# Beyond The Fermi’s Golden Rule: Discrete-Time Decoherence Of Quantum
Mesoscopic Devices Due To Bandlimited Quantum Noise

Evgeny A. Polyakov

###### Abstract

We are at the midst of second quantum revolution where the mesoscopic quantum
devies are actively employed for technological purposes. Despite this fact,
the description of their real-time dynamics beyond the Fermi’s golden rule
remains a formiddable theoretical problem. This is due to the rapid spread of
entanglement within the degrees of freedom of the surrounding environment.
This is accompanied with a quantum noise (QN) acting on the mesoscopic device.
In this work we propose a possible way out: to exploit the fact that this QN
is usually bandlimited. This is because its spectral density is often
contained in peaks of localized modes and resonances, and may be constrained
by bandgaps. Inspired by the Kotelnikov sampling theorem from the theory of
classical bandlimited signals, we put forward and explore the idea that when
the QN spectral density has effective bandwidth $B$, the quantum noise becomes
a discrete-time process, with an elementary time step $\tau\propto B^{-1}$.
After each time step $\tau$, one new QN degree of freedom (DoF) gets coupled
to the device for the first time, and one new QN DoF get irreversibly
decoupled. Only a bounded number of QN DoFs are significantly coupled at any
time moment. We call these DoFs the Kotelnikov modes. As a result, the real-
time dissipative quantum motion has a natural structure of a discrete-time
matrix product state, with a bounded bond dimension. This yields a
microscopically derived collision model. The temporal entanglement entropy
appears to be bounded (area-law scaling) in the frame of Kotelnikov modes. The
irreversibly decoupled modes can be traced out as soon as they occur during
the real-time evolution. This leads to a novel bandlimited input-output
formalism and to quantum jump Monte Carlo simulation techniques for real-time
motion of open quantum systems. We illustrate this idea on a spin-boson model.

##  I Introduction

The last decade has witnessed a rapid develoment of quantum technology [1].
This technology is concerned with measurement, control, communication, and
computation based on quantum mesoscopic degrees of freedom. The motivation
behind this development is two-fold. The first is miniaturization: the
technological devices on a nanometer scale must be designed according to the
laws of quantum physics. The second one is the promise to achieve a
significant supremacy over the technology based on the laws of classical
physics [1]. The role of the quantum mesoscopic degrees of freedom can be
played by e.g. selected electron levels of trapped ions [2], semiconductor
quantum dots [3], superconducting circuits [4], spin of electrons in nitrogen-
vacancy centers [5], Majorana zero modes [6]. In any physical device, these
dedicated degrees of freedom are always coupled to an infinite number of other
degrees of freedom. The latter are called the environment. This way we arrive
to the model of open quantum system (OQS) [7], which is thus a topic of active
research. Part of the environment can be used to control the open quantum
system (e.g. transmission line resonator in superconducting circuits [8, 9]).
The other part is responsible for irreversible scattering of the quantum
information (decoherence) [7, 9]. The latter is the major obstacle to the
creation of scalable quantum computers [9], which stimulates the research on
decoherence phenomena.

In the field of condensed matter the open quantum system is referred to as the
quatum impurity model (QIM), especially in the solid state context. The role
of QIM may be played by a single lattice site, localized spin, or some other
defect. QIM provides a model of solid state mesoscopic quantum devices [3].
Besides that, at low temperatures QIM develops complicated many-body
correlations with the itinerant electrons. The latter play the role of the
environment, and QIM becomes an open quantum system. These correlations
significantly affect the transport properties of materials [10] and properties
of quantum devices [11]. Currently there is an active investigation of non-
equilibrium properties of QIM [12, 13], in particular under periodical driving
[14].

Finally, besides the technological importance, the model of open quantum
system plays important role in the microscopic foundations of non-equilibrium
thermodynamics [15, 16, 17]: how the time arrow emerges from the reversible
microscopic dynamics [18, 19]; statistics of work, heat, and entropy
production [17, 16, 20, 21, 22, 23]; how the thermodynamics is related to
quantum trajectories [24, 25, 26, 27]; the studies of quantum heat engines
[28, 29, 30].

All of the above shows that at the moment the actual problem is to study the
non-stationary real-time motion of OQS/QIM, in order to clarify open questions
in the research fields mentioned above.

The description of real-time motion of OQS/QIM is clear and simple when we can
adopt the Fermi’s golden rule (the so-called Markovian approximation [7]): OQS
experiences sudden transitions (quantum jumps) $\omega_{1}\to\omega_{2}$
between the energy eigenstates $\left|\omega_{1}\right\rangle$ and
$\left|\omega_{2}\right\rangle$ with a rate $\Gamma_{1\to
2}=\left|\left\langle\omega_{2}\left|\widehat{V}\right|\omega_{1}\right\rangle\right|^{2}J\left(\omega_{1}-\omega_{2}\right)2\pi/\hbar$.
Here $\widehat{V}$ is coupling to the environment, and
$J\left(\omega_{1}-\omega_{2}\right)$ is the effective density of degrees of
freedom (DoFs) of the environment seen by OQS. The resulting stochastic
dynamics is described by the so-called Lindblad master equation [7]. Due to
such a clear probabilistic picture, this Markovian approximation is well
understood nowadays. It is a workhorse in describing the quantum optics
experiments [31] and the physical mechanisms of decoherence in qubits [32].
There exist efficient numerical simulation techniques [7]. The Markovian
approximation is also used in the simulations of quantum heat engines [33] and
in foundations of non-equilibrium quantum thermodynamics [34]. The Fermi’s
golden rule corresponds to a white (delta-correlated in time) quantum noise
[35, 36].

The real-time motion of OQS becomes a formiddable theoretical problem outside
the applicability of the Fermi’s golden rule (the so-called non-Markovian
regime). This happens when the spectral density $J\left(\omega\right)$ of the
environment is not flat, but on the contrary, is structured. The structured
$J\left(\omega\right)$ naturally occurs in quantum devices which are based on
solid state impurities. For example, in the nitrogen-vacancy centers in the
diamond [37], the impurity (defect) is coupled to the environment of acoustic
phonons. The corresponding $J\left(\omega\right)$ develops a number of peaks
corresponding to vibrational resonances and localized modes. Analogous
situation occurs for impurities in semiconductors [38]. The structured
$J\left(\omega\right)$ also occurs in the photonic band gap materials [39]
where $J\left(\omega\right)$ is organized into allowed and forbidden
($J\left(\omega\right)\approx 0$) bands. Finally, there is experimental
progress in creating artificial environments with a structured band-gapped
$J\left(\omega\right)$ [40]. The motivation is to introduce effects into the
motion of OQS which are beyond the Markovian approximation e.g. the
information backflow [41]. One hopes to exploit such non-Markovian effects as
a resource for the quantum technologies (the so-called reservoir enegineering)
[42].

The deep reason behind the difficulty of non-Markovian regime is the apparent
absence of the full-fledged quantum jumps. Indeed, the Fermi’s golden rule
provides us with quantum jumps which are completed and irreversible elements
of classical reality. As a result, they have a probabilistic interpretation,
which leads to visual and formal beauty (the divisibility of quantum Markovian
motion and the concept of dynamical semigroups) [7], and to efficient Monte-
Carlo simulation methods [7]. On the contrary, beyound the Markovian
approximation, every excitation which is emitted by OQS can always be
reabsorbed back (return to OQS) after a time interval $t$ with some amplitude
$C_{\textrm{q}}\left(t\right)$. This amplitude slowly decreases as a certain
inverse power $C_{\textrm{q}}\left(t\right)\propto t^{-p}$, $p>0$ [43, 44,
45]. As a result, one fails to introduce the concept of full-fledged non-
Markovian quantum jumps [46, 47]: they appear to be never complete in finite
time. For any finite time, there is a non-nebligible amplitude that the
excitation will return. Therefore, the entanglement rapidly propagates through
the surrounding enviromnent, with a combinatorial growth of the complexity of
real-time quantum motion (the so-called temporal entanglement barrier [44, 48,
49, 50]).

