
of random pure quantum states describing systems in local thermal equilibrium,
which we refer to as local thermal pure quantum ($\ell\mathrm{TPQ}$) states.
This is a generalization of thermal pure quantum (TPQ) states Bocchieri and
Loinger (1959); Hams and De Raedt (2000); Popescu  _et al._ (2006); Sugiura
and Shimizu (2012, 2013); Monnai and Sugita (2014); Hyuga  _et al._ (2014);
Endo  _et al._ (2018) randomly sampled from the Hilbert space to reproduce
thermal equilibrium behaviors. We show that $\ell\mathrm{TPQ}$ states have a
numerical advantage in computing hydrodynamic expectation values Sugiura and
Shimizu (2012, 2013); Hyuga  _et al._ (2014); Endo  _et al._ (2018) and give
the equivalent results with the conventional method based on the LG ensemble
in the large fluid-cell limit.

We establish all the fundamental hydrodynamic behaviors from the
$\ell\mathrm{TPQ}$ states,being equipped with a numerical efficiency.
Specifically, we prove the Green-Kubo formula Green (1954); Nakano (1956);
Kubo (1957), the second law of thermodynamics and the quantum fluctuation
theorem Kurchan (2000); Tasaki (2000); Esposito  _et al._ (2009); Campisi  _et
al._ (2011); Sagawa ; Iyoda  _et al._ (2017); Funo  _et al._ (2018), as well
as the constitutive relation of hydrodynamics itself. We remark that our
formulation applies to both the norelativistic and relativistic regimes, and
therefore potentially covers a variety of phenomena from low to high energy
scales. As a concrete demonstration, applying the $\ell\mathrm{TPQ}$-state
formulation to a one-dimensional quantum spin chain (see Refs. Gemmer  _et
al._ (2006); Dhar (2008); Dubi and Di Ventra (2011); Steinigeweg  _et al._
(2014a); Lepri (Ed.); Bernard and Doyon (2016); Bertini  _et al._ (2021) and
references therein for a review on recent theoretical approaches to thermal
transport in low-dimensional systems), we numerically investigate the
hydrodynamic relaxation starting from a $\ell\mathrm{TPQ}$ state, and confirm
the Fourier’s law for thermal conduction.

Local thermal pure quantum ($\ell\mathrm{TPQ}$) state.— We first introduce the
$\ell\mathrm{TPQ}$ state. Suppose that the system possesses a set of conserved
currents $\hat{\mathcal{J}}^{\mu}_{~{}a}(t)$ satisfying

$\partial_{\mu}\hat{\mathcal{J}}^{\mu}_{~{}a}(t)\equiv\partial_{0}\hat{\mathcal{J}}^{0}_{~{}a}(t)+\partial_{i}\hat{\mathcal{J}}^{i}_{~{}a}(t)=0,$
(1)

where $\mu,\nu,\cdots$ denote spacetime indices, and $a,b,\cdots$ label
conserved charges such as energy, momentum, and particle numbers111 Summations
over the repeated indices for Greek letters ($\mu,\nu,\cdots$) are assumed
throughout the paper. The repeated indices for Latin letters is assumed to
contain the spatial integral, e.g.
$\lambda^{a}\hat{c}_{a}\equiv\int\mathrm{d}^{d}x\lambda^{a}(\bm{x})\hat{c}_{a}(\bm{x})$
where $d$ denotes the spatial dimension. . Throughout the paper, we use the
Heisenberg picture, and Eq. (1) gives the equation of motion for conserved
charge densities $\hat{c}_{a}(t)\equiv\hat{\mathcal{J}}^{0}_{~{}a}(t)$. Then,
we consider the system in local thermal equilibrium, which is described by
local thermodynamic parameters $\lambda^{a}$ conjugate to average charge
densities $\langle\hat{c}_{a}\rangle$. We assume that each fluid cell contains
many degrees of freedom so that thermodynamic parameters are smooth functions
of spatial coordinates. In other words, we identify the current operators
$\hat{\mathcal{J}}^{\mu}_{~{}a}$ in Eq. (1) as coarse-graining ones over a
fluid-cell volume $V_{\mathrm{cell}}$.

To describe the locally thermalized system, we introduce the
$\ell\mathrm{TPQ}$ state at time $t$ by

$\ket{\lambda_{t};t}\equiv\sum_{\alpha}z_{\alpha}\mathrm{e}^{-\frac{1}{2}\hat{K}[\lambda_{t};t]}\ket{\alpha_{t}},$
(2)

with $\hat{K}[\lambda_{t};t]\equiv\lambda^{a}(t)\hat{c}_{a}(t)$, any
orthonormal basis state vector $\ket{\alpha_{t}}$ at time $t$, and a complex
random variable
$z_{\alpha}\equiv(z_{\alpha}^{\prime}+\mathrm{i}z_{\alpha}^{\prime\prime})/\sqrt{2}$
whose real/imaginary parts are taken from the normal distribution. Note that
we distinguish two arguments in $\hat{K}[\lambda_{t};t]$; one for
thermodynamic parameters $\lambda_{t}$, whose suffix denotes its configuration
$\lambda_{t}=\lambda(t)$ at time $t$, while the second argument $t$ represents
the time argument of charge densities as Heisenberg operators. When the system
only contains the energy as the conserved charge, $\hat{K}$ reads
$\hat{K}[\lambda_{t};t]=\int\mathrm{d}^{d}x\beta(t,\bm{x})\hat{h}(t,\bm{x})$
with the local inverse temperature $\beta(t,\bm{x})$ and energy density
operator $\hat{h}(t,\bm{x})$, for instance. The normalization of the
$\ell\mathrm{TPQ}$ state is defined by
$Z_{\ell\mathrm{TPQ}}[\lambda_{t}]\equiv\braket{\lambda_{t};t}{\lambda_{t};t}$,
whose logarithm is identified as a thermodynamic functional. Using these, we
define the entropy-functional operator for the $\ell\mathrm{TPQ}$ state as

$\hat{S}[\lambda_{t};t]=\hat{K}[\lambda_{t};t]+\log
Z_{\ell\mathrm{TPQ}}[\lambda_{t}],$ (3)

and the average value over the $\ell\mathrm{TPQ}$ state as

$\langle\hat{\mathcal{O}}\rangle^{\ell\mathrm{TPQ}}_{\lambda_{t}}\equiv\frac{1}{Z_{\ell\mathrm{TPQ}}[\lambda_{t}]}\braket{\lambda_{t};t}{\hat{\mathcal{O}}}{\lambda_{t};t}.$
(4)

Hydrodynamics and the quantum fluctuation theorem.— Based on the
$\ell\mathrm{TPQ}$ state, we derive hydrodynamics and the quantum fluctuation
theorem. The hydrodynamic equation is regarded as the averaged Eq. (1) over a
certain initial density operator $\hat{\rho}_{0}$. Here, we put a crucial
assumption on the initial state: the system is in the $\ell\mathrm{TPQ}$ state
at initial time $t_{0}$ parametrized by $\lambda^{a}_{t_{0}}$. Starting from
this initial $\ell\mathrm{TPQ}$ state, we shall consider the subsequent time-
evolution of charge densities described by

$\partial_{\mu}\langle\hat{\mathcal{J}}^{\mu}_{~{}a}(t)\rangle=0\quad\mathrm{with}\quad\langle\hat{\mathcal{J}}^{\mu}_{~{}a}(t)\rangle\equiv\langle\hat{\mathcal{J}}^{\mu}_{~{}a}(t)\rangle^{\ell\mathrm{TPQ}}_{\lambda_{t_{0}}},$
(5)

for $t\geq t_{0}$. To make Eq. (5) a closed set of equations, we need a
constitutive relation, which expresses
$\langle\hat{\mathcal{J}}^{i}_{~{}a}(t)\rangle$ by the dynamical variable
$\langle\hat{c}_{a}(t)\rangle=\langle\hat{\mathcal{J}}^{0}_{~{}a}(t)\rangle$.
Thus, our problem is to find the constitutive relation.

The vital point here is that we have a solution to this problem if we replace
the initial $\ell\mathrm{TPQ}$-state average with the LG-ensemble average over
the density operator:

$\hat{\rho}_{\mathrm{LG}}[\lambda_{t_{0}};t_{0}]\equiv\frac{1}{Z_{\mathrm{LG}}[\lambda_{t_{0}}]}\mathrm{e}^{-\hat{K}[\lambda_{t_{0}};t_{0}]},$
(6)

where we defined the partition functional for the LG ensemble as
$Z_{\mathrm{LG}}[\lambda_{t_{0}}]\equiv\mathop{\mathrm{Tr}}\mathrm{e}^{-\hat{K}[\lambda_{t_{0}};t_{0}]}$.
We will express the LG average as
$\langle\hat{\mathcal{O}}\rangle^{\mathrm{LG}}_{\lambda_{t}}\equiv\mathop{\mathrm{Tr}}\big{(}\hat{\rho}_{\mathrm{LG}}[\lambda_{t};t]\hat{\mathcal{O}}\big{)}$.
Thus, it is enough to show the equivalence between the $\ell\mathrm{TPQ}$ and
LG averages for the purpose of deriving hydrodynamics from the
$\ell\mathrm{TPQ}$ state.

Following the similar treatment of Refs. Sugiura and Shimizu (2012, 2013);
Hyuga  _et al._ (2014); Endo  _et al._ (2018), we can indeed show such
equivalence holds as convergence in probability $\xrightarrow{P}$ w.r.t. the
random variable $z_{a}$ Sup :

$\displaystyle Z_{\ell\mathrm{TPQ}}[\lambda_{t}]$
$\displaystyle\xrightarrow{P}Z_{\mathrm{LG}}[\lambda_{t}],$ (7)
$\displaystyle\langle\hat{\mathcal{O}}\rangle^{\ell\mathrm{TPQ}}_{\lambda_{t}}$
$\displaystyle\xrightarrow{P}\langle\hat{\mathcal{O}}\rangle^{\mathrm{LG}}_{\lambda_{t}},$
(8)

where we took the large size limit of all fluid cells
$V_{\mathrm{cell}}\to\infty$. Therefore, the LG-based derivation developed in
Refs. Hayata  _et al._ (2015); Hongo (2017, 2019); Hongo and Hattori (2021)
also provides a derivation of hydrodynamics from the $\ell\mathrm{TPQ}$ state
in the large fluid-cell limit.

We here summarize some consequences of Eqs. (7)-(8). Within a first-order
derivative expansion supplemented by the Markovian approximation for current-
current correlators, the equivalence leads to the constitutive relation

$\begin{split}\langle\hat{\mathcal{J}}^{i}_{~{}a}(t)\rangle&=\langle\hat{\mathcal{J}}^{i}_{~{}a}(t)\rangle^{\ell\mathrm{TPQ}}_{\lambda_{t}}+L_{ab}^{ij}(t)\partial_{j}\lambda^{b}_{t}+O(\partial^{2}),\end{split}$
(9)

where transport coefficients $L^{ij}_{ab}$ is given by the Green-Kubo formula
with the $\mathrm{TPQ}$ state:

$\displaystyle L_{ab}^{ij}(t)$
$\displaystyle=\beta_{t}\int_{-\infty}^{t}\mathrm{d}t^{\prime}\int\mathrm{d}^{d}x^{\prime}\int_{0}^{1}\mathrm{d}\tau\big{\langle}\widetilde{\delta}\hat{\mathcal{J}}^{i}_{~{}a,\tau}(x)\widetilde{\delta}\hat{\mathcal{J}}^{j}_{~{}b}(x^{\prime})\big{\rangle}^{\ell\mathrm{TPQ}}_{\lambda_{t}}$
$\displaystyle\simeq\frac{\beta_{t}}{4V}\int_{-\infty}^{\infty}\mathrm{d}t^{\prime}\big{\langle}\\{\widetilde{\delta}\hat{\mathsf{J}}^{i}_{~{}a}(t),\widetilde{\delta}\hat{\mathsf{J}}^{j}_{~{}b}(t^{\prime})\\}\big{\rangle}_{\lambda_{t}}^{\mathrm{TPQ}},$
(10)

where we defined
$\langle\hat{\mathcal{O}}\rangle^{\mathrm{TPQ}}_{\lambda_{t}}$ as a
grandcanonical TPQ average parametrized by $\lambda_{t}$ Sugiura and Shimizu
(2012, 2013); Hyuga  _et al._ (2014); Endo  _et al._ (2018). Here we defined a
projected operator
$\widetilde{\delta}\hat{\mathcal{O}}\equiv(1-\hat{\mathcal{P}})\delta\hat{\mathcal{O}}$
using the $\ell\mathrm{TPQ}$ version of Mori’s projection Mori (1965);
Kawasaki and Gunton (1973):

