$missing\text{\,}\mathrm{missing}$

††thanks: These authors contributed equally to this work:
<EMAIL_ADDRESS><EMAIL_ADDRESS>These authors
contributed equally to this work:
<EMAIL_ADDRESS><EMAIL_ADDRESS>
# Experimentally probing Landauer’s principle in the quantum many-body regime

Stefan Aimet  Dahlem Centre for Complex Quantum Systems, Freie Universität
Berlin, 14195 Berlin, Germany  Mohammadamin Tajik  Vienna Center for Quantum
Science and Technology, Atominstitut, TU Wien, 1020 Vienna, Austria  Gabrielle
Tournaire  Dahlem Centre for Complex Quantum Systems, Freie Universität
Berlin, 14195 Berlin, Germany  Department of Physics and Astronomy, and
Stewart Blusson Quantum Matter Institute, University of British Columbia,
V6T1Z1 Vancouver, Canada  Philipp Schüttelkopf  Vienna Center for Quantum
Science and Technology, Atominstitut, TU Wien, 1020 Vienna, Austria  João
Sabino  Vienna Center for Quantum Science and Technology, Atominstitut, TU
Wien, 1020 Vienna, Austria  Spyros Sotiriadis  Institute of Theoretical and
Computational Physics, University of Crete, 71003 Heraklion, Greece  Dahlem
Centre for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin,
Germany  Giacomo Guarnieri  Dipartimento di Fisica, Università di Pavia, 27100
Pavia, Italy  Dahlem Centre for Complex Quantum Systems, Freie Universität
Berlin, 14195 Berlin, Germany  Jörg Schmiedmayer  Vienna Center for Quantum
Science and Technology, Atominstitut, TU Wien, 1020 Vienna, Austria  Jens
Eisert  Dahlem Centre for Complex Quantum Systems, Freie Universität Berlin,
14195 Berlin, Germany

###### Abstract

Landauer’s principle bridges information theory and thermodynamics by linking
the entropy change of a system during a process to the average energy
dissipated to its environment. Although typically discussed in the context of
erasing a single bit of information, Landauer’s principle can be generalised
to characterise irreversibility in out-of-equilibrium processes, such as those
involving complex quantum many-body systems. Specifically, the relationship
between the entropy change of the system and the energy dissipated to its
environment can be decomposed into changes in quantum mutual information and a
difference in relative entropies of the environment. Here we experimentally
probe Landauer’s principle in the quantum many-body regime using a quantum
field simulator of ultracold Bose gases. Employing a dynamical tomographic
reconstruction scheme, we track the temporal evolution of the quantum field
following a global mass quench from a Klein-Gordon to a Tomonaga-Luttinger
liquid model and analyse the information-theoretic contributions to Landauer’s
principle for various system-environment partitions of the composite system.
Our results agree with theoretical predictions, interpreted using a semi-
classical quasiparticle picture. Our work demonstrates the potential of
ultracold atom-based quantum field simulators to experimentally investigate
quantum thermodynamics.

Information theory and thermodynamics constitute foundational pillars of
modern technology, underpinning our understanding of modern computers and heat
engines, respectively. While these disciplines may seem distinct, they are
intricately connected. This connection is encapsulated in Landauer’s
principle, first articulated in 1961 [1]. Landauer understood that erasing a
bit of information in a computer is not for free but is instead accompanied by
a minimal energy cost dissipated to the environment. Subsequently, various
experimental studies [2, 3, 4] have confirmed this lower bound on energy
dissipation for near-reversible bit erasure. Logical operations thus imply
irreversibility, posing a fundamental theoretical limitation to the design of
any small-scale energy-efficient information-processing technologies [5].
However, extending beyond the case of bit erasure, recent influential work [6]
has generalised the link between information theory and thermodynamics. Using
a quantum statistical mechanics framework, their work reinterprets Landauer’s
principle to relate the entropy change of a system to the energy dissipated to
its environment in very general out-of-equilibrium processes. This
relationship can be quantified by a measure of process irreversibility [7].
Such an extension is thus particularly relevant for quantum many-body systems,
where contributions to irreversibility remain an area of active research,
notably in phenomena such as equilibration and thermalisation [8, 9, 10, 11].

In this work, we experimentally probe Landauer’s principle for an out-of-
equilibrium process in the quantum many-body regime by measuring the time
evolution of quantum information-theoretic quantities that characterise
irreversibility. We present a crisp information-inspired interpretation of
correlations present in states of quantum many-body systems in terms of
entropic expressions.

Concretely, we consider a system-environment composite initially in the state
$\varrho_{\rm SE}(0)$, evolving under a global unitary time evolution $U$ to
the state $\varrho_{\rm SE}(t)=U\varrho_{\rm SE}(0)U^{\dagger}$. The reduced
state of the system $\rm S$ (environment $\rm E$) is $\varrho_{\rm
S(E)}(t)=\mbox{Tr}_{E(S)}[\varrho_{\rm SE}(t)]$. First, let us assume that
there are no initial correlations between the system and environment, and the
environment is in a thermal state $\gamma_{\rm E}^{\beta_{\rm
E}}=e^{-\beta_{\rm E}H_{\rm E}}/\mbox{Tr}[e^{-\beta_{\rm E}H_{\rm E}}]$ at
inverse temperature $\beta_{\rm E}$ with respect to its Hamiltonian $H_{\rm
E}$. In this case, the entropy production $\Sigma(t)$, which is a measure of
irreversibility [12, 6, 7], can be decomposed as

$\Sigma(t):=I_{\rm SE}(t)+D(\varrho_{\rm E}(t)||\gamma_{\rm E}^{\beta_{\rm
E}})\;.$ (1)

Here, the quantum relative entropy

$D(\varrho_{\rm E}(t)||\gamma_{\rm E}^{\beta_{\rm E}})=\mbox{Tr}[\varrho_{\rm
E}(t)\log{\varrho_{\rm E}(t)}]-\mbox{Tr}[\varrho_{\rm E}(t)\log{\gamma_{\rm
E}^{\beta_{\rm E}}}]\;,$ (2)

quantifies the deviation of the environment from its initial thermal state.
The quantum mutual information

$I_{\rm
SE}(t)=S(\varrho_{\mathrm{S}}(t))+S(\varrho_{\mathrm{E}}(t))-S(\varrho_{\mathrm{SE}}(t))$
(3)

measures the system-environment correlations, where

$S(\varrho)=-\mbox{Tr}[\varrho\log{\varrho}]$ (4)

denotes the von Neumann entropy of a state $\varrho$.

Figure 1: Schematic of the experimental protocol. a, The experimental system
consists of two tunnelling-coupled ultracold ${}^{87}\text{Rb}$ gases, with a
single-particle tunnelling rate $J$, initially prepared in an initial state
described by a global thermal state of the massive Klein-Gordon Hamiltonian.
By ramping up a barrier between the condensates, a global mass quench is
performed, and the condensates evolve independently under the post-quench
Tomonaga-Luttinger liquid Hamiltonian for $t\geq 0$. b, The atomic clouds are
released, and they interfere as they expand. We obtain integrated 2D atomic
density via absorption imaging, from which the relative phase profiles are
obtained. As an c, Using the measured phase-phase correlations, we dynamically
reconstruct the full state. By successively shifting the observation window,
we fit the covariance matrix $\Gamma(t)$ for different times $t$. The
covariance matrices of system $\rm S$ and environment $\rm E$ are defined
accordingly.

Now we move on to a more general scenario where the system and the environment
are initially correlated, and the state of the environment deviates from
thermal equilibrium. We thereby introduce the generalised entropy production
$\Delta\Sigma:=\Sigma(t)-\Sigma(0)$ where the initial correlations and
deviations from thermal equilibrium are accounted for by a resource cost
$\Sigma(0)=I_{\rm SE}(0)+D(\varrho_{\rm E}(0)||\gamma_{\rm E}^{\beta_{\rm
E}})$ [13], with $\beta_{\rm E}$ being an effective inverse temperature [14,
15]. Landauer’s principle can now be formulated [13] as an equality as

$\Delta\Sigma=\beta_{\rm E}\Delta E_{\rm E}+\Delta S=\Delta I+\Delta D\;,$ (5)

relating the system’s entropy change $\Delta S:=S(\varrho_{\rm
S}(t))-S(\varrho_{\rm S}(0))$ to the energy dissipated to the environment
$\Delta E_{\rm E}:=\mbox{Tr}[(\varrho_{\rm E}(t)-\varrho_{\rm E}(0))H_{\rm
E}]$. Alternatively, $\Delta\Sigma$ also decomposes into the change of quantum
mutual information, $\Delta I=I_{\rm SE}(t)-I_{\rm SE}(0)$, expressing how
much the system-environment correlations change along the out-of-equilibrium
process, as well as the term $\Delta
D=D(\varrho_{\mathrm{E}}(t)||\gamma_{\mathrm{E}}^{\beta_{\mathrm{E}}})-D(\varrho_{\mathrm{E}}(0)||\gamma_{\mathrm{E}}^{\beta_{\mathrm{E}}})$
that quantifies how much both the initial and final state of the environment
are different from the reference equilibrium thermal state. We provide a more
thorough motivation for the generalised entropy production $\Delta\Sigma$ in
the Methods section.

Expressed in this form, Landauer’s principle serves as a means of tracking
changes in information-theoretic quantities, which compose generalised entropy
production $\Delta\Sigma$ in two different ways via a single equation. Herein
lies the power at the heart of thermodynamics: just as knowing the precise
microscopic degrees of freedom, such as the position and momentum of each
particle in a classical gas, is not useful for drawing meaningful conclusions
about its dynamics, knowing the density matrix of a quantum many-body system
is similarly not descriptive. In traditional thermodynamics, macroscopic
variables like changes in pressure or volume effectively characterise the
dynamics. Similarly, analysing the information-theoretic quantities in
Landauer’s principle allows for a simple characterisation of the
irreversibility in the out-of-equilibrium dynamics of a quantum many-body
system.

So far, there has been no systematic experimental pursuit of probing
Landauer’s principle in the quantum many-body regime. Here, we address this
gap by measuring the time evolution of the different terms in Eq. (5) in a
$(1+1)\mathrm{D}$ quantum field simulator of tunnelling-coupled ultracold Bose
gases. This experimental platform has proven to be a suitable testbed for
simulating one-dimensional quantum field theories. Previous studies using this
setup have examined the area law of quantum mutual information in thermal
equilibrium [16] and the propagation of second- [17] or higher-order
correlations [18].

In this experiment, two parallel ultracold clouds of ${}^{87}\text{Rb}$ atoms
(illustrated in Fig. 1) are confined in highly anisotropic magnetic traps
produced by an atom chip [19]. In the axial direction ($z$), the clouds are
confined in a parabolic trap superposed with an optical dipole potential to
produce hard walls, while in the radial directions ($x,\,y$), by a double-well
trap with an adjustable barrier in between, created by RF dreessing [20]. The
single-particle tunnelling rate can be modified by the amplitude of a radio-
frequency field created by two parallel wires on the atom chip.

The bosonic quantum field operator for each condensate can be written using
the phase-density representation as
$\psi_{n}(z)=\sqrt{\rho_{n}(z)}e^{i\theta_{n}(z)}$, where $\theta_{n}$ and
$\rho_{n}$ denote the phase and density of the respective condensate, indexed
by $n=1,2$. We prepare the Bose-Einstein condensates in a global thermal state
of a massive _Klein-Gordon Hamiltonian_

$\displaystyle H_{\rm KG}$ $\displaystyle=\int_{0}^{L}dz\bigg{[}g_{\rm
1D}\delta\rho^{2}(z)+\frac{\hbar^{2}n_{\rm
1D}}{4m}(\partial_{z}\varphi(z))^{2}$ (6) $\displaystyle\hskip
113.81102pt+\hbar Jn_{\rm 1D}\varphi^{2}(z)\bigg{]}$

which models low-energy phononic excitations. Here, $\hbar$ is the reduced
Planck constant, $m$ is the atomic mass, $g_{\rm 1D}$ is the effective one-
dimensional atomic interaction strength, $L=49\,$\mathrm{\SIUnitSymbolMicro
m}$$ is the axial length of the condensates, $n_{\rm 1D}\approx
70\,$\mathrm{\SIUnitSymbolMicro m}$^{-1}$ is the average linear density and
$J\approx 2\pi\times 0.8\,$\mathrm{H}\mathrm{z}$$ is the tunnelling rate
introduced before. The operators $\varphi(z)=\theta_{1}(z)-\theta_{2}(z)$ and
$\delta\rho(z)=[\rho_{1}(z)-\rho_{2}(z)]/2$ represent the relative phase and
relative density, respectively, and satisfy the commutation relation
$[\varphi(z),\delta\rho(z^{\prime})]=i\delta(z-z^{\prime})$.

To initiate out-of-equilibrium dynamics in the experimental system and measure
the evolution of information-theoretic quantities, we rapidly quench $J$ to
zero by ramping up the barrier between the condensates within approximately
$2\,$\mathrm{ms}$$. This change corresponds to a global mass quench of the
Klein-Gordon Hamiltonian. The condensates then evolve independently under the
post-quench Tomonaga-Luttinger liquid Hamiltonian [21, 22] ($H_{\mathrm{KG}}$
with $J=0$) for times $t$, up to $65\,$\mathrm{ms}$$.

