# Quantum scrambling in a toy model of photodetectors

Yubao Liu  Center for Gravitational Experiment, Hubei Key Laboratory of
Gravitation and Quantum Physics, School of Physics, Huazhong University of
Science and Technology, Wuhan, 430074, China  Haixing Miao
<EMAIL_ADDRESS>State Key Laboratory of Low Dimensional Quantum
Physics, Department of Physics, Tsinghua University, Beijing, China  Yanbei
Chen<EMAIL_ADDRESS>Burke Institute of Theoretical Physics, California
Institute of Technology, Pasadena, CA, 91125, USA  Yiqiu Ma
<EMAIL_ADDRESS>Center for Gravitational Experiment, Hubei Key Laboratory
of Gravitation and Quantum Physics, School of Physics, Huazhong University of
Science and Technology, Wuhan, 430074, China

###### Abstract

Quantum measurement is a process that involves the interaction between a
quantum system and a macroscopic measurement apparatus containing many degrees
of freedom. The photodetector is such an apparatus with many electrons
interacting with the incoming quantum photon. Therefore the incoming photon
will spread and get scrambled in the photodetector, that is, the operator of
the initial incoming local photons will grow and becomes highly non-local
through the interaction process. Investigating this scrambling process in
detail is useful for understanding the interaction between the quantum system
and the measurement apparatus. In this paper, we study the quantum scrambling
process in an effective toy model of photodetectors in three different
physical scenarios, by numerically simulating the evolution of the out-of-time
correlators (OTOC). In particular, the integrability of the effective model is
explored through level spacing statistics, and the effect of the
spatially/temporarily distributed disorders on the system evolution is
carefully investigated using the OTOC. Looking into these detailed dynamical
processes paves the way to the quantum simulation and manipulation of
photondetector, which would provide insights into understanding the wave-
function collapse processes.

##  I Introduction

The evolution of physics in the early 20th century fundamentally changed human
understanding of nature, of which quantum mechanics profoundly changes our
epistemology of physical reality. However, one cloud that shades the sunshine
of quantum mechanics is the measurement process, during which the wavefunction
of a quantum system “collapses" when the system interacts with the apparatus
Schlosshauer2005 ; Wheeler1984 ; Wigner1963 ; Bohr1928 ; Zurek2003 ;
Landau1958 . This wavefunction collapse process is at the heart of the
probabilistic nature of quantum mechanics. It is currently treated as an axiom
of quantum mechanics which can not be derived from the Schroedinger equation.
Various interpretations are proposed, but none of them is completely
conclusive. For example, Copenhagen’s interpretation introduces a cut on the
quantum world and classical world to explain the wavefunction collapse
Bohr1928 ; Heisenberg1930 ; Bohm1952a , without a clear definition of the
quantum/classical boundary. Hidden-variable interpretation, such as De
Broglie-Bohm theory Bohm1952b ; Bohm1952c ; Bohm1963 , takes a different
worldview 111De Broglie-Bohm theory is a nonlocal hidden-variable theory,
therefore does not contradict to the violation of Bell’s inequality.. It
attributes the probabilistic nature of quantum phenomenon to the sub-quantum
hidden variables, which obey a deterministic law. In the De Broglie-Bohm
theory, the uncertainty of the measurement result comes from the complicated
interaction via Bohm’s “quantum potential" between the guiding wave of the
quantum system and the macroscopic apparatus which consists of many degrees of
freedom Bohm1952b ; Bohm1952c . Everett’s relative-state formulation (often
referred to as the “many-world interpretation”) does not have collapses, but
simply keeps all alternative possibilities in the quantum state of the
universe Everett1957 ; Dewitt1970 ; DeWitt1973 , while such a branching is
very counter-intuitive although it is a consistent theory. Another possibility
(maybe conceptually the simplest possibility) is a “full-quantum worldview".
It means that (1) the world (system and apparatus) is completely quantum-
mechanical, and the world’s state evolution can be completely described by the
unitary evolution; (2) there is no mysterious hidden variable or world-
branching; (3) the wave function collapse is a phenomenon due to the
complicated interaction between the system and the apparatus. This work
devotes to the exploration of this “full-quantum worldview".

One unavoidable question in all quantum measurement interpretation is the
interaction between the to-be-measured system and the apparatus, of which the
latter contains a large number of degrees of freedom (a “macroscopic system")
Bohr1928 ; Landau1958 . A typical example of a quantum measurement system is
the photo-detector or photo-multiplier tube Feynman1988 , see the left panel
of Fig. 1. A single photon knocks out an electron in the first photo-sensitive
electrode (dynode), which is accelerated to the second dynode due to the
biased electric field between these two dynodes. The accelerated electron will
hit the second dynode and generate more electrons (e.g. two electrons), which
will be accelerated to the third dynode, and so on. This avalanche process
generates a macroscopic current, which corresponds to one of the eigenstates
of the initial photon.

Figure 1: A toy model of a photo-detector or a photo-multiplier tube. The left
panel schematically shows the photon-electric avalanche process, where the
electron triggered by a single photon at the first dynode will be amplified to
be a macroscopic photon-electric current. The right panel is an effective toy
model of the photon-detection process.

In an accompanying paper Hu2022 , an effective toy model that manifests this
avalanche process is discussed and shown in Fig. 1. The work presented here is
its continuation that analyse the evolution of quantum information in this
effective model. In this model, the translational degrees of freedom of the
electrons in the photo-detector are mimicked by discrete two-level qubits,
where the electrons knocked out from different dynode are represented by the
qubits at different layers. These layers are coupled via a Hamiltonian as:

$\begin{split}\hat{H}=&\sum_{i}\hbar\frac{\omega_{i}}{2}\hat{\sigma}^{i}_{Z}+\hbar
g_{1}\hat{\sigma}_{-}^{A}\hat{\sigma}_{+}^{B_{1}}\hat{\sigma}_{+}^{B_{2}}\\\
&+\hbar
g_{2}\hat{\sigma}_{-}^{B_{1}}\hat{\sigma}_{+}^{C_{1}}\hat{\sigma}_{+}^{C_{2}}+\hbar
g_{3}\hat{\sigma}_{-}^{B_{2}}\hat{\sigma}_{+}^{C_{3}}\hat{\sigma}_{+}^{C_{4}}+\cdots+{\rm
h.c.}\,,\end{split}$ (1)

where the $\omega_{i}$ is the eigenfrequency of the $i$-th qubit, which
fluctuates for a real detector. The $g$-coefficients are the three-qubit
interaction strength, which is decided by the microscopic interaction details
of the electrons and dynodes thereby being also random. The
$\hat{\sigma}_{\pm}=(\hat{\sigma}_{x}\pm i\hat{\sigma}_{y})/2$ are the raising
and lower operators of the qubits. It is important to note that this is a
Hamiltonian containing randomly fluctuating parameters (details are presented
in the next section). Our accompanying paper discussed the simulation of the
above model using near-term quantum computers Hu2022 .