$\hat{\mathcal{P}}\hat{\mathcal{O}}(t)\equiv\delta\hat{c}_{a}(t)\frac{\delta}{\delta
c_{a}(t)}\langle\hat{\mathcal{O}}\rangle^{\ell\mathrm{TPQ}}_{\lambda_{t}},$
(11)

together with
$\delta\hat{\mathcal{O}}(t)\equiv\hat{\mathcal{O}}(t)-\langle\hat{\mathcal{O}}(t)\rangle^{\ell\mathrm{TPQ}}_{\lambda_{t}}$
and
$\hat{\mathcal{O}}_{\tau}(t)\equiv\mathrm{e}^{\hat{K}[\lambda_{t};t]\tau}\hat{\mathcal{O}}(t)\mathrm{e}^{-\hat{K}[\lambda_{t};t]\tau}$.
In the second line of Eq. (10), we neglected the higher-order derivative
correction and defined total currents
$\hat{\mathsf{J}}^{i}_{~{}a}(t)\equiv\int\mathrm{d}^{d}x\hat{\mathcal{J}}^{i}_{~{}a}(t,\bm{x})$
with the anti-commutator
$\\{\hat{A},\hat{B}\\}\equiv\hat{A}\hat{B}+\hat{B}\hat{A}$ and total volume
$V\equiv\int\mathrm{d}^{d}x$. Note that the constitutive relation (9) is
expressed by the parameter $\lambda_{t}^{a}=\\{\beta_{t},\cdots\\}$ at time
$t$, which is defined by the matching condition
$\langle\hat{c}_{a}(t)\rangle=\langle\hat{c}_{a}(t)\rangle^{\ell\mathrm{TPQ}}_{\lambda_{t}}$.
Thanks to one-to-one correspondence between $\lambda^{a}_{t}$ and
$\langle\hat{c}_{a}(t)\rangle$ (apart from the first-order phase transition),
Eq. (9) indeed expresses $\langle\hat{\mathcal{J}}^{i}_{~{}a}(t)\rangle$ by
$\langle\hat{c}_{a}(t)\rangle$. Then, once we evaluate
$\langle\hat{\mathcal{J}}^{i}_{~{}a}(t)\rangle^{\ell\mathrm{TPQ}}_{\lambda_{t}}$
following from the thermodynamic functional $\log
Z_{\ell\mathrm{TPQ}}[\lambda_{t}]$ (see e.g., Hayata  _et al._ (2015); Hongo
(2017, 2019); Hongo and Hattori (2021)) and the Green-Kubo formula (10), Eq.
(9) closes the averaged conservation law (5), which completes an
$\ell\mathrm{TPQ}$-based derivation of the first-order hydrodynamics.

In addition to hydrodynamic equations, we can also show the quantum
fluctuation theorem and the second law of thermodynamics for the
$\ell\mathrm{TPQ}$ state within a small error. For that purpose, let us
introduce the generating function of the total entropy production from initial
time $t_{0}$ to arbitrary time $t(>t_{0})$ with forward/backward evolution:

$\displaystyle G_{F}^{\ell\mathrm{TPQ}}(z)$
$\displaystyle=\braket{\hat{\mathcal{U}}^{\dagger}(t)\mathrm{e}^{\mathrm{i}z\hat{S}[\lambda_{t};t_{0}]}\hat{\mathcal{U}}(t)\mathrm{e}^{-\mathrm{i}z\hat{S}[\lambda_{t_{0}};t_{0}]}}_{\lambda_{t_{0}}}^{\ell\mathrm{TPQ}},$
(12) $\displaystyle G_{B}^{\ell\mathrm{TPQ}}(z)$
$\displaystyle=\braket{\widetilde{\mathcal{U}}(t)\mathrm{e}^{\mathrm{i}z\hat{S}[\tilde{\lambda}_{t_{0}};t]}\widetilde{\mathcal{U}}^{\dagger}(t)\mathrm{e}^{-\mathrm{i}z\hat{S}[\tilde{\lambda}_{t};t]}}_{\tilde{\lambda}_{t}}^{\ell\mathrm{TPQ}},$
(13)

where $\hat{\mathcal{U}}(t)$ is the time evolution operator. By using a
combined parity and time-reversal $\Theta\equiv\mathsf{P}\mathsf{T}$, we also
defined $\mathsf{P}\mathsf{T}$-transformed time evolution operator and
parameter by
$\widetilde{\mathcal{U}}(t)\equiv\Theta\hat{\mathcal{U}}(t)\Theta^{-1}$ and
$\tilde{\lambda}_{t}\equiv\epsilon^{a}\lambda^{a}(t,-\bm{x})$ with
$\epsilon^{a}$ being a $\mathsf{P}\mathsf{T}$-eigenvalue of charge densities
$\hat{c}_{a}$, respectively. We can prove the second law of thermodynamics and
quantum fluctuation theorem as Sup

$\displaystyle\langle\hat{S}[\lambda_{t};t]-\hat{S}[\lambda_{t_{0}};t_{0}]\rangle$
$\displaystyle\geq O(\mathrm{e}^{-A_{1}V_{\mathrm{cell}}}),$ (14)
$\displaystyle\left|G_{F}^{\ell\mathrm{TPQ}}(z)-G^{\ell\mathrm{TPQ}}_{B}(-z+\mathrm{i})\right|$
$\displaystyle=O(\mathrm{e}^{-A_{2}V_{\mathrm{cell}}}),$ (15)

where $A_{1}$ and $A_{2}$ denote certain numerical factors. Note that the
violation of both the fluctuation theorem and the second law is exponentially
small with respect to the volume of fluid-cell $V_{\mathrm{cell}}$, which is
the same as Eqs. (7)-(8).

Application to spin chain.— We now apply the developed formalism to a one-
dimensional nonintegrable lattice half-integer spin system, whose Hamiltonian
reads

$\begin{split}\hat{H}=\sum_{n=1}^{N}\big{[}&J_{z}\sigma_{n}^{z}\sigma_{n+1}^{z}+D(\sigma_{n}^{z}\sigma_{n+1}^{x}-\sigma_{n}^{x}\sigma_{n+1}^{z})\\\
&+\Gamma\sigma_{n}^{x}+B\sigma_{n}^{z}\big{]},\end{split}$ (16)

where we impose the periodic boundary condition. The energy is the only
conserved quantity of this system so that we find the energy-current operator
satisfies the averaged conservation law

$\partial_{t}\langle\hat{h}(t,n)\rangle+\nabla_{x}\langle\hat{J}_{E}(t,n)\rangle=0,$
(17)

over the initial $\ell\mathrm{TPQ}$ state $\ket{\beta_{t_{0}};t_{0}}$ with
$\nabla_{x}\langle\hat{J}_{E}(t,n)\rangle\equiv\langle\hat{J}_{E}(t,n+1/2)\rangle-\langle\hat{J}_{E}(t,n-1/2)\rangle$.

In accordance with our formal derivation, one can expect the long-time/length
behavior of the energy current is governed by the hydrodynamic equation. In
the present setup, the constitutive relation within the leading-order
derivative expansion leads to the Fourier’s law:

$\langle\hat{J}_{E}(t,n-1/2)\rangle=-\kappa_{\mathrm{Fourier}}(t,n)\nabla_{x}T(t,n-1/2)$
(18)

where we defined $\nabla_{x}T(t,n-1/2)\equiv T(t,n)-T(t,n-1)$ with
$T(t,n)\equiv 1/\beta(t,n)$. The proportional coefficient
$\kappa_{\mathrm{Fourier}}(t,n)$ is identified as the thermal conductivity. On
the other hand, we can also compute the thermal conductivity from the Green-
Kubo formula:

$\kappa(t,n)=\frac{\beta(t,n)^{2}}{4N}\\!\\!\lim_{\tau\to\infty}\\!\int_{-\tau}^{\tau}\\!\\!\mathrm{d}t\langle\\{\delta\hat{\mathsf{J}}_{E}(t),\delta\hat{\mathsf{J}}_{E}(t_{0})\\}\rangle^{\mathrm{TPQ}}_{\beta(t,n)},$
(19)

with the total energy current
$\hat{\mathsf{J}}_{E}(t)\equiv\sum_{n=1}^{N}\hat{J}_{E}(t,n)$. Our derivation
tells us that $\kappa_{\mathrm{Fourier}}$ should agree with $\kappa$ in the
large fluid-cell limit. Thus, we can quantitatively evaluate the validity of
the hydrodynamic description by comparing the values of
$\kappa_{\mathrm{Fourier}}(t,n)$ and $\kappa(t,n)$, which are independently
evaluated from Eqs. (18)-(19).

A numerical implementation of the $\ell\mathrm{TPQ}$-based simulation is
realized in a similar way with the canonical TPQ state Sugiura and Shimizu
(2012, 2013); Hyuga  _et al._ (2014); Endo  _et al._ (2018). Expanding the
exponential operator in Eq. (2), we first prepare the initial
$\ell\mathrm{TPQ}$ state as

$\begin{split}&\ket{\beta_{t_{0}};t_{0}}\equiv\mathrm{e}^{-\frac{1}{2}\sum_{m=1}^{N}\beta(t_{0},m)l_{m}}\sum_{k=0}^{\infty}\frac{1}{2^{k}k!}\\\
&\qquad\times\left(\sum_{n=1}^{N}\beta(t_{0},n)\big{[}l_{n}-\hat{h}(t_{0},n)\big{]}\right)^{k}\sum_{\alpha}z_{\alpha}\ket{\alpha_{t_{0}}},\end{split}$
(20)

where $l_{n}$ is a certain number larger than the maximum eigenvalue of
$\hat{h}(t_{0},n)$, and $\beta(t_{0},n)$ provides an initial distribution of
the local temperature. Note that the position dependence of the local
temperature does not allow us to express the $\ell\mathrm{TPQ}$ state by the
microcanonical counterpart, as is the case for global thermal equilibrium
Sugiura and Shimizu (2012, 2013); Hyuga  _et al._ (2014); Endo  _et al._
(2018). Starting from this initial state, we numerically investigate the time
evolution by solving the Schrödinger equation with the Runge-Kutta method Sup
.

Figure 1: The time-evolutions of (a) local temperature, and (b) energy current
of the model (16) with parameters $(J_{z},D,\Gamma,B)=(0.2,0.2,0.2,-0.2)$.

Figure 1 shows the time-evolution of the local temperature and energy current
starting from an inhomogeneous temperature profile:
$\beta(t_{0},n)=\bar{\beta}+\Delta\beta\sin(2\pi(n-1)/N)$ with
$\bar{\beta}=1.5$ and $\Delta\beta=0.2$. Here we imposed the periodic boundary
condition with $N=20$. It is worth emphasizing that we perform the coarse-
graining along both spatial and temporal directions: We set the fluid-cell
size to contain $5$ spins or so and take the temporal average over $\tau=10$
Sup . We clearly sees the diffusive relaxation of inhomogeneous temperature
(or equivalently energy density). Furthermore, we can find a tendency of the
energy current responding to the temperature gradient, which is compatible
with Fourier’s law (18). Note, however, that we cannot conclude that the
system really obeys Eqs. (18)-(19) at this point because we have not shown the
thermal conductivity, appearing in Eq. (18) indeed matches with that evaluated
from the Green-Kubo formula (19).

To microscopically confirm the hydrodynamic relaxation, we evaluate the
coefficient of Fourier’s law (18) and compare it with the thermal conductivity
$\kappa(t,n)$ independently evaluated from the Green-Kubo formula (19). If the
system is really in the hydrodynamic regime, these quantities should coincide
with each other. The result is given in Fig. 2. Taking account of the
temperature window, we compute the Green-Kubo formula for three values of
temperatures (see shaded regions in Fig. 2). Figure 2 (b) then indicates that
the values of Fourier’s law coefficient (18) stay comparable regions with
those obtained from the Green-Kubo formula, though they fluctuate depending on
temporal and spatial positions. The agreement may be surprisingly good if we
recall our fluid-cell contains just $5$ spins or so, and the deviation is
probably the finite-volume effect. In short, we conclude that the hydrodynamic
description is confirmed to be a good approximation on the basis of the
Schrödinger equation.

Figure 2:  Numerical results for (a) Green-Kubo formula (19), and (b)
comparison of $\kappa_{\mathrm{Fourier}}(t,n)$ with the thermal conductivity
evaluated from the Green-Kubo formula. The parameters are the same as in Fig.
1.