Figure 2: Time evolution of different quantities in Landauer’s principle. The
quantities of Landauer’s principle are shown as a function of time for
subregion size ratios a, $L_{\rm S}/L=0.14$ and b, $L_{\rm S}/L=0.43$.
Experimental data is represented by circles with error bars marking the 68%
confidence intervals obtained via bootstrapping. The shaded areas show the 68%
confidence interval for the theoretical predictions, considering the
uncertainty in the estimated temperature and tunnelling rate. The experimental
data is in agreement with the quantum field theory simulation results using
Neumann boundary conditions.  Figure 3: Scaling of different quantities in
Landauer’s principle. The quantities of Landauer’s principle are shown as a
function of subregion size for various times $ct/L=0.19,0.38,0.57$ (see Fig. 2
for more details on the error bars and shaded areas). The experimental data is
in agreement with the quantum field theory simulation results using Neumann
boundary conditions.

For every time step $t$, we turn off all the traps and let the atoms fall
freely for $15.6\,$\mathrm{ms}$$. The clouds will expand and interfere,
resulting in absorption pictures similar to the one shown in Fig. 1b, allowing
us to measure the spatially resolved relative phase $\varphi(z)$ between them.
Since the detection process is destructive, the measurements are repeated to
gather statistics. In our current experimental setup, relative density
fluctuations $\delta\rho$ are not directly measurable, which prompted the
development of a _dynamical tomographic reconstruction technique_ [23, 16] to
access all of the elements of the covariance matrix

$\Gamma(t)=\begin{pmatrix}\Gamma_{\varphi\varphi}(t)&\Gamma_{\varphi\rho}(t)\\\
\Gamma_{\rho\varphi}(t)&\Gamma_{\rho\rho}(t)\\\ \end{pmatrix}\;.$ (7)

Here, the elements are defined as
$[\Gamma_{\varphi\varphi}(t)]_{m,n}=\langle\varphi(z_{m},t)\varphi(z_{n},t)\rangle$,
$[\Gamma_{\rho\rho}(t)]_{m,n}=\langle\delta\rho(z_{m},t)\delta\rho(z_{n},t)\rangle$,
and
$[\Gamma_{\varphi\rho}(t)]_{m,n}=[\Gamma_{\rho\varphi}(t)]^{T}_{m,n}=\langle\frac{1}{2}\\{\varphi(z_{m},t),\delta\rho(z_{n},t)\\}\rangle$,
all on a discrete grid with $N$ pixels ($m,n\in\\{1,\ldots,N\\}$), which are
determined by resolution constraints of our imaging system, thus introducing
an ultraviolet (UV) cutoff.

The phase–phase correlations for different evolution times after the quench
can be measured directly from the extracted relative phase profiles. Assuming
that the short-time dynamics after the quench are governed by the
_Tomonaga–Luttinger liquid_ (TLL) Hamiltonian, as time progresses, the initial
eigenmodes of the relative density transform into the phase quadrature, and
the phase quadrature transforms into the relative density. This transformation
allows us to extract information about these eigenmodes by fitting the initial
second-order correlation functions of phase–density, and density–density with
the observed evolution of the phase–phase correlations in momentum space.

The schematics in Fig. 1c illustrate our dynamical tomographic reconstruction
scheme. We follow the technique in Ref. [16] for various input intervals with
varying starting points. By scanning the starting points of these input
intervals throughout the trapping times, we can reconstruct the full
covariance matrix at every time $t$, as depicted in Fig. 1. The interval
length ($32.5\,$\mathrm{ms}$$ close to $L/c\approx 27\,$\mathrm{ms}$$) is
selected to be sufficiently long for the slowest eigenmode to acquire enough
dynamical phase for a stable reconstruction, yet short enough to prevent mode
interactions from affecting the reconstruction. A detailed overview of this
reconstruction process is provided in the Methods section.

The quadratic form of the pre- and post-quench Hamiltonians allows us to work
within the framework of Gaussian quantum information theory [24, 25]. In this
framework, the covariance matrix $\Gamma$ captures all the accessible
information about the state of the composite system during the dynamics, from
which we can extract the information-theoretic quantities of interest in Eq.
(5). As has been demonstrated in various experiments [26, 27, 28, 29, 30], the
quadratic approximation of the Hamiltonian accurately captures the dynamics
for the considered time scales. Even if the true dynamics deviate from the
Gaussian regime and violate Tomonaga-Luttinger liquid theory, a Gaussian
extremality argument presented in the Supplementary Material justifies the
Gaussian tomography scheme and provides bounds on the information-theoretic
quantities. Thus, based on our technique of dynamical tomographic
reconstruction of covariance matrices, Landauer’s principle can also be
meaningfully experimentally probed for interacting models.

Having experimentally reconstructed the post-quench time evolution of the
covariance matrices, we partition the one-dimensional field of length $L$ into
two distinct subregions and split the covariance matrix accordingly (Fig. 1c).
Since the quantum field is isolated from its surroundings, one subregion
serves as the system $\mathrm{S}$ with length $L_{\mathrm{S}}$, while the
other subregion functions as the environment $\mathrm{E}$ with length
$L_{\mathrm{E}}=L-L_{\mathrm{S}}$. We probe Landauer’s principle (Eq. (5)) for
various system-environment bipartitions by calculating $\beta_{\rm E}\Delta
E_{\rm E},\Delta S,\Delta I$ and $\Delta D$. The results are presented in Fig.
2 as a function of time for different subregion size ratios and in Fig. 3 as a
function of subregion size for different times. The time scales are shown in
units of $ct/L$, where $c=\sqrt{g_{\rm 1D}n_{\rm 1D}/m}\approx
1.8\,$\mathrm{\SIUnitSymbolMicro m}$/$\mathrm{ms}$$ is the speed of sound.
Overall, a very good fit of the data (circles) is obtained compared to the
theoretical calculations (shaded areas), which use the lowest $N=7$ modes and
consider the imaging resolution of the experiment. The low-lying modes already
capture the dynamics of the continuum theory very well. The error bars
representing the $68\%$ confidence intervals are obtained via bootstrapping
[31] with 999 samples, considering the uncertainty in the tunnelling rate and
the estimated initial global temperature. The effective inverse temperature of
the environment $\beta_{\rm E}$ with respect to the post-quench Tomonaga-
Luttinger liquid Hamiltonian is computed using field-theoretic simulations by
constructing a global thermal state of the Klein-Gordon Hamiltonian with the
estimated initial global temperature from the experimental data.

First, we examine the decomposition of the generalized entropy production
$\Delta\Sigma$ into $\beta_{\rm E}\Delta E_{\rm E}$ and $\Delta S$. The term
associated with the energy dissipated to the environment, $\beta_{\rm E}\Delta
E_{\rm E}$, reveals that a small amount of energy flows into (or out of) the
environment for small (or large) systems around $ct/L_{\rm S}=1$. However,
this term contributes only minimally to the irreversible post-quench dynamics
of the quantum field. In contrast, the entropy change of the system, $\Delta
S$, dominates the dynamics and shows a clear linear increase up to $ct/L_{\rm
S}<1$, followed by more pronounced growth.

In comparison, the second decomposition involving $\Delta I$ and $\Delta D$
sheds a slightly different focus on the out-of-equilibrium dynamics. The
largest contribution to $\Delta\Sigma$, the change in quantum mutual
information, $\Delta I$, exhibits similar behaviour to $\Delta S$ but with a
higher magnitude. This increased magnitude is accounted for by the additional
entropy change in the environment. However, $\Delta I$ competes with the term
proportional to the change of the environment, $\Delta D$, which accounts for
both the entropic and energetic changes to the environment. This makes $\Delta
D$ a particularly interesting quantity to characterise the out-of-equilibrium
dynamics of the environment experimentally. In our case, $\Delta D$ increases
negatively, mirroring otherwise the behaviour of $\Delta S$, due to the small
energetic contribution.

To interpret the results, we need to consider the effects of Neumann and
Dirichlet boundary conditions on the quantum field, as shown with field-
theoretic simulations in Fig. 4. These boundary conditions may be contrasted
with the curved backgrounds giving rise to effective boundary conditions
discussed in Refs. [17, 32]. The experimental system exhibits Neumann boundary
conditions ($\partial_{z}\varphi(z)|_{z=0,L}=0$) due to the vanishing particle
current at the edges. Plots contrasting the scaling with subregion size are
shown in the Supplementary Material.

However, we first consider simulations for Dirichlet boundary conditions
($\varphi(z)|_{z=0,L}=0$), which are simpler to understand. Here, the well-
established semi-classical quasiparticle picture [33] offers a clear and
intuitive model for the post-quench dynamics of global mass quenches [34, 35],
depicting the propagation of short-range initial correlations via
ballistically moving quasiparticles. Following the homogeneous global quench,
the initial Klein-Gordon thermal state becomes a non-equilibrium state with
excess energy relative to the post-quench Tomonaga-Luttinger liquid
Hamiltonian, which governs the time evolution. This initial state acts as a
source of quasiparticle pairs emitted globally from every point across the
length of the composite system. Each pair in the bulk consists of two
quasiparticles correlated with each other and moving in opposite directions at
the same speed. For our set of parameters, due to the finite initial
correlation length, there exists a small localised region where quasiparticles
are correlated.

In this quasiparticle picture, the change of quantum mutual information
$\Delta I$ between the system and environment is proportional to the number of
pairs of quasiparticles that are shared between the two different subregions.
The spread of correlations of short-range interacting models is described by a
linear effective light cone originating at the system-environment boundary.
The increase in correlations is thus proportional to the section of the light
cone, or in other words, it is proportional to the distance between the
quasiparticles of the pair emitted at the system-environment boundary (Fig.
4c). If the system is smaller than the environment, once the first
quasiparticle of this boundary pair reaches the edge of the system at
$ct/L_{\rm S}=1$ and undergoes reflection, the section of the system-
environment boundary pair light cone remains constant. The behaviour of
$\Delta I$ transitions to a plateau value. From a system perspective, the
composite system appears to have locally equilibrated to a generalised Gibbs
ensemble steady state [36, 27, 37]. For finite-size composite systems, the
plateau eventually ends again at $ct/L_{\rm E}=1$ when the other quasiparticle
of the boundary pair reaches the edge. The distance between the two
quasiparticles decreases until they meet again at $ct/L=1$, explaining the
phenomenon of recurrences [30, 38] (with respect to the correlation
functions).

Figure 4: Quasiparticle picture interpretation based on quantum field theory
simulations. The time evolution of the quantities in Landauer’s principle is
shown for c, Neumann boundary conditions (BC) (as relevant in the experiment),
and d, Dirichlet boundary conditions, using quantum field-theoretic
simulations. We interpret the post-quench dynamics of the global mass quench
using a semi-classical quasiparticle picture, as discussed in the main text.
a,-b, The energetic dynamics can be explained by the differing energy density
of the initial state at the edges compared to the bulk. e, The linear increase
in correlations for $ct/L_{\rm S}<1$, followed by a plateau, and a linear
decrease for $ct/L_{\rm E}>1$, can be explained by the linear effective light
cone originating at the system-environment boundary. The effect of the zero-
mode, present in Neumann boundary conditions, is not captured by the
quasiparticle picture for $ct/L_{\rm S}>1$.

On the other hand, $\Delta I$ behaves differently for Neumann boundary
conditions. Neumann boundary conditions introduce a zero mode, whose variance
does not evolve harmonically like the other momentum modes, but quadratically
(phase diffusion [39, 40]). The global fluctuations of the zero mode
contribute to entropic quantities [41] and as they increase with time after a
quench they eventually dominate over the contribution of other modes. Since
the quasiparticle picture cannot capture this zero-mode feature, its
predictions for $ct/L_{\rm S}>1$, when a plateau value should be reached, need
to be modified under the present Neumann boundary conditions. The above
features of the dynamics also apply to the change of system entropy $\Delta
S$. In addition, the zero mode restricts the validity of the tomography scheme
to $ct/L<1$.

The energetic contribution, $\beta_{\rm E}\Delta E_{\rm E}$, can also be
explained using the quasiparticle picture. Initially, $\beta_{\rm E}\Delta
E_{\rm E}$ remains constant because quasiparticles traveling between the
system and the environment carry equal energy, resulting in zero net energy
flux. However, a finite-size energy flow becomes apparent when quasiparticles
from the edge of the system cross the system-environment boundary. The
translational invariance of the homogeneous quench is broken due to the higher
energy density at the edges of the composite system for Neumann boundary
conditions (the reverse is true for Dirichlet boundary conditions, where the
energy density is lower at the edges). This edge region has a size of the
order of the initial correlation length. The resulting behaviour of the change
of the environment $\Delta D$ reflects both the entropic and energetic fluxes.