In literature there are many attemps to solve the puzzle of non-Markovian
real-time motion. Nowadays the most activity is concentrated on combining the
Feynman-Vernon influence functional [51] with the tensor network techniques
[52, 50, 48, 14, 53, 54, 55]. At the heart of these attemps is a bruteforce
numerical compression of many-body quantum states using smart linear algebra
machinery and the SVD of high-rank tensors.

In this work we continue another line of research: our aim is to find a clear
visual physical principle, which could replace/extend the Fermi’s golden rule
in the non-Markovian regime. We started in [45].where we have employed the
time-domain wavefunction to identify how OQS/QIM forgets about the previously
emitted quanta. We have found that all the emitted spectral content is
forgotten except progressively small vicinities of frequencies $\omega_{k}$
where $J\left(\omega_{k}\right)$ has sharp features (e.g. band edges, sharp
peaks). In [44] we have implemented this approach in a numerical method and
conjectured that the time domain should be employed in a properly formulated
real-time renormalization group (RTRG). Such RTRG was proposed in [56] which
was recently published as [43]. The present paper further develops this
approach by taking into account the spectral properties of the quantum noise.

This work is structured as follows. In sec. II we present the model of open
quantum system and state the problem considered. Then in sec. III we introduce
the concept of bandlimited quantum noise and bear ananlogies with the
classical Kotelnikov sampling. In sec. IV we introduce the quantum analog of
Kolelnikov sampling in the context of interaction quench. The resulting
discrete-time model, simulation techniques, and physical picture are
described. In sec. V we derive the discrete-time model and provide a numerical
recipe to find its coupling constants. We conclude in VI. There are two
appendices with some details.

##  II The model

In this work we study the real time motion of a small open quantum system with
a Hamiltonian $\widehat{H}_{\textrm{s}}$. It is surrounded by a quantum
environment $\widehat{H}_{\textrm{b}}$. We assume there is some local site in
the environment which is represented by annihilation $\widehat{a}_{0}$ and
creation $\widehat{a}_{0}^{\dagger}$ operators,
$\left[\widehat{a}_{0},\widehat{a}_{0}^{\dagger}\right]_{\xi}=1$, with
$\xi=\pm 1$ depending on the statistics of the environment’s excitations
(fermions or bosons). OQS is coupled to this site via some operators
$\widehat{V}$, $\widehat{V}^{\dagger}$. The joint Hamiltonian for this model
is

$\widehat{H}=\widehat{H}_{\textrm{s}}+\widehat{V}^{\dagger}\widehat{a}_{0}+\widehat{V}\widehat{a}_{0}^{\dagger}+\widehat{H}_{\textrm{b}}.$
(1)

Here for simplicity we assume the bilinear coupling between the OQS operators
$\widehat{V},\,\,\widehat{V}^{\dagger}$ and between the environment’s
$\widehat{a}_{0}$ and $\widehat{a}_{0}^{\dagger}$.

We study the joint quantum state $\left|\Psi\left(t\right)\right\rangle$ of
OQS and environment. The state $\left|\Psi\left(t\right)\right\rangle$ may
appear as a result of the interaction quench. That is, we assume that
initially the environment is in its ground state
$\left|0\right\rangle_{\textrm{b}}$, and OQS is in some state
$\left|\psi\right\rangle_{\textrm{s}}$:
$\left|\Psi\left(0\right)\right\rangle=\left|\psi\right\rangle_{\textrm{s}}\otimes\left|0\right\rangle_{\textrm{b}}$.
Then at $t=0$ we couple OQS to the environment and track the resulting
evolution.

In this work we consider a specific form of the environment, a semiinfinite
chain of sites

$\widehat{H}_{\textrm{b}}=\sum_{j=0}^{\infty}\left\\{\varepsilon_{j}\widehat{a}_{j}^{\dagger}\widehat{a}_{j}+h_{j}\widehat{a}_{j}^{\dagger}\widehat{a}_{j+1}+h_{j}\widehat{a}_{j+1}^{\dagger}\widehat{a}_{j}\right\\},$
(2)

with
$\left[\widehat{a}_{k},\widehat{a}_{l}^{\dagger}\right]_{\xi}=\delta_{kl}$.
Here the couling site operators $\widehat{a}_{0}$ and
$\widehat{a}_{0}^{\dagger}$ appear as the first sites of the chain.

##  III Bandlimited quantum noise. Analogy to the classical Kotelnikov
sampling theorem

A good physical principle requires good assumpitions. In this work we assume
that the spectral density $J\left(\omega\right)$ of the environment is
effectively bounded to some frequency range $\left[0,B\right]$, see Fig. 1.

Figure 1: In this work we consider the quantum environment whose spectral
density $J\left(\omega\right)$ is effectively bounded to some frequency range
$\left[0,B\right]$, but otherwise can be arbitrary complex.

In most cases (e.g., environment with non-interacting quasiparticles) this
means that the quantum field emitted into the environment also has a
bandlimited spectrum. From the discipline of classical signal processing we
know the Kotelnikov sampling theorem [57], which is the following.

Figure 2: The classical Kotelnikov sampling theorem states that a) if a signal
$z\left(t\right)$ has a spectral density $J_{\textrm{cl}}\left(\omega\right)$
which is limited to a frequency range $\left[-B,B\right]$, then b) the signal
$z\left(t\right)$ is completely determined (without loss of information) by a
sequence of its values
$\ldots,\,z\left(t_{-1}\right),z\left(t_{0}\right),z\left(t_{1}\right),\ldots
z\left(t_{k}\right)\ldots$, at discrete times moments which are spaced
$\tau=1/2B$ seconds apart.

Consider a random classical signal $z\left(t\right)$. For each time moment $t$
the value of the signal $z\left(t\right)$ is a complex number. Its time
correlator is

$C_{\textrm{cl}}\left(t\right)=\overline{z^{*}\left(t\right)z\left(0\right)},$
(3)

where the averaging $\overline{\left[\cdot\right]}$ is over the signal
samples. The spectral density $J_{\textrm{cl}}\left(\omega\right)$ of the
signal is provided by the causal Fourier transform of the correlator

$J_{\textrm{cl}}\left(\omega\right)=\textrm{Re}\intop_{0}^{+\infty}C_{\textrm{cl}}\left(t\right)e^{i\omega
t-0t}.$ (4)

Suppose that the signal turns out to have a finite band halfwidth $B$ , see
Fig. 2, a). Then the signal can be faithfully (without loss of information)
represented by its values at discrete time moments $t_{k}=k\tau$, $\tau=1/2B$,
with integer $k$, Fig. 2 b):

$z\left(t\right)=\sum_{k=-\infty}^{\infty}z\left(t_{k}\right)\frac{\sin\left(2\pi
B\left(t-t_{k}\right)\right)}{2\pi B\left(t-t_{k}\right)}.$ (5)

The message of the Kotelnikov sampling is that in each finite time interval
$t$ there is only a finite number $m\left(t\right)\propto TB$ of significant
DoFs of the signal. These DoFs are given by the sinc basis functions,
$\kappa_{k}\left(t\right)=\sin\left(2\pi B\left(t-t_{k}\right)\right)/2\pi
B\left(t-t_{k}\right)$. Each DoF $\kappa_{k}$ is in one-to-one correspondence
with its time moment $t_{k}=k\tau$.

Now we return to the quantum case. Analogously to the classical correlator eq.
(3), we introduce the correlator of the zero-point quantum noise [43, 44, 58],

$C_{\textrm{q}}\left(t\right)=\prescript{}{\textrm{b}}{\left\langle
0\right|}\left[\widehat{a}_{0}\left(t\right),\widehat{a}_{0}^{\dagger}\left(0\right)\right]_{\xi}\left|0\right\rangle_{\textrm{b}}$
(6)

for the vacuum state $\left|0\right\rangle_{\textrm{b}}$ of environment. Here
we employ the partial interaction picture with respect to the free motion of
environment,
$\widehat{a}_{0}\left(t\right)=\exp\left(it\widehat{H}_{\textrm{b}}\right)\widehat{a}_{0}\exp\left(-it\widehat{H}_{\textrm{b}}\right)$.