We finally show the result on the second law of thermodynamics and the quantum
fluctuation theorem. Figure 3 demonstrates the time-dependence of the entropy
production (main figure) and the quantum fluctuation theorem (inset). The
solid line in Fig. 3 for the entropy production is from a direct evaluation,
and data are evaluated from the slopes of the generating functions
$G_{F}^{\ell\mathrm{TPQ}}(z)$ and $G_{B}^{\ell\mathrm{TPQ}}(\mathrm{i}-z)$,
which are shown in the inset by solid and dotted lines, respectively. They
take positive values in the entire time regions, which confirms the second law
of thermodynamics. Moreover, one sees that $G_{F}^{\ell\mathrm{TPQ}}(z)$
indeed works as a generating function of the entropy production. However, due
to uncertainty of the fluctuation theorem coming from the finiteness of the
fluid cell as shown in Eq. (15), $G_{B}^{\ell\mathrm{TPQ}}(\mathrm{i}-z)$ (and
its slope) at a later time will deviate from $G_{F}^{\ell\mathrm{TPQ}}(z)$
(see also the inset of Fig. 3). Although we cannot analytically evaluate
dependences of the deviation on the time and $z$, this behavior looks similar
to those obtained in Ref. Iyoda  _et al._ (2017, 2021).

Figure 3:  A numerical result for the entropy production (14) [inset: the
quantum fluctuation theorem (15)]. The parameters are the same as in Fig. 1.

Concluding remarks.— In this Letter, we have formulated hydrodynamics based on
a special class of random pure quantum states, which we call the
$\ell\mathrm{TPQ}$ state. We have shown the equivalence of the
$\ell\mathrm{TPQ}$-state formalism to that of the LG ensemble in the large
fluid-cell limit in Eqs. (7)-(8), from which we have derived the hydrodynamic
equations (9)-(10), the second law of thermodynamics (14), and the quantum
fluctuation theorem (15). Using the $\ell\mathrm{TPQ}$ state, we have
performed the numerical simulation of one-dimensional nonintegrable spin chain
model. From a careful analysis, we have numerically confirmed that the
observed thermal diffusion allows the hydrodynamic description with the
Fourier’s law and the Green-Kubo formula (Figs. 1-2). The validity of the
second law of thermodynamics and quantum fluctuation theorem has also been
confirmed (Fig. 3).

Let us comment on limitations of the $\ell\mathrm{TPQ}$-state formulation both
from the conceptual and practical viewpoints. First of all, the present
formulation employs a $\ell\mathrm{TPQ}$ state mainly motivated by its
numerical efficiency. However, a $\ell\mathrm{TPQ}$ state is a randomly
sampled state to reproduce local thermodynamics for isolated quantum systems.
Therefore, a $\ell\mathrm{TPQ}$ state is not directly related to the
eigenstate thermalization hypothesis (ETH) Deutsch (1991); Srednicki (1994);
Rigol  _et al._ (2008); Steinigeweg  _et al._ (2014b); Kim  _et al._ (2014);
D’Alessio  _et al._ (2016); Dymarsky  _et al._ (2018); Yoshizawa  _et al._
(2018); Deutsch (2018); Noh  _et al._ (2020), as the latter concerns energy
eigenstates that are responsible for quantum ergodicity of Hamiltonian
dynamics. We thus believe that it is crucial to construct a pure state
formulation based on the ETH in establishing a more solid foundation for
hydrodynamics of isolated quantum systems, which is an important future issue.

Another limitation of the $\ell\mathrm{TPQ}$-state formulation is set by the
dimension of the Hilbert space of bosonic systems. Since we do not assume the
relativistic or nonrelativistic nature of systems, the $\ell\mathrm{TPQ}$
formulation is, in principle, applicable to any quantum system including the
quark-gluon plasma or relativistic/nonrelativistic bose gas. Although a
current computational resource does not allow us to perform the
$\ell\mathrm{TPQ}$ simulation for such bosonic systems with a huge degrees of
freedom, it would be of immediate interest to perform the $\ell\mathrm{TPQ}$
simulation for fermionic systems.

There are also interesting prospects that can be clarified along the line of
this Letter. While we performed a $\ell\mathrm{TPQ}$ simulation for the system
with the single conserved quantity, it is interesting to realize the
simulation with multiple conserved charges. In particular, the existence of
the momentum density in low-dimensional systems is interersting, because it is
expected to drastically change conventional hydrodynamics due to a large
hydrodynamic fluctuation Lepri (Ed.). Furthemore, it would be an interesting
direction to extend our $\ell\mathrm{TPQ}$ framework so as to apply thermal
transports for quantum integrable systems described by generalized
hydrodynamics Castro-Alvaredo  _et al._ (2016); Bertini  _et al._ (2016);
Doyon and Yoshimura (2017); De Nardis  _et al._ (2018); Ruggiero  _et al._
(2020). We left them as future works (some will be reported in the full paper
Tsutsui  _et al._ ).

###### Acknowledgements.

The authors thank Shunsuke Furukawa, Taiki Morinaga, Keiji Saito, and Shin-
ichi Sasa for useful discussions. S. T. was supported by the RIKEN Special
Postdoctoral Researchers Program. M.H. was supported by the US Department of
Energy, Office of Science, Office of Nuclear Physics under Award No. DE-
FG0201ER41195. TS is supported by JSPS KAKENHI Grant Number JP19H05796. TS is
also supported by institute of AI and Beyond of the University of Tokyo. This
work was partially supported by the RIKEN iTHEMS Program, in particular iTHEMS
STAMP working group.

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Supplemental Materials: Quantum hydrodynamics from local thermal pure states

##  Appendix A Proofs of equivalence in the large fluid-cell limit

We here give proofs of equivalence between $\ell\mathrm{TPQ}$ formulation and
the local Gibbs (LG) ensemble method, which includes a proof of the second law
of thermodynamics and quantum fluctuation theorem in the $\ell\mathrm{TPQ}$
formulation.

###  A.1 Proof of Eqs. (7) and (8)

We here give a proof of Eqs. (7)-(8), which is similarly accomplished, as is
the case with the global thermal equilibrium Sugiura and Shimizu (2012, 2013);
Hyuga  _et al._ (2014); Endo  _et al._ (2018). For that purpose, we rely on
Chebyshev’s inequality:

$\mathrm{Prob}_{z}\Big{[}\left|\mathcal{O}-\langle\mathcal{O}\rangle_{z}\right|\geq\epsilon\Big{]}\leq\frac{\langle\left(\mathcal{O}-\langle\mathcal{O}\rangle_{z}\right)^{2}\rangle_{z}}{\epsilon^{2}}=\frac{\langle\mathcal{O}^{2}\rangle_{z}-\langle\mathcal{O}\rangle_{z}^{2}}{\epsilon^{2}}.$
(S1)

Let us first define the average over the random variable
$z_{\alpha}=(z_{\alpha}^{\prime}+\mathrm{i}z_{\alpha}^{\prime\prime})/\sqrt{2}$
by $\langle\mathcal{O}(z)\rangle_{z}$. Since both real and imaginary parts are
independently sampled from the standard normal distribution, they satisfy,
e.g.,

$\langle z_{\alpha}\rangle_{z}=0,\quad\langle
z_{\alpha_{1}}^{\ast}z_{\alpha_{2}}\rangle_{z}=\delta_{{\alpha_{1}}{\alpha_{2}}},\quad\langle|z_{\alpha}|^{4}\rangle_{z}=2,\quad\langle|z_{\alpha_{1}}|^{2}|z_{\alpha_{2}}|^{2}\rangle_{z}=1,\quad\langle
z_{\alpha_{1}}^{\ast}z_{\alpha_{2}}^{\ast}z_{\alpha_{3}}z_{\alpha_{4}}\rangle_{z}=0,$
(S2)

where $\delta_{{\alpha_{1}}{\alpha_{2}}}$ denotes the Kronecker delta, and the
last two equations hold for ${\alpha_{1}}\neq{\alpha_{2}}$. With the help of
Eqs. (S2), we can show the following key identity for a state
$\ket{\psi_{0}}\equiv\sum_{\alpha}z_{\alpha}\ket{{\alpha}}$:

$\Big{\langle}\left(\bra{\psi_{0}}\hat{A}\ket{\psi_{0}}-\big{\langle}\bra{\psi_{0}}\hat{A}\ket{\psi_{0}}\big{\rangle}_{z}\right)\left(\bra{\psi_{0}}\hat{B}\ket{\psi_{0}}-\big{\langle}\bra{\psi_{0}}\hat{B}\ket{\psi_{0}}\big{\rangle}_{z}\right)\Big{\rangle}_{z}=\mathop{\mathrm{Tr}}\big{(}\hat{A}\hat{B}\big{)},$
(S3)

where $\hat{A}$ and $\hat{B}$ denote arbitrary operators.

Proof of Eq. (7): In order to show the convergence of the partition functional
(7), we use Chebyshev’s inequality (S1) for
$\mathcal{O}=Z_{\ell\mathrm{TPQ}}[\lambda_{t}]/Z_{\mathrm{LG}}[\lambda_{t}]=\braket{\lambda_{t};t}{\lambda_{t};t}/\big{\langle}\braket{\lambda_{t};t}{\lambda_{t};t}\big{\rangle}_{z}$.
Recalling the definition of the partition functional for the
$\ell\mathrm{TPQ}$ state and LG ensemble
$Z_{\mathrm{LG}}[\lambda_{t}]\equiv\mathop{\mathrm{Tr}}\mathrm{e}^{-\hat{K}[\lambda_{t};t]}$,
we find the identity (S3) with
$\hat{A}=\hat{B}=\mathrm{e}^{-\hat{K}[\lambda_{t};t]}$ provides the right-
hand-side of Chebyshev’s inequality thanks to

$\begin{split}\bra{\psi_{0}}\mathrm{e}^{-\hat{K}[\lambda_{t};t]}\ket{\psi_{0}}&=\braket{\lambda_{t};t}{\lambda_{t};t}=Z_{\ell\mathrm{TPQ}}[\lambda_{t}],\\\
\big{\langle}\bra{\psi_{0}}\mathrm{e}^{-\hat{K}[\lambda_{t};t]}\ket{\psi_{0}}\big{\rangle}_{z}&=\big{\langle}\braket{\lambda_{t};t}{\lambda_{t};t}\big{\rangle}_{z}=Z_{\mathrm{LG}}[\lambda_{t}].\end{split}$
(S4)

As a consequence, we obtain

$\Big{\langle}\left(\frac{\braket{\lambda_{t};t}{\lambda_{t};t}}{\big{\langle}\braket{\lambda_{t};t}{\lambda_{t};t}\big{\rangle}_{z}}-1\right)^{2}\Big{\rangle}_{z}=\frac{Z_{\mathrm{LG}}[2\lambda_{t}]}{(Z_{\mathrm{LG}}[\lambda_{t}])^{2}}=\exp\left[-2\int_{\Lambda}\mathrm{d}^{d}x\beta_{t}\Big{(}p(\beta_{t},\mu_{t}^{a})-p(2\beta_{t},\mu_{t}^{a})+O(\partial^{2})\Big{)}\right],$
(S5)

where, to show the second equality, we used the leading-order expression of
the Massieu-Planck functional $\log Z_{\mathrm{LG}}[\lambda_{t}]$ for a
parity-preserving fluid obtained in Refs. Hayata  _et al._ (2015); Hongo
(2017, 2019). Here $\beta_{t}=\beta(t,\bm{x})$ denotes the local inverse
temperature conjugate to the energy density, $\mu_{t}^{a}=\mu^{a}(t,\bm{x})$
local chemical potentials for charge densities $\hat{J}_{~{}a}^{0}$ attached
to internal symmetries of the system labeled by indices $a$. We explicitly put
the size of the fluid cell $\Lambda$. Substituting this result into
Chebyshev’s inequality, we obtain

$\mathrm{Prob}_{z}\left[\left|\frac{Z_{\ell\mathrm{TPQ}}[\lambda_{t}]}{Z_{\mathrm{LG}}[\lambda_{t}]}-1\right|\geq\epsilon\right]\leq\frac{1}{\epsilon^{2}}\exp\left[-2\int_{\Lambda}\mathrm{d}^{d}x\beta_{t}\Big{(}p(\beta_{t},\mu_{t}^{a})-p(2\beta_{t},\mu_{t}^{a})+O(\partial^{2})\Big{)}\right].$
(S6)

The vital point here is that the pressure of a fluid cell $p(\beta,\mu^{a})$
is the decreasing function with respect to $\beta$, and satisfies
$p(\beta,\mu^{a})-p(2\beta,\mu^{a})>0$. Therefore, taking the large size limit
of the fluid cell $V_{\mathrm{cell}}\equiv\int_{\Lambda}\mathrm{d}^{d}x$, we
see that the probability of finding the difference between
$Z_{\ell\mathrm{TPQ}}[\lambda_{t}]$ and $Z_{\mathrm{LG}}[\lambda_{t}]$ will be
exponentially small. This completes the proof of Eq. (7).