In this study, we have experimentally probed Landauer’s principle in the
quantum many-body regime following a global mass quench in an ultracold atom-
based quantum field simulator. By reconstructing the dynamics of the state of
the composite system, we examined the information-theoretic quantities related
by Landauer’s principle, which we interpret using a semi-classical
quasiparticle picture. Our approach underscores the general utility of
Landauer’s principle for characterising the irreversibility of out-of-
equilibrium dynamics in quantum many-body systems. The way we express
correlations in terms of entropic quantities with an information-theoretic
meaning can be viewed as a vehicle to capture many-body correlations in non-
equilibrium. The Gaussian extremality argument may offer a pathway to extend
our methodology to capture features of out-of-equilibrium processes in
interacting models, despite the challenges posed by non-Gaussian effects.
Looking forward, progress has already been made towards investigating a local
quench involving two Bose-Einstein condensates of different temperatures being
joined together [42]. This protocol represents a crucial primitive towards
developing a quantum field thermal machine [43], potentially allowing for
Landauer erasure as a mechanism to reduce the entropy of a subregion in the
quantum many-body regime, hence functioning as an effective cooling mechanism.
Our current work demonstrates the potential of ultracold one-dimensional gases
as testbeds for quantum thermodynamics in the many-body regime, where
complexity, quantum effects, and finite-size effects play a crucial role.

###### Acknowledgements.

We thank Bernhard Rauer for conducting the experimental measurements. In
addition, we are grateful to Marcus Huber, Patrick Emonts, Ivan Kukuljan,
Mauro Paternostro, Philippe Faist, Florent Goulette, Marek Gluza, and Igor
Mazets for helpful discussions and comments. This work has been supported by
the DFG Research Unit FOR 2724 on ‘Thermal machines in the quantum world’, the
FQXi, the Quantum Flagship (‘Millenion’ and ‘PasQuans2’), the Einstein
Research Unit, the BMBF, Berlin Quantum, and the ERC-AdGs ‘Emergence in
Quantum Physics’ and ‘Delineating the boundary between the computational power
of quantum and classical devices’. S.S. acknowledges support by the European
Union’s Horizon 2020 research and innovation program under the Marie
Skłodowska-Curie Grant Agreement No. 101030988.

## Methods

### Landauer’s principle

Consider a system $\mathrm{S}$ and an environment $\mathrm{E}$ with a
bipartite Hilbert space
$\mathcal{H}=\mathcal{H}_{\mathrm{S}}\otimes\mathcal{H}_{\mathrm{E}}$. For an
initial product state
$\varrho_{\mathrm{SE}}=\varrho_{\mathrm{S}}\otimes\gamma_{\mathrm{E}}^{\beta_{\mathrm{E}}}$,
where the environment is in a thermal state $\gamma_{\mathrm{E}}^{\beta_{\rm
E}}=e^{-\beta_{\rm E}H_{\rm
E}}/\mbox{Tr}{[e^{-\beta_{\mathrm{E}}H_{\mathrm{E}}}]}$, the entropy
production under global unitary dynamics is given by [7]

$\Sigma(t)=I_{\rm
SE}(t)+D(\varrho_{\mathrm{E}}(t)||\gamma_{\mathrm{E}})=D(\varrho_{\rm
SE}(t)||\varrho_{\mathrm{S}}(t)\otimes\gamma_{\mathrm{E}}^{\beta_{\rm E}})\;.$
(8)

Since it is expressed as a quantum relative entropy, $\Sigma$ is non-negative
due to Klein’s inequality [44], which is a manifestation of the second law of
thermodynamics. More generally, as in our work, we start with a resourceful
initial state that includes both initial correlations $I_{\rm SE}(0)$ and
athermality of the environment
$D(\varrho_{\mathrm{E}}(0)||\gamma_{\mathrm{E}}^{\beta_{\rm E}})$. While
defining temperature for a non-equilibrium state is ambiguous, we determine
the effective inverse temperature of the environment $\beta_{\rm E}$ by
demanding that the environment’s energy is equal to that of a Gibbs state,
such that $\mbox{Tr}[\varrho_{\rm E}H_{\rm
E}]\equiv\mbox{Tr}[\gamma_{\mathrm{E}}^{\beta_{\rm E}}H_{\rm E}]$ [45]. This
choice also minimises the athermality term. Thus, we define the resource cost
of the initial state [13]

$\Sigma(0)=I_{\rm
SE}(0)+D(\varrho_{\mathrm{E}}(0)||\gamma_{\mathrm{E}}^{\beta_{\rm
E}})=D(\varrho_{\rm SE}(0)||\varrho_{\mathrm{S}}(0)\otimes\gamma_{\rm
E}^{\beta_{\rm E}}).$ (9)

Although there is no unique consensus on entropy production for initial non-
equilibrium environment states [46] and initial correlations [47, 48], we
define Landauer’s principle Eq. (5) in terms of generalised entropy production
$\Delta\Sigma:=\Sigma(t)-\Sigma(0)$ as

$\Delta\Sigma:=\beta_{\mathrm{E}}\Delta E_{\rm E}+\Delta S=\Delta I+\Delta
D\;.$ (10)

This can be derived similarly to Ref. [6]. One caveat is that $\Delta\Sigma$
may not necessarily be non-negative, which is a necessary criterion for a
strictly valid definition of entropy production. However, this makes sense as
$\Delta\Sigma$ may be considered a finite-difference version of the entropy
production rate
$\Delta\Sigma=\Sigma(t)-\Sigma(0)=\int_{0}^{t}dt^{\prime}\frac{d\Sigma(t^{\prime})}{dt^{\prime}}$,
which can also be negative [49]. Thus, $\Delta\Sigma$ intuitively measures the
”extra” irreversibility associated with the protocol, excluding the initial
resources.

### Experimental implementation

The experimental setup consists of a pair of tunnelling-coupled one-
dimensional ${}^{87}\text{Rb}$ quasicondensates, following a protocol
implemented previously [16]. The magnetic fields required to trap the gas in a
double-well potential are generated by an atom chip [19], achieving transverse
trapping frequencies of $\omega_{\perp}/2\pi=1.4\text{kHz}$. After preparing a
thermal equilibrium state at a temperature of $49\text{nK}$, all traps are
turned off, and absorption imaging of the atoms is performed during a free-
fall time of $15.6\text{ms}$ to measure the two-dimensional atomic density
distributions. This procedure enables us to detect the relative phase between
the condensates across the length of the quasicondensate using matter
interferometry (Fig. 1b). Due to the destructive nature of the detection
process, many repetitions of the experiment are necessary to accumulate
numerous phase correlation images throughout the time evolution, thereby also
ensuring statistically accurate expectation values.

### Quantum field simulation

We have the Klein-Gordon Hamiltonian as a model for our quantum field
simulator

$H_{\rm KG}=\int_{0}^{L}\\!\\!dz\\!\\!\left[g_{\rm 1D}\delta\rho(z)^{2}+\hbar
Jn_{\rm 1D}\varphi(z)^{2}+\frac{\hbar^{2}n_{\rm
1D}}{4m}(\partial_{z}\varphi(z))^{2}\right]\;,$ (11)

where the length of the composite system is set to
$L=49\,$\mathrm{\SIUnitSymbolMicro m}$$ with a mean density of $n_{\rm
1D}=70\,$\mathrm{\SIUnitSymbolMicro m}$^{-1}$. The coupling strength between
the particles is chosen as $g_{\rm 1D}=8.594\times
10^{-39}\text{kg}\cdot\text{m}^{3}\cdot\text{s}^{-2}$, and we have an atomic
mass of $m=1.433\times 10^{-25}\text{kg}$. The initial single-particle
tunnelling rate is $J=2\pi\times 0.76\text{Hz}$. The post-quench evolution is
described by the _Tomonaga-Luttinger liquid Hamiltonian_

$H_{\rm TLL}=\int_{0}^{L}dz\left[g_{\rm
1D}(\delta\rho(z))^{2}+\frac{\hbar^{2}n_{\rm
1D}}{4m}(\partial_{z}\varphi(z))^{2}\right]\;.$ (12)

### Dynamical tomographic reconstruction of covariance matrices

The tomography scheme for reconstructing the initial thermal covariance matrix
is detailed in Refs. [23, 16]. Here, we extend this method to dynamically
reconstruct the state over time. We provide a brief overview below. Given
Neumann boundary conditions $\partial_{z}\varphi(z)|_{z=0,L}=0$, the relative
phase $\varphi(z)$ and relative density fluctuations $\delta\rho(z)$ can be
expanded in Fourier space in terms of cosine eigenfunctions

$\displaystyle
f_{k}^{\varphi}(z)=\begin{cases}2\left(\frac{1}{k\hbar\pi}\sqrt{\frac{mg_{\rm
1D}}{n_{\rm 1D}}}\right)^{1/2}\cos{\left(k\frac{\pi}{L}z\right)}&\text{if
}k>0,\\\ 1&\text{if }k=0,\end{cases}$ (13) $\displaystyle
f_{k}^{\rho}(z)=\begin{cases}\frac{1}{L}\left(k\hbar\pi\sqrt{\frac{n_{\rm
1D}}{mg_{\rm 1D}}}\right)^{1/2}\cos{\left(k\frac{\pi}{L}z\right)}&\text{if
}k>0,\\\ \frac{1}{L}&\text{if }k=0.\end{cases}$

The Hamiltonian of the post-quench Tomonaga-Luttinger liquid Eq. (12) is then
diagonalised as (indices omitted for brevity from now on)

$H=\frac{\hbar
u}{2}\delta\rho_{0}^{2}+\sum_{k=1}^{\infty}\frac{\hbar\omega_{k}}{2}[\delta\rho_{k}^{2}+\varphi_{k}^{2}]$
(14)

with $u:=\frac{2g_{\rm 1D}}{\hbar L}$ and

$\omega_{k}:=\sqrt{\frac{g_{\rm 1D}n_{\rm 1D}}{m}}k\frac{\pi}{L}\;.$ (15)

The interaction strength $g_{\rm 1D}$ is given by

$g_{\rm 1D}:=\hbar\omega_{\perp}a_{s}\frac{2+3a_{s}n_{\rm 1D}}{(1+2a_{s}n_{\rm
1D})}\;,$ (16)

where $a_{s}=5.2\text{nm}$ is the three-dimensional scattering length.
Consequently, the time evolution of modes ($k>0$) in Fourier space is

$\displaystyle\varphi_{k}(t)=\varphi_{k}(0)\cos{(\omega_{k}t)}+\delta\rho_{k}(0)\sin{(\omega_{k}t)},$
(17)
$\displaystyle\delta\rho_{k}(t)=\delta\rho_{k}(0)\cos{(\omega_{k}t)}-\varphi_{k}(0)\sin{(\omega_{k}t)},$

with non-harmonic time evolution of the zero mode

$\displaystyle\varphi_{0}(t)=ut\delta\rho_{0}(t)+\varphi_{0}(0),$ (18)
$\displaystyle\delta\rho_{0}(t)=\delta\rho_{0}(0)=\rm const\,.$

As mentioned in the main text and shown in Fig. 1, the covariance matrices can
be dynamically tomographically reconstructed by measuring the relative phase
profiles using matter interferometry at various times. Subsequently, the
relative phase correlations are computed as
$\Phi^{2}_{m,n}(t)=\langle(\varphi(z_{m},t)-\varphi(z_{0},t))(\varphi(z_{n},t)-\varphi(z_{0},t))\rangle$
with $z_{0}$ as a reference position. We then expand in Fourier space, see Eq.
(13), and expand the time evolved operators to express the measured phase
correlation function in terms of the covariance matrix elements prior to the
time evolution

$\displaystyle\Phi^{2}_{m,n}(t)=\langle(\varphi(z_{m},t)-\varphi(z_{0},t))(\varphi(z_{n},t)-\varphi(z_{0},t))\rangle$
(19)
$\displaystyle=\sum_{k,l=1}^{N-1}f_{k,l}^{m,n}\langle\varphi_{k}(t)\varphi_{l}(t)\rangle$

where

$f_{k,l}^{m,n}=(f_{k}^{\varphi}(z_{m})-f_{k}^{\varphi}(z_{0}))(f_{l}^{\varphi}(z_{n})-f_{l}^{\varphi}(z_{0}))$
(20)

and so we see that the covariance matrix $\Gamma(t)$ can be obtained by
fitting the coefficients $f_{k,l}^{m,n}$ using the phase-space correlations

$\displaystyle\Phi^{2}_{m,n}(t+\Delta
t)=\sum_{k,l=1}^{N-1}f_{k,l}^{m,n}\cos{(\omega_{k}\Delta
t)}\cos{(\omega_{l}\Delta t)}\Gamma_{\varphi\varphi;k,l}(t)$ (21)
$\displaystyle+\sum_{k,l=1}^{N-1}f_{k,l}^{m,n}\sin{(\omega_{k}\Delta
t)}\sin{(\omega_{l}\Delta t)}\Gamma_{\rho\rho;k,l}(t)$
$\displaystyle+\sum_{k,l=1}^{N-1}(f_{k,l}^{m,n}+f_{l,k}^{m,n})\cos{(\omega_{k}\Delta
t)}\sin{(\omega_{l}\Delta t)}\Gamma_{\varphi\rho;k,l}(t)\;.$

By gathering statistics, we employ the same method as in Ref. [16], but simply
shift the window of observation.

We obtain the covariance matrix in momentum space for the first $N$ modes that
give physical (positive) occupation numbers. The imaging system is
additionally accounted for by a Gaussian point spread function with standard
deviation of $3\mu m$ to estimate the initial state. In this way, we can also
confirm the global initial thermal state and extract a pre-quench temperature.
We use discrete cosine eigenfunctions to convert the momentum space
correlators to real-space correlators, from which we compute the quantities of
interest.