Analogously to the classical spectral density eq. (4), the spectral density of
the zero-point quantum noise (which coincides with the spectral density of the
environment) $J\left(\omega\right)$ is also a Fourier transform [43, 44]

$J\left(\omega\right)=\intop_{0}^{+\infty}dtC_{\textrm{q}}\left(t;0\right)e^{i\omega
t-0t}.$ (7)

If $J\left(\omega\right)$ is bandlimited, we expect that, analogously to the
classical case, the quantum noise should be faithfully represented by a
discrete time process.

##  IV Kotelnikov sampling of bandlimited quantum noise: the resulting model

In this section we present the resulting model of quantum dissipative motion
under bandlimited quantum noise. It is ready for use in calculations of
specific problems. We also discuss some physical implications. In the
following sections we justify this model and present the algorithm how to
calculate the numerical values of the model constants.

###  IV.1 Forward quantum Kotelnikov sampling

Notice that the statement of the real-time problem is slightly different from
that of the original classical Kotelnikov theorem eq.(5): we have the initial
value problem instead of the signal on the whole time axis. Therefore, we need
a causal variant of the Kotelnikov theorem, which we call the forward
Kotelnikov sampling, see appendix A According to the latter, we expect that
the degrees of freedom of the environment can be arragned in a sequence,
$\kappa_{k}^{\textrm{in}},$ $k=1,2,3\ldots$ by some change of frame. This
sequence can be chosen so that $\kappa_{k}^{\textrm{in}}$ couple to OQS
sequentially, one after the other, at discrete time moments
$t_{k}^{\textrm{in}}.$ Let us denote $m_{\textrm{in}}\left(t\right)$ the
number of modes which are already coupled to OQS at the time moment $t$. The
function $m_{\textrm{in}}\left(t\right)$ is piecewise constant and increases
by jumps at the time moments $t_{k}^{\textrm{in}}$ when a new mode is coupled.
For convenience of notation we define $m_{\textrm{in}}\left(t\right)$ to be
left-continuous:
$m_{\textrm{in}}\left(t_{k}^{\textrm{in}}+0\right)=m_{\textrm{in}}\left(t_{k}^{\textrm{in}}\right)=m_{\textrm{in}}\left(t_{k}^{\textrm{in}}-0\right)+1$.
Let us discuss the expected rate of such coupling events. The interaction
quench at $t=0$ (i.e. sudden switching-on of the coupling
$\widehat{V}^{\dagger}\widehat{a}_{0}+\widehat{V}\widehat{a}_{0}^{\dagger}$)
is a broadband event. Therefore, we expect that near $t=0$ the modes
$\kappa_{k}^{\textrm{in}}$ will appear at a high rate. But fast enough the
high-frequency part of the quench spectrum will become decoupled [45, 43], and
we reach the Kotelnikov asymptocic rate
$t_{k+1}^{\textrm{in}}-t_{k}^{\textrm{in}}\propto\tau=1/B$, see Fig. 3.

Figure 3: When the quantum noise has a finite bandwidth $B$, its degrees of
freedom also appear at discrete time moments $\tau\propto 1/B$. Here we plot
the number $m_{\textrm{in}}\left(t\right)$ of coupled incoming quantum
Kotelnikov modes $\kappa_{k}^{\textrm{in}}$ as a function of time. The case of
semicircle spectral density $J\left(\omega\right)$ of semiinfinite bosonic
chain with hopping $h$ and on-site energy $\varepsilon$, eq.
(LABEL:eq:environment_chain). Here the spectral density occupies a band
$\left[\varepsilon-2h,\varepsilon+2h\right]$. Black line: $\varepsilon=1$,
$h=0.1$. Blue line: $\varepsilon=1$, $h=0.05$. New significant modes occurs at
discrete time moments. The rate of occurence of new modes is asymptotically
constant and proportional to the bandwidth. At small times the rate of
occurence is high since the initial coupling quench is a wideband event.

We call $\kappa_{k}^{\textrm{in}}$ the incoming Kotelnikov modes.

In the following sections we provide a recipe how to find
$\kappa_{k}^{\textrm{in}}$. If we find these $\kappa_{k}^{\textrm{in}}$, then
the joint state $\left|\Psi\left(t\right)\right\rangle$ is found to
effectively evolve according to the Schrodinger equation

$i\partial_{t}\left|\Psi\left(t\right)\right\rangle\\\
=\left\\{\widehat{H}_{\textrm{s}}+\sum_{l=1}^{m_{\textrm{in}}\left(t\right)}\left(\widehat{V}^{\dagger}\chi_{l}^{\textrm{in}}\left(t\right)\widehat{\kappa}_{l}^{\textrm{in}}+\widehat{V}\chi_{l}^{\textrm{in}*}\left(t\right)\widehat{\kappa}_{l}^{\textrm{in}\dagger}\right)\right\\}\left|\Psi\left(t\right)\right\rangle,$
(8)

where we introduce the creation $\widehat{\kappa}_{k}^{\textrm{in}\dagger}$
and annihilation $\widehat{\kappa}_{k}^{\textrm{in}}$ operators corresdonding
to the mode $\kappa_{k}^{\textrm{in}},$ and these modes are independent DoFs:
$\left[\widehat{\kappa}_{k}^{\textrm{in}},\widehat{\kappa}_{l}^{\textrm{in}\dagger}\right]_{\xi}=\delta_{kl}$.
The coupling constant

$\chi_{l}^{\textrm{in}}\left(t\right)=\left[\widehat{a}_{0}\left(t\right),\widehat{\kappa}_{l}^{\textrm{in}\dagger}\right]_{\xi}$
(9)

is the amplitude to absorb a quantum from the mode $\kappa_{l}^{\textrm{in}}$
at time $t$. The time-dependent constant
$\chi_{l}^{\textrm{in}}\left(t\right)$ is a model parameter, which should be
found only once for a given $J\left(\omega\right)$.

In Fig. 4 we provide an example calculation for the model

$\widehat{H}\left(t\right)=\varepsilon_{\textrm{s}}\widehat{\sigma}_{+}\widehat{\sigma}_{-}+\widehat{\sigma}_{x}f\left(t\right)+h\widehat{\sigma}_{+}\widehat{a}_{0}+h\widehat{\sigma}_{-}\widehat{a}_{0}^{\dagger}\\\
+\sum_{i=0}^{\infty}\left\\{\varepsilon\widehat{a}_{i}^{\dagger}\widehat{a}_{i}+h\widehat{a}_{i+1}^{\dagger}\widehat{a}_{i}+h\widehat{a}_{i}^{\dagger}\widehat{a}_{i+1}\right\\}.$
(10)

Here the driving classical force $f\left(t\right)=0.1\cos t$. The chain
parameters are $h=0.05$ and $\varepsilon=1$. The qubit transition frequency
$\varepsilon_{\textrm{s}}=1$. The Schrodinger equation in the frame of
incoming modes, eq. (8), yields the same results as the numerically exact
solution.

Figure 4: The Schrodinger equation in the frame of incoming Kotelnikov modes
eq. (8) yields the same results as a direct numerically exact solution of the
Schrodinger equation. Here we consider the model of a driven qubit coupled to
a semiinfinite bosonic chain, eq. (10). Upper panel: black curve is the
numerically exact solution in a Hilbert space which is truncated in the number
of bosonic sites $n=7$ and in the total occupation of bosonic chain $N=14$ .
The parameters $n$ and $N$ where varied until convergence on the presented
time interval. Yellow dotted curve is the solution of the same problem in the
frame of incoming Kotelnikov modes, eq. (8). Lower panel: the plot of
$m_{\textrm{in}}\left(t\right)$ \- the number of incoming Kotelnikov modes
which are present in the Schrodinger equation at the time moment $t$.

The evolution under eq. (8) results in a discrete progressive spread of
entanglement which we call the forward Kotelnikov lightcone.

It is important that the DoFs
$\widehat{\kappa}_{l}^{\textrm{in}},\widehat{\kappa}_{l}^{\textrm{in}\dagger}$are
physical DoFs of the environment, which are related to the original ones
$\widehat{a}_{l},\widehat{a}_{l}^{\dagger}$ via some Bogoliubov transform (the
change of frame) $W$,

$\left[\begin{array}[]{c}\widehat{\kappa}_{1}^{\textrm{in}}\\\
\widehat{\kappa}_{2}^{\textrm{in}}\\\
\vdots\end{array}\right]=W\left[\begin{array}[]{c}\widehat{a}_{1}\\\
\widehat{a}_{2}\\\ \vdots\end{array}\right]$ (11)

for some unitary matrix $W$. Therefore, we have a full access to the
observables of the environment.