Proof of Eq. (8): Let us move on to the proof for Eq. (8). The average value
over $\ell\mathrm{TPQ}$ and LG distribution are given by

$\langle\hat{\mathcal{O}}\rangle^{\ell\mathrm{TPQ}}_{\lambda_{t}}=\frac{\bra{\lambda_{t};t}\hat{\mathcal{O}}\ket{\lambda_{t};t}}{\braket{\lambda_{t};t}{\lambda_{t};t}}\quad\mathrm{and}\quad\langle\hat{\mathcal{O}}\rangle^{\mathrm{LG}}_{\lambda_{t}}=\frac{\big{\langle}\bra{\lambda_{t};t}\hat{\mathcal{O}}\ket{\lambda_{t};t}\big{\rangle}_{z}}{\big{\langle}\braket{\lambda_{t};t}{\lambda_{t};t}\big{\rangle}_{z}}.$
(S7)

Noting
$\langle\hat{\mathcal{O}}\rangle^{\mathrm{LG}}_{\lambda_{t}}\neq\big{\langle}\langle\hat{\mathcal{O}}\rangle^{\ell\mathrm{TPQ}}_{\lambda_{t}}\big{\rangle}_{z}$,
we divide the proof into two steps: First, we evaluate the difference between
$\langle\hat{\mathcal{O}}\rangle^{\ell\mathrm{TPQ}}_{\lambda_{t}}$ and
$\big{\langle}\langle\hat{\mathcal{O}}\rangle^{\ell\mathrm{TPQ}}_{\lambda_{t}}\big{\rangle}_{z}$,
and second, we evaluate the difference between
$\big{\langle}\langle\hat{\mathcal{O}}\rangle^{\ell\mathrm{TPQ}}_{\lambda_{t}}\big{\rangle}_{z}$
and $\langle\hat{\mathcal{O}}\rangle^{\mathrm{LG}}_{\lambda_{t}}$.

As a first step, we use Chebyshev’s inequality for
$\langle\hat{\mathcal{O}}\rangle^{\ell\mathrm{TPQ}}_{\lambda_{t}}=\bra{\lambda;t}\hat{\mathcal{O}}\ket{\lambda_{t};t}/\braket{\lambda_{t};t}{\lambda_{t};t}$.
In order to evaluate the right-hand-side of that inequality, we expand it as

$\begin{split}\Big{\langle}\Big{(}\langle\hat{\mathcal{O}}\rangle^{\ell\mathrm{TPQ}}_{\lambda_{t}}-\big{\langle}\langle\hat{\mathcal{O}}\rangle^{\ell\mathrm{TPQ}}_{\lambda_{t}}\big{\rangle}_{z}\Big{)}^{2}\Big{\rangle}_{z}&=\Bigg{\langle}\left(\frac{\bra{\lambda_{t};t}\hat{\mathcal{O}}\ket{\lambda_{t};t}}{\braket{\lambda_{t};t}{\lambda_{t};t}}-\Big{\langle}\frac{\bra{\lambda_{t};t}\hat{\mathcal{O}}\ket{\lambda_{t};t}}{\braket{\lambda_{t};t}{\lambda_{t};t}}\Big{\rangle}_{z}\right)^{2}\Bigg{\rangle}_{z}\\\
&=\frac{\big{\langle}(\delta
f)^{2}\big{\rangle}_{z}}{(Z_{\mathrm{LG}}[\lambda_{t}])^{2}}-\frac{2\langle\hat{\mathcal{O}}\rangle^{\mathrm{LG}}_{\lambda_{t}}\big{\langle}\delta
f\delta
g\big{\rangle}_{z}}{(Z_{\mathrm{LG}}[\lambda_{t}])^{2}}+\frac{\big{(}\langle\hat{\mathcal{O}}\rangle^{\mathrm{LG}}_{\lambda_{t}}\big{)}^{2}\big{\langle}(\delta
g)^{2}\big{\rangle}_{z}}{(Z_{\mathrm{LG}}[\lambda_{t}])^{2}}+O(\delta^{3}),\end{split}$
(S8)

where we defined

$\begin{split}\delta
f&=\bra{\lambda_{t};t}\hat{\mathcal{O}}\ket{\lambda_{t}t;t}-\big{\langle}\bra{\lambda_{t};t}\hat{\mathcal{O}}\ket{\lambda_{t};t}\big{\rangle}_{z},\\\
\delta
g&=\braket{\lambda_{t};t}{\lambda_{t};t}-\big{\langle}\braket{\lambda_{t};t}{\lambda_{t};t}\big{\rangle}_{z}.\end{split}$
(S9)

Since we have already evaluated $\big{\langle}(\delta g)^{2}\big{\rangle}_{z}$
in Eq. (S5), we only need to evaluate the first two terms in Eq. (S8);
$\big{\langle}(\delta f)^{2}\big{\rangle}_{z}$ and $\big{\langle}\delta
f\delta g\big{\rangle}_{z}$.

Let us first evaluate $\big{\langle}(\delta f)^{2}\big{\rangle}_{z}$. By the
use of the identity (S3) with
$\hat{A}=\hat{B}=\mathrm{e}^{-\frac{1}{2}\hat{K}[\lambda_{t};t]}\hat{\mathcal{O}}\mathrm{e}^{-\frac{1}{2}\hat{K}[\lambda_{t};t]}$,
we obtain

$\big{\langle}(\delta
f)^{2}\big{\rangle}_{z}=\mathop{\mathrm{Tr}}\left(\mathrm{e}^{-\hat{K}[\lambda_{t};t]}\hat{\mathcal{O}}\mathrm{e}^{-\hat{K}[\lambda_{t};t]}\hat{\mathcal{O}}\right).$
(S10)

Since $\hat{K}[\lambda_{t};t]$ is an Hermitian operator, taking trace with its
eigenstate $\ket{n}$ result in

$\big{\langle}(\delta
f)^{2}\big{\rangle}_{z}=\sum_{n,m}\mathrm{e}^{-K_{n}[\lambda_{t};t]}\mathrm{e}^{-K_{m}[\lambda_{t};t]}\bra{n}\hat{\mathcal{O}}\ket{m}\bra{m}\hat{\mathcal{O}}\ket{n}\leq\sum_{n,m}\mathrm{e}^{-K_{n}[\lambda_{t};t]}\mathrm{e}^{-K_{m}[\lambda_{t};t]}\bra{n}\hat{\mathcal{O}}\ket{m}\bra{m}\hat{\mathcal{O}}\ket{n},$
(S11)

where $K_{n}[\lambda_{t};t]$ denotes the eigenvalue of
$\hat{K}[\lambda_{t};t]$, and we used the completeness for the eigenstate of
$\hat{K}[\lambda_{t};t]$ to show the first equality. The inequality results
from the inequality of arithmetic and geometric means: $\sqrt{ab}\leq(a+b)/2$.
The right-hand side of this inequality can be further rewritten as

$\sum_{n,m}\mathrm{e}^{-K_{n}[\lambda_{t};t]}\mathrm{e}^{-K_{m}[\lambda_{t};t]}\bra{n}\hat{\mathcal{O}}\ket{m}\bra{m}\hat{\mathcal{O}}\ket{n}=Z_{\mathrm{LG}}[2\lambda_{t}]\langle\hat{\mathcal{O}}^{2}\rangle^{\mathrm{LG}}_{2\lambda_{t}},$
(S12)

and we now obtain the following inequality for $\big{\langle}(\delta
f)^{2}\big{\rangle}_{z}$:

$\big{\langle}(\delta f)^{2}\big{\rangle}_{z}\leq
Z_{\mathrm{LG}}[2\lambda_{t}]\langle\hat{\mathcal{O}}^{2}\rangle^{\mathrm{LG}}_{2\lambda_{t}}.$
(S13)

To evaluate $\big{\langle}\delta f\delta g\big{\rangle}_{z}$, we simply apply
the identity (S3) with
$\hat{A}=\mathrm{e}^{-\frac{1}{2}\hat{K}[\lambda_{t};t]}\hat{\mathcal{O}}\mathrm{e}^{-\frac{1}{2}\hat{K}[\lambda_{t};t]}$
and $\hat{B}=\mathrm{e}^{-\hat{K}[\lambda_{t};t]}$:

$\big{\langle}\delta f\delta
g\big{\rangle}_{z}=\mathop{\mathrm{Tr}}\left(\mathrm{e}^{-2\hat{K}[\lambda_{t};t]}\hat{\mathcal{O}}\right)=Z_{\mathrm{LG}}[2\lambda_{t}]\langle\hat{\mathcal{O}}\rangle^{\mathrm{LG}}_{2\lambda_{t}}.$
(S14)

Substituting Eqs. (S5), (S13) and (S14) into Eq. (S8), we eventually obtain an
upper bound of the variance as

$\begin{split}\Big{\langle}\Big{(}\langle\hat{\mathcal{O}}\rangle^{\ell\mathrm{TPQ}}_{\lambda_{t}}-\big{\langle}\langle\hat{\mathcal{O}}\rangle^{\ell\mathrm{TPQ}}_{\lambda_{t}}\big{\rangle}_{z}\Big{)}^{2}\Big{\rangle}_{z}&\leq\frac{Z_{\mathrm{LG}}[2\lambda_{t}]}{(Z_{\mathrm{LG}}[\lambda_{t}])^{2}}\left[\langle\hat{\mathcal{O}}^{2}\rangle^{\mathrm{LG}}_{2\lambda_{t}}-2\langle\hat{\mathcal{O}}\rangle^{\mathrm{LG}}_{\lambda_{t}}\langle\hat{\mathcal{O}}\rangle^{\mathrm{LG}}_{2\lambda_{t}}+\big{(}\langle\hat{\mathcal{O}}\rangle^{\mathrm{LG}}_{\lambda_{t}}\big{)}^{2}\right]\\\
&=\frac{Z_{\mathrm{LG}}[2\lambda_{t}]}{(Z_{\mathrm{LG}}[\lambda_{t}])^{2}}\left[\big{\langle}\big{(}\hat{\mathcal{O}}-\langle\hat{\mathcal{O}}\rangle^{\mathrm{LG}}_{2\lambda_{t}}\big{)}^{2}\big{\rangle}^{\mathrm{LG}}_{2\lambda_{t}}+\big{(}\langle\hat{\mathcal{O}}\rangle^{\mathrm{LG}}_{\lambda_{t}}-\langle\hat{\mathcal{O}}\rangle^{\mathrm{LG}}_{2\lambda_{t}}\big{)}^{2}\right],\end{split}$
(S15)

which results in the following inequality:

$\mathrm{Prob}_{z}\left[\left|\langle\hat{\mathcal{O}}\rangle^{\ell\mathrm{TPQ}}_{\lambda_{t}}-\big{\langle}\langle\hat{\mathcal{O}}\rangle^{\ell\mathrm{TPQ}}_{\lambda_{t}}\big{\rangle}_{z}\right|\geq\epsilon\right]\leq\frac{\mathrm{e}^{-2\int_{\Lambda}\mathrm{d}^{d}x\beta_{t}[p(\beta_{t},\mu_{t}^{a})-p(2\beta_{t},\mu_{t}^{a})])}}{\epsilon^{2}}\left[\big{\langle}\big{(}\hat{\mathcal{O}}-\langle\hat{\mathcal{O}}\rangle^{\mathrm{LG}}_{2\lambda_{t}}\big{)}^{2}\big{\rangle}^{\mathrm{LG}}_{2\lambda_{t}}+\big{(}\langle\hat{\mathcal{O}}\rangle^{\mathrm{LG}}_{\lambda_{t}}-\langle\hat{\mathcal{O}}\rangle^{\mathrm{LG}}_{2\lambda_{t}}\big{)}^{2}\right].$
(S16)

As a consequence, for any operator $\hat{\mathcal{O}}$ whose two variations
$\big{\langle}\big{(}\hat{\mathcal{O}}-\langle\hat{\mathcal{O}}\rangle^{\mathrm{LG}}_{2\lambda_{t}}\big{)}^{2}\big{\rangle}^{\mathrm{LG}}_{2\lambda_{t}}$
and
$\big{(}\langle\hat{\mathcal{O}}\rangle^{\mathrm{LG}}_{\lambda_{t}}-\langle\hat{\mathcal{O}}\rangle^{\mathrm{LG}}_{2\lambda_{t}}\big{)}^{2}$
do not show an exponentially large behavior with respect to the volume of
fluid cells $V_{\mathrm{cell}}$, the inequality brings about the following
result on convergence:

$\langle\hat{\mathcal{O}}\rangle^{\ell\mathrm{TPQ}}_{\lambda_{t}}\xrightarrow{P}\big{\langle}\langle\hat{\mathcal{O}}\rangle^{\ell\mathrm{TPQ}}_{\lambda_{t}}\big{\rangle}_{z},$
(S17)

which gives the first step of the proof.