### Probing Landauer’s principle from covariance matrices

In this section, we delineate how the quantities in Landauer’s principle, as
expressed in Eq. (5), are computed using the full covariance matrix $\Gamma$
in real space, defined on a grid of $N$ pixels. First, the von Neumann entropy
$S(\Gamma)$ is computed using the symplectic eigenvalues $\lambda_{n}$ of the
covariance matrix $\Gamma$ [24]. These eigenvalues correspond to the
eigenvalues of $i\Omega\Gamma$, where

$\Omega:=\begin{pmatrix}0&\mathbb{1}_{N}\\\ -\mathbb{1}_{N}&0\end{pmatrix}$
(22)

is the standard kernel of the symplectic form capturing the canonical
commutation relations. The expression for the von Neumann entropy $S(\Gamma)$
of a Gaussian state is given by

$S(\Gamma)=\sum_{n=1}^{N}\left[(\lambda_{n}+\frac{1}{2})\ln{(\lambda_{n}+\frac{1}{2})}-(\lambda_{n}-\frac{1}{2})\ln{(\lambda_{n}-\frac{1}{2})}\right].$
(23)

Given two full covariance matrices corresponding to different times and
partitioning it into system and environment subspace covariance matrices (Fig.
1c), this allows us to compute the entropy change of the system $\Delta S$ and
the mutual information $\Delta I$.

Next, we illustrate how to calculate the energy of the environment with
respect to the post-quench Tomonaga-Luttinger liquid Hamiltonian. To obtain
the correct environment Hamiltonian, we need to proceed with care. The field
corresponding to the environment is discretely defined with $N_{\rm
E}=N(L_{\rm E}/L)$ pixels. A naive approach would discretise the continuous
Hamiltonian defined over the length of the environment, but this nearest-
neighbour phase coupling Hamiltonian would yield a non-linear dispersion
relation, inadequately describing the physics of the environment for the
continuum model used in the experiment. Instead, we start from the environment
Hamiltonian of the continuous theory, diagonalise it to obtain a momentum
decomposition of infinitely many modes, and then truncate the Hamiltonian [50]
of the environment to a form with a linear dispersion relation, retaining only
$N_{\rm E}$ modes

$H_{\rm E}=\frac{\hbar u_{\rm E}}{2}(\delta\rho^{\rm
E}_{0})^{2}+\sum_{k=1}^{N_{\rm E}-1}[(\delta\rho_{k}^{\rm
E})^{2}+(\varphi_{k}^{\rm E})^{2}]$ (24)

with $u_{\rm E}:=\frac{2g_{\rm 1D}}{\hbar L_{\rm E}}$ and $\omega_{k}^{\rm
E}:=\sqrt{\frac{g_{\rm 1D}n_{\rm 1D}}{m}}k\frac{\pi}{L_{\rm E}}$. Converting
back to real-space operators, we derive an effective resolution scale-
dependent Hamiltonian

$\displaystyle H_{\rm E}=g_{\rm 1D}\Delta z\sum_{n=N_{\rm
S}}^{N-1}\delta\rho(z_{n})^{2}$ (25) $\displaystyle+\frac{\hbar^{2}n_{\rm
1D}}{4m\Delta z}\sum_{n,m=N_{\rm
S}}^{N-1}\varphi(z_{n})\varphi(z_{m})\frac{2\pi^{2}}{N_{\rm
E}^{3}}\kappa_{n,m}^{\rm E}\;,$

where

$\displaystyle\kappa_{n,m}^{\rm E}$ $\displaystyle:=$
$\displaystyle\sum_{k=1}^{N_{\rm E}-1}\Bigg{\\{}k^{2}\cos{(k\frac{\pi}{N_{\rm
E}}(n-N_{\rm S}+\frac{1}{2}))}$ $\displaystyle\times$
$\displaystyle\cos{(k\frac{\pi}{N_{\rm E}}(m-N_{\rm
S}+\frac{1}{2}))}\Bigg{\\}}$

encodes the long-range coupling of the effective Hamiltonian. Here, $\Delta
z:=L/N$ is the pixel size, $N_{\rm S(E)}$ is the number of pixels of the
system (environment).

The energy of the environment can then be calculated [24] as
$\mbox{Tr}[\varrho_{\rm E}H_{\rm E}]=\mbox{Tr}[\Gamma^{\rm E}\tilde{H}_{\rm
E}]$, where $\tilde{H}_{\rm E}$ is the matrix expression related to $H_{\rm
E}=X^{T}\tilde{H}_{\rm E}X$, with
$X=(\varphi(z_{N_{S}}),\dots,\varphi(z_{N-1}),\delta\rho(z_{N_{S}}),\dots,\delta\rho(z_{N-1}))^{T}$.
Thus, to calculate the energetic contribution $\beta_{\rm E}\Delta E_{\rm E}$,
we need to obtain the inverse temperature of the environment. As indicated in
the previous section and detailed in Ref. [16], we have obtained the inverse
temperature of the composite system with respect to the pre-quench Klein-
Gordon Hamiltonian experimentally. Subsequently, we construct the thermal
state corresponding to this temperature in the continuum limit simulation. We
obtain the inverse temperature of the environment $\beta_{\rm E}$ for
different subregion sizes, by demanding the environment’s energy to match that
of a Gibbs state. We then interpolate the profile for each pixel. We note that
the post-quench temperature is slightly higher than the temperature measured
with respect to the pre-quench Hamiltonian and $\beta_{\rm E}>0$ is nearly
homogeneous across different subregion sizes.

To calculate the energy of a Gibbs state for the environment in the Tomonaga-
Luttinger liquid model, we begin with the partition function $Z_{\rm
E}(\beta_{\rm E})=Z^{\rm zm}_{\rm E}(\beta_{\rm E})Z^{\rm ho}_{\rm
E}(\beta_{\rm E})$ [41]. Due to the compactified nature of the field, the zero
mode partition function can be expressed in terms of the Jacobi-theta
function. Given our experimental parameters, it approximates to

$Z^{\rm zm}_{\rm E}(\beta_{\rm
E})=\sum_{n=-\infty}^{\infty}e^{-\frac{\beta_{\rm E}\hbar u_{\rm
E}}{2}n^{2}}\approx\sqrt{\frac{2\pi}{\beta_{\rm E}\hbar u_{\rm E}}}\;.$ (27)

The harmonic oscillator partition function is

$Z_{\rm E}^{\rm ho}(\beta_{\rm E})=\prod_{k=1}^{N_{\rm
E}-1}\frac{1}{2\sinh{(\beta\hbar\omega^{\rm E}_{k}/2)}}\;.$ (28)

The energy is then $E_{\rm E}(\beta_{\rm E})=-\partial_{\beta_{\rm
E}}[\log{Z_{\rm E}(\beta_{\rm E})}]=E_{\rm E}^{\rm zm}(\beta_{\rm E})+E_{\rm
E}^{\rm ho}(\beta_{\rm E})$. The zero-mode contribution to the energy is

$\displaystyle E^{\rm zm}_{\rm E}=\frac{1}{Z_{\rm 0}^{\rm
E}}\sum_{n=-\infty}^{\infty}e^{-\beta_{\rm E}\frac{\hbar u_{\rm
E}}{2}n^{2}}\left(\frac{\hbar u_{\rm E}}{2}n^{2}\right)\;,$ (29)

which approximates to $1/(2\beta_{\rm E})$ for our experimental parameters.
The harmonic oscillator contribution to the energy is

$E_{\rm E}^{\rm ho}=\sum_{k=1}^{N_{\rm E}-1}\frac{\hbar\omega_{k}^{\rm
E}}{2}\coth{(\beta_{\rm E}\hbar\omega_{k}^{\rm E}/2)}\;.$ (30)

Consequently, we can express then quantum relative entropy of the environment
state $\varrho_{\rm E}$ with respect to a Gibbs state at inverse temperature
$\beta_{\rm E}$ as

$\displaystyle D(\varrho_{\rm E}||\gamma_{\rm E}^{\beta_{\rm
E}})=-S(\varrho_{\rm E})+\beta_{\rm E}\mbox{Tr}[\varrho_{\rm E}H_{\rm
E}]+\log{Z_{\rm E}(\beta_{\rm E})}$ (31)

and we obtain $\Delta D$ accordingly.

Supplementary Material

### Continuum limit and role of boundary conditions

In this section we compare the effect of Neumann boundary conditions
($\partial_{z}\varphi(z)|_{z=0,L}=0$) and Dirichlet boundary conditions
($\varphi(z)|_{z=0,L}=0$) on the information-theoretic quantities within
Landauer’s principle Eq. (5) in the continuum limit. The quantum field is
discretised on a sufficiently fine grid to ensure convergence, achieved when
higher momentum modes no longer affect the results significantly. The
simulation parameters closely match experimental values, assuming a
homogeneous density profile $n_{\rm 1D}$.

Figure S1: Scaling of different quantities in Landauer’s principle in the
continuum limit for Neumann and Dirichlet boundary conditions. Quantum field
theory simulations are used in the continuum limit, similar to Fig. 3, but
without accounting for experimental inhomogeneities. a)-d) Neumann boundary
conditions and e)-h) Dirichlet boundary conditions.

The plots showing the scaling with subregion size at various times are
presented in Fig. S1. The scaling of $\beta_{\rm E}\Delta E_{\rm E}$ can be
explained by the initial symmetric energy density profile, which is higher at
the edges than in the bulk for Neumann boundary conditions and lower for
Dirichlet boundary conditions (Fig. 4a-b). The dominant contribution for the
considered quench, $\Delta I$, shows scaling symmetrically about $L_{\rm
S}/L=0.5$ by construction. For Dirichlet boundary conditions, small systems
(or environments) do not increase in correlations with its environment for
$ct/L_{\rm S}>1$ as correlations reach a plateau phase in time (Fig. 4e).
Neumann boundary conditions, however, actually yield $\Delta I$ to be higher
for the case of a small system than for the case of system and environment
being of equal size. As discussed in the main text, the zero mode causes the
results for Dirichlet boundary conditions, which are well described by the
quasiparticle picture, to diverge from those for Neumann boundary conditions
after $ct/L_{\rm S}>1$.

### Exploiting the extremality of Gaussian states

The validity of the Gaussian tomography scheme has been confirmed by measuring
the normalised averaged connected fourth-order correlation functions, as
demonstrated in previous works [18, 51, 16]. However, leveraging properties of
Gaussian quantum information, we argue that the information-theoretic
quantities in Landauer’s principle can also be estimated using our second-
moment tomography scheme, even for time evolution under a non-Gaussian model
with significant higher-order correlation functions. This justifies the
potential use of our experimental protocol under the lens of Landauer’s
principle for more complex protocols and interacting models.

To begin, we define $G$ as the Gaussian state that shares the same second
moments as the true state $\varrho$ that may be non-Gaussian, reflecting a
preparation involving interactions. Central to the argument are several
extremality properties of Gaussian states. Firstly, Gaussian states $G$ have
the largest von Neumann entropy $S(G)\geq S(\varrho)$ among all states
$\varrho$ with a given covariance matrix. Secondly, we can make use of the
property

$\mbox{Tr}[\varrho(t)\log{G}(0)]=\mbox{Tr}[G(t)\log{G}(0)]\;,$ (S1)

as any Gaussian state can be written as a matrix exponent of a quadratic
polynomial, thereby causing the matrix logarithm of the relative entropy to
cancel out. Thirdly, Ref. [52] has demonstrated that the quantum conditional
entropy is also maximised for Gaussian states, and the inequality

$S(\varrho_{\rm S}(t))-S(\varrho_{\rm SE}(t))\geq S(G_{\rm S}(t))-S(G_{\rm
SE}(t))$ (S2)

is true. This is less obvious, as the expression under consideration is a
difference of von Neumann entropies. The generalised entropy production of the
true dynamics $\Delta\Sigma:=\beta_{\rm E}\Delta E_{\rm E}+\Delta S=\Delta
I+\Delta D$ can be written as

$\displaystyle\Delta\Sigma:=\Delta I+\Delta D$ (S3)
$\displaystyle=S(\varrho_{\rm S}(t))+S(\varrho_{\rm E}(t))-S(\varrho_{\rm
SE}(t))-S(\varrho_{\rm S}(0))-S(\varrho_{\rm E}(0))+S(\varrho_{\rm SE}(0))$
$\displaystyle-S(\varrho_{\rm E}(t))-\mbox{Tr}[\varrho_{\rm
E}(t)\log{\gamma_{\rm E}^{\beta_{\rm E}}}]+S(\varrho_{\rm
E}(0))+\mbox{Tr}[\varrho_{\rm E}(0)\log{\gamma_{\rm E}^{\beta_{\rm E}}}]$
$\displaystyle=S(\varrho_{\rm S}(t))-S(\varrho_{\rm
S}(0))-\mbox{Tr}[\varrho_{\rm E}(t)\log{\gamma_{\rm E}^{\beta_{\rm
E}}}]+\mbox{Tr}[\varrho_{\rm E}(0)\log{\gamma_{\rm E}^{\beta_{\rm E}}}]$
$\displaystyle=\Delta S-\mbox{Tr}[\varrho_{\rm E}(t)\log{\gamma_{\rm
E}^{\beta_{\rm E}}}]+\mbox{Tr}[\varrho_{\rm E}(0)\log{\gamma_{\rm
E}^{\beta_{\rm E}}}]\;,$

where we have used the unitary invariance of the von Neumann entropy. Clearly,
the generalised entropy production $\Delta\Sigma^{G}:=\beta_{\rm E}\Delta
E_{\rm E}^{G}+\Delta S^{G}=\Delta I^{G}+\Delta D^{G}$ assuming an initially
Gaussian state and assuming Gaussian dynamics is given by