###  IV.2 Backward quantum Kotelnikov sampling

Since the quantum mechanics is time-reversal symmetric, we expect that the
time-reversed process is also happening: at time intervals $\propto 1/B$ some
modes $\kappa_{l}^{\textrm{out}}$ get irreversibly decoupled from OQS. We call
them the outgoing Kotelnikov modes. Here we should distinguish between the two
cases: (i) the modes which never become coupled to OQS, and (ii) the modes
which couple to OQS but after some time irreversibly decouple. We are not
interested in (i): we discard them from our model. However we need to track
the evolution of (ii). This consideration determines the way we find
$\kappa_{l}^{\textrm{out}}$: the mode $\kappa_{l}^{\textrm{out}}$ which
decouples at time moment $t_{l}^{\textrm{out}}$ should be a linear combination
of
$\kappa_{1}^{\textrm{in}}\ldots\kappa_{m\left(t_{l}^{\textrm{out}}\right)}^{\textrm{in}}$.
In other words, the newly decoupled mode should emerge in the subspace of
modes that coupled to OQS by the time $t_{l}^{\textrm{out}}$. This is because
the modes which are not incduled in this linear combination have not yet been
coupled to OQS.

This picture leads to the following iterative construction of the stream of
outgoing modes. The first outgoing mode $\kappa_{1}^{\textrm{out}}$ is found
via some Bogoliubov transform (the change of frame)

$\left[\begin{array}[]{c}\kappa_{1}^{\textrm{out}\dagger}\\\
\widehat{\kappa}_{1}^{\textrm{rel}\dagger}\\\ \vdots\\\
\widehat{\kappa}_{m\left(t_{1}^{\textrm{out}}\right)-1}^{\textrm{rel}\dagger}\end{array}\right]=U_{1}\left[\begin{array}[]{c}\widehat{\kappa}_{1}^{\textrm{in}\dagger}\\\
\vdots\\\
\kappa_{m\left(t_{1}^{\textrm{out}}\right)}^{\textrm{in}\dagger}\end{array}\right]$
(12)

where $U_{1}$ is $m_{\textrm{in}}\left(t_{1}^{\textrm{out}}\right)\times
m_{\textrm{in}}\left(t_{1}^{\textrm{out}}\right)$ unitary matrix. Here
$m_{\textrm{in}}\left(t_{1}^{\textrm{out}}\right)$ is the number of modes
which have coupled before the time of the first decoupling
$t_{1}^{\textrm{out}}$. We place the new outgoing mode in the first component
of the vector on the left hand side. Observe that the remaining
$m_{\textrm{in}}\left(t_{1}^{\textrm{out}}\right)-1$ modes can also be rotated
somehow. We call them the relevant modes since they are coupled to OQS thus
important for the future evolution. We mark them with a superscript “rel”. The
subsequent outgoing modes are found recurrently. Namely, by the time moment
$t_{k}^{\textrm{out}}$ we have $k-1$ outgoing modes
$\kappa_{1}^{\textrm{out}}\ldots\kappa_{k-1}^{\textrm{out}}$ and
$r\left(t_{k}^{\textrm{out}}\right)$ relevant modes
$\widehat{\kappa}_{1}^{\textrm{rel}}\ldots\widehat{\kappa}_{r\left(t_{k}^{\textrm{out}}\right)}^{\textrm{rel}}$,
which obviously satisfy

$r\left(t_{k}^{\textrm{out}}\right)+k-1=m_{\textrm{in}}\left(t_{k}^{\textrm{out}}\right).$
(13)

Then the newly formed outgoing mode is a linear combination of relevant modes:

$\left[\begin{array}[]{c}\widehat{\kappa}_{k}^{\textrm{out}\dagger}\\\
\widehat{\kappa}_{1}^{\textrm{rel}\prime\dagger}\\\ \vdots\\\
\widehat{\kappa}_{r\left(t_{k}^{\textrm{out}}\right)-1}^{\textrm{rel}\prime\dagger}\end{array}\right]=U_{k}\left[\begin{array}[]{c}\widehat{\kappa}_{1}^{\textrm{rel}\dagger}\\\
\vdots\\\
\widehat{\kappa}_{r\left(t_{k}^{\textrm{out}}\right)}^{\textrm{rel}\dagger}\end{array}\right],$
(14)

were $U_{k}$ is $r\left(t_{k}^{\textrm{out}}\right)\times
r\left(t_{k}^{\textrm{out}}\right)$ unitary matrix. We get a picture in which
the coupling and decoupling events are interspersed, see Fig. …., c). In the
course of a coupling event at $t=t_{k}^{\textrm{in}}$, the number of relevant
modes increases by one:
$r\left(t_{k}^{\textrm{out}}+0\right)=r\left(t_{k}^{\textrm{out}}\right)=r\left(t_{k}^{\textrm{out}}-0\right)+1$,
with the new mode
$\widehat{\kappa}_{r\left(t_{k}^{\textrm{out}}\right)+1}^{\textrm{rel}}\equiv\kappa_{m\left(t_{1}^{\textrm{out}}\right)}^{\textrm{in}}$.
In the course of decoupling event $t=t_{k}^{\textrm{out}}$, we rotate the
frame, eq. (14).

An important feature of this picture is that all these events have a finite
duration $\propto\tau=1/B$. Therefore, we can do all the rotations smoothly in
conjunction with the Hamiltonian evolution. Namely, let us introduce a time
interval $\left[t_{k}^{*},t_{k}^{\textrm{out}}\right]$, where $t_{k}^{*}$ is
the time of the last event (whatever it is) before $t_{k}^{\textrm{out}}$. All
the time axis $\left[0,+\infty\right)$ becomes covered with such intervals:
every time moment $t$ belongs to some
$\left[t_{k}^{*},t_{k}^{\textrm{out}}\right]$. We can introduce the generator
$D\left(t\right)$ of
$U_{k}=\exp\left(-i\left(t_{k}^{\textrm{out}}-t_{k}^{*}\right)D\left(t\right)\right)$.
Here for convenience of notation we define $D\left(t\right)$ as a piecewise
constant function on intervals $\left[t_{k}^{*},t_{k}^{\textrm{out}}\right]$.
Then the joint state $\left|\Psi\left(t\right)\right\rangle$ evolves on
$\left[t_{k}^{*},t_{k}^{\textrm{out}}\right]$ according to the Schrodinger
equation

$i\partial_{t}\left|\Psi\left(t\right)\right\rangle=\widehat{H}_{\textrm{eff}}\left(t\right)\left|\Psi\left(t\right)\right\rangle,$
(15)

with the effective Hamiltonian in the moving frame

$\widehat{H}_{\textrm{eff}}\left(t\right)=\widehat{H}_{\textrm{s}}+\sum_{l=1}^{r\left(t\right)}\left(\widehat{V}^{\dagger}\chi_{l}^{\textrm{rel}}\left(t\right)\widehat{\kappa}_{l}^{\textrm{rel}}+\widehat{V}\chi_{l}^{\textrm{rel}*}\left(t\right)\widehat{\kappa}_{l}^{\textrm{rel}\dagger}\right)\\\
-\sum_{kl=1}^{r\left(t\right)}D_{lk}\left(t\right)\widehat{\kappa}_{k}^{\textrm{rel}\dagger}\widehat{\kappa}_{l}^{\textrm{rel}}.$
(16)

Here the first line describes the interaction of OQS with $r$ relevant modes,
and the coupling constants

$\chi_{l}^{\textrm{rel}}\left(t\right)=\left[\widehat{a}_{0}\left(t\right),\widehat{\kappa}_{l}^{\textrm{rel}\dagger}\right]_{\xi}$
(17)

represent the amplitude to absorb a quantum from the relevant mode
$\kappa_{l}^{\textrm{rel}}$. The second line of eq. (16) gradually exercises
the frame rotation eq. (14).