Thanks to Eq. (S5) and (S14), the second step is mostly finished if we
evaluate the average difference by expanding it as follows:

$\begin{split}\left|\big{\langle}\langle\hat{\mathcal{O}}\rangle^{\ell\mathrm{TPQ}}_{\lambda_{t}}\big{\rangle}_{z}-\langle\hat{\mathcal{O}}\rangle^{\mathrm{LG}}_{\lambda_{t}}\right|&=\frac{1}{(Z_{\mathrm{LG}}[\lambda_{t}])^{2}}\left|-\big{\langle}\delta
f\delta
gf\big{\rangle}_{z}+\langle\hat{\mathcal{O}}\rangle^{\mathrm{LG}}_{\lambda_{t}}\big{\langle}(\delta
g)^{2}\big{\rangle}_{z}\right|\\\
&=\frac{Z_{\mathrm{LG}}[2\lambda_{t}]}{(Z_{\mathrm{LG}}[\lambda_{t}])^{2}}\left|\langle\hat{\mathcal{O}}\rangle^{\mathrm{LG}}_{2\lambda_{t}}-\langle\hat{\mathcal{O}}\rangle^{\mathrm{LG}}_{\lambda_{t}}\right|\\\
&=\exp\left(-2\int_{\Lambda}\mathrm{d}^{d}x\beta_{t}[p(\beta_{t},\mu_{t}^{a})-p(2\beta_{t},\mu_{t}^{a})])\right)\left|\langle\hat{\mathcal{O}}\rangle^{\mathrm{LG}}_{2\lambda_{t}}-\langle\hat{\mathcal{O}}\rangle^{\mathrm{LG}}_{\lambda_{t}}\right|.\end{split}$
(S18)

We thus conclude that any operator $\hat{\mathcal{O}}$ not showing the
exponentially large fluctuation, the difference between
$\big{\langle}\langle\hat{\mathcal{O}}\rangle^{\ell\mathrm{TPQ}}_{\lambda_{t}}\big{\rangle}_{z}$
and $\langle\hat{\mathcal{O}}\rangle^{\mathrm{LG}}_{\lambda_{t}}$ is
exponentially small. This completes the proof of Eq. (8).

###  A.2 Proof of the Eqs. (14) and (15)

Although the second law of thermodynamics follows from the quantum fluctuation
theorem (15), we provide independent proofs of them.

Proof of Eq. (14): The second law of thermodynamics can be easily shown by
rewriting the entropy production as follows:

$\begin{split}\langle\hat{\Sigma}[t,t_{0};\lambda]\rangle&\equiv\langle\hat{S}[\lambda_{t};t]\rangle^{\ell\mathrm{TPQ}}_{\lambda_{t_{0}}}-\langle\hat{S}[\lambda_{t_{0}};t_{0}]\rangle^{\ell\mathrm{TPQ}}_{\lambda_{t_{0}}}\\\
&=\underbrace{\langle\hat{S}[\lambda_{t};t]\rangle^{\ell\mathrm{TPQ}}_{\lambda_{t_{0}}}-\langle\hat{S}[\lambda_{t};t]\rangle^{\mathrm{LG}}_{\lambda_{t_{0}}}}_{(a)}-\underbrace{\big{(}\langle\hat{S}[\lambda_{t_{0}};t_{0}]\rangle^{\ell\mathrm{TPQ}}_{\lambda_{t_{0}}}-\langle\hat{S}[\lambda_{t_{0}};t_{0}]\rangle^{\mathrm{LG}}_{\lambda_{t_{0}}}\big{)}}_{(b)}+\underbrace{\langle\hat{S}[\lambda_{t};t]\rangle^{\mathrm{LG}}_{\lambda_{t_{0}}}-\langle\hat{S}[\lambda_{t_{0}};t_{0}]\rangle^{\mathrm{LG}}_{\lambda_{t_{0}}}}_{(c)}.\end{split}$
(S19)

Using the equivalence between the $\ell\mathrm{TPQ}$ state and LG ensemble, we
can estimate the first and second terms to be exponentially small:
$(a)=O(\mathrm{e}^{-B_{1}V_{\mathrm{cell}}})$ and
$(b)=O(\mathrm{e}^{-B_{2}V_{\mathrm{cell}}})$ with some constants $B_{1}$ and
$B_{2}$. Furthermore, the last term is shown to be positive: $(c)\geq 0$ (See
Refs. Hayata  _et al._ (2015); Hongo (2019); Hongo and Hattori (2021)). This
results in Eq. (14).

Proof of Eq. (15): To prove the quantum fluctuation theorem (15) for the
$\ell\mathrm{TPQ}$ state, we first recall that for the LG ensemble Hongo
(2019). For that purpose, let us introduce the generating function for the
entropy production under the forward/backward evolution by

$\displaystyle G_{F}^{\mathrm{LG}}(z)$
$\displaystyle\equiv\mathop{\mathrm{Tr}}\left(\hat{\rho}_{\mathrm{LG}}[\lambda_{t_{0}};t_{0}]\hat{\mathcal{U}}^{\dagger}(t)\mathrm{e}^{\mathrm{i}z\hat{S}[\lambda_{t};t_{0}]}\hat{\mathcal{U}}(t)\mathrm{e}^{-\mathrm{i}z\hat{S}[\lambda_{t_{0}};t_{0}]}\right),$
(S20) $\displaystyle G_{B}^{\mathrm{LG}}(z)$
$\displaystyle\equiv\mathop{\mathrm{Tr}}\left(\hat{\rho}_{\mathrm{LG}}[\tilde{\lambda}_{t};t]\widetilde{\mathcal{U}}(t)\mathrm{e}^{\mathrm{i}z\hat{S}[\tilde{\lambda}_{t_{0}};t]}\widetilde{\mathcal{U}}^{\dagger}(t)\mathrm{e}^{-\mathrm{i}z\hat{S}[\tilde{\lambda}_{t};t]}\right),$
(S21)

where $\hat{\mathcal{U}}(t)$ is a time evolution operator. By using a combined
parity and time-reversal $\Theta\equiv\mathsf{P}\mathsf{T}$, we also defined
$\mathsf{P}\mathsf{T}$-transformed time evolution operator and parameter by
$\widetilde{\mathcal{U}}(t)\equiv\Theta\hat{\mathcal{U}}(t)\Theta^{-1}$ and
$\tilde{\lambda}_{t}\equiv\epsilon^{a}\lambda^{a}(t,-\bm{x})$ with
$\epsilon^{a}$ being a $\mathsf{P}\mathsf{T}$-eigenvalue of charge densities
$\hat{c}_{a}$, respectively. Here, we introduced the LG distribution for the
density operator $\hat{\rho}_{\mathrm{LG}}[\lambda_{t};t]$ and the entropy
functional operator $\hat{S}_{\mathrm{LG}}[\lambda_{t};t]$ by

$\hat{\rho}_{\mathrm{LG}}[\lambda_{t};t]\equiv\mathrm{e}^{-\hat{S}_{\mathrm{LG}}[\lambda_{t};t]}\quad\mathrm{and}\quad\hat{S}_{\mathrm{LG}}[\lambda_{t};t]\equiv\hat{K}[\lambda_{t};t]+\log
Z_{\mathrm{LG}}[\lambda_{t}].$ (S22)

The generating function in Eqs. (S20) and (S21) are shown to satisfy the
quantum fluctuation theorem Hongo (2019):

$G^{\mathrm{LG}}_{F}(z)=G^{\mathrm{LG}}_{B}(-z+\mathrm{i}).$ (S23)

In order to show the quantum fluctuation theorem for the $\ell\mathrm{TPQ}$
state, we then introduce the reference generating function by

$\overline{G}_{F}(z)\equiv\mathop{\mathrm{Tr}}\left(\hat{\rho}_{\mathrm{LG}}[\lambda_{t_{0}};t_{0}]\mathrm{e}^{\mathrm{i}z\hat{S}_{\ell\mathrm{TPQ}}[\lambda_{t};t]}\mathrm{e}^{-\mathrm{i}z\hat{S}_{\ell\mathrm{TPQ}}[t_{0};\lambda_{t_{0}}]}\right).$
(S24)

By the use of this reference generating function with the triangle inequality,
we can evaluate the difference between $G^{\ell\mathrm{TPQ}}_{F}(z)$ and
$G^{\mathrm{LG}}_{F}(z)$ as follows:

$\begin{split}\left|G^{\ell\mathrm{TPQ}}_{F}(z)-G^{\mathrm{LG}}_{F}(z)\right|&=\left|G^{\ell\mathrm{TPQ}}_{F}(z)-\overline{G}_{F}(z)-G^{\mathrm{LG}}_{F}(z)+\overline{G}_{F}(z)\right|\\\
&\leq\underbrace{\left|G^{\ell\mathrm{TPQ}}_{F}(z)-\overline{G}_{F}(z)\right|}_{(A)}+\underbrace{\Big{|}G^{\mathrm{LG}}_{F}(z)+\overline{G}_{F}(z)\Big{|}}_{(B)}.\end{split}$
(S25)

Defining $\Delta\Psi[\lambda_{t},\lambda_{t_{0}}]\equiv\log
Z[\lambda_{t}]-\log Z[\lambda_{t_{0}}]$ for both $\ell\mathrm{TPQ}$ and LG
states, and using the error estimation for the partition functional, we can
estimate the second term in the second line as

$\begin{split}(B)&=\left|\mathop{\mathrm{Tr}}\left(\hat{\rho}_{\mathrm{LG}}[\lambda_{t_{0}};t_{0}]\mathrm{e}^{\mathrm{i}z\hat{K}[\lambda_{t};t]}\mathrm{e}^{-\mathrm{i}z\hat{K}[\lambda_{t_{0}};t_{0}]}\right)\left(\mathrm{e}^{\mathrm{i}z\Delta\Psi_{\mathrm{LG}}[\lambda_{t},\lambda_{t_{0}}]}-\mathrm{e}^{\mathrm{i}z\Delta\Psi_{\ell\mathrm{TPQ}}[\lambda_{t},\lambda_{t_{0}}]}\right)\right|\\\
&=\left|\mathop{\mathrm{Tr}}\left(\hat{\rho}_{\mathrm{LG}}[\lambda_{t_{0}};t_{0}]\mathrm{e}^{\mathrm{i}z\hat{K}[\lambda_{t};t]}\mathrm{e}^{-\mathrm{i}z\hat{K}[\lambda_{t_{0}};t_{0}]}\right)\mathrm{e}^{\mathrm{i}z\Delta\Psi_{\ell\mathrm{TPQ}}[\lambda_{t},\lambda_{t_{0}}]}\left(\exp\left[O(\mathrm{e}^{-C_{1}V_{\mathrm{cell}}})\right]-1\right)\right|\\\
&=O(\mathrm{e}^{-C_{1}V_{\mathrm{cell}}}).\end{split}$ (S26)

Since the first term in Eq. (S25) is given by the difference of the averaged
operator over the $\ell\mathrm{TPQ}$ state and LG distribution, we can
evaluate it as

$\begin{split}(A)&=\left[\langle\mathrm{e}^{\mathrm{i}z\hat{S}_{\ell\mathrm{TPQ}}[\lambda_{t};t]}\mathrm{e}^{-\mathrm{i}z\hat{S}_{\ell\mathrm{TPQ}}[\lambda_{t_{0}};t_{0}]}\rangle^{\ell\mathrm{TPQ}}_{\lambda_{t_{0}}}-\langle\mathrm{e}^{\mathrm{i}z\hat{S}_{\ell\mathrm{TPQ}}[\lambda_{t};t]}\mathrm{e}^{-\mathrm{i}z\hat{S}_{\ell\mathrm{TPQ}}[\lambda_{t_{0}};t_{0}]}\rangle^{\mathrm{LG}}_{\lambda_{t_{0}}}\right]\\\
&=O(\mathrm{e}^{-C_{2}V_{\mathrm{cell}}}),\end{split}$ (S27)

where we assumed that the variation of the averaged operator does not show the
exponentially large behavior. Substituting Eqs. (S26)-(S27) into Eq. (S25), we
obtain