$\displaystyle\Delta\Sigma^{G}:=\Delta I^{G}+\Delta D^{G}$ (S4)
$\displaystyle=\Delta S^{G}-\mbox{Tr}[G_{\rm E}(t)\log{\gamma_{\rm
E}^{\beta_{\rm E}}}]+\mbox{Tr}[G_{\rm E}(0)\log{\gamma_{\rm E}^{\beta_{\rm
E}}}]$

where $\Delta E_{\rm E}^{G}=\mbox{Tr}[(G_{\rm E}(t)-G_{\rm E}(0))H_{\rm E}]$,
$\Delta S^{G}=S(G_{\rm S}(t))-S(G_{\rm S}(0))$, $\Delta I^{G}=I^{G}_{\rm
SE}(t)-I^{G}_{\rm SE}(0)$ and $\Delta D^{G}=D(G_{\rm E}(t)||\gamma_{\rm
E}^{\beta_{\rm E}})-D(G_{\rm E}(0)||\gamma_{\rm E}^{\beta_{\rm E}})$. By
utilising the property in Eq. (S1) and considering that $\gamma_{\rm
E}^{\beta_{\rm E}}$ is Gaussian we can write

$\Delta\Sigma=\Delta S-\mbox{Tr}[G_{\rm E}(t)\log{\gamma_{\rm E}^{\beta_{\rm
E}}}]+\mbox{Tr}[G_{\rm E}(0)\log{\gamma_{\rm E}^{\beta_{\rm E}}}]=\Delta
S-\Delta S^{G}+\Sigma^{G}\;.$ (S5)

This also implies that

$\Delta E_{\rm E}=\Delta E_{\rm E}^{G}$ (S6)

demonstrating that covariance matrices are entirely sufficient for capturing
the energetic dynamics, even when the true dynamics are not Gaussian. This is
important, as in this way, one can meausure second moments experimentally and
still get bounds for the true state that may show signatures of interactions.

To proceed further, we must assume that the initial state post-quench is also
Gaussian, i.e., $\varrho_{SE,S,E}=G_{SE,S,E}$. At this step, we have not made
the additional assumption of Gaussian dynamics to bound the quantities in
Landauer’s principle. By the maximality of entropy for Gaussian states, we
have $\Delta S\leq\Delta S^{G}$, we quickly deduce that the generalised
entropy production is upper-bounded by the Gaussian value, i.e.,
$\Delta\Sigma\leq\Delta\Sigma^{G}$. Similarly, we infer that the change in
mutual information is upper-bounded by the Gaussian value [52], i.e., $\Delta
I\leq\Delta I^{G}$. Utilising Eq. (S1), we find that the change in relative
entropy is lower-bounded by the Gaussian value, i.e.,

$\Delta D\geq\Delta D^{G}\;.$ (S7)

However, in both cases, these inequalities alone do not provide quantitative
information on the extent to which these quantities dominate each other in the
true state. To gain a more precise understanding of the relative contributions
of mutual information and relative entropy in this context, a more
comprehensive theoretical model and advanced experimental methods are required
for an experiment modelled by an arbitrary interacting theory.

## References

  * Landauer [1961] R. Landauer, Irreversibility and heat generation in the computing process, IBM J. Res. and Dev. 5, 183 (1961).
  * Bérut  _et al._ [2012] A. Bérut, A. Arakelyan, A. Petrosyan, S. Ciliberto, R. Dillenschneider, and E. Lutz, Experimental verification of Landauer’s principle linking information and thermodynamics, Nature 483, 187 (2012).
  * Hong  _et al._ [2016] J. Hong, B. Lambson, S. Dhuey, and J. Bokor, Experimental test of Landauer’s principle in single-bit operations on nanomagnetic memory bits, Science Adv. 2, e1501492 (2016).
  * Yan  _et al._ [2018] L. L. Yan, T. P. Xiong, K. Rehan, F. Zhou, D. F. Liang, L. Chen, J. Q. Zhang, W. L. Yang, Z. H. Ma, and M. Feng, Single-atom demonstration of the quantum Landauer principle, Phys. Rev. Lett. 120, 210601 (2018).
  * Lloyd [2000] S. Lloyd, Ultimate physical limits to computation, Nature 406, 1047 (2000).
  * Reeb and Wolf [2014] D. Reeb and M. M. Wolf, An improved Landauer principle with finite-size corrections, New J. Phys. 16, 103011 (2014).
  * Landi and Paternostro [2021] G. T. Landi and M. Paternostro, Irreversible entropy production: From classical to quantum, Rev. Mod. Phys. 93, 035008 (2021).
  * Polkovnikov  _et al._ [2011] A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore, Colloquium: Nonequilibrium dynamics of closed interacting quantum systems, Rev. Mod. Phys. 83, 863 (2011).
  * Eisert  _et al._ [2015] J. Eisert, M. Friesdorf, and C. Gogolin, Quantum many-body systems out of equilibrium, Nature Phys. 11, 124 (2015).
  * Gogolin and Eisert [2016] C. Gogolin and J. Eisert, Equilibration, thermalisation, and the emergence of statistical mechanics in closed quantum systems, Rep. Prog. Phys. 79, 056001 (2016).
  * Abanin  _et al._ [2019] D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Colloquium: Many-body localization, thermalization, and entanglement, Rev. Mod. Phys. 91, 021001 (2019).
  * Esposito  _et al._ [2010] M. Esposito, K. Lindenberg, and C. Van den Broeck, Entropy production as correlation between system and reservoir, New J. Phys. 12, 013013 (2010).
  * Mondal  _et al._ [2023] S. Mondal, A. Bhattacharyya, A. Ghoshal, and U. Sen, Modified Landauer’s principle: How much can the Maxwell’s demon gain by using general system-environment quantum state?, (2023), arXiv:2309.09678 .
  * Kliesch  _et al._ [2014] M. Kliesch, C. Gogolin, M. Kastoryano, A. Riera, and J. Eisert, Locality of temperature, Phys. Rev. X 4, 031019 (2014).
  * Lipka-Bartosik  _et al._ [2023] P. Lipka-Bartosik, M. Perarnau-Llobet, and N. Brunner, Operational definition of the temperature of a quantum state, Phys. Rev. Lett. 130, 040401 (2023).
  * Tajik  _et al._ [2023a] M. Tajik, I. Kukuljan, S. Sotiriadis, B. Rauer, T. Schweigler, F. Cataldini, J. Sabino, F. Møller, P. Schüttelkopf, S.-C. Ji, _et al._ , Verification of the area law of mutual information in a quantum field simulator, Nature Phys. 19, 1022–1026 (2023a).
  * Tajik  _et al._ [2023b] M. Tajik, M. Gluza, N. Sebe, P. Schüttelkopf, F. Cataldini, J. Sabino, F. Møller, S.-C. Ji, S. Erne, G. Guarnieri, _et al._ , Experimental observation of curved light-cones in a quantum field simulator, Proc. Natl. Acad. Sci. USA 120, e2301287120 (2023b).
  * Schweigler  _et al._ [2021] T. Schweigler, M. Gluza, M. Tajik, S. Sotiriadis, F. Cataldini, S.-C. Ji, F. S. Møller, J. Sabino, B. Rauer, J. Eisert, _et al._ , Decay and recurrence of non-Gaussian correlations in a quantum many-body system, Nature Phys. 17, 559 (2021).
  * Folman  _et al._ [2000] R. Folman, P. Krüger, D. Cassettari, B. Hessmo, T. Maier, and J. Schmiedmayer, Controlling cold atoms using nanofabricated surfaces: Atom chips, Phys. Rev. Lett. 84, 4749 (2000).
  * Hofferberth  _et al._ [2006] S. Hofferberth, I. Lesanovsky, B. Fischer, J. Verdu, and J. Schmiedmayer, Radiofrequency-dressed-state potentials for neutral atoms, Nature Phys. 2, 710 (2006).
  * Luttinger [1963] J. Luttinger, An exactly soluble model of a many-fermion system, J. Math. Phys. 4, 1154 (1963).
  * Mora and Castin [2003] C. Mora and Y. Castin, Extension of bogoliubov theory to quasicondensates, Phys. Rev. A 67, 053615 (2003).
  * Gluza  _et al._ [2020] M. Gluza, T. Schweigler, B. Rauer, C. Krumnow, J. Schmiedmayer, and J. Eisert, Quantum read-out for cold atomic quantum simulators, Commun. Phys. 3, 1 (2020).
  * Weedbrook  _et al._ [2012] C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, Gaussian quantum information, Rev. Mod. Phys. 84, 621 (2012).
  * Eisert and Plenio [2003] J. Eisert and M. B. Plenio, Introduction to the basics of entanglement theory in continuous-variable systems, Int. J. Quant. Inf. 1, 479 (2003).
  * Langen  _et al._ [2013] T. Langen, R. Geiger, M. Kuhnert, B. Rauer, and J. Schmiedmayer, Local emergence of thermal correlations in an isolated quantum many-body system, Nature Phys. 9, 640 (2013).
  * Langen  _et al._ [2015] T. Langen, S. Erne, R. Geiger, B. Rauer, T. Schweigler, M. Kuhnert, W. Rohringer, I. E. Mazets, T. Gasenzer, and J. Schmiedmayer, Experimental observation of a generalized gibbs ensemble, Science 348, 207 (2015).
  * Yang  _et al._ [2017] B. Yang, Y.-Y. Chen, Y.-G. Zheng, H. Sun, H.-N. Dai, X.-W. Guan, Z.-S. Yuan, and J.-W. Pan, Quantum criticality and the tomonaga-luttinger liquid in one-dimensional bose gases, Phys. Rev. Lett. 119, 165701 (2017).
  * Schweigler  _et al._ [2017] T. Schweigler, V. Kasper, S. Erne, I. Mazets, B. Rauer, F. Cataldini, T. Langen, T. Gasenzer, J. Berges, and J. Schmiedmayer, Experimental characterization of a quantum many-body system via higher-order correlations, Nature 545, 323 (2017).
  * Rauer  _et al._ [2018] B. Rauer, S. Erne, T. Schweigler, F. Cataldini, M. Tajik, and J. Schmiedmayer, Recurrences in an isolated quantum many-body system, Science 360, 307 (2018).
  * Efron and Tibshirani [1986] B. Efron and R. Tibshirani, Bootstrap methods for standard errors, confidence intervals, and other measures of statistical accuracy, Stat. Sc. 1, 54 (1986).
  * Flesch  _et al._ [2008] A. Flesch, M. Cramer, I. P. McCulloch, U. Schollwöck, and J. Eisert, Probing local relaxation of cold atoms in optical superlattices, Phys. Rev. A 78, 033608 (2008).
  * Calabrese and Cardy [2005] P. Calabrese and J. Cardy, Evolution of entanglement entropy in one-dimensional systems, J. Stat. Mech. Theor. Exp. 2005, P04010 (2005).
  * Calabrese and Cardy [2016] P. Calabrese and J. Cardy, Quantum quenches in $1+1$-dimensional conformal field theories, J. Stat. Mech. Theor. Exp. 2016, 064003 (2016).
  * Di Giulio and Tonni [2020] G. Di Giulio and E. Tonni, On entanglement Hamiltonians of an interval in massless harmonic chains, J. Stat. Mech. Theor. Exp. 2020, 033102 (2020).
  * Sotiriadis and Calabrese [2014] S. Sotiriadis and P. Calabrese, Validity of the GGE for quantum quenches from interacting to noninteracting models, J. Stat. Mech. Theor. Exp. 2014, P07024 (2014).
  * Calabrese [2018] P. Calabrese, Entanglement and thermodynamics in non-equilibrium isolated quantum systems, Phys. A: Stat. Mech. Appl. 504, 31 (2018).
  * Modak  _et al._ [2020] R. Modak, V. Alba, and P. Calabrese, Entanglement revivals as a probe of scrambling in finite quantum systems, J. Stat. Mech. Theor. Exp. 2020, 083110 (2020).
  * Lewenstein and You [1996] M. Lewenstein and L. You, Quantum phase diffusion of a Bose-Einstein condensate, Phys. Rev. Lett. 77, 3489 (1996).
  * Jo  _et al._ [2007] G.-B. Jo, Y. Shin, S. Will, T. A. Pasquini, M. Saba, W. Ketterle, D. E. Pritchard, M. Vengalattore, and M. Prentiss, Long phase coherence time and number squeezing of two Bose-Einstein condensates on an atom chip, Phys. Rev. Lett. 98, 030407 (2007).
  * Michel and Srednicki [2016] B. Michel and M. Srednicki, Entanglement entropy and boundary conditions in 1+ 1 dimensions, (2016), arXiv:1612.08682 .
  * Ventura Sabino [2024] J. D. Ventura Sabino,  _Quantum thermal machines with cold atoms_ , Ph.D. thesis, Technische Universität Wien (2024).
  * Gluza  _et al._ [2021] M. Gluza, J. a. Sabino, N. H. Ng, G. Vitagliano, M. Pezzutto, Y. Omar, I. Mazets, M. Huber, J. Schmiedmayer, and J. Eisert, Quantum field thermal machines, PRX Quantum 2, 030310 (2021).
  * Klein [1931] O. Klein, Zur quantenmechanischen Begründung des zweiten Hauptsatzes der Wärmelehre, Zeitschrift für Physik 72, 767 (1931).
  * Strasberg and Winter [2021] P. Strasberg and A. Winter, First and second law of quantum thermodynamics: A consistent derivation based on a microscopic definition of entropy, PRX Quantum 2, 030202 (2021).
  * Santos  _et al._ [2019] J. P. Santos, L. C. Céleri, G. T. Landi, and M. Paternostro, The role of quantum coherence in non-equilibrium entropy production, npj Quantum Inf. 5, 23 (2019).
  * Bera  _et al._ [2017] M. N. Bera, A. Riera, M. Lewenstein, and A. Winter, Generalized laws of thermodynamics in the presence of correlations, Nature Comm. 8, 2180 (2017).
  * Jiang  _et al._ [2018] K.-X. Jiang, Y.-M. Li, and H. Fan, Improved Landauer’s principle and generalized second law of thermodynamics with initial correlations and non-equilibrium surrounding environments, (2018), arXiv:1803.06125 .
  * Strasberg and Esposito [2019] P. Strasberg and M. Esposito, Non-Markovianity and negative entropy production rates, Phys. Rev. E 99, 012120 (2019).
  * Ott  _et al._ [2024] R. Ott, T. V. Zache, M. Prüfer, S. Erne, M. Tajik, H. Pichler, J. Schmiedmayer, and P. Zoller, Hamiltonian learning in quantum field theories, (2024), arXiv:2401.01308 .
  * Gluza  _et al._ [2022] M. Gluza, T. Schweigler, M. Tajik, J. Sabino, F. Cataldini, F. S. Møller, S.-C. Ji, B. Rauer, J. Schmiedmayer, J. Eisert, _et al._ , Mechanisms for the emergence of gaussian correlations, SciPost Phys. 12, 113 (2022).
  * Eisert and Wolf [2005] J. Eisert and M. M. Wolf, Gaussian quantum channels, (2005), arXiv:quant-ph/0505151 .