The equation eq. (16) is ready for applications. The constants of the model,
$\chi_{l}^{\textrm{rel}}\left(t\right)$ and $D_{kl}\left(t\right)$, should be
found only once for a given $J\left(\omega\right)$. The recipe is presented in
the following sections.

In Fig. 5 we present example calculation with eqs. (15)-(16) of the same model
eq. (10). The exact solution when available coincides with our model. The
numbers of coupled modes $m_{\textrm{in}}\left(t\right),$ emerged outgoing
modes $m_{\textrm{out}}\left(t\right)$, and the relevant modes
$r\left(t\right)$ are presented on Fig. 6.

Figure 5: The plot of the qubit occupation for the model eq. (10). The
Schrodinger equation in the frame of incoming/relevant/outgoing Kotelnikov
modes eq. (15)-(16) yields the same results as a direct numerically exact
solution of the Schrodinger equation. We consider the same case as in Fig. 4.
The outgoing modes were traced out via the quantum jump Monte Carlo simulation
of section IV.5. The Fock space of relevant modes was truncated at a maximal
occupation of 6 quanta.

The appearance of the irreversibly decoupled modes results in a discrete
progressive disentanglement structure which we call the backward Kotelnikov
lightcone.

###  IV.3 Tracing out the irreversibly decoupled modes

The merit of the presented model is that the irreversibly decoupled modes can
be traced out as soon as they appear during the real time motion:

$\widehat{\rho}_{\textrm{rel}}\left(t\right)=\textrm{Tr}_{\kappa_{1}^{\textrm{out}}\ldots\kappa_{k}^{\textrm{out}}}\left\\{\left|\Psi\left(t\right)\right\rangle\left\langle\Psi\left(t\right)\right|\right\\},$
(18)

for all $t_{k}^{\textrm{out}}<t$. Here the subscript “rel” denotes the fact
that the density matrix $\widehat{\rho}_{\textrm{rel}}\left(t\right)$ depends
only on the quantum numbers of OQS and of $r\left(t\right)$ relevant modes.
This is a reduced description of real-time motion of bounded complexity with
respect to the time, with a closed equation of motion

$\partial_{t}\widehat{\rho}_{\textrm{rel}}\left(t\right)=-i\left[\widehat{\rho}_{\textrm{rel}}\left(t\right),\widehat{H}_{\textrm{eff}}\left(t\right)\right].$
(19)

###  IV.4 Number of relevant modes saturates

In our model the decoupling process is a time-reversed coupling process, and
vice versa, see section ….. Therefore, we expect them to have the same
asymptotic rate $\propto B$: the number $m_{\textrm{in}}\left(t\right)$ of
coupled and the number $m_{\textrm{out}}\left(t\right)$ of decoupled modes
grow with the same rate. As a result, their difference, the number of relevant
modes $r\left(t\right)$ (see eq. (13)), is asymptotically constant:
$r\left(t\right)\propto r$, see Fig. 6. Therefore the von Neumann equation
(19) provides a representation of real-time motion with bounded number of DOFs
on wide time scales.

Figure 6: The number of coupled modes $m_{\textrm{in}}\left(t\right)$ and the
number of outgoing modes $m_{\textrm{out}}\left(t\right)$ grow in time with
the same asymptotic rate $\propto B$, where $B$ is the bandwidth of the
spectral density $J\left(\omega\right)$. Therefore, the number of relevant
modes
$r\left(t\right)=m_{\textrm{in}}\left(t\right)-m_{\textrm{out}}\left(t\right)$
is asymptotically constant in time. The case of semicircle spectral density
$J\left(\omega\right)$ of semiinfinite bosonic chain with hopping $h$ and on-
site energy $\varepsilon$, eq. (LABEL:eq:environment_chain). Here the spectral
density occupies a band $\left[\varepsilon-2h,\varepsilon+2h\right]$. Here
$\varepsilon=1$, $h=0.05$. The relative significance treshold
$r_{\textrm{cut}}=10^{-4}$.

###  IV.5 Monte Carlo sampling of quantum jumps

There is alternative to the partial trace in eq. (18). When a new outgoing
mode $\kappa_{k}^{\textrm{out}}$ is formed at $t=t_{k}^{\textrm{out}}$, we can
apply the von Neumann measurement model [59]: the mode
$\kappa_{k}^{\textrm{out}}$ is an auxiliary probe which was previously coupled
to OQS. It is entangled with OQS according to the Schmidt decomposition

$\left|\Psi\left(t_{k}^{\textrm{out}}\right)\right\rangle\\\
=\sum_{p}c_{p}\left(k\right)\left|\Psi_{\textrm{collapsed}}^{\left(p\right)}\left(t_{k}^{\textrm{out}}\right)\right\rangle_{\textrm{rel}}\otimes\left|\textrm{jump}^{\left(p\right)}\left(t_{k}^{\textrm{out}}\right)\right\rangle_{\kappa_{k}^{\textrm{out}}},$
(20)

were we consider a biparticle system: one part labeled “rel” is the open
system with relevant modes, and the second part labeled
$\kappa_{k}^{\textrm{out}}$ is the newly formed outgoing mode. Since the probe
$\kappa_{k}^{\textrm{out}}$ is irreversibly decoupled, the entanglement
structure (20) is invariant under the future evolution. Then, according to the
von Neumann measurement model, we can interpret eq. (20) as a $p$th quantum
jump at time $t=t_{k}^{\textrm{out}}$,

$\left|\Psi\left(t_{k}^{\textrm{out}}\right)\right\rangle\to\left|\Psi_{\textrm{collapsed}}^{\left(p\right)}\left(t_{k}^{\textrm{out}}\right)\right\rangle_{\textrm{rel}},$
(21)

with a probability $P_{k}\left(p\right)=\left|c_{p}\left(k\right)\right|^{2}$.
These jumps are irreversible and non-Markovian since they happen during a
finite time interval $t_{k}^{\textrm{out}}-t_{k}^{\textrm{*}}\propto 1/B$.

At a time $t$ there are $m_{\textrm{out}}\left(t\right)$ decoupled modes. Each
of them is accompanied with a quantum jump, which are obtained by a recurrent
application of the von Neumann rule eq. (20). Therefore, before the time $t$
there are $m_{\textrm{out}}\left(t\right)$ jumps. They are characterized by a
history $h$ of choices,
$h=\left(p_{1},p_{2},\ldots,p_{k}\right)=\left(p_{k}\right)_{k:\,t_{k}^{\textrm{out}}\leq
t}$ , which occurs with the probability $P\left(p_{1}p_{2}\ldots
p_{k}\right)=\prod_{k:\,t_{k}^{\textrm{out}}\leq
t}\left|c_{p_{k}}\left(k\right)\right|^{2}$. Then the reduced density matrix
eq. (18) is computed as an average over the histories $h$ of jumps before $t$,

$\widehat{\rho}_{\textrm{rel}}\left(t\right)=\\\
\overline{\left[\left|\Psi_{\textrm{collapsed}}^{\left(h\right)}\left(t\right)\right\rangle_{\textrm{rel}}\,\,\,\prescript{}{\textrm{rel}}{\left\langle\Psi_{\textrm{collapsed}}^{\left(h\right)}\left(t\right)\right|}\right]}_{\textrm{over
all }h}.$ (22)

Here by
$\left|\Psi_{\textrm{collapsed}}^{\left(h\right)}\left(t\right)\right\rangle_{\textrm{rel}}$
we denote a particular trajectory of jumps,

$\left|\Psi\left(t_{1}^{\textrm{out}}\right)\right\rangle\to\left|\Psi_{\textrm{collapsed}}^{\left(p_{1}\right)}\left(t_{1}^{\textrm{out}}\right)\right\rangle_{\textrm{rel}},\\\
\left|\Psi_{\textrm{collapsed}}^{\left(p_{1}\right)}\left(t_{2}^{\textrm{out}}\right)\right\rangle_{\textrm{rel}}\to\left|\Psi_{\textrm{collapsed}}^{\left(p_{1}p_{2}\right)}\left(t_{2}^{\textrm{out}}\right)\right\rangle_{\textrm{rel}},\\\
\left|\Psi_{\textrm{collapsed}}^{\left(p_{1}p_{2}\right)}\left(t_{3}^{\textrm{out}}\right)\right\rangle_{\textrm{rel}}\to\left|\Psi_{\textrm{collapsed}}^{\left(p_{1}p_{2}p_{3}\right)}\left(t_{3}^{\textrm{out}}\right)\right\rangle_{\textrm{rel}},\\\
\textrm{etc}.$ (23)

This procedure results in a novel Monte Carlo simulation technique of real-
time motion. It has a benefit with respect to the density matrix formulation
(18)-(19) since the density matrix has a quadratically higher dimension than
that of wavefunction,
$\dim\widehat{\rho}_{\textrm{rel}}\left(t\right)=\left(\dim\left|\Psi_{\textrm{collapsed}}^{\left(h\right)}\left(t\right)\right\rangle_{\textrm{rel}}\right)^{2}$.