$\left|G^{\ell\mathrm{TPQ}}_{F}(z)-G^{\mathrm{LG}}_{F}(z)\right|\leq
O(\mathrm{e}^{-C_{3}V_{\mathrm{cell}}}),$ (S28)

which states the equivalence of the generating function for the entropy
production for the forward evolution. On the other hand, a similar analysis
also works for $G_{B}^{\ell\mathrm{TPQ}}(z)-G_{B}^{\mathrm{LG}}(z)$, and thus,
we have,

$\left|G^{\ell\mathrm{TPQ}}_{B}(-z+\mathrm{i})-G^{\mathrm{LG}}_{B}(-z+\mathrm{i})\right|\leq
O(\mathrm{e}^{-C_{4}V_{\mathrm{cell}}}),$ (S29)

Therefore, combining Eq. (S23) with Eqs. (S28)-(S29), we can evaluate

$\begin{split}&\left|G^{\ell\mathrm{TPQ}}_{F}(z)-G^{\ell\mathrm{TPQ}}_{B}(-z+\mathrm{i})\right|\\\
=&\left|G^{\ell\mathrm{TPQ}}_{F}(z)-G_{F}^{\mathrm{LG}}(z)-\Big{(}G_{B}^{\ell\mathrm{TPQ}}(-z+\mathrm{i})-G_{B}^{\mathrm{LG}}(-z+\mathrm{i})\Big{)}+G_{F}^{\mathrm{LG}}(z)-G^{\mathrm{LG}}_{B}(-z+\mathrm{i})\right|\\\
\leq&\left|G^{\ell\mathrm{TPQ}}_{F}(z)-G_{F}^{\mathrm{LG}}(z)\right|+\left|G_{B}^{\ell\mathrm{TPQ}}(-z+\mathrm{i})-G_{B}^{\mathrm{LG}}(-z+\mathrm{i})\right|\\\
=&O(\mathrm{e}^{-A_{2}}V_{\mathrm{cell}}),\end{split}$ (S30)

where the inequality in the third line again follows from the triangle
inequality. This gives the quantum fluctuation theorem for the
$\ell\mathrm{TPQ}$ state.

##  Appendix B Detail of numrerical implementation

We here summarize a detail of our numerical simulation based on the
$\ell\mathrm{TPQ}$ formulation such as (i) the concrete definition of the
(smeared) energy density/current operator, (ii) how to prepare the
$\ell\mathrm{TPQ}$ state and the equation of state, and (iii) two ways to
evaluate the thermal conductivity, and (iv) a numerical confirmation of the
fluctuation theorem and the second law of thermodynamics.

###  B.1 Energy density/current operator and their smearing

Let us first write the Hamiltonian in the following form:

$\displaystyle\hat{H}=\sum_{n}\hat{h}(n),\quad\hat{h}(n)=\mathcal{H}_{n,n+1}+\mathcal{H}_{n},$
(S31)

where the energy density operator $\hat{h}(n)$ is decomposed into the sum of
the nearest-neighbor term $\mathcal{H}_{n,n+1}$ and the on-site term
$\mathcal{H}_{n}$. Then, we shall derive the energy current operator
$\hat{J}_{E}(n-1/2)$. In order to determine the form of $\hat{J}_{E}(n-1/2)$,
we require that the time evolution of the Hamiltonian density should be
written as the discrete divergence of the energy current operator as

$\displaystyle\frac{\partial\hat{h}(t,n)}{\partial
t}=-\big{(}\hat{J}_{E}(t,n+1/2)-\hat{J}_{E}(t,n-1/2)\big{)}.$ (S32)

On the other hand, the Heisenberg equation for the Hamiltonian density
operator reads

$\displaystyle\frac{\partial\hat{h}(t,n)}{\partial
t}=\mathrm{i}\big{[}\hat{H},\hat{h}(t,n)\big{]}$ (S33)

Comparing these two equations, we identify the energy current as follows:

$\displaystyle\hat{J}_{E}(n-1/2)=\mathrm{i}\left[\mathcal{H}_{n-1,n},\mathcal{H}_{n,n+1}+\mathcal{H}_{n}\right].$
(S34)

Thus, the concrete form of the energy density operator $\hat{h}(n)$ and energy
current operator $\hat{J}_{E}(n-1/2)$ for the system defined by the
Hamiltonian (16) is given by

$\displaystyle\hat{h}(n)=$ $\displaystyle
J_{z}\sigma_{n}^{z}\sigma_{n+1}^{z}+D(\sigma_{n}^{z}\sigma_{n+1}^{x}-\sigma_{n}^{x}\sigma_{n+1}^{z})+\Gamma\sigma_{n}^{x}+B\sigma_{n}^{z}$
(S35) $\displaystyle\hat{J}_{E}(n-1/2)=$
$\displaystyle-2D^{2}\left(\sigma_{n-1}^{x}\sigma_{n}^{y}\sigma_{n+1}^{z}-\sigma_{n-1}^{z}\sigma_{n}^{y}\sigma_{n+1}^{x}\right)+4DJ_{z}\sigma_{n-1}^{z}\sigma_{n}^{y}\sigma_{n+1}^{z}$
$\displaystyle-2\Gamma
J_{z}\sigma_{n-1}^{z}\sigma_{n}^{y}+2D\Gamma\sigma_{n-1}^{x}\sigma_{n}^{y}+2DB\sigma_{n-1}^{z}\sigma_{n}^{y}.$
(S36)

Let us then define a smearing of local operators. To extract the hydrodynamic
behavior, we define a smeared operator of an arbitrary local operator
$\hat{\mathcal{O}}(n)$ by

$\displaystyle\bar{\mathcal{O}}(n)=\sum_{n=1}^{N}w_{n-m}\hat{\mathcal{O}}(m),$
(S37)

with an appropriate weight function $w_{n-m}$ which is a positive and has a
peak at $n=m$. We also require that $w_{n}$ satisfies the periodic boundary
condition $w_{n-N}=w_{n}$. This requirement is needed in our setup to define a
smeared energy current operator in a consistent manner. To see this, we apply
Eq. (S37) to the local Hamiltonian:

$\displaystyle\frac{\partial\bar{h}(t,n)}{\partial
t}=\sum_{m=1}^{N}w_{n-m}\frac{\partial\hat{h}(t,m)}{\partial
t}=-\left(\sum_{m=1}^{N}w_{n+1-(m+1/2)}\hat{J}_{E}(t,m+1)-\bar{J}_{E}(t,n-1/2)\right)$
(S38)

The first term of the right hand side is rewritten as

$\displaystyle\sum_{m=1}^{N}w_{n+1-(m+1)}\hat{J}_{E}(t,m+1/2)=\bar{J}_{E}(t,n+1/2)-(w_{n}-w_{n-N})\hat{J}_{E}(t,1/2).$
(S39)

Thus, we find that the smeared current operator holds the continuity equation

$\displaystyle\frac{\partial\bar{h}(t,n)}{\partial
t}=-\left(\bar{J}_{E}(t,n+1/2)-\bar{J}_{E}(t,n-1/2)\right)$ (S40)

thanks to the periodicity $w_{n-N}=w_{n}$. In our numerical simulations, we
choose the weight function $w_{n}$ as

$\displaystyle
w_{n}=\frac{1}{\sum_{m=1}^{N}e^{\frac{1}{\sigma^{2}}\cos\frac{2\pi
m}{N}}}e^{\frac{1}{\sigma^{2}}\cos\frac{2\pi n}{N}},$ (S41)

where $\sigma^{2}$ is a positive parameter controlling the smearing length. In
the thermodynamic limit with $N\to\infty$, the typical width of this function
is given by $1-I_{1}(1/\sigma^{2})/I_{0}(1/\sigma^{2})$, where $I_{n}(x)$ is
the modified Bessel function of the first kind. In the $1/\sigma^{2}\to\infty$
limit for a fixed $N$, the weight function converges to the Kronecker delta.
For our purpose, the width of the weight function, regarded as the effective
size of the fluid cell $\Lambda$, should be taken so that $1\ll\Lambda\ll N$.
Typical shapes of the weight function is shown in Fig. S1. For instance, the
number of spins in the fluid cell is about 5 when $1/\sigma^{2}=5$.

Figure S1: The weight function Eq. (S41).

###  B.2 How to prepare $\ell\mathrm{TPQ}$ state and equation of state

#### Construction of the $\ell\mathrm{TPQ}$ state.

Let us first review a way to implement a canonical TPQ state which is defined
by

$\displaystyle\ket{\beta}\equiv\exp\left(\frac{-N\beta\hat{h}}{2}\right)\sum_{\alpha}z_{\alpha}\ket{\alpha},$
(S42)

where $\hat{h}=\hat{H}/N$ is the Hamiltonian density and $\ket{\alpha}$ is a
random vector in the Hilbert space. We expand the canonical TPQ state by
microcanonical TPQ states as

$\displaystyle\ket{\beta}=\mathrm{e}^{-N\beta
l/2}\sum_{k=0}^{\infty}R_{k}\ket{\psi_{k}}\quad\mathrm{with}\quad\ket{\psi_{k}}=\frac{\ket{k}}{\braket{k}{k}},\quad\ket{k}\equiv(l-\hat{h})^{k}\sum_{\alpha}z_{\alpha}\ket{\alpha}\quad\mathrm{and}\quad
R_{k}(\beta)$ $\displaystyle=\frac{(N\beta/2)^{k}\braket{k}{k}}{k!},$ (S43)

where $l$ is a certain number larger than the maximum eigenvalue of $\hat{h}$.
One can show that $R_{k}$ has a sharp peak at a certain $k=k^{*}$, whose
magnitude is controlled by the inverse temperature Sugiura and Shimizu (2012,
2013); Hyuga  _et al._ (2014); Endo  _et al._ (2018). Thanks to this property,
it is justified to truncate the expansion at a certain order
$k=k_{\text{max}}$. Then, we define $k_{\text{max}}$ by

$\displaystyle\frac{R_{k_{\text{max}}}}{\max_{k}R_{k}}>10^{-t},$ (S44)

for a given tolerance parameter $t$. In our numerical simulations, we take
$t=20$ and confirm that the expectation values of observables are well
converged.