# The mean field approximation and disentanglement

Eyal Buks  Andrew and Erna Viterbi Department of Electrical Engineering,
Technion, Haifa 32000, Israel

###### Abstract

The mean field approximation becomes applicable when entanglement is
sufficiently weak. We explore a nonlinear term that can be added to the
Schrödinger equation without violating unitarity of the time evolution. We
find that the added term suppresses entanglement, without affecting the
evolution of any product state. The dynamics generated by the modified
Schrödinger equation is explored for the case of a two-spin 1/2 system. We
find that for this example the added term strongly affects the dynamics when
the Hartmann Hahn matching condition is nearly satisfied.

##  I Introduction

Consider a system made of two subsystems. Let $A=A^{{\dagger}}$
($B=B^{{\dagger}}$) be a measurable of the first (second) subsystem. In the
mean field approximation breuer2002theory ; Drossel_217 ; Hicke_024401  it is
assumed that $\left\langle AB\right\rangle=\left\langle
A\right\rangle\left\langle B\right\rangle$ (angle brackets denote an
expectation value). This approximation is valid when entanglement is
sufficiently small, and it becomes exact for any product state. Let
$Q\left(\left|\psi\right\rangle\right)$ be the level of entanglement of a
given ket state vector $\left|\psi\right\rangle$. There are several different
ways to quantify entanglement Wootters_2245 , however, none is linear in
$\left|\psi\right\rangle$, i.e. generally
$Q\left(\left|\psi_{1}\right\rangle+\left|\psi_{2}\right\rangle\right)\neq
Q\left(\left|\psi_{1}\right\rangle\right)+Q\left(\left|\psi_{2}\right\rangle\right)$.
On the other hand, the Schrödinger equation is linear in
$\left|\psi\right\rangle$. Moreover, the Gorini Kossakowski Sudarshan Lindblad
equation (GKSL) master equation Fernengel_385701 ; Lindblad_119  is linear in
the density operator $\rho$. Hence, a process which gives rise to
disentanglement only, without affecting the dynamics of product states, cannot
be properly described with time evolution that is generated either by
Schrödinger or GKSL equations.

Here we consider a modified Schrödinger equation, which includes a nonlinear
term that suppresses entanglement. The proposed equation can be constructed
for any physical system whose Hilbert space has finite dimensionality.

Previously proposed nonlinear terms that can be added to the Schrödinger
equation are reviewed in Rembielinski_012027 . Weinberg has considered a class
of nonlinear Schrödinger equations, for which combining subsystems is possible
Weinberg_336 ; Weinberg_61 . The time evolution of the probability density
generated by a nonlinear Schrödinger equation has been studied using the
Fokker-Planck equation in Doebner_397 . Gauge inveriance in nonlinear
Schrödinger equations has been explored in Doebner_3764 . In most previous
proposals, the purpose of the added nonlinear terms is to generate a
spontaneous collapse Tumulka_821 ; Gisin_2259 ; Grigorenko_1459 ;
Bialynicki_62 ; Hossenfelder_139 .

On the hand, here we propose an added nonlinear term that generates
disentanglement, i.e. it suppresses entanglement without affecting the
dynamics of product states. We explore the effect of the added disentanglement
nonlinear term for the case of a two-spin 1/2 system. With externally applied
driving the two-spin 1/2 system can become unstable Levi_053516 ; Buks_052217
when the Hartmann Hahn matching condition Hartmann1962 ; Yang_1  is nearly
satisfied. We find that in the same region the added nonlinear disentangling
term has a relatively large effect on the dynamics.

##  II The Schmidt decomposition

Consider a system composed of two subsystems labeled as ’1’ and ’2’,
respectively. The dimensionality of the Hilbert spaces of both subsystems,
which is denoted by $N_{1}$ and $N_{2}$, respectively, is assumed to be
finite. The system is in a normalized pure state vector
$\left|\psi\right\rangle$ given by

$\left|\psi\right\rangle=\mathcal{K}_{1}C\otimes\mathcal{K}_{2}^{\mathrm{T}}\;,$
(1)

where $C$ is a $N_{1}\times N_{2}$ matrix having entries $C_{k_{1},k_{2}}$,
matrix transposition is denoted by $\mathrm{T}$, the raw vectors
$\mathcal{K}_{1}$ and $\mathcal{K}_{1}$ are given by

$\displaystyle\mathcal{K}_{1}$
$\displaystyle=\left(\left|k_{1}\right\rangle_{1},\left|k_{2}\right\rangle_{1},\cdots,\left|k_{N_{1}}\right\rangle_{1}\right)\;,$
(2) $\displaystyle\mathcal{K}_{2}$
$\displaystyle=\left(\left|k_{1}\right\rangle_{2},\left|k_{2}\right\rangle_{2},\cdots,\left|k_{N_{2}}\right\rangle_{2}\right)\;,$
(3)

and $\left\\{\left|k_{1}\right\rangle_{1}\right\\}$
($\left\\{\left|k_{2}\right\rangle_{2}\right\\}$) is an orthonormal basis
spanning the Hilbert space of subsystem ’1’ (’2’).

The purity $P_{1}$ ($P_{2}$) is defined by
$P_{1}=\operatorname{Tr}\rho_{1}^{2}$ ($P_{2}=\operatorname{Tr}\rho_{2}^{2}$),
where $\rho_{1}=\operatorname{Tr}_{2}\rho$
($\rho_{2}=\operatorname{Tr}_{1}\rho$) is the reduced density operator of the
first (second) subsystems. By employing the Schmidt decomposition one finds
that $P_{1}=P_{2}\equiv P$, and that the level of entanglement $Q$, which is
defined by $Q=1-P$, is given by [see Eq. (8.107)) in Buks_QMLN ]

$Q=2{\displaystyle\sum\limits_{k_{1}^{\prime}<k_{1}^{\prime\prime}}}{\displaystyle\sum\limits_{k_{2}^{\prime}<k_{2}^{\prime\prime}}}\left|\left\langle\Psi_{k_{1}^{\prime},k_{1}^{\prime\prime},k_{2}^{\prime},k_{2}^{\prime\prime}}\right.\left|\psi\right\rangle\right|^{2}\;,$
(4)

where the state
$\left\langle\Psi_{k_{1}^{\prime},k_{1}^{\prime\prime},k_{2}^{\prime},k_{2}^{\prime\prime}}\right|$,
which depends on on the matrix $C$ corresponding to a given state
$\left|\psi\right\rangle$, is given by (note that
$\left\langle\Psi_{k_{1}^{\prime},k_{1}^{\prime\prime},k_{2}^{\prime},k_{2}^{\prime\prime}}\right|$
is not normalized)

$\left\langle\Psi_{k_{1}^{\prime},k_{1}^{\prime\prime},k_{2}^{\prime},k_{2}^{\prime\prime}}\right|=C_{{}_{k_{1}^{\prime\prime},k_{2}^{\prime\prime}}}\left\langle
k_{1}^{\prime},k_{2}^{\prime}\right|-C_{k_{1}^{\prime\prime},k_{2}^{\prime}}\left\langle
k_{1}^{\prime},k_{2}^{\prime\prime}\right|\;.$ (5)

Note that $Q=0$ for a product state, and that $Q$ is time independent when the
subsystems are decoupled (i.e. their mutual interaction vanishes).

As an example, consider a two spin 1/2 system (i.e. $N_{1}=N_{2}=2$) in a pure
state $\left|\psi\right\rangle$ given by

$\left|\psi\right\rangle=a\left|--\right\rangle+b\left|-+\right\rangle+c\left|+-\right\rangle+d\left|++\right\rangle\;.$
(6)

For this case Eq. (4) yields
$Q=2\left|\left\langle\Psi\right.\left|\psi\right\rangle\right|^{2}$, where
[for this case the sum in Eq. (4) contains a single term with
$k_{1}^{\prime}=-$, $k_{1}^{\prime\prime}=+$, $k_{2}^{\prime}=-$ and
$k_{2}^{\prime\prime}=+$]

$\left\langle\Psi\right|=d\left\langle--\right|-c\left\langle-+\right|\;,$ (7)

hence $Q=2\left|ad-bc\right|^{2}$. The following holds $Q\leq 1/2$ provided
that $\left|\psi\right\rangle$ is normalized Wootters_2245 .

##  III Disentanglement

As will be shown below, entanglement can be suppressed by adding appropriate
nonlinear terms to the Schrödinger equation. Consider a modified Schrödinger
equation for the ket vector $\left|\psi\right\rangle$ having the form

$\frac{\mathrm{d}}{\mathrm{d}t}\left|\psi\right\rangle=\left(-i\hbar^{-1}\mathcal{H}+\gamma_{\mathrm{D}}M_{\mathrm{D}}\right)\left|\psi\right\rangle\;,$
(8)

where $\hbar$ is the Planck’s constant, $\mathcal{H}$ is the Hamiltonian, the
rate $\gamma_{\mathrm{D}}$ is positive, the operator $M_{\mathrm{D}}$ is given
by

$M_{\mathrm{D}}=-\sqrt{\frac{\left\langle\Psi\right.\left|\Psi\right\rangle}{1-\left\langle\mathcal{P}\right\rangle}}\left(\mathcal{P}-\left\langle\mathcal{P}\right\rangle\right)\;,$
(9)

the projection operator $\mathcal{P}$ is given by

$\mathcal{P}=\frac{\left|\Psi\right\rangle\left\langle\Psi\right|}{\left\langle\Psi\right.\left|\Psi\right\rangle}\;,$
(10)

the expectation value $\left\langle\mathcal{P}\right\rangle$ is given by

$\left\langle\mathcal{P}\right\rangle=\frac{\left\langle\psi\right|\mathcal{P}\left|\psi\right\rangle}{\left\langle\psi\right.\left|\psi\right\rangle}=\frac{\left|\left\langle\Psi\right.\left|\psi\right\rangle\right|^{2}}{\left\langle\Psi\right.\left|\Psi\right\rangle\left\langle\psi\right.\left|\psi\right\rangle}\;,$
(11)

where $\left|\Psi\right\rangle$ is a given ket vector.

Note that $M_{\mathrm{D}}\left|\psi\right\rangle=0$ provided that
$\left\langle\Psi\right.\left|\psi\right\rangle=0$, thus the added term has no
effect when $\left|\psi\right\rangle$ is orthogonal to
$\left|\Psi\right\rangle$. On the other hand, any product state
$\left|\psi\right\rangle$ is orthogonal to all the vectors
$\left|\Psi_{k_{1}^{\prime},k_{1}^{\prime\prime},k_{2}^{\prime},k_{2}^{\prime\prime}}\right\rangle$
given by Eq. (5). Thus, any added term having the form
$\gamma_{\mathrm{D}}M_{\mathrm{D}}$, which is constructed based on a vector
$\left|\Psi\right\rangle$ that is one of the vectors
$\left|\Psi_{k_{1}^{\prime},k_{1}^{\prime\prime},k_{2}^{\prime},k_{2}^{\prime\prime}}\right\rangle$
given by Eq. (5), has no effect when $\left|\psi\right\rangle$ represents a
product state.