Actually the Fig. 5 was computed with this technique. The averaging was done
over $36000$ randomly generated jump histories. In Fig. 7

Figure 7: An example of a single jump history: the qubit observable (OQS)
experiences discontinuities due to quantum jumps. The latter happen at time
moments $t_{k}^{\textrm{out}}$ when $m_{\textrm{out}}\left(t\right)$ increases
by 1. The model is eq. (10). The Fock space of relevant modes was truncated at
a maximal occupation of 6 quanta.

we present an example of single random history of jumps. It is seen that the
qubit observable behaves discontinuously at the times of jumps (when
$m_{\textrm{out}}\left(t\right)$ increases by 1).

This scheme is also of conceptual interest since it represents the environment
as a measurement apparatus which autonomously selects the time moment of
measurement and the preferred basis, without the intervention of the human
experimenter. The classical reality (a statistical ensemble of jump histories)
is encoded in the emerging invariant entanglement structure.

###  IV.6 Bounded population of relevant modes

The outgoing modes $\kappa_{k}^{\textrm{out}}$ carry away quantum excitations.
Therefore we have two fluxes of quanta: the incoming flux $j^{\textrm{in}}$ of
quanta which are emitted by OQS into the relevant $\kappa_{l}^{\textrm{rel}}$
modes, and the outgoing flux $j^{\textrm{out}}$ quanta which are irreversibly
carried away by $\kappa_{k}^{\textrm{out}}$. On a physical grounds it is
natural to conjecture a balance between these two fluxes. Our calculation
supports this conjecture, Fig. 8.

Figure 8: For the model of a qubit coupled to a waveguide eq. (10). Here we
present a single random history of quantum jumps. Upper plot: total occupation
of relevant modes $\kappa_{k}^{\textrm{rel}}$. Lower plot: total occupation of
outgoing modes $\kappa_{k}^{\textrm{out}}$. We see that the occupation of
relevant modes is bounded. Since the number of relevant modes is bounded, and
their population is bounded, then the Fock space of relevant modes can be
truncated by keeping only a finite number $d$ of basis states. As a result,
the quantum evolution becomes a matrix product state whose bond dimension $d$
is bounded with time.

###  IV.7 Geometric arrow of time

Suppose we know that the environment is in some global equilibrium state
$\textrm{$\widehat{\rho}$}_{\textrm{b}}^{\textrm{eq}}$, and that OQS was
coupled to the environment at some time moment in the past. Then our streams
of incoming and outgoing Kotelnikov modes define an arrow of time. Indeed, at
any time moment $t$ the future incoming modes
$\kappa_{k}^{\textrm{in}},$$t_{k}^{\textrm{in}}>t$, did not have an
opportunity to significantly interact with OQS. Therefore, the partial density
matrix of the incoming stream should coincide with the equilibrium one at all
times:

$\textrm{$\widehat{\rho}$}_{\textrm{in}}\left(t\right)\equiv\textrm{Tr}_{\textrm{all}\kappa_{k}^{\textrm{in}}:\,t_{k}^{\textrm{in}}\leq
t}\textrm{Tr}_{\textrm{s}}\left\\{\textrm{$\widehat{\rho}$}\left(t\right)\right\\}\\\
=\textrm{Tr}_{\textrm{all}\kappa_{k}^{\textrm{in}}:\,t_{k}^{\textrm{in}}\leq
t}\left\\{\textrm{$\widehat{\rho}$}_{\textrm{b}}^{\textrm{eq}}\right\\}\equiv\textrm{$\widehat{\rho}$}_{\textrm{in}}^{\textrm{eq}},$
(24)

where $\textrm{$\widehat{\rho}$}\left(t\right)$ is the joint density of OQS
and environment due to the coupling quench. At the same time, the outgoing
stream carries away the disturbance due to the coupling quench, therefore its
partical density matrix in general differs from the equilibrium one:

$\textrm{$\widehat{\rho}$}_{\textrm{out}}\left(t\right)\equiv\textrm{Tr}_{\textrm{all}\kappa_{k}^{\textrm{out}}:\,t_{k}^{\textrm{out}}\geq
t}\textrm{Tr}_{\textrm{s}}\left\\{\textrm{$\widehat{\rho}$}\left(t\right)\right\\}\\\
\neq\textrm{Tr}_{\textrm{all}\kappa_{k}^{\textrm{out}}:\,t_{k}^{\textrm{out}}\geq
t}\left\\{\textrm{$\widehat{\rho}$}_{\textrm{b}}^{\textrm{eq}}\right\\}\equiv\textrm{$\widehat{\rho}$}_{\textrm{out}}^{\textrm{eq}}.$
(25)

We show in section …. that under the time reversal
$\textrm{$\widehat{\rho}$}_{\textrm{in}}\left(t\right)$ and
$\textrm{$\widehat{\rho}$}_{\textrm{out}}\left(t\right)$ are swapped. However,
in the latter case
$\textrm{$\widehat{\rho}$}_{\textrm{in}}\left(t\right)\neq\textrm{$\widehat{\rho}$}_{\textrm{in}}^{\textrm{eq}}$,
so that we detect the “wrong” time direction.

We call this arrow of time geometric because it is revealed by a specific
arrangement of the microscopic degrees of freedom. By itself, this arrangement
is time-symmetric ($\kappa_{k}^{\textrm{out}}$ and $\kappa_{k}^{\textrm{in}}$
swap under the time reversal). However the boundary condition (global
equilibrium of uncoupled environment at $t=-\infty$) is asymmetric, which
leads to the emergence of the preferred time direction.

###  IV.8 Entropy of the ensemble of classical records

The frame of outgoing modes $\kappa_{k}^{\textrm{out}}$ reveals the emerging
asymptotically invariant entanglement structure:

$\left|\Psi\left(t\right)\right\rangle=\sum_{p_{1}\ldots
p_{k}}c_{p_{1}}\left(1\right)\ldots
c_{p_{k}}\left(k\right)\left|\Psi_{\textrm{collapsed}}^{\left(p_{1}\ldots
p_{k}\right)}\left(t\right)\right\rangle_{\textrm{rel}}\\\
\otimes\left|\textrm{jump}^{\left(p_{1}\right)}\left(t_{1}^{\textrm{out}}\right)\right\rangle_{\kappa_{1}^{\textrm{out}}}\ldots\otimes\left|\textrm{jump}^{\left(p_{k}\right)}\left(t_{k}^{\textrm{out}}\right)\right\rangle_{\kappa_{k}^{\textrm{out}}}$
(26)

for $t_{k}^{\textrm{out}}\leq t\leq t_{k+1}^{\textrm{out}}$. As we have
discussed, this structure encodes the classical ensemble of jump histories
$\left(p_{1}\ldots p_{k}\right)$ with probabilities

$P\left(p_{1}p_{2}\ldots p_{k}\right)=\prod_{k:\,t_{k}^{\textrm{out}}\leq
t}\left|c_{p_{k}}\left(k\right)\right|^{2}.$ (27)

The entropy of such ensemble is

$S_{\textrm{jump}}\left(\Psi\left(t\right)\right)=-\sum_{p_{1}\ldots
p_{k}}P\left(p_{1}\ldots p_{k}\right)\ln P\left(p_{1}\ldots p_{k}\right).$
(28)

Informally we can say that for a given pure state
$\left|\Psi\left(t\right)\right\rangle$, the entropy
$S_{\textrm{jump}}\left(\Psi\left(t\right)\right)$ quantifies how much
classical reality was generated in this pure state due to non-Markovian
decoherence.