In this setup, the collapse of quantum state takes place in a simple form: an
exponentially decaying wavefunction gets collapsed into a pulse at a
particular moment.

$|\gamma\rangle=\int dte^{-\lambda t}|t\rangle\rightarrow|t_{0}\rangle$ (2)

with probability density $p(t_{0})\propto e^{-2\lambda t_{0}}$. The eigenstate
$|t_{0}\rangle$ corresponds to the avalanche taking place at time $t_{0}$. In
this paper, we will not exactly recover the behavior, but will provide
arguments and preliminary numerical evidence that such evolution will take
place. In our case, it is the particular realizations for the $g_{j}$’s that
will control when the collapse will take place. Our dynamics does not solve
the collapse problem, before further issues will be addressed. We will outline
those issues at the end of this paper.

Before returning to the issue of quantum-state collapse, in the main part of
this paper, we focus on the dynamics of our photodetector toy model. This toy
model, which captures the avalanche property of the photodetector, is a many-
body system with random parameters. Therefore it is interesting to investigate
how the quantum information carried by the initial photon spreads into the
measurement apparatus and gets scrambled in this network, which will be
carefully discussed and simulated in the later sections. The so-called out-of-
time correlator (OTOC) and Holevo information are useful tools to study the
scrambling of the quantum information carried by the initial photon in a many-
body system Larkin1969 ; Swingle2018 ; Braumuller2022 . In this work, we will
simulate the evolution of OTOC and also the Holevo information of our model.
Moreover, we also explore the integrability of our toy model by analysing the
eigenvalue level spacing statistics, and the results strongly indicate that
our system is integrable, even though there is an avalanching behaviour.

The structure of this paper is the following. Section II discusses the toy
model, its Hilbert space structures and the primary picture of quantum
information scrambling in this system. In Section III, the integrability of
the model is studied using level spacing statistics. Section IV devotes to
studying the evolution of the out-of-time correlators in different
environmental settings. Section V investigates the quantum information
scrambling by simulating the Holevo information. Section VII concludes the
paper.

##  II Model Hamiltonian and the structure of network Hilbert space

###  II.1 Model descriptions

We have briefly introduced our model Hamiltonian in Section I. Here we give a
more detailed discussion and comments on this model. The main features of our
toy model of a quantum measurement process using a photo-multiplier tube can
be summarised as follows.

(1) The avalanching process in our model spreads the quantum information from
a small corner of the Hilbert space to a much larger portion of the Hilbert
space (with a larger number of excitation) which means the process is
practically irreversible (or equivalently, the reversing time is very long).
This reminds us of the Liouville flow of a classical complex system that
diffuses in the phase space.

(2) Our toy model is a unitary system. This unitary setup not only fits the
“full-quantum worldview”, but also matches the real photo-detection process
shown in Fig. 1. The basic physical processes in Fig. 1 contain the
acceleration of the electron and the collision of the accelerated electron
with the dynode. (i) The electron acceleration happens during a very short
time scale $\tau_{\rm fly}\sim l_{D}/v_{e}\sim\sqrt{ml_{D}^{2}/eU_{D}}\sim
20\,{\rm ps}$ (for the inter-spacing of dynodes $l_{D}\sim 0.1\,{\rm mm}$, and
the voltage between neighbouring dynodes to be $U_{D}=100$ V, a real photo-
multiplier has a nanosecond rise-up time Hamamatsu2007 ) and it is reasonable
to assume that the space between different dynodes is an ideal vacuum. This
means that the electron acceleration process is unitary. (ii) The collision of
the accelerated electrons with the dynode also happens in a very short time
interval $\tau_{\rm col}\sim\hbar/E_{e\rm in}\sim 10^{-17}\,{\rm s}$ for a 100
eV incoming electron as an example. During this interaction process, the
energies of the electrons in the dynode can have fluctuations, which manifest
as the fluctuation of the phase matching in Eq. 1. For example, in the
interaction picture:

$\hbar
g_{2}\hat{\sigma}_{-}^{B_{1}}\hat{\sigma}_{+}^{C_{1}}\hat{\sigma}_{+}^{C_{2}}\rightarrow\hbar
g_{2}\hat{\sigma}_{-}^{B_{1}}\hat{\sigma}_{+}^{C_{1}}\hat{\sigma}_{+}^{C_{2}}e^{-i(\Delta_{B1}-\Delta_{C1}-\Delta_{C2})t},$
(3)

where $\Delta_{B1}-\Delta_{C1}-\Delta_{C2}\equiv\delta\Delta_{g_{2}}$ is
fluctuating due to the environmental effects in the dynode. However, since the
interaction time is very short, it is a good approximation to think that the
quantum information carried by the electrons can hardly spread into the dynode
environment (this time scale can be estimated as the electric charge
equilibrium relaxation time scale $\sim 10^{-15}$ s, which is orders of
magnitude longer than the $\tau_{\rm col}$). Therefore the phase-matching
fluctuation can be treated as a parameter fluctuation in the Hamiltonian while
the system is still unitary. In our model, we assume the frequency detuning
$\delta\Delta_{gi}$ at each interaction vertex has a random fluctuation, and
satisfies a homogenous distribution within
$[-\sigma_{\Delta},+\sigma_{\Delta}]$, in which the $\sigma_{\Delta}$ is
denoted as the disorder strength.

(3) Besides the dynode environmental effect, our toy model also ignores the
possible Coulomb interaction between the electrons during the acceleration
process, which is complicated and leads to the exchange of quantum information
among these electrons, especially at the late stage of the avalanche process
where the electron density becomes high. Since the electron flying time
$\tau_{\rm fly}$ is very short, we assume that this Coulomb interaction effect
can be ignored.

(4) For a practical quantum measurement process, the macroscopic current at
the later avalanche stage will be recorded by the observer. Taking into
account this recording process means that this toy model must be treated in an
open system, which is beyond the current scope of this work. Here we only
focus on the unitary evolution of our avalanching network.