Using Eqs. (9) and (10) one finds that (note that
$\mathcal{P}^{2}=\mathcal{P}$)

$\displaystyle\left\langle\psi\right|M_{\mathrm{D}}\left|\psi\right\rangle$
$\displaystyle=0\;,$ (12)
$\displaystyle\left\langle\Psi\right|M_{\mathrm{D}}\left|\psi\right\rangle$
$\displaystyle=-\sqrt{\left\langle\Psi\right.\left|\Psi\right\rangle\left(1-\left\langle\mathcal{P}\right\rangle\right)}\left\langle\Psi\right.\left|\psi\right\rangle\;,$
(13)
$\displaystyle\left\langle\psi\right|M_{\mathrm{D}}^{2}\left|\psi\right\rangle$
$\displaystyle=\left|\left\langle\Psi\right.\left|\psi\right\rangle\right|^{2}\;.$
(14)

The relation (12) implies that $M_{\mathrm{D}}\left|\psi\right\rangle$ is
orthogonal to $\left|\psi\right\rangle$, and thus the unitarity condition,
which reads [see Eq. (8)]
$0=\left(\mathrm{d}/\mathrm{d}t\right)\left\langle\psi\right.\left|\psi\right\rangle=i\hbar^{-1}\left\langle\psi\right|\mathcal{H}^{{\dagger}}-\mathcal{H}\left|\psi\right\rangle+2\gamma_{\mathrm{D}}\operatorname*{Re}\left\langle\psi\right|M_{\mathrm{D}}\left|\psi\right\rangle$,
is satisfied provided that the Hamiltonian is Hermitian,
i.e.$\mathcal{H}^{{\dagger}}=\mathcal{H}$ (henceforth it is assumed that
$\mathcal{H}$ is Hermitian).

The modified Schrödinger equation (8) yields a modified Heisenberg equation
given by

$\frac{\mathrm{d}}{\mathrm{d}t}\left\langle\psi\right|O\left|\psi\right\rangle=\frac{\left\langle\psi\right|\left[O,\mathcal{H}\right]\left|\psi\right\rangle}{i\hbar}+\gamma_{\mathrm{D}}\left\langle\psi\right|\left\\{O,M_{\mathrm{D}}\right\\}\left|\psi\right\rangle\;,$
(15)

where $O=O^{{\dagger}}$ is a given observable,that does not explicitly depend
on time, and where
$\left\\{O,M_{\mathrm{D}}\right\\}=OM_{\mathrm{D}}+M_{\mathrm{D}}O$. For the
case where $O=\mathcal{P}$ Eq. (15) yields [see Eq. (13)]

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\frac{\left|\left\langle\Psi\right.\left|\psi\right\rangle\right|^{2}}{\left\langle\Psi\right.\left|\Psi\right\rangle}$
$\displaystyle=\frac{\left\langle\psi\right|\left[\mathcal{P},\mathcal{H}\right]\left|\psi\right\rangle}{i\hbar}$
$\displaystyle-2\gamma_{\mathrm{D}}\sqrt{\left\langle\Psi\right.\left|\Psi\right\rangle\left(1-\left\langle\mathcal{P}\right\rangle\right)}\frac{\left|\left\langle\Psi\right.\left|\psi\right\rangle\right|^{2}}{\left\langle\Psi\right.\left|\Psi\right\rangle}\;.$
(16)

An upper bound can be derived for the first term on the right hand side of Eq.
(16) using the uncertainty principle
$\left|\left\langle\left[\mathcal{P},\mathcal{H}\right]\right\rangle\right|^{2}\leq
4\left\langle\left(\mathcal{P}-\left\langle\mathcal{P}\right\rangle\right)^{2}\right\rangle\left\langle\left(\mathcal{H}-\left\langle\mathcal{H}\right\rangle\right)^{2}\right\rangle\leq\left\langle\left(\mathcal{H}-\left\langle\mathcal{H}\right\rangle\right)^{2}\right\rangle$
[note that
$\left\langle\left(\mathcal{P}-\left\langle\mathcal{P}\right\rangle\right)^{2}\right\rangle=\left\langle\mathcal{P}^{2}\right\rangle-\left\langle\mathcal{P}\right\rangle^{2}$,
$\mathcal{P}^{2}=\mathcal{P}$, $0\leq\left\langle\mathcal{P}\right\rangle\leq
1$ and $0\leq 4x\left(1-x\right)\leq 1$ for $0\leq x\leq 1$]. The second term
on the right hand side of Eq. (16) represents a rotation of the ket vector
$\left|\psi\right\rangle$ away from the ket vector $\left|\Psi\right\rangle$.
This rotation gives rise to disentanglement when $\left|\Psi\right\rangle$ is
chosen to be one of the vectors
$\left|\Psi_{k_{1}^{\prime},k_{1}^{\prime\prime},k_{2}^{\prime},k_{2}^{\prime\prime}}\right\rangle$
given by Eq. (5).

##  IV Two-spin system

As an example, consider two two-level systems (TLS) having a mutual coupling
that is characterized by a coupling coefficient $g$. The first TLS, which is
labelled as ’$\mathrm{a}$’, has a relatively low angular frequency
$\omega_{\mathrm{a}}$ in comparison with the angular frequency
$\omega_{\mathrm{b}}$ of the second TLS, which is labelled as ’$\mathrm{b}$’,
and which is externally driven. The Hamiltonian $\mathcal{H}$ of the closed
system is given by

$\displaystyle\mathcal{H}$
$\displaystyle=\omega_{\mathrm{a}}S_{\mathrm{az}}+\omega_{\mathrm{b}}S_{\mathrm{bz}}+\frac{\omega_{1}\left(S_{\mathrm{b+}}+S_{\mathrm{b-}}\right)}{2}$
$\displaystyle+g\hbar^{-1}\left(S_{\mathrm{a+}}+S_{\mathrm{a-}}\right)S_{\mathrm{bz}}\;,$

where the driving amplitude and angular frequency are denoted by $\omega_{1}$
and $\omega_{\mathrm{p}}=\omega_{\mathrm{b}}+\Delta$, respectively ($\Delta$
is the driving detuning), the operators $S_{\mathrm{a\pm}}$ are given by
$S_{\mathrm{a\pm}}=S_{\mathrm{ax}}\pm iS_{\mathrm{ay}}$, and the rotated
operators $S_{\mathrm{b\pm}}$ are given by
$S_{\mathrm{b\pm}}=\left(S_{\mathrm{bx}}\pm iS_{\mathrm{by}}\right)e^{\pm
i\omega_{\mathrm{p}}t}$. In the basis
$\left\\{\left|--\right\rangle,\left|-+\right\rangle,\left|+-\right\rangle,\left|++\right\rangle\right\\}$
the matrix representation of the Hamiltonian is given by $\hbar\Omega$, where
the matrix $\Omega$ is given by

$\Omega=\left(\begin{array}[c]{cccc}\frac{-\omega_{\mathrm{a}}-\omega_{\mathrm{b}}}{2}&\frac{\omega_{1}}{2}e^{i\omega_{\mathrm{p}}t}&-\frac{g}{2}&0\\\
\frac{\omega_{1}}{2}e^{-i\omega_{\mathrm{p}}t}&\frac{-\omega_{\mathrm{a}}+\omega_{\mathrm{b}}}{2}&0&\frac{g}{2}\\\
-\frac{g}{2}&0&\frac{\omega_{\mathrm{a}}-\omega_{\mathrm{b}}}{2}&\frac{\omega_{1}}{2}e^{i\omega_{\mathrm{p}}t}\\\
0&\frac{g}{2}&\frac{\omega_{1}}{2}e^{-i\omega_{\mathrm{p}}t}&\frac{\omega_{\mathrm{a}}+\omega_{\mathrm{b}}}{2}\end{array}\right)\;.$
(18)

The unitary transformations $U_{1}=1_{\mathrm{a}}\otimes u_{\mathrm{b1}}$ and
$U_{2}=1_{\mathrm{a}}\otimes u_{\mathrm{b2}}$ are successively employed below,
where $1_{\mathrm{a}}$ is the identity operator of TLS a. For the first one,
which transforms TLS b to a frame rotated at the angular driving frequency
$\omega_{\mathrm{p}}$, the matrix representation of $u_{\mathrm{b1}}$ is given
by

$u_{\mathrm{b1}}\dot{=}\left(\begin{array}[c]{cc}e^{\frac{i\omega_{\mathrm{p}}t}{2}}&0\\\
0&e^{-\frac{i\omega_{\mathrm{p}}t}{2}}\end{array}\right)\;,$ (19)

and the corresponding transformed matrix $\Omega^{\prime}$ is given by [see
Eq. (18) and Eq. (6.329) in Buks_QMLN ]

$\Omega^{\prime}=\left(\begin{array}[c]{cccc}\frac{-\omega_{\mathrm{a}}+\Delta}{2}&\frac{\omega_{1}}{2}&-\frac{g}{2}&0\\\
\frac{\omega_{1}}{2}&\frac{-\omega_{\mathrm{a}}-\Delta}{2}&0&\frac{g}{2}\\\
-\frac{g}{2}&0&\frac{\omega_{\mathrm{a}}+\Delta}{2}&\frac{\omega_{1}}{2}\\\
0&\frac{g}{2}&\frac{\omega_{1}}{2}&\frac{\omega_{\mathrm{a}}-\Delta}{2}\end{array}\right)\;.$
(20)

For the second transformation the matrix representation of $u_{\mathrm{b2}}$
is given by

$u_{\mathrm{b2}}\dot{=}\left(\begin{array}[c]{cc}\cos\frac{\theta}{2}&\sin\frac{\theta}{2}\\\
-\sin\frac{\theta}{2}&\cos\frac{\theta}{2}\end{array}\right)\;,$ (21)

where

$\tan\theta=-\frac{\omega_{1}}{\Delta}\;,$ (22)

and the corresponding transformed matrix $\Omega^{\prime\prime}$ is given by
[see Eq. (20)]

$\Omega^{\prime\prime}=\left(\begin{array}[c]{cccc}\frac{-\omega_{\mathrm{a}}+\omega_{\mathrm{R}}}{2}&0&-\frac{g\Delta}{2\omega_{\mathrm{R}}}&\frac{g\omega_{1}}{2\omega_{\mathrm{R}}}\\\
0&\frac{-\omega_{\mathrm{a}}-\omega_{\mathrm{R}}}{2}&\frac{g\omega_{1}}{2\omega_{\mathrm{R}}}&\frac{g\Delta}{2\omega_{\mathrm{R}}}\\\
-\frac{g\Delta}{2\omega_{\mathrm{R}}}&\frac{g\omega_{1}}{2\omega_{\mathrm{R}}}&\frac{\omega_{\mathrm{a}}+\omega_{\mathrm{R}}}{2}&0\\\
\frac{g\omega_{1}}{2\omega_{\mathrm{R}}}&\frac{g\Delta}{2\omega_{\mathrm{R}}}&0&\frac{\omega_{\mathrm{a}}-\omega_{\mathrm{R}}}{2}\end{array}\right)\;,$
(23)

where $\omega_{\mathrm{R}}$, which is given by

$\omega_{\mathrm{R}}=\sqrt{\omega_{1}^{2}+\Delta^{2}}\;,$ (24)

is the Rabi angular frequency.

The matrix $\Omega^{\prime\prime}$ in the limit where the TLSs are decoupled
is denoted by $\Omega_{0}^{\prime\prime}$, i.e.
$\Omega_{0}^{\prime\prime}=\lim_{g\rightarrow 0}\Omega^{\prime\prime}$. The
first and forth energy eigenvalues of $\Omega_{0}^{\prime\prime}$ become
degenerate when the Hartmann Hahn matching condition
$\omega_{\mathrm{a}}=\omega_{\mathrm{R}}$ is satisfied. Consequently, the
effect of the coupling becomes relatively strong when
$\omega_{\mathrm{a}}\simeq\omega_{\mathrm{R}}$. In this region the problem can
be simplified by employing a truncation approximation into the subspace
spanned by the transformed states $\left|--\right\rangle$ and
$\left|++\right\rangle$ [first and forth vectors of the basis that is used for
constructing the matrix $\Omega^{\prime\prime}$ given by Eq. (23)]. In this
approximation the $4\times 4$ matrix $\Omega^{\prime\prime}$ is replaced by
the $2\times 2$ truncated matrix $\Omega_{\mathrm{T}}^{\prime\prime}$, which
is given by

$\Omega_{\mathrm{T}}^{\prime\prime}=\left(\begin{array}[c]{cc}\frac{\omega_{\mathrm{R}}-\omega_{\mathrm{a}}}{2}&\frac{g\omega_{1}}{2\omega_{\mathrm{R}}}\\\
\frac{g\omega_{1}}{2\omega_{\mathrm{R}}}&-\frac{\omega_{\mathrm{R}}-\omega_{\mathrm{a}}}{2}\end{array}\right)=\bm{\omega}\cdot\bm{\sigma}\;,$
(25)

where the effective magnetic field vector $\bm{\omega}$ is given by

$\bm{\omega}=\left(\frac{g\omega_{1}}{2\omega_{\mathrm{R}}},0,\frac{\omega_{\mathrm{R}}-\omega_{\mathrm{a}}}{2}\right)\equiv\left(\omega_{x},0,\omega_{z}\right)\;,$
(26)

and the components of the Pauli matrix vector
$\bm{\sigma}=\left(\sigma_{x},\sigma_{y},\sigma_{z}\right)$ are given by

$\sigma_{x}=\left(\begin{array}[c]{cc}0&1\\\
1&0\end{array}\right),\;\sigma_{y}=\left(\begin{array}[c]{cc}0&-i\\\
i&0\end{array}\right),\;\sigma_{z}=\left(\begin{array}[c]{cc}1&0\\\
0&-1\end{array}\right)\;.$ (27)

The notation given by Eq. (6) is employed below for the transformed state
$\left|\psi^{\prime\prime}\right\rangle$. It is henceforth assumed that
$\left|\psi^{\prime\prime}\right\rangle$ is normalized, i.e.
$\left|a\right|^{2}+\left|d\right|^{2}=1$ (note that $b=c=0$ for the
truncation approximation). The polarization vector $\mathbf{P}$ is given by
$\mathbf{P}=\left(\left\langle\psi\right|\sigma_{x}\left|\psi\right\rangle,\left\langle\psi\right|\sigma_{y}\left|\psi\right\rangle,\left\langle\psi\right|\sigma_{z}\left|\psi\right\rangle\right)$,
where $P_{x}=d^{\ast}a+a^{\ast}d$, $P_{y}=i\left(d^{\ast}a-a^{\ast}d\right)$
and $P_{z}=\left|a\right|^{2}-\left|d\right|^{2}$. Note that
$\mathbf{P}\cdot\mathbf{P}=1$ provided that
$\left|\psi^{\prime\prime}\right\rangle$ is normalized.