One can generalize the above procedure in a straightforward manner to the
implementation of $\ell\mathrm{TPQ}$ states. For that purpose, we first define

$\displaystyle\ket{\\{\beta(n)\\}}\equiv\exp\left(\frac{-\sum_{n=1}^{N}\beta(n)\hat{h}(n)}{2}\right)\sum_{\alpha}z_{\alpha}\ket{\alpha},$
(S45)

where $\hat{h}_{n}$ is the local Hamiltonian density. Expanding the
exponential operator, we find

$\displaystyle\ket{\\{\beta(n)\\}}$
$\displaystyle=\mathrm{e}^{-\frac{1}{2}\sum_{n}\beta(n)l_{n}}\exp\left(\frac{\sum_{n=1}^{N}\beta(n)(l_{n}-\hat{h}(n))}{2}\right)\sum_{\alpha}z_{\alpha}\ket{\alpha}$
(S46)
$\displaystyle=\mathrm{e}^{-\frac{1}{2}\sum_{n}\beta(n)l_{n}}\sum_{k}\frac{1}{2^{k}k!}\left(\sum_{n=1}^{N}\beta(n)(l_{n}-\hat{h}(n))\right)^{k}\sum_{\alpha}z_{\alpha}\ket{\alpha}$
(S47)
$\displaystyle=\mathrm{e}^{-\frac{1}{2}\sum_{n}\beta(n)l_{n}}\sum_{k}\frac{\bar{\beta}^{k}N^{k}}{2^{k}k!}\left(\frac{1}{N\bar{\beta}}\sum_{n=1}^{N}\beta(n)(l_{n}-\hat{h}(n))\right)^{k}\sum_{\alpha}z_{\alpha}\ket{\alpha},$
(S48)

where $l_{n}$ is a certain number larger than the maximum eigenvalue of
$\hat{h}$, and $\bar{\beta}=\frac{1}{N}\sum_{n=1}^{N}\beta(n)$ is the average
inverse temperature. We note that the state appeared in Eq. (S48)

$\displaystyle\ket{k,\\{\beta(n)\\}}\equiv\left(\frac{1}{N\bar{\beta}}\sum_{n=1}^{N}\beta(n)(l_{n}-\hat{h}(n))\right)^{k}\sum_{\alpha}z_{\alpha}\ket{\alpha}$
(S49)

can be regarded as a generalization of the microcanonical TPQ state $\ket{k}$:
Indeed, we can show $\ket{k,\\{\beta(n)\\}}=\ket{k}$ if
$\beta=\beta_{1}=\dots=\beta_{N}$, $l=l_{1}=\dots,l_{N}$. Then, we obtain the
expression the $\ell\mathrm{TPQ}$ state as follows:

$\displaystyle\ket{\\{\beta(n)\\}}=e^{-N\beta
l/2}\sum_{k=0}^{\infty}R_{k}(\\{\beta(n)\\})\ket{\psi_{k},\\{\beta(n)\\}},$
(S50)
$\displaystyle\quad\mathrm{with}\quad\ket{\psi_{k},\\{\beta(n)\\}}=\frac{\ket{k,\\{\beta(n)\\}}}{\braket{k,\\{\beta(n)\\}}{k,\\{\beta(n)\\}}},\quad
R_{k}(\\{\beta(n)\\})=\frac{(N\beta/2)^{k}\braket{k,\\{\beta(n)\\}}{k,\\{\beta(n)\\}}}{k!}.$
(S51)

We confirmed that $R_{k}(\\{\beta(n)\\})$ also has a sharp peak at a certain
$k=k^{*}$ so that we can truncate the summation over $k$ in Eq. (S50) in a
same manner as the canonical TPQ states. Equations (S48)-(S51) define a way to
prepare the $\ell\mathrm{TPQ}$ state starting from an arbitrary random vector
$\ket{a}$ in the Hilbert space.

#### The energy-temperature relation.

In the main article, we introduce the local temperature according to the
matching condition
$\langle\bar{h}(t,n)\rangle=\langle\bar{h}(t,n)\rangle^{\ell\mathrm{TPQ}}_{\lambda_{t}}$.
Within the derivative expansion, this condition is equivalent to use the
thermodynamic relation to determine the local temperature from the energy
density.

One can use the equation of state as such a thermodynamic relation, which
enables us to relate the energy density and local temperature. Using the
canonical TPQ states, the total energy of the system is calculated as

$\displaystyle
E\equiv\frac{\braket{\beta}{\hat{H}}{\beta}}{\braket{\beta}{\beta}}.$ (S52)

This relation gives the one-to-one correspondence between the energy and the
inverse temperature as long as first order phase transitions do not occur. We
calculate the total energy for several inverse temperatures
$\beta=0.0,0.1,\cdots,3.0$ for the system size $N=20$. The coupling constants
are set to $J_{z}=\Gamma=-B=D=0.2$. The results is shown in Fig. S2.

Figure S2: The equation of state for the Hamiltonian Eq. (16) with the system
size $N=20$. The total energy is calculated at $\beta=0.0,0.1,\cdots,3.0$ and
they are represented by circles. The solid line is the interpolation of them.

###  B.3 Two ways to evaluate thermal conductivity

#### Thermal conductivity from the Fourier law.

While the formal arguments rely on the Heisenberg picture, we perform
numerical simulations based on the Schrödinger picture. The expectation value
of an operator $\hat{O}$ at time $t$ is calculated by

$\displaystyle\braket{\hat{O}(t)}=\frac{\braket{\psi(t)}{\hat{O}}{\psi(t)}}{\braket{\psi(0)}{\psi(0)}},$
(S53) $\displaystyle\mathrm{i}\frac{\partial}{\partial t}\ket{\psi(t)}$
$\displaystyle=\hat{H}\ket{\psi(t)},\quad\ket{\psi(0)}=\ket{\\{\beta(n)\\}}.$
(S54)

Unless otherwise noted, the number of spins is taken as $N=20$, and the
coupling constants are $J_{z}=\Gamma=-B=D=0.2$ so that the system is
nonintegrable. We set the initial temperature profile of the
$\ell\mathrm{TPQ}$ state as

$\displaystyle\beta(n)=\bar{\beta}+\Delta\beta\sin(2\pi(n-1)/N),\quad
n=1,\dots,N,$ (S55)

with $\bar{\beta}=1.5$ and $\Delta\beta=0.2$. We numerically solve the
Schrödinger equation by the fourth order Runge-Kutta method with a time slice
$\Delta t=0.01$, and calculated the local energy and the energy current. The
smearing is also performed by using the weight function given in Eq. (S41)
with $1/\sigma^{2}=5$. The local temperature is estimated from the local
energy through the relation in equilibrium shown in Fig. S2 assuming that
derivative corrections are negligible.

If the system is well described by hydrodynamics, we can extract the thermal
conductivity from the Fourier law Eq. (18). We introduce temporally averaged
operator by

$\displaystyle\tilde{O}(t)$
$\displaystyle\equiv\frac{1}{\tau}\int_{t}^{t+\tau}\mathrm{d}t^{\prime}\bar{O}(t^{\prime}).$
(S56)

The time evolution of temporally averaged quantities are shown in Fig. S3 (and
Fig. 1 in the main text), where $\tau=10$. Again, the local temperature is
estimated by using the equation of state in equilibrium as
$\tilde{T}(t,n)=T(\braket{\tilde{h}(t,n)})$. (In the main text, the tilde is
omitted for the simplicity of notation.)

Figure S3: The time evolution of the local energy, the local temperature and
the energy current.

In Fig. S4, we show the time evolution of $\nabla\tilde{T}(t,n)$,
$\braket{\tilde{J}_{E}(t,n-1/2)}$ and $\kappa_{\text{Fourier}}(t,n)$. We see
that the typical equilibration time is $t_{\text{eq}}\sim 100$ at which there
are no net energy current nor temperature gradient.

We remark that evaluation of the thermal conductivity
$\kappa_{\text{Fourier}}(t,n)$ from the Fourier law is difficult at a large-
time limit $t\geq t_{\text{eq}}$ or inappropriate site around $n=6$ and $16$.
This is because both temperature gradient $|\nabla\tilde{T}(t,n)|$ and energy
current $\braket{\tilde{J}_{E}(t,n-1/2)}$ takes a small value in those setups
so that numerical evaluations suffer from large uncertainty. Thus, we can
rigorously estimate the thermal conductivity from the data at $t\ll
t_{\text{eq}}$ and at the node position of the temperature profile. In the
main article, we show the thermal conductivity at $n=1,11$ in Fig. 2.

Figure S4: The time evolution of the energy current, the temperature gradient,
and the thermal conductivity at $n=1,\cdots,5$.

#### Thermal conductivity from the Green-Kubo formula.

On the basis of the Green-Kubo formula, the thermal conductivity is given by

$\displaystyle\kappa(t,n)=\frac{\beta_{n}(t)}{4N}\int_{-\infty}^{\infty}\mathrm{d}t\langle\\{\hat{\mathsf{J}}_{E}(t),\hat{\mathsf{J}}_{E}(0)\\}\rangle^{\mathrm{TPQ}}_{\beta_{n}}\simeq\frac{\beta}{4N}\int_{-\infty}^{\infty}\mathrm{d}t\langle\\{\hat{\mathsf{J}}_{E}(t),\hat{\mathsf{J}}_{E}(0)\\}\rangle^{\mathrm{TPQ}}_{\beta}$
(S57)

where $\hat{\mathsf{J}}_{E}(t)=\sum_{n=1}^{N}\hat{J}^{n}_{E}(t)$. In the
second equality, we neglect the spatial and temporal dependence of
temperature, which gives rise to the higher-order derivative correction.
Conceptually, the Green-Kubo formula does not require a fully microscopic time
resolution but only a hydrodynamic one. Therefore, we also replace the bare
correlation function
$\langle\\{\hat{\mathsf{J}}_{E}(t),\hat{\mathsf{J}}_{E}(0)\\}\rangle^{\mathrm{TPQ}}_{\beta}$
with the temporally smeared one, which leads to

$\displaystyle
G(t)\equiv\frac{1}{t_{h}^{2}}\int_{t}^{t+t_{h}}\mathrm{d}s\int_{0}^{t_{h}}\mathrm{d}s^{\prime}\langle\\{\hat{\mathsf{J}}_{E}(s),\hat{\mathsf{J}}_{E}(s^{\prime})\\}\rangle^{\mathrm{TPQ}}_{\beta}.$
(S58)

An appropriate time scale $t_{h}$ is estimated from the damping time of the
bare correlation function
$\langle\\{\hat{\mathsf{J}}_{E}(t),\hat{\mathsf{J}}_{E}(0)\\}\rangle^{\mathrm{TPQ}}_{\beta}$.
Below, we set $t_{h}=10$ which is numerically computed. The thermal
conductivity is now given by

$\displaystyle\kappa(t,n)\simeq\frac{\beta}{2}\left(C(\infty)-C(-\infty)\right)\quad\mathrm{with}\quad
C(t)=\frac{1}{2N}\int_{0}^{t}\mathrm{d}t^{\prime}G(t^{\prime}).$ (S59)

Since $C(t)$ is defined by the symmetric Green’s function, we can show it is
equal to $-C(-t)$ within a small error, which is actually confirmed
numerically. Thus, we reach the following simple linear response result in the
$\ell\mathrm{TPQ}$ formulation:

$\displaystyle\kappa(t,n)\simeq\beta C(\infty)\equiv\kappa$ (S60)

The left panel of Fig. S5 demonstrates a typical behavior of the function
$C(t)$, from which one can confirm that $C(t)$ converges when $t\gtrsim 100$.
The value at infinitely late time $C(\infty)$ is estimated by the average of
$C(t)$ from $t=100$ to $200$. The resulting thermal conductivity at each
$\beta$ is shown in the left panel of Fig. S5, which corresponds to the result
shown in Fig. 2 in the main text.

Figure S5:  (Left) Solid line represents $C(t)$ at $\beta=1.5$. The dotted
line shows the late time average of $C(t)$. (Right) The $\beta$-dependence of
the thermal conductivity.

###  B.4 Numerical evaluation of generating functions in fluctuation theorem

Let us finally present a numerical method to compute the generating
functional. When the Hamiltonian does not depend on time explicitly, we can
rewrite the generating functions for the total entropy production Eqs.
(12)-(13) as follows:

$\displaystyle G_{F}^{\ell\mathrm{TPQ}}(z;t)$
$\displaystyle=\frac{\mathrm{e}^{\mathrm{i}z\log(Z_{\ell\mathrm{TPQ}}[\\{\tilde{\beta}(t,n)\\}]/Z_{\ell\mathrm{TPQ}}[\\{\tilde{\beta}(0,n)\\}])}}{Z_{\text{lTPQ}}[\\{\tilde{\beta}(0,n)\\}]}\Braket{\\{\tilde{\beta}(0,n)\\}}{\mathrm{e}^{\mathrm{i}Ht}\mathrm{e}^{\mathrm{i}z\hat{K}[\\{\tilde{\beta}(t,n)\\}]}\mathrm{e}^{-\mathrm{i}\hat{H}t}\mathrm{e}^{-\mathrm{i}z\hat{K}[\\{\tilde{\beta}(0,n)\\}]}}{\\{\tilde{\beta}(0,n)\\}},$
(S61) $\displaystyle G_{B}^{\ell\mathrm{TPQ}}(\mathrm{i}-z;t)$
$\displaystyle=\frac{\mathrm{e}^{-\mathrm{i}z^{*}\log(Z_{\ell\mathrm{TPQ}}[\\{\tilde{\beta}(t,n)\\}]/Z_{\ell\mathrm{TPQ}}[\\{\tilde{\beta}(0,n)\\}])}}{Z_{\ell\mathrm{TPQ}}[\\{\tilde{\beta}(0,n)\\}]}$
$\displaystyle\times\Braket{\\{\tilde{\beta}(t,n)\\}}{\mathrm{e}^{-\mathrm{i}Ht}\mathrm{e}^{-\hat{K}[\\{\tilde{\beta}(0,n)\\}]}\mathrm{e}^{\mathrm{i}\hat{K}[\\{\tilde{\beta}(0,n)\\}]z^{*}}\mathrm{e}^{\mathrm{i}\hat{H}t}\mathrm{e}^{-\mathrm{i}\hat{K}[\\{\tilde{\beta}(t,n)\\}]z^{*}}\mathrm{e}^{\hat{K}[\\{\tilde{\beta}(t,n)\\}]}}{\\{\tilde{\beta}(t,n)\\}}^{*}.$
(S62)

Here $\ket{\\{\tilde{\beta}(t,n)\\}}$ is an $\ell\mathrm{TPQ}$ state, whose
temperature profile is chosen to be that at time $t$. We compute these
quantities based on the following steps:

  1. (i)

Make an $\ell\mathrm{TPQ}$ state $\ket{\\{\beta(0,n)\\}}$ whose temperature
profile is set by the initial condition (S55).