###  II.2 Relationship with other models

Figure 2: Lvovsky’s understanding of a quantum measurement process, in which
he pointed out the scrambling behaviour of quantum photon-detector. This
figure is re-drawed from Lvovsky2018 .

(1) It is worth noting that it is A. I. Lvovsky who first noticed the
scrambling behaviour of a photodetector Lvovsky2018 , the figure which we
borrowed from his book is re-drawn in Fig. 2. The difference between Lvovsky’s
scrambling model and our work is the following: Lvovsky’s model consists of an
ideal projective measurement on the basis $|\nu_{i}\rangle$ and then scrambler
as a classical device that randomly maps the eigenstates to the detector’s
outputs. Therefore in Lvovsky’s model, the scrambling process is entirely
classical. However, the scrambling process in our model is inherent in the
qubit-network unitary evolution with a random Hamiltonian.

(2) This model is reminiscent of the famous Fermi-Pasta-Ulam-Tsingou (FPUT)
Fermi1955 ; Dauxois2008  model in classical nonlinear dynamics, which is
widely used in understanding the thermalisation problem Berman2005 . By
studying a chain of $N$ non-linear interacting harmonic oscillators with the
initial condition that the energy is stored in one of the harmonic
oscillators, FPUT tried to demonstrate that the irreversibility of a
thermalisation process can emerge from local reversibility. FPUT found a
quasi-periodic appearance of specific states which is the so-called “ _Fermi-
Pasta-Ulam-Tsingou recurrence_ ". Our toy model is similar in the sense that
the harmonic oscillators in the FPUT model are replaced by the qubits, and the
FPUT recurrence will also exhibit in our model (see Section III). Note that
despite the similarity of our model and the FPUT model in the interaction
form, our model is inherently linear since quantum mechanics is linear.

(3) The topology of our model may remind the reader about the Cayley tree or
Bethe lattice, which has been studied in Anderson1980 ; Shapiro1983 . Though
the similarity in the configuration topology, the Hamiltonian mainly consists
of near-site hopping and random chemical potential, which is different from
our model since it does not have avalanche behaviour. The Anderson model was
used to explain the electron conductivity in solid materials with disorders.
Moreover, the randomness in our model could be time-dependent.

###  II.3 Reduced network Hilbert space

Suppose we have $\mathcal{N}$ qubits in this network, and initially only the
first qubit is occupied, which corresponds to the excitation of the photo-
electron in the first dynode. The dimension of the Hilbert space of the entire
network is $2^{\mathcal{N}}$, which means we should simulate
$2^{\mathcal{N}}\times 2^{\mathcal{N}}$ matrix evolution. However, our three-
qubit interaction Hamiltonian has a special conservation quantity:

$\bar{N}=\sum_{j}\sum^{N_{j}}_{i_{j}=1}\frac{\hat{n}_{i_{j},j}}{2^{j-1}},$ (4)

where $j$ is the index of different layers and the $i_{j}$ labels the $i$-th
site of the $j_{\rm th}$ layer and $\hat{n}$ is the occupation operator. For
example, we have $\bar{N}=1$ when our network is triggered by an excitation on
the first qubit, which means only a very small portion of the Hilbert space of
the network will be involved. All those states that violate this conservation
law will not participate in the network evolution. The dimension $D$ of the
conserved subspace of the $j$-th layer satisfies the following recurrence
equation:

$D[N_{j}]=D[N_{j}-1]^{2}+1.$ (5)

For example, for 6 layers with 63 qubits, the interested reduced Hilbert space
has a dimension of 458330, which is only a $\sim 10^{-15}$ part of the full
Hilbert space. This structure greatly simplifies our numerical simulation.

Figure 3: Graphic representation of the Hamiltonian matrix in the conserved
subspace for different network sizes. The red spots represent different basis
states of the system and the links represent the interactions that connect
these basis states.

In order to study the quantum information scrambling in our toy model, we need
to construct an algorithm to relate the reduced network Hilbert space and the
full Hilbert space.

For presenting the general formula for relating the reduced network Hilbert
space and the full network Hilbert space, some definitions are given as
follows. First, we define the $\tilde{|i\rangle}_{N}$ as the $i_{\rm th}$
basis in the _reduced Hilbert space_ of a $N$-layers configuration. Then we
define an operator $\hat{O}_{N}(i)$ acts on the zero-excitation ground state
$|0\rangle^{\otimes 2^{\mathcal{N}}-1}$ and generates a basis
$\tilde{|i\rangle}_{N}$ in the reduced Hilbert space:

$\tilde{|i\rangle}_{N}=\hat{O}_{N}(i)|0\rangle^{\otimes 2^{\mathcal{N}}-1}$
(6)

It is shown in Fig. 4 that the $N$-layers structure can be split into two
$N-1$-layers configurations with a top qubit. We can now define an operator
$\hat{O}^{R/L}_{N-1}(i)$ acts on the right/left $N-1$ sub-layers of the
original $N$-layers structure, and generates the state with qubit excitation
in a similar way as $\hat{O}_{N-1}(i)$. Then it is crucial to find the order
of the basis of the reduced Hilbert space with excitation
$\hat{O}^{R}_{N-1}(m)$ and $\hat{O}^{L}_{N-1}(m)$ in the _$N-1$ sub-layer
structures_ in the reduce Hilbert space of the _$N$ -layer structure_. The
algorithm that fulfills this task is by using the recursive formula:

$\begin{split}&\hat{O}_{N}((m-1)D[N-1]+n+1)=\\\
&\begin{cases}\hat{\sigma}^{1}_{+},&m=1,n=0;\\\
\hat{O}^{R}_{N-1}(m)\hat{O}^{L}_{N-1}(n),&m,n\in\\{1,...,D[N-1]\\};\end{cases}\end{split}$
(7)

where the $\hat{\sigma}^{1}_{+}$ is the flip operator of the top qubit. To
understand this algorithm, we can determine the order in the $N$-layer
structure as follows (see Fig. 4): for the $m_{\rm th}$ basis of the left
$N-1$ sub-layer to step forward to the $m_{\rm th}+1$ state, one need to go
through all $D[N-1]$ basis of the right sublayer. This raises the order index
of the corresponding basis of the $N$-layer structure by $D[N-1]$. The
iteration leads to a $(m-1)D[N-1]$ index, adding by an $n+1$ denotes the
states of the right sub-layer and the top qubit.