##  V Variational principle and numerical scheme

In this section we provide a recipe how to compute the constants
$\chi_{l}^{\textrm{rel}}\left(t\right)$ and $D_{lk}\left(t\right)$ of the
model eq. (15)-(16). First we introduce a variational principle for the real
time motion which yields the computational scheme which is presented at the
end of this section.

We begin by noting that the classical Kotelnikov theorem eq. (5) can be
reformulated in a statistical form: on every finite time interval $T$ there is
a finite number $m\left(t\right)\propto BT$ of independent statistically
significant DoFs for a signal with effective bandwidth $B$. See appendix A.
Also, this statistical reformulation yields the desired causal variant of the
Kotelnikov theorem.

###  V.1 Measure of retarded statistical significance

Returning to the coupling quench of OQS, we face the problem how to identify
the statistically significant DoFs of environment on a time interval
$\left[0,t\right]$. We need to identify these DoFs before sovling the many
body problem eq.(1). The simplest trick we can do is to replace OQS in eq. (1)
with a classical white noise $\xi\left(t\right)$:

$\widehat{H}\to\widehat{H}_{\xi+}\left(t\right)=\xi\left(t\right)^{*}\widehat{a}_{0}\left(t\right)+\xi\left(t\right)\widehat{a}_{0}^{\dagger}\left(t\right)$
(29)

(here we consider a specific case of bosonic environment),
$\overline{\xi^{*}\left(t\right)\xi\left(t^{\prime}\right)}=\delta\left(t-t^{\prime}\right)$.
Since the white noise $\xi\left(t\right)$ has infinite bandwidth and uniform
spectral density, it efficiently lights up all the available DoFs of the
environment. The intensity of such interaction is modulated with
$J\left(\omega\right)$. Then, given a trial DoF $\phi$, with the corresponding
creation operator $\widehat{\phi}^{\dagger}=\sum_{i=1}^{\infty}\left\langle
i\left|\phi\right.\right\rangle\widehat{a}_{i}^{\dagger}$, we can estimate its
statistical significance in the following way. We compute its average
occupation $n_{+}\left(t;\phi\right)$ by averaging over the ensemble of noisy
evolutions of the environment
$\left|\Psi_{\xi+}\left(t\right)\right\rangle_{\textrm{b}}$ under the
Hamiltonian eq. (29),

$n_{+}\left(t;\phi\right)=\left\langle\phi\right|\widehat{\rho}_{+}\left(t\right)\left|\phi\right\rangle,$
(30)

where the one-body reduced density matrix $\widehat{\rho}_{+}\left(t\right)$
is $\left\langle
i\right|\widehat{\rho}_{+}\left(t\right)\left|j\right\rangle=\overline{\left\langle\Psi_{\xi+}\left(t\right)\right|\widehat{a}_{i}^{\dagger}\widehat{a}_{j}\left|\Psi_{\xi+}\left(t\right)\right\rangle}$.
Here for each noise realization we solve the Schrodinger equation
$i\partial_{t}\left|\Psi_{\xi+}\left(t\right)\right\rangle_{\textrm{b}}=\widehat{H}_{\xi+}\left(t\right)$
with the vacuum initial condition
$\left|\Psi_{\xi+}\left(0\right)\right\rangle_{\textrm{b}}=\left|0\right\rangle_{\textrm{b}}$.
If $n_{+}\left(t;\phi\right)$ is negligible, then no other open system can
excite this DoF, and $\phi$ can be safely discarded. Indeed, unlike white
noise, OQS interacts only with those DoFs which are close to its transition
frequencies. Therefore, OQS is a less efficient illuminator than the white
noise.

We call $n_{+}\left(t;\phi\right)$ the measure of retarded statistical
significance, i.e. the statistical significance of $\phi$ for the past quantum
motion up to time $t$. The modes $\phi$ with negligible
$n_{+}\left(t;\phi\right)$ can be discarded from the Hamiltonian
$\widehat{H}_{\textrm{b}}$ when solving the Schrodinger equation on
$\left[0,t\right]$.

In appendix B we show that $\widehat{\rho}_{+}\left(t\right)$ has a simple
closed solution

$\left\langle
i\right|\widehat{\rho}_{+}\left(t\right)\left|j\right\rangle=\intop_{0}^{t}d\tau\phi_{i}^{*}\left(\tau\right)\phi_{j}\left(\tau\right),$
(31)

where $\phi_{j}\left(\tau\right)$ is a free one-particle wavepacket in the
environment, which is initially located on the site $0$ of the environment
(which is coupled to OQS), $\phi_{j}\left(0\right)=\delta_{j0}$, and at later
times its free motion under $\widehat{H}_{\textrm{b}}$ is given by a first-
quantized Schrodinger equation

$\partial_{\tau}\phi_{j}\left(\tau\right)=i\varepsilon_{j}\phi_{j}\left(\tau\right)+ih_{j}\phi_{j+1}\left(\tau\right)+ih_{j-1}\phi_{j-1}\left(\tau\right).$
(32)

These equations can be easily solved on a computer .

The measure $n_{+}\left(t;\phi\right)$ is a monotonically increasing function
of $t$, which follows from eq. (31). Therefore, given a significance treshold
$r_{\textrm{cut}}$, there is a unique time moment
$t_{\phi}^{\textrm{in}}\in\left[0,\infty\right]$ such that before
$t_{\phi}^{\textrm{in}}$ the mode $\phi$ is effectively decoupled,

$n_{+}\left(t;\phi\right)<r_{\textrm{cut}}\,\,\textrm{for}\,\,t<t_{\phi}^{\textrm{in}},$
(33)

and becomes coupled after this time moment,

$n_{+}\left(t;\phi\right)>r_{\textrm{cut}}\,\,\textrm{for}\,\,t>t_{\phi}^{\textrm{in}}.$
(34)

If the mode is always significant, we formally put $t_{\phi}^{\textrm{in}}=0$.
If the mode is never significant, we formally put
$t_{\phi}^{\textrm{in}}=+\infty.$

###  V.2 Measure of advanced statistical significance

Let us assume that we consider the quantum motion on a maximal time interval
$\left[0,T\right]$. Analogously to the previous section, we can introduce the
measure of statistical significance for the future quantum motion
$n_{-}\left(t;\phi\right)$ on the time interval $\left[t,T\right]$. We call
$n_{-}\left(t;\phi\right)$ the measure of advanced statistical significance.

In order to define $n_{-}\left(t;\phi\right)$, we again employ the white noise
trick. We consider the ensemble of noisy evolutions of the environment
$\left|\Psi_{\xi-}\left(\tau\right)\right\rangle_{\textrm{b}}$, but this time
the noise is switched on only after $\tau=t:$

$\widehat{H}_{\xi-}\left(\tau\right)=\theta\left(\tau-t\right)\left\\{\xi\left(\tau\right)^{*}\widehat{a}_{0}\left(t\right)+\xi\left(\tau\right)\widehat{a}_{0}^{\dagger}\left(t\right)\right\\}.$
(35)

We solve the Schrodinger equation
$i\partial_{\tau}\left|\Psi_{\xi-}\left(\tau\right)\right\rangle_{\textrm{b}}=\widehat{H}_{\xi-}\left(\tau\right)$
with the vacuum initial condition
$\left|\Psi_{\xi-}\left(0\right)\right\rangle_{\textrm{b}}=\left|0\right\rangle_{\textrm{b}}$.
Then, given a trial DoF $\phi$, with the corresponding creation operator
$\widehat{\phi}^{\dagger}=\sum_{i=1}^{\infty}\left\langle
i\left|\phi\right.\right\rangle\widehat{a}_{i}^{\dagger}$, we can estimate its
advanced statistical significance as average occupation

$n_{-}\left(t;\phi\right)=\left\langle\phi\right|\widehat{\rho}_{-}\left(t\right)\left|\phi\right\rangle,$
(36)

where the one-body reduced density matrix $\widehat{\rho}_{-}\left(t\right)$
is $\left\langle
i\right|\widehat{\rho}_{-}\left(t\right)\left|j\right\rangle=\overline{\left\langle\Psi_{\xi-}\left(t\right)\right|\widehat{a}_{i}^{\dagger}\widehat{a}_{j}\left|\Psi_{\xi-}\left(t\right)\right\rangle}$.
The modes $\phi$ with negligible $n_{-}\left(t;\phi\right)$ can be discarded
from the Hamiltonian $\widehat{H}_{\textrm{b}}$ when solving the Schrodinger
equation after the time $t$.