In the truncation approximation $M_{\mathrm{D}}$ is replaced by the $2\times
2$ matrix $M_{\mathrm{DT}}$, which is given by [see Eqs. (7) and (9)]

$M_{\mathrm{DT}}=\left(\begin{array}[c]{cc}-\left|d\right|^{2}&0\\\
0&\left|a\right|^{2}\end{array}\right)=\frac{\left|a\right|^{2}-\left|d\right|^{2}}{2}-\frac{\sigma_{z}}{2}\;.$
(28)

The following holds
$\left\langle\psi^{\prime\prime}\right|M_{\mathrm{DT}}^{2}\left|\psi^{\prime\prime}\right\rangle=\left|ad\right|^{2}=Q/2$,
where $Q$ is the level of entanglement [recall that
$\left|a\right|^{2}+\left|d\right|^{2}=1$, and compare with Eq. (14)].

With the help of the identity

$\left(\bm{\sigma}\cdot\mathbf{a}\right)\left(\bm{\sigma}\cdot\mathbf{b}\right)=\mathbf{a}\cdot\mathbf{b}+i\bm{\sigma}\cdot\left(\mathbf{a}\times\mathbf{b}\right)\;,$
(29)

where $\mathbf{a}$ and $\mathbf{b}$ are given vectors, one finds that [recall
the vector identity
$\mathbf{A}\cdot\left(\mathbf{B}\times\mathbf{C}\right)=\mathbf{B}\cdot\left(\mathbf{C}\times\mathbf{A}\right)=\mathbf{C}\cdot\left(\mathbf{A}\times\mathbf{B}\right)$,
and see Eqs. (15), (25) and (28)]

$\frac{\mathrm{d}\mathbf{P}}{\mathrm{d}t}=2\bm{\omega}\times\mathbf{P}+\gamma_{\mathrm{D}}\mathbf{V}_{\mathrm{D}}\;,$
(30)

where the vector $\mathbf{V}_{\mathrm{D}}$ is given by [note that
$\left\\{\sigma_{x},\sigma_{z}\right\\}=\left\\{\sigma_{y},\sigma_{z}\right\\}=0$
and $\left\\{\sigma_{z},\sigma_{z}\right\\}=2$, see Eq. (29)]

$\mathbf{V}_{\mathrm{D}}=P_{z}\mathbf{P}-\mathbf{\hat{z}}\;,$ (31)

where $\mathbf{\hat{z}}$ is a unit vector in the $z$ direction. The following
holds $\mathbf{P}\cdot\mathbf{V}_{\mathrm{D}}=0$ [compare with Eq. (12)] and
$\mathbf{V}_{\mathrm{D}}\cdot\mathbf{V}_{\mathrm{D}}=1-P_{z}^{2}=4\left|ad\right|^{2}=2Q$
[recall that $\mathbf{P}\cdot\mathbf{P}=1$ and
$P_{z}=\left|a\right|^{2}-\left|d\right|^{2}$, and compare with Eq. (14)]. For
a normalized $\mathbf{P}$ the following holds
$\mathbf{V}_{\mathrm{D}}=\left(\mathbf{\hat{z}}\times\mathbf{P}\right)\times\mathbf{P}$,
hence Eq. (30) can be rewritten as

$\frac{\mathrm{d}\mathbf{P}}{\mathrm{d}t}=\left(2\bm{\omega}+\gamma_{\mathrm{D}}\mathbf{\hat{z}}\times\mathbf{P}\right)\times\mathbf{P}\;.$
(32)

Note that when $\omega$$\mathbf{\parallel\hat{z}}$ the equation of motion (32)
is similar to the Landau–Lifshitz equation for the time evolution of the
magnetization vector of a ferromagnet. However, the condition
$\omega$$\mathbf{\parallel\hat{z}}$, which is satisfied only when the driving
amplitude $\omega_{1}$ vanishes [see Eq. (26)], is inconsistent with the
assumption that the Hartmann Hahn matching condition is nearly satisfied.

Let
$\mathbf{P}_{0}=\left(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta\right)$
be a fixed point of Eq. (32) (i.e. $\mathrm{d}\mathbf{P}/\mathrm{d}t=0$ at
$\mathbf{P}_{0}$). Using Eq. (32) one finds that the angle $\theta$ for the
fixed point $\mathbf{P}_{0}$ can be found by solving

$\frac{\gamma_{\mathrm{D}}^{2}}{4\left(\omega_{x}^{2}+\omega_{z}^{2}\right)}=\frac{\sin\left(\theta_{\mathrm{H}}-\theta\right)\sin\left(\theta_{\mathrm{H}}+\theta\right)}{\sin^{2}\theta\cos^{2}\theta}\;,$
(33)

where

$\tan\theta_{\mathrm{H}}=\frac{\omega_{x}}{\omega_{z}}\;.$ (34)

According to Eq. (33), in the absence of disentanglement, i.e. when
$\gamma_{\mathrm{D}}=0$, the vector $\mathbf{P}_{0}$ is parallel to the vector
$\omega$ (26), whereas in the opposite limit of strong disentanglement, i.e.
when $\gamma_{\mathrm{D}}^{2}\gg\omega_{x}^{2}+\omega_{z}^{2}$, the vector
$\mathbf{P}_{0}$ becomes nearly parallel to $\mathbf{\hat{z}}$ (i.e. the state
represented by the fixed point $\mathbf{P}_{0}$ nearly becomes a product
state).

##  V Summary

In summary, the modified Schrödinger equation suppresses entanglement without
violating unitarity. Future study will be devoted to the rich nonlinear
dynamics that is generated by the added disentanglement term. Some outstanding
questions, which were left outside the scope of the current manuscript, are
briefly mentioned below. How a given system should be divided into two (or
perhaps more) subsystems? In traditional quantum mechanics such a division is
generally not unique. How to determine the values of the disentanglement
rates? The hypothesis that these rates remain finite even when the subsystems
become decoupled is likely to be inconsistent with the principle of causality
Bassi_055027 ; Jordan_022101  (recall that in traditional quantum mechanics
the level of entanglement $Q$ becomes time independent when the subsystems are
decoupled). Recently, it was shown that when a condition, called ’convex
quasilinearity’ is satisfied by a given nonlinear Schrödinger equation,
violation of the causality principle becomes impossible Rembielinski_012027 .
Upper bound imposed upon the disentanglement rates can be derived from
experimental observations of quantum entanglement.

The existence of quantum entanglement has been conclusively demonstrated in
many different physical systems. On the other hand, entanglement can be held
responsible for a fundamental self-inconsistency related to the quantum to
classical transition Penrose_4864 ; Leggett_939 ; Leggett_022001 , which was
first introduced by Schrödinger Schrodinger_807  (this self-inconsistency is
commonly known as the problem of quantum measurement). Exploring possible
mechanisms of disentanglement may help resolving this long-standing self-
inconsistency.

##  VI Acknowledgments

We thank Ido Kaminer and Eliahu Cohen for useful discussions. This work is
supported by the Israeli Science foundation.

## References

  * (1) Heinz-Peter Breuer, Francesco Petruccione, et al.,  The theory of open quantum systems,  Oxford University Press on Demand, 2002.
  * (2) Barbara Drossel,  “What condensed matter physics and statistical physics teach us about the limits of unitary time evolution”,  Quantum Studies: Mathematics and Foundations, vol. 7, no. 2, pp. 217–231, 2020.
  * (3) C Hicke and MI Dykman,  “Classical dynamics of resonantly modulated large-spin systems”,  Physical Review B, vol. 78, no. 2, pp. 024401, 2008.
  * (4) William K Wootters,  “Entanglement of formation of an arbitrary state of two qubits”,  Physical Review Letters, vol. 80, no. 10, pp. 2245, 1998.
  * (5) Bernd Fernengel and Barbara Drossel,  “Bifurcations and chaos in nonlinear lindblad equations”,  Journal of Physics A: Mathematical and Theoretical, vol. 53, no. 38, pp. 385701, 2020.
  * (6) Goran Lindblad,  “On the generators of quantum dynamical semigroups”,  Communications in Mathematical Physics, vol. 48, no. 2, pp. 119–130, 1976.
  * (7) Jakub Rembieliński and Paweł Caban,  “Nonlinear evolution and signaling”,  Physical Review Research, vol. 2, no. 1, pp. 012027, 2020.
  * (8) Steven Weinberg,  “Testing quantum mechanics”,  Annals of Physics, vol. 194, no. 2, pp. 336–386, 1989.
  * (9) Steven Weinberg,  “Precision tests of quantum mechanics”,  in THE OSKAR KLEIN MEMORIAL LECTURES 1988–1999, pp. 61–68. World Scientific, 2014.
  * (10) H-D Doebner and Gerald A Goldin,  “On a general nonlinear schrödinger equation admitting diffusion currents”,  Physics Letters A, vol. 162, no. 5, pp. 397–401, 1992.
  * (11) H-D Doebner and Gerald A Goldin,  “Introducing nonlinear gauge transformations in a family of nonlinear schrödinger equations”,  Physical Review A, vol. 54, no. 5, pp. 3764, 1996.
  * (12) Roderich Tumulka,  “A relativistic version of the ghirardi–rimini–weber model”,  Journal of Statistical Physics, vol. 125, no. 4, pp. 821–840, 2006\.
  * (13) Nicolas Gisin,  “A simple nonlinear dissipative quantum evolution equation”,  Journal of Physics A: Mathematical and General, vol. 14, no. 9, pp. 2259, 1981.
  * (14) AN Grigorenko,  “Measurement description by means of a nonlinear schrodinger equation”,  Journal of Physics A: Mathematical and General, vol. 28, no. 5, pp. 1459, 1995.
  * (15) Iwo Bialynicki-Birula and Jerzy Mycielski,  “Nonlinear wave mechanics”,  Annals of Physics, vol. 100, no. 1-2, pp. 62–93, 1976.
  * (16) Sabine Hossenfelder and Tim Palmer,  “Rethinking superdeterminism”,  Frontiers in Physics, vol. 8, pp. 139, 2020.
  * (17) Roei Levi, Sergei Masis, and Eyal Buks,  “Instability in the hartmann-hahn double resonance”,  Phys. Rev. A, vol. 102, pp. 053516, Nov 2020.
  * (18) Eyal Buks and Dvir Schwartz,  “Stability of the grabert master equation”,  Physical Review A, vol. 103, no. 5, pp. 052217, 2021.
  * (19) SR Hartmann and EL Hahn,  “Nuclear double resonance in the rotating frame”,  Physical Review, vol. 128, no. 5, pp. 2042, 1962.
  * (20) Pengcheng Yang, Martin B Plenio, and Jianming Cai,  “Dynamical nuclear polarization using multi-colour control of color centers in diamond”,  EPJ Quantum Technology, vol. 3, pp. 1–9, 2016.
  * (21) Eyal Buks,  Quantum mechanics - Lecture Notes,  http://buks.net.technion.ac.il/teaching/, 2020.
  * (22) Angelo Bassi and Kasra Hejazi,  “No-faster-than-light-signaling implies linear evolution. a re-derivation”,  European Journal of Physics, vol. 36, no. 5, pp. 055027, 2015.
  * (23) Thomas F. Jordan,  “Assumptions that imply quantum dynamics is linear”,  Phys. Rev. A, vol. 73, pp. 022101, Feb 2006.
  * (24) Roger Penrose,  “Uncertainty in quantum mechanics: faith or fantasy?”,  Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 369, no. 1956, pp. 4864–4890, 2011.
  * (25) A. J. Leggett,  “Experimental approaches to the quantum measurement paradox”,  Found. Phys., vol. 18, pp. 939–952, 1988.
  * (26) A. J. Leggett,  “Realism and the physical world”,  Rep. Prog. Phys., vol. 71, pp. 022001, 2008.
  * (27) E. Schrodinger,  “Die gegenw artige situation in der quantenmechanik”,  Naturwissenschaften, vol. 23, pp. 807, 1935.