For realising this recurrence algorithm, the $\hat{O}_{2}(i)$ is needed as an
initial condition. In this paper, we choose $\hat{O}_{2}(i)$ by following way,

$\begin{split}\tilde{|1\rangle}_{2}&=\hat{O}_{2}(1)|0\rangle^{\otimes
3}=|1\rangle_{1}|0\rangle_{2}|0\rangle_{3},\\\
\tilde{|2\rangle}_{2}&=\hat{O}_{2}(2)|0\rangle^{\otimes
3}=|0\rangle_{1}|1\rangle_{2}|1\rangle_{2},\end{split}$ (8)

where $|1/0\rangle_{i}$ represents that the $i_{\rm th}$ qubit is one/no
excitation.

Figure 4: Graphic illustration of the algorithm to relate the states in the
reduced network Hilbert space and the full Hilbert space. Note that the blue
spot here represent the qubit in the tree, not the basis states as in Fig. 3

To simulate the operator evolution later, we need to construct the operator
representation in the reduced Hilbert space. Note that some local operators
can not be represented in reduced Hilbert space in a closed way, because some
states are out of the reduced Hilbert space. For example, The local operator
$\hat{\sigma}_{+}^{1}$ in two layers case, acting on the state
$|0\rangle_{0}|1\rangle_{2}|1\rangle_{3}$, will transfer it to
$|1\rangle_{1}|1\rangle_{2}|1\rangle_{3}$, which is out of the reduced Hilbert
space. Fortunately, there exist some operators of which the representation
always stays in the reduced Hilbert space, such as $\hat{\sigma}_{i}^{z}$,
which will be used in constructing the out-of-time correlator later on. The
formula of the matrix elements are given as:

$\begin{split}\tilde{\langle
m|}\hat{\sigma}_{i}^{z}\tilde{|n\rangle}=\begin{cases}0,&m\neq n;\\\
1,&m=n,{\rm Tr}_{\bar{i}}(\tilde{|m\rangle}\tilde{\langle
m|})=|1\rangle_{i}\langle 1|_{i};\\\ -1,&m=n,{\rm
Tr}_{\bar{i}}(\tilde{|m\rangle}\tilde{\langle m|})=|0\rangle_{i}\langle
0|_{i};\end{cases}\end{split}$ (9)

where ${\rm Tr}_{\bar{i}}(...)$ is tracing over the other qubits complementary
to the $i_{\rm th}$ qubit.

###  II.4 Quantum scrambling

Quantum information of the incoming photon is encoded in the first qubit with
the initial state as $|\psi\rangle=\alpha|0\rangle+\beta|1\rangle$, where the
coefficients $\alpha,\beta$ carry the initial unscrambled quantum information.
Quantum information scrambling means that once this qubit interacts with the
other qubit in the network that mimics the photo-detector, one can not recover
the $\alpha,\beta$ from any _local measurement_ of this qubit network. A
simple illustration is that the quantum state at time $t$ can have the
structure as:

$\begin{split}&|\psi(t)\rangle=\alpha|0\rangle_{1}\otimes\prod_{ij}|0_{ij}\rangle+\beta|\psi\rangle_{\rm
scr},\quad{\rm with}\\\ &|\psi\rangle_{\rm
scr}=c_{0}|1\rangle\otimes\prod_{ij}|0_{ij}\rangle+...+\sum_{m}c_{m}|0\rangle\otimes|...1_{4j}...\rangle_{m}\\\
&\qquad\qquad+\sum_{n}c_{n}|0\rangle\otimes|...0_{4j}...\rangle_{n}+...,\end{split}$
(10)

where the $|\psi\rangle_{\rm scr}$ represents the state with the avalanching
process, and $|...1/0_{4j}...\rangle$ represents the state where the $j_{\rm
th}$ site of the 4th layer is occupied/unoccupied.

Measuring the local site $4j$ leads to the probability of occupied and
unoccupied $4j$-site as $P_{1}=|\beta|^{2}\sum_{m}|c_{m}|^{2}$ and
$P_{0}=|\alpha|^{2}+\beta\sum_{n}|c_{n}|^{2}$. We also have
$|\alpha|^{2}+|\beta|^{2}=1$ and the normalisation condition for the
coefficients in $|\psi\rangle_{\rm scr}$. These equations are not sufficient
to solve the $|\alpha|^{2},|\beta|^{2}$, which demonstrates that local
measurement can not recover the initial quantum information and therefore the
detector network scrambles the quantum information.

Taking into account the time-dependent randomness of the qubit level, after a
sufficiently long evolution time, as we shall see in the later simulations,
the $|\psi\rangle_{\rm scr}$ evolves into a heavily scrambled state (note that
the corresponding density matrix is proportional to a unitary matrix), but
still unitary. Therefore the total quantum state is:

$|\psi(t)\rangle=\alpha|0\rangle_{1}\otimes\prod_{ij}|0_{ij}\rangle+\beta|\psi(t\rightarrow\infty)\rangle_{\rm
scr},$ (11)

which is a scrambled macroscopic Schroedinger cat state. In the next section,
we will use the out-of-time correlator to quantitatively study the quantum
scrambling in our toy model.

##  III Level Statistics and integrability

Before introducing our simulation result, it is worth investigating the
integrability of our model. In the structure of the reduced Hilbert space
(corresponds to the conserved quantity $\bar{N}=1$) shown in Fig. 3, the link
between different red spots physically represents a non-zero matrix element
that connect these basis-states due to the interaction kernels. Therefore in
the reduced Hilbert space, our model (triggered by one excitation on the first
layer) can be rewritten as:

$\hat{H}_{\rm
reduced}=\sum_{i}E_{\tilde{i}}|\tilde{i}\rangle\langle\tilde{i}|+\sum_{\langle
i,j\rangle}g_{ij}|\tilde{i}\rangle\langle\tilde{j}|,$ (12)

where $\langle i,j\rangle$ means summation over all allowed connections in the
graph of Fig. 3. This bilinear Hamiltonian is diagonalisable, therefore one
can obtain its eigenvalue and level spacing statistics defined as Pal2010 ;
Oganesyan2007 :

$r_{n}=\frac{{\rm min}[\delta_{n},\delta_{n+1}]}{{\rm
max}[\delta_{n},\delta_{n+1}]},$ (13)

where $\delta_{n}=|E_{n}-E_{n-1}|$ is the energy difference of the adjacent
levels. We compute the probability distribution of $r_{n}$ using a five-layer
model with $667$ basis-states for the ideal case and the case with random
phase distribution, obtaining Poisson distributions for both cases shown in
Fig. 5. For the model with more layers, e.g. 6 layers, the number of basis
would increase to $458330$, and our current computational resources can not
diagonalise a matrix of this size. However, our result on the 5-layer system
still indicates strongly that our system is integrable. Therefore, although
our system has a avalanching beahviour that involves an exponentially growing
number of operators, this Poission distribution means that our system seems to
be not chaotic.