  2. (ii)

Compute $\mathrm{e}^{-\mathrm{i}Ht}\ket{\\{\beta(0,n)\\}}$.

  3. (iii)

Compute spatially smeared and temporally averaged local energies
$\\{\braket{\tilde{h}(t,n)}\\}$.

  4. (iv)

Compute local temperatures $\\{\tilde{\beta}(t,n)\\}$ using the energy-
temperature relation.

  5. (v)

Make an auxiliary $\ell\mathrm{TPQ}$ state $\ket{\\{\tilde{\beta}(t,n)\\}}$.

Exponential operators like
$\mathrm{e}^{\mathrm{i}z\hat{K}[\\{\tilde{\beta}(t,n)\\}]}$ and
$\mathrm{e}^{-\mathrm{i}z\hat{K}[\\{\tilde{\beta}(0,n)\\}]}$ are regarded as a
time evolution operator. Action of these operators to pure states are computed
by fourth order Runge-Kutta method.

The entropy production can be computed by three independent ways. One way is
to compute the expectation value of the entropy operator Eq. (3) directly. The
others are to compute the gradient of
$\mathop{\mathrm{Im}}G^{\ell\mathrm{TPQ}}_{F}(z)$ and
$\mathop{\mathrm{Im}}G^{\ell\mathrm{TPQ}}_{B}(\mathrm{i}-z)$ at $z=0$.
Concretely, they are given by

$\displaystyle\Sigma_{F}\equiv\mathop{\mathrm{Im}}\left.\frac{\mathrm{d}G_{F}^{\ell\mathrm{TPQ}}(z)}{\mathrm{d}(\mathrm{i}z)}\right|_{z=0},\quad\Sigma_{B}\equiv\mathop{\mathrm{Im}}\left.\frac{\mathrm{d}G_{B}^{\ell\mathrm{TPQ}}(\mathrm{i}-z)}{\mathrm{d}(\mathrm{i}z)}\right|_{z=0}.$
(S63)

In Fig. S6, we demonstrate the estimation of the entropy production from
$G_{F}^{\ell\mathrm{TPQ}}$. At each time, we confirm that the generating
function behaves as a linear function around the origin, which allows us to
evaluate the entropy production according to Eq. (S63). In the figure, the
solid line shows the fitting result using the data at
$z=0.00,0.01,\dots,0.03$. The choice of the fitting range is insensitive to
the result. The fitting of $G_{B}^{\ell\mathrm{TPQ}}(\mathrm{i}-z)$ can be
performed in the same manner.

Figure S6: The generating function $G_{F}^{\ell\mathrm{TPQ}}$ at $t=20$. The
solid line stands for the fitting function.

Unlike the standard quantum fluctuation theorem, $\Sigma_{F}$ agrees with
$\Sigma_{B}$ only in the large fluid-cell limit. We numerically confirm that
the difference between $\Sigma_{F}$ and $\Sigma_{B}$ scales as
$\mathrm{e}^{-N}$, where $N$ is the system size. In Fig, S7, we show the
scaling behavior at $t=20$.

Figure S7: The difference between $\Sigma_{F}$ and $\Sigma_{B}$ at $t=20$.

We also check that the difference tends to vanish by taking the random vector
average, which is consistent with our proof of equivalence. We compute the
entropy production for 5 random vectors and take the random vector average. In
Fig. S8, the solid line and the closed symbols denote the averaged ones while
the dotted line and open symbols do the $\ell\mathrm{TPQ}$ result for the
entropy production; namely, computed by a single random vector. Here,
numerical simulations are performed for $N=12$. This result is consistent with
the fact that $\Sigma_{F}$ coincides with $\Sigma_{B}$ for the standard mixed
state (or canonical ensemble) formulation.

Figure S8: The entropy production for $N=12$. The dotted line and the open
symbols show the entropy production obtained for a single random vector. The
solid line and the closed symbols show the average over 5 independent samples.


# Machine Learning Pipelines: Provenance, Reproducibility and FAIR Data
Principles

Sheeba Samuel Frank Löffler Birgitta König-Ries
Heinz-Nixdorf Chair for Distributed Information Systems
Michael Stifel Center Jena
Friedrich Schiller University Jena
{sheeba.samuel, frank.loeffler<EMAIL_ADDRESS>
###### Abstract

Machine learning (ML) is an increasingly important scientific tool supporting
decision making and knowledge generation in numerous fields. With this, it
also becomes more and more important that the results of ML experiments are
reproducible. Unfortunately, that often is not the case. Rather, ML, similar
to many other disciplines, faces a reproducibility crisis. In this paper, we
describe our goals and initial steps in supporting the end-to-end
reproducibility of ML pipelines. We investigate which factors beyond the
availability of source code and datasets influence reproducibility of ML
experiments. We propose ways to apply FAIR data practices to ML workflows. We
present our preliminary results on the role of our tool, ProvBook, in
capturing and comparing provenance of ML experiments and their reproducibility
using Jupyter Notebooks.

##  1 Introduction

Over the last few years, advances in artificial intelligence and machine
learning (ML) have led to their use in numerous applications. With more and
more decision making and knowledge generation being based on ML, it becomes
increasingly important, that ML experiments are reproducible. Only
reproducible results are trustworthy and a suitable basis for future work.
Unfortunately, similar to other disciplines, ML faces a “reproducibility
crisis” [1, 2]. In this paper, we investigate which factors contribute to this
crisis and propose first solutions to address some of them. We have conducted
an initial study among domain experts for a better understanding of the
requirements for reproducibility of ML experiments. Based on the results from
the study and the current scenario in the field of ML, we propose the
application of FAIR data practices [3] in end-to-end ML pipelines. We use the
ontologies to achieve interoperability of scientific experiments. We
demonstrate the use of ProvBook to capture and compare the provenance of
executions of ML pipelines through Jupyter Notebooks. Tracking the provenance
of the ML workflow is needed for other scientists to understand how the
results are derived. Along with the provenance, the version for each
provenance item needs to be maintained for the end-to-end reproducibility of
an ML pipeline.

##  2 The situation: Characteristics of Machine Learning Experiments and their
Reproducibility

An ML pipeline consists of a series of ordered steps used to automate the ML
workflow. Even though the general workflow is the same for most ML
experiments, there are many activities, tweaks, and parameters that are
involved which require proper documentation. We conducted an internal study
among 15 domain experts to understand what is needed to achieve
reproducibility of ML experiments. We asked questions regarding the
reproducibility of results, the challenges in reproducing published results
and the factors required for describing experiments for reproducibility in the
field of ML. We present here some relevant challenges and problems faced by
scientists in reproducing published results of others: (1) Unavailability,
incomplete, outdated or missing parts of source code (2) Unavailability of
datasets used for training and evaluation (3) Unavailability of a reference
implementation (4) Missing or insufficient description of hyperparameters that
need to be set or tuned to obtain the exact results (5) Missing information on
the selection of the training, test and evaluation data (6) Missing
information of the required packages and their version (7) Tweaks performed in
the code not mentioned in the paper (8) Missing information in methods and the
techniques used, e.g., batch norm or regularization techniques (9) Lack of
documentation of preprocessing steps including data preparation and cleaning
(10) Difficulty in reproducing training of large neural networks due to
hardware requirements. All the participants mentioned that if ML experiments
are properly described with all the entities of the experiments and their
relationships between each other, it will benefit them not only in the
reproducibility of results but also for comparison to other competing methods
(baseline). The results of this survey are available
online111https://github.com/Sheeba-Samuel/MLSurvey.

##  3 Towards a solution: Applying FAIR data principles to ML

The FAIR data principles not only apply to research data but also to the
tools, algorithms, and workflows that lead to the results. This aids to
enhance transparency, reproducibility, and reuse of the research pipeline. In
the context of the current reproducibility crisis in ML, there is a definite
need to explore how the FAIR data principles and practices can be applied in
this field. In this paper, we focus on two of the four principles, namely
Interoperability and Reusability which we equate with reproducibility. To
implement FAIR principles regarding interoperability, it is important that
there is a common terminology to describe, find and share the research process
and datasets. Describing ML workflows using ontologies could, therefore, help
to query and answer competency questions like: (1) Which hyperparameters were
used in one particular run of the model? (2) Which libraries and their
versions are used in validating the model? (3) What is the execution
environment of the ML pipeline? (4) How many training runs were performed in
the ML pipeline? (5) What is the allocation of samples for training, testing
and validating the model? (6) What are the defined error bars? (7) Which are
the measures used for evaluating the model? (8) Which are the predictions made
by the model?

In previous work, we have developed the REPRODUCE-ME ontology which is
extended from PROV-O and P-Plan [2]. REPRODUCE-ME introduces the notions of
Data, Agent, Activity, Plan, Step, Setting, Instrument, and Materials, and
thus models the general elements of scientific experiments required for their
reproducibility. Work is in progress to extend this ontology to include ML
concepts which scientists consider important according to our survey. We also
aim to be compliant with existing ontologies like ML-Schema [4] and MEX
vocabulary [5]. With REPRODUCE-ME, the ML pipeline developed through Jupyter
Notebooks can be described in an interoperable manner.

##  4 Achieving Reproducibility using ProvBook

Building an ML pipeline requires constant tweaks in the algorithms and models
and parameter tuning. Training of the ML model is conducted through trial and
error. The role of randomness in ML experiments is big and its use is common
in steps like data collection, algorithm, sampling, etc. Several runs of the
model with the same data can generate different results. Thus, repeating and
reproducing results and reusing pipelines is difficult.

The use of Jupyter Notebooks is rapidly increasing as they allow scientists to
perform many computational activities including statistical modeling, machine
learning, etc. They support computational reproducibility by allowing users to
share code along with documentation and results. However, the surveys [6] on
Jupyter Notebooks point out the need of provenance information of the
execution of these notebooks. To overcome this problem, we developed ProvBook
[2]. With ProvBook, users can capture, store, describe and compare the
provenance of different executions of Jupyter notebooks. This allows users to
compare the results from the original author with their own results from
different executions. ProvBook provides the difference in the result of the ML
pipeline from the original author of the Jupyter notebook in GitHub with the
result from our execution using ProvBook. Even though the code and data remain
the same in both the executions, there is a subtle difference in the result.
In ML experiments, users need to figure out the reason behind different
results due to modification in data or models or because of a random sample.
Therefore, it is important to describe the data being used, the code and
parameters of the model, the execution environment to know how the results
have been derived. ProvBook helps in achieving this reproducibility level by
providing the provenance of each run of the model along with the execution
environment.

## Acknowledgments

The authors thank the Carl Zeiss Foundation for the financial support of the
project “A Virtual Werkstatt for Digitization in the Sciences (K3)” within the
scope of the program-line “Breakthroughs: Exploring Intelligent Systems” for
“Digitization – explore the basics, use applications”.

## References

  * [1] Matthew Hutson.  Artificial intelligence faces reproducibility crisis.  Science, 359(6377):725–726, 2018.
  * [2] Sheeba Samuel.  A provenance-based semantic approach to support understandability, reproducibility, and reuse of scientific experiments.  PhD thesis, Friedrich-Schiller-Universität Jena, 2019.
  * [3] Mark D Wilkinson et al.  The FAIR Guiding Principles for scientific data management and stewardship.  Scientific data, 3, 2016.
  * [4] Gustavo Correa Publio et al.  ML-schema: Exposing the semantics of machine learning with schemas and ontologies.  CoRR, abs/1807.05351, 2018.
  * [5] Diego Esteves et al.  MEX vocabulary: a lightweight interchange format for machine learning experiments.  In Proceedings of the 11th International Conference on Semantic Systems, SEMANTICS 2015, Vienna, Austria, September 15-17, 2015, pages 169–176, 2015.
  * [6] João Felipe Pimentel et al.  A large-scale study about quality and reproducibility of jupyter notebooks.  In Proceedings of the 16th International Conference on Mining Software Repositories, pages 507–517, USA, 2019.