Figure 5: Level spacing in our model with five layers (667 basis-states in
total). Upper panel: the ideal case when the phase matching in each
interaction kernel is perfect. Lower panel: the phase matching takes a random
number distributed among interaction kernels with $\sigma_{\Delta}=5g$. It is
clear that the level spacing in our system follows a Poisson distribution
which indicates that our system is integrable.

##  IV Simulation Results: Evolution of the toy model

_Out-of-time correlator_ (OTOC) is initially introduced by Larkin and
Ovchinnikov in the 1960s where they used the quasi-classical method to study
the electron behaviour in a superconductor with impurities Larkin1969 . Its
importance was widely appreciated in recent years when its relevance to chaos
in the quantum complex system was revealed Swingle2018 ; Hashimoto2017 ;
Braumuller2022 ; Maldacena2016 . The OTOC $F_{ij}(t)$ in a quantum complex
system is defined as:

$F_{ij}(t)=\langle\psi|\hat{W}_{i}^{\dagger}(t)\hat{V}_{j}^{\dagger}\hat{W}_{i}(t)\hat{V}_{j}|\psi\rangle,$
(14)

where $\ket{\psi}$ is the initial state of the system at $t=0$,
$\hat{W}_{i},\hat{V}_{j}$ are two operators acting on the site $i,j$ at $t=0$,
respectively. The $\hat{W}_{i}(t)$ is the time-evolving operators defined as
$\hat{W}_{i}(t)=e^{i\hat{H}t}\hat{W}_{i}e^{-i\hat{H}t}$. The physical meaning
of OTOC can be understood from its connection to the operator commutation
square:

$\begin{split}C_{ij}(t)&=\bra{\psi}[\hat{W}_{i}(t),\hat{V}_{j}]^{\dagger}[\hat{W}_{i}(t),\hat{V}_{j}]\ket{\psi}\\\
&=2(1-{\rm Re}[F_{ij}(t)]).\end{split}$ (15)

where the OTOC is real for Hermitian operators $\hat{W}_{i},\hat{V}_{j}$.
Scrambling in a quantum complex system can be understood as the growth of the
initial “simple” operators in terms of size and complexity. This operator
growth means the initial operator correlates with more and more operators in
the system, and it is in this way that the quantum information spreads in the
system. For example, initially commuting operators
$[\hat{W}_{i},\hat{V}_{j}]=0$ becomes non-commuting when
$\hat{W}_{i}\rightarrow e^{i\hat{H}t}\hat{W}_{i}e^{-i\hat{H}t}$, that makes
the $C_{ij}(t)\neq 0$ and ${\rm Re}[F_{ij}(t)]<1$. The operator correlation
diminishes when $C_{ij}(t)\rightarrow 0$, or $F_{ij}(t)\rightarrow 1$.

For our toy model, the Pauli operators at different sites
$\hat{\sigma}^{z}_{i}$ is a natural choice for the $\hat{W}$ and $\hat{V}$
operators, because the dynamical process in this toy model mainly consists of
the excitation and de-excitation of the qubits. In particular, we choose
$\hat{V}=\hat{\sigma}^{z}_{1}(t=0)$ which is the Pauli-$z$ operator for the
first qubit and describes the electron knocked out from the first dynode by
the incoming photon, and $\hat{W}(t)=\hat{\sigma}^{z}_{i}(t)$ which is the
Pauli-$z$ operator of the qubit in other layers.

Besides the OTOC, the mean excitation
$\langle\psi|(1+\hat{\sigma}^{z}_{i})/2|\psi\rangle$ can also be used to
describe the evolution of the toy model. In the following, we numerically plot
the dynamical evolution of the OTOC and the mean excitation of our toy model
in Fig. 1. We try to understand the behaviour of our model in three different
physical scenarios: (1) The ideal case where no randomness exists in our
model. (2) There will be time-independent randomness of the qubit energy
levels in our model. and (3) There will be time-dependent randomness of the
qubit energy levels. It is important to note that both the (1) and (2)
scenarios in later sections IV.1 and IV.2 in principle should satisfy the
quantum Poincare recurrence theorem Percival1961 ; Bocchieri1957 ;
Schulman1978  (note that this should be distinguished from the FPUT recurrence
Berman2005 ). Poincare recurrence theorem states that for a time-independent
quantum system with discrete energy eigenstates, after a sufficiently long but
finite time, return to a quantum state that is exactly equal to the initial
quantum state Zermelo1896 ; Rauer2018 ; Essler2016 . However, this condition
would be broken if the system Hamiltonian becomes time-dependent, as we will
show in the next paragraph. In this case, the quantum Poincare recurrence will
be broken and the system undergoes an ergodic evolution, which leads to the
heavily scrambled Schroedinger cat state.

###  IV.1 Fermi-Pasta-Ulam-Tsingou recurrence

Figure 6: FPUT recurrence of the ideal model without qubit level fluctuations.
The upper panel is the space-time evolution of the mean occupation number of
each layer simulated for a different time duration, where the ballistic
propagation of the excitation and its reflection on the boundary are
demonstrated. The lower panel is the evolution of the mean occupation of the
first layer for a longer time scale. The features of quasi-periodicity and the
super-recurrence are very similar to what happens in the classical Fermi-
Pasta-Ulam-Tsingou model. The corresponding OTOC has the same behaviour.

We first consider the ideal case described by the Hamiltonian Eq. (1) with a
perfect environment, which means the energy level of the qubit in the dynode
does not fluctuate hence the additional phase in Eq. (3) is zero. Initially,
the qubit in the first layer is excited by an incoming photon. Then the mean
excitation and the OTOC induced by the excitation spreads _ballistically_ over
the system with velocity $v\propto g$, which will be affected by the disorders
introduced later. Larger interaction strength means a quicker propagation of
the quantum information into the layers. When the excitation/OTOC propagates
to the boundary of the system, the flow of quantum information will _quasi_
-periodically bounce back and forth later on without information scrambling as
shown in Fig. 6, which is similar to the FPUT recurrence in classical dynamics
Fermi1955 . In Fig. 6, the evolution of the averaged excitation is shown for
both short and long time scales. The results exhibit a clear FPUT recurrence
structure in a relatively short time scale and super-recurrence structure in a
longer time scale Tuck1972 ; Pace_2019 .

###  IV.2 “Localisation" of quantum information.

The environmental effect is characterised by the fluctuating qubit energy
levels, which will significantly affect the recurrence structure shown above.
We first discuss the case when there is a time-independent energy level
fluctuations randomly distributed among all qubits, with the OTOC evolution
shown in Fig. 7 for one specific random qubit level realisation.

The randomness of “chemical potential" creates a phenomenon similar to the
Anderson localisation or many-body localisation discussed in the quantum
thermalisation problem Anderson1958 ; Altman2018 ; Alet2018 ; Abanin2019 . The
out-of-time correlator is also used in studying many-body system with
randomness Huang2017 ; Swingle2017 ; Slagle2017 ; Kim2014 ; Fan2017 ; Chen2016
. Anderson localisation is studied by imposing the Anderson model on a lattice
system (even for the Cayley tree similar to our model), where the Hamiltonian
contains a hopping term between neighbouring sites and a random qubit energy
level:

$\hat{H}_{\rm
Anderson}=\sum_{i}\hbar\frac{\omega_{i}}{2}\hat{\sigma}_{z}^{i}+\hbar
g_{1}\hat{\sigma}_{-}^{A}(\hat{\sigma}_{+}^{B_{1}}+\hat{\sigma}_{+}^{B_{2}})+...,$
(16)

where the mean occupation number is conserved during the interaction process
for those hopping interactions. The system evolves as a wave propagating in a
random medium. The randomness of the energy level will halt the spreading of
the initial quantum information over different sites, which corresponds to the
Anderson localisation Jackson2012 .

The difference between the Anderson model Hamiltonian and our toy model brings
an important difference in their evolution, despite their similarity.
Basically, the upper panel of Fig. 9 shows that, for an occupation at the
$i_{\rm th}$ layer to efficiently go to the $j_{\rm th}$ site at the
$i+2$-layer, the phase matching of interactions
$\propto\hat{\sigma}^{i,j}_{-}\hat{\sigma}^{i+1,j}_{+}$ and
$\propto\hat{\sigma}^{i+1,j}_{-}\hat{\sigma}^{i+2,j}_{+}$ must be satisfied,
with the probability $p_{\rm for}=p^{2}$, where $p$ is the probability for the
phase matching of each interaction kernel. The probability $p_{\rm back}$ for
the backward process is the same.

Figure 7: OTOC evolution of the toy model with time-independent qubit level
fluctuations among different sites, _for one realisation of the disorder_. The
disorder strength is chosen to be $\sigma_{\Delta}=5g$. The upper panel is the
space-time evolution of the OTOC simulated for a different time duration,
where the “localisation" of the quantum information can be seen. The Lower
panel is the OTOC evolution for different layers. It shows that the OTOC of
the last two layers does not decrease to zero, and most of the quantum
information is localised within the first and third layers.

Figure 8: OTOC evolution of the toy model with time-independent qubit level
fluctuations among different sites, _averaged over $10$ realisations of the
randomness_. The upper left panel is the space-time evolution of OTOC on a
short time scale with disorder strength $\sigma_{\Delta}=5g$, and it is clear
that the quantum information propagation is blocked even at the early time.
The upper right panel is the OTOC evolution for different layers. The
mid/lower panel is the evolution of the averaged OTOC between the first qubit
and a qubit in the fifth layer $F_{15}(t)$ with different disorder strengthes
$\sigma_{\Delta}$ in the short/long time scale.

Our toy model behaves differently, which can be understood as follows (see the
lower panel of Fig. 9). In the Anderson model, the hopping happens between
neighbouring sites with a conserved occupation number. The hopping in our
model accompanies the avalanche process, where the excitation numbers are not
conserved. This creates an important difference which is crucial to
understanding irreversibility: For the forward process described by an
interaction kernel as $\sim
g\hat{\sigma}_{(-)}^{i}\hat{\sigma}_{(+)}^{i+1,j}\hat{\sigma}_{(+)}^{i+1,j+1}e^{-i\Delta\phi}$,
i.e. annihilating the occupied state in the $i_{\rm th}$ layer while creating
the double excitations of the $(i+1)_{\rm th}$ layer, the phase matching error
is $\Delta\phi=(\omega_{i}-\omega_{i+1,j}-\omega_{i+1,j+1})t$. If we assume
the probability for the phase-matching is $p$ at all interaction kernels, then
the probability for this forward process to be phase-matched is $p_{\rm
for}\sim p\times p_{o}$, where $p_{o}$ is the probability of an occupied site.
The backward process described by $\sim
g\hat{\sigma}_{(+)}^{i}\hat{\sigma}_{(-)}^{i+1,j}\hat{\sigma}_{(-)}^{i+1,j+1}e^{i\Delta\phi}$,
is a process where the double excitation at the $(i+1)_{\rm th}$-layer
annihilated while creating a single excitation at the $i_{\rm th}$-layer. In
this case, the condition of doubly excited $(i+1)_{\rm th}$-layer can
efficiently happen with the probability $p_{\rm back}\sim p\times
p_{o}^{2}<p_{\rm for}$. Therefore the randomness in our model creates a weaker
localisation of quantum information compared to the Anderson model.
Furthermore, Fig. 8 demonstrates the OTOC evolution averaged over 20
realisations of random qubit levels of the network, where the wiggles in the
Fig. 7 are suppressed.

Fig. 8 shows how the evolution of the correlation between the first qubit and
the qubit in the last layer (e.g. $F_{15}(t)$) depends on the disorder
strength $\sigma_{\Delta}$. The results show that the $F_{15}(t)$ increases
(or the correlation decreases) with the increasing of the disorder strength
$\sigma_{\Delta}$, which is also a sign of the localisation.

Figure 9: Phase matching fluctuation induces the system irreversibility. The
upper panel: the Anderson model imposed on the qubit lattice in a Cayley tree,
where the probability for the phase matching of the forward process and the
backward process is the same. The lower panel is on our toy model. The
randomly fluctuating phase matching error favours the forward process rather
than the backward process in terms of interaction efficiency.

###  IV.3 Temporally random qubit levels.

The more realistic scenario is that the qubit energy levels are fluctuating in
time and randomly distributed among different sites. The corresponding
behaviour of the OTOC is plotted in Fig. 10. This time-dependent disorder
breaks the localisation in the previous section and the quantum information
and the excitations will accumulate on the last two layers at the later
evolution stage.

In the interaction picture, the evolution operator of a typical interaction
kernel (e.g. $A-B_{1}-B_{2}$) can be written as:

$\hat{U}_{I}(\tau,0)={\rm
exp}\left[-ig\int_{0}^{\tau}\left(e^{i\int_{0}^{\tilde{\tau}}\epsilon(\tau^{\prime})d\tau^{\prime}}\hat{\sigma}_{-}^{A}\hat{\sigma}_{+}^{B1}\hat{\sigma}_{+}^{B2}+h.c\right)d{\tilde{\tau}}\right],$
(17)

where $\epsilon(t)=\Delta_{A}(t)-\Delta_{B1}(t)-\Delta_{B2}(t)$, which is set
to fluctuate around zero. For illustration, the distribution of the $\epsilon$
is set to be an uniform distribution within
$[-\sigma_{\Delta},+\sigma_{\Delta}]$ and the time-dependence manifests as a
revision of all the $\epsilon_{i}$ in the network by every time interval
$\Delta t$. In the numerical simulation, we reduce the integration to a
discrete summation as shown in the upper panel of Fig. 10, and the simulation
result is plotted in Fig. 10 with the same disorder strength
$\sigma_{\Delta}=5g$ but different time variation interval $\Delta t=2/g$ and
$\Delta t=0.2/g$. For the small $\Delta t$ and disorder strength satisfying
$\Delta t\sigma_{\Delta}\ll 1$, the influence of the time-dependent phase
mismatch in Eq. 17 is very weak, therefore the system will still exhibits
recurrence behaviour as in the ideal case. On the contrary, for the larger
$\Delta t$ and disorder strength satisfying $\Delta t\sigma_{\Delta}\geq 1$,
the recurrence phenomenon disappears and the information will be trapped on
the last two layers, since the quantum information flow backward encounters a
different realisation of the disorder. In the limit $\Delta t\to\infty$, the
evolution will return to the previous model with time-independent disorders.
Note that the time variation scale difference mainly affect the evolution at
the relatively early stage, at the long time tail the evolution of OTOCs
converge to a stable value, since the quantum information has been strongly
scrambled in the system.

Figure 10: Evolution of the averaged out-of-time correlator $\bar{F}_{1j}(t)$
of four layers quantum photo-detector toy model, where the $j$ labels the
qubit located at the left boundary of each layer and the qubit level
fluctuates both spatially and temporally. The result shown here is averaged
over 10 different realisations. Upper panel: one example showing how we choose
a realisation of the time-dependent disorder, in the units of time variations
scale. Middle panel: the color bar represents the value of the OTOC. The upper
left/right figures shows the space-time evolution of the OTOC in the
long/short time scale. Lower panel: The long-time evolution of OTOC when $j$
runs over different layers.

It is also interesting to investigate the (un-averaged) OTOC for different
realisations, which is shown in Fig. 11. In particular, we want to explore how
the time-dependent disorder in the network affect the arrival of the initial
quantum information on the first qubit site to the last layer, with a
significantly amplified occupation number by the avalanche process. As we
mentioned in the beginning of this section, a more negative OTOC means a
stronger correlation of the operators, this leads us to set the
$F_{1j}\leq-0.5$ as a threshold to judge the arrival of quantum information on
the $j_{\rm th}$ layer. To avoid the recurrence effect by the information
propagate backward from the last layer, Fig. 11 only shows the early stage of
the OTOC evolution and the results indicates that there will be a stronger
scattering of the arrival time when the time variation scale is larger.

Figure 11: Evolution of the out-of-time correlator $\bar{F}_{14}(t)$ of four
layers quantum photo-detector toy model, for 10 different realisations of
time-dependent random detuning $\epsilon(t)$. The upper and lower panels
correspond to different time variation scale $\Delta t$. We set the
$F_{15}<-0.5$ as the threshold for the arrival of quantum information on the
last layer of the network. Note that the arrival time of the quantum
information to the last layer for different realisations also randomly
distributes. We only show the early evolution stage, which is less affected by
the backward propagating information.

##  V Evolution of the Holevo information

As pointed out in Xu2022 , treating the squared commutator $C_{ij}(t)$ or the
OTOC $F_{ij}(t)$ as an indicator of the quantum information propagation when
the initial state is a simple maximumly mixed state. The relation between OTOC
and the mutual information is not well-established when the initial state is
not maximumly mixed. Therefore, in this section, we further investigate the
quantum information evolution using the Holevo information, which is used to
characterise the accessible information (or maximum mutual information)
between two separate agents in a complex system.

The formula of Holevo information is Dong2022 :

$\chi=S(\sum_{i=1}^{n}p_{i}\hat{\rho}_{i})-\sum_{i=1}^{n}p_{i}S(\hat{\rho}_{i}),$
(18)

where $S(\hat{\rho})$ is the von Neumann entanglement entropy with definition
$S(\hat{\rho})=-{\rm Tr}(\hat{\rho}{\rm Log}\hat{\rho})$. Holevo information
can be interpreted as the measurement of the difference of different quantum
states in $\\{\hat{\rho}_{1},\hat{\rho}_{2},...,\hat{\rho}_{n}\\}$. As a
simple example, if $p_{1}=p_{2}=1/2$, then $\chi=1$ for
$\hat{\rho}_{1}=|0\rangle\langle 0|$, $\hat{\rho}_{2}=|1\rangle\langle 1|$ and
$\chi=0$ for $\hat{\rho}_{1}=\hat{\rho}_{2}$. Now we consider a local operator
$\hat{O}_{i}$ acting on $i_{\rm th}$ qubit and the density matrices
$\hat{\rho}$ and $\hat{\rho}^{\prime}$ are defined to describe the state
evolution from the initial network state $|\psi\rangle$ and
$\hat{O}_{i}|\psi\rangle$ respectively. To characterise how this local
operation on the $i_{\rm th}$-site affects the $j_{\rm th}$-site, we define
the $\tilde{\rho}_{j}$ and $\tilde{\rho}^{\prime}_{j}$ as the reduced density
matrix of the $j_{\rm th}$-qubit before and after the local operation,
respectively. In this case, the Holevo information can be written as:

$\chi_{ij}(t)=S\left[\frac{\tilde{\rho}_{j}(t)+\tilde{\rho}_{j}^{\prime}(t)}{2}\right]-\frac{S(\tilde{\rho}_{j}(t))+S(\tilde{\rho}^{\prime}_{j}(t))}{2},$
(19)

of which the evolution is plotted in Fig. 12, and exhibits the same behaviour
as the OTOC.

Figure 12: Evolution of the Holevo information $\chi_{ij}(t)$ of a five layer
quantum photon-detector toy model, where the $j$ takes the qubit located at
the left boundary of each layer. The upper panel describes the Holevo
information when the qubit level fluctuation is time independent with
$\sigma_{\Delta}=5g$, while the lower panel describes the case when the qubit
level fluctuation is time-dependent.

##  VI Connections to quantum-state Collapse, Conceptual Issues and Outlook

The key issue with the quantum measurement process is that identical initial
states lead to different phenomenological outcomes. Unless we take the
relative-state (or many-world) point of view, we will need to provide
additional information in order to explain why this particular outcomes is
achieved at this particular trial of the “same experiment”.

###  VI.1 Connection and Conceptual Issues

From numerical simulations, we have seen that: (i) the values of $g_{j}$
controls the time at which the avalanche toward lower layers takes place, (ii)
if the number of layers can be extended, information will tend not to flow
back to higher layers. This supports the notion that internal mechanisms of
the detector supplies that additional information. The complexity of the
device also prevents further observers from modifying the result of quantum
measurement. However, as one attempts further to solve the “measurement
problem” this way, one encounters two serious challenges: the non-locality
problem, and the Schroedinger-Cat problem.

In the non-locality problem, if we use two of our detectors to probe entangled
photons at space-like separations, then each of their internal states, i.e.,
the $g_{j}$’s will have to influence the outcome of the other experiment. In
this way, we seem to have constructed a superluminal communcations device. In
the cat problem, if we prepare the atom at a superposition state
$\alpha|g\rangle+\beta|e\rangle$, then from quantum mechanics we already have
a superposition of two states, one with avalanche, the other without. For this
reason, it is possible that while this many-body model resembles the quantum
collapse, it does not in fact explain it. Nevertheless, we shall try to
outline some loopholes below.

###  VI.2 Loopholes

For the non-locality problem, superluminal communication requires the observer
to be in total control of the device, which is not practical. If the observer
is able to divulge the measurement result, then additional influence from the
same channel can also affect the measurement result. Furthermore, if the
observer is then going to use the measurement result to create logical
inconsistencies, back-action from the follow-up device will not allow
inconsistencies to take place. This has been argued for in the context of
closed time-like curves by Novikov novikov1992time , Friedman et al.
friedman1990cauchy  in the 1990s, and more recently by Tobar and Costa
tobar2020reversible .

For the cat problem, we need to realize that the state-preparation process is
just another measurement process that took place before the “official”
measurement starts. In this way, a “superposition state” for the system being
prepared is really a collection of possible outcomes in the preparation
devices, instead of a superposition state for the joint system that consist of
the preparation and measurement devices. A more fundamental treatment of the
preparation-measurement procedure should incorporate both the preparation and
measurement devices, and in this situation the cat paradox may not arise.

###  VI.3 Outlook

Finally, to move toward understanding realistic quantum measurement processes,
we need to model the entire environment of the quantum system. Consider a
single atom that is prepared into an excited state and radiates a photon,
which can either be detected by one of several measurement detectors, or leave
toward infinity. The atom also had to be monitored by preparation detectors,
whose signal must indicate that the atom is present and at the correct initial
quantum state. One will need to elaborate how the $g_{j}$’s within the
measuring devices, as well as those within the preparation devices, together
reproduces the spontaneous emission rate of the atom, as well as the correct
branching ratio toward each of the measuremnt detectors. This seems highly
non-trivial; from our numerical simulations, it seems that if $g_{j}$’s are
appropriately tuned, then the “collapse process” can be accelerated
indefinitely. Here we should be reminded of Wheeler and Feynman’s work on
absorber theory of radiation Wheeler1945 , in which the spontaneous emission
rate is always recovered even in presence of absorbers that seem to have high
susceptibility to radiation. One way to understand this is that radiation
damping within the device will not allow it to respond to the emitter’s
radiation faster than rate that will make it emit more than the spontaneous
emission rate.

##  VII Conclusion and Discussions

In this paper, we investigated the quantum information scrambling in a qubit-
network toy model of the quantum measurement process, which resembles the
typical photo-detector that generates the macroscopic measurable photo-current
through the avalanche process. Our system is studied based on a full-unitary
evolution model, where the environmental effect is taken into account through
the fluctuation of the qubit levels. The eigenvalue level spacing statistics
of our model in the reduced Hilbert space is numerically computed and the
result exhibits a Poission distribution, which indicates that our system is a
integrable system in the reduced Hilbert space. The evolution of the out-of-
time correlator and the Holevo information is simulated in different physical
scenarios. We found that there is localisation of the quantum information in
our system with time-independent disorder, which is similar to the Anderson
localisation in condensed matter physics. When the disorder becomes time-
dependent, our simulation shows that the localisation will disappear and the
quantum information finally would be kept in the last several layers in our
model.

The interpretation of the quantum measurement process is a long-standing issue
at the foundation of physics. One key question is: where does the unitary
process end, and the device start, which is related to the interaction
between, for example, the photon and the measurement device. Even though the
device is highly complex, it still falls into the current scope of quantum
many-body physics. Furthermore, the energy scale of most quantum measurement
processes is well within the typical energy scale of quantum electrodynamics,
which makes it possible that no new physics, hidden at different energy scales
(e.g. predicted by Bohm as “sub-quantum" structure in Bohm1963 ), play a
critical role to this process — although it had also been argued that gravity
may play a role in this process Penrose1996 .

The modern development of quantum technologies, in particular quantum
information and computing technology, provide the possibility to simulate the
details of the quantum measurement process, by customer-designing a table-top
quantum measurement apparatus, observing and even manipulating the details of
the measurement process. Though far from providing an answer to the quantum
measurement problem, the purpose of this work is a hope to trigger the
theoretical and experimental research in this direction in the current and
future quantum era, on the various quantum simulator platforms such as neutral
atoms, superconducting qubits and ion-trap. This could be a step forward to
understanding the mystery of quantum measurement or “wave function collapse".

###### Acknowledgements.

Y.L. and Y.M. thank Yinghai Wu, Xiaopeng Li and Daiqin Su for very helpful
discussions. Y.M. also thank Shun Wang for the discussion and his
encouragement. Y.M. thanks Jingtao Lü for the conversation about
thermalisation in solid-state physics. Y.M. is supported by the start-up fund
provided by Huazhong University of Science and Technology. H.M. is supported
by the State Key Laboratory of Low Dimensional Quantum Physics and the start-
up fund from Tsinghua University. Y.C. is supported by Simons Foundation
(Award Number 568762) .

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