Analogously to appendix B, one can show that
$\widehat{\rho}_{-}\left(t\right)$ has a simple closed solution

$\left\langle
i\right|\widehat{\rho}_{-}\left(t\right)\left|j\right\rangle=\intop_{t}^{T}d\tau\phi_{i}^{*}\left(\tau\right)\phi_{j}\left(\tau\right),$
(37)

where $\phi_{j}\left(\tau\right)$ is the same as in eq. (32). This equation
can also be easily solved on a computer.

The measure $n_{-}\left(t;\phi\right)$ is a monotonically decreasing function
of $t$, which follows from eq. (37). Therefore, given a significance treshold
$r_{\textrm{cut}}$, there is a unique time moment
$t_{\phi}^{\textrm{out}}\in\left[0,\infty\right]$ such that after
$t_{\phi}^{\textrm{out}}$ the mode $\phi$ is irreversibly decoupled,

$n_{-}\left(t;\phi\right)<r_{\textrm{cut}}\,\,\textrm{for}\,\,t>t_{\phi}^{\textrm{out}},$
(38)

and is coupled before this time moment,

$n_{-}\left(t;\phi\right)>r_{\textrm{cut}}\,\,\textrm{for}\,\,t<t_{\phi}^{\textrm{out}}.$
(39)

If the mode is always significant, we formally put
$t_{\phi}^{\textrm{out}}=+\infty$. If the mode is never significant, we
formally put $t_{\phi}^{\textrm{out}}=0.$

###  V.3 Streams of incoming and outgoing Kotelnikov modes

Having defined meaningful, easily computable measures of significance, the
problem of finding the streams of incoming $\kappa_{l}^{\textrm{in}}$ and
outgoing $\kappa_{l}^{\textrm{out}}$ Kotelnikov modes becomes purely
technical.

####  V.3.1 Incoming Kotelnikov modes

Given a maximal time interval $\left[0,T\right]$, we define the stream of
incoming $\kappa_{l}^{\textrm{in}}$ modes, $l=1\ldots
m_{\textrm{in}}\left(T\right),$ as a sequence of independent DoFs such that
(i) it is the set of all independent DoFs which are significant at $t=T$. That
is, for any other trial independent DoF $\phi$ we will necessarily have
$n_{+}\left(T;\phi\right)<r_{\textrm{cut}}$. (ii) It is the slowest coupling
sequence. That is, we can always rotate the frame via some unitary matrix $U$
and arrange the set $\kappa_{l}^{\textrm{in}}$ as another sequence
$\kappa_{l}^{\textrm{in}\prime}=\sum_{k}U_{lk}\kappa_{k}^{\textrm{in}}$. Then
the slowest coupling means that the sequence $\kappa_{l}^{\textrm{in}}$
couples at a sequence of times $t_{k}^{\textrm{in}}$ which are the most
delayed: $t_{k}^{\textrm{in}}\geq t_{k}^{\textrm{in}\prime}$, where
$t_{k}^{\textrm{in}\prime}$ are the coupling time for any other arrangement
$\kappa_{l}^{\textrm{in}\prime}$. We want to have (i) in order to have a
faithful representation of the quantum motion. We want to have (ii) because we
want to slow down the growth of complexity as much as possible. It is evident
that the lowest bound for the growth rate is nothing but the Kotelnikov
sampling rate $\propto B$.

####  V.3.2 Outgoing Kotelnikov modes

Given a maximal time interval $\left[0,T\right]$, we define the stream of
outgoing $\kappa_{l}^{\textrm{out}}$ modes, $l=1\ldots
m_{\textrm{in}}\left(T\right),$ as a sequence of independent DoFs such that
(i) it is the set of all independent DoFs which are significant at $t=T$. That
is, for any other trial independent DoF $\phi$ we will necessarily have
$n_{+}\left(T;\phi\right)<r_{\textrm{cut}}$. (ii) It is the fastest decoupling
sequence. That is, we can always rotate the frame via some unitary matrix $U$
and arrange the set $\kappa_{l}^{\textrm{out}}$ as another sequence
$\kappa_{l}^{\textrm{out}\prime}=\sum_{k}U_{lk}\kappa_{k}^{\textrm{out}}$.
Then the fastest decoupling means that the sequence
$\kappa_{l}^{\textrm{out}}$ decouples at a sequence of times
$t_{k}^{\textrm{out}}$ which are the earlies times: $t_{k}^{\textrm{out}}\leq
t_{k}^{\textrm{out}\prime}$, where $t_{k}^{\textrm{out}\prime}$ are the
coupling time for any other arrangement $\kappa_{l}^{\textrm{out}\prime}$.
Again, we want to have (i) in order to have a faithful representation of the
quantum motion. We want to have (ii) because we want to speed up the decay of
complexity as much as possible. It is evident that the maximal decay rate is
determined by bandwidth, i.e. it is the Kotelnikov sampling rate $\propto B$.

####  V.3.3 Time reversal symmetry between the streams of incoming and
outgoing modes

These definitions are manifestly time reversal invariant. Let is define the
time reversal operator $\mathcal{T}$ on the time interval $\left[0,T\right]$:
$\mathcal{T}\left[\phi\left(t\right)\right]=\phi^{*}\left(T-t\right)$. Then
$\mathcal{T}\widehat{\rho}_{\pm}\left(t\right)\mathcal{T}=\widehat{\rho}_{\mp}^{*}\left(T-t\right)$,
and the measures of statistical significance swap:
$n_{\pm}\left(t;\phi\right)=n_{\mp}\left(T-t;\mathcal{T}\phi\right)$. The
slowest coupling becomes the fastest decoupling, and vice versa, so that the
incoming and the outgoing Kotelnikov modes also swap:
$\mathcal{T}\kappa_{l}^{\textrm{in}}=\kappa_{m_{\textrm{in}}\left(T\right)-l+1}^{\textrm{out}}$,
with
$t_{l}^{\textrm{in}}=T-t_{m_{\textrm{in}}\left(T\right)-l+1}^{\textrm{out}}$.
Observe that the initial condition for eq. (32) is not time reversal
invariant. But this does not matter: the time reversal results in a unitarily
displaced initial condition. That is, the frame of the whole environment is
just unitarily displaced, but the model constants
$\chi_{l}^{\textrm{rel}}\left(t\right)$ and $D_{lk}\left(t\right)$ in eq. (16)
are invariant under such a change, and
$\widehat{H}_{\textrm{eff}}\left(t\right)$ is the same.

####  V.3.4 Composition of incoming and outgoing streams

As we pointed out in sec. IV.2, we are only interested in the outgoing modes
which were previously coupled to OQS. Therefore, we impose an additional
constraint on the definition of outgoing modes: (iii)
$\kappa_{k}^{\textrm{out}}$ should be a linear combination of all
$\kappa_{l}^{\textrm{in}}$ with $t_{l}^{\textrm{in}}\leq
t_{k}^{\textrm{out}}$. This definition is not required for correctness of the
model. But it is required to achieve the balance of complexity: due to (iii)
the number of relevant modes saturates, Fig. 6. Observe that formally this
requirement breaks the strict time reversal symmetry between the incoming and
outgoing streams:
$\mathcal{T}\kappa_{l}^{\textrm{in}}\neq\kappa_{m_{\textrm{in}}\left(T\right)-l+1}^{\textrm{out}}.$
However this symmetry holds in a more general sense,
$\mathcal{T}\kappa_{l}^{\textrm{in}}=\kappa_{m_{\textrm{in}}\left(T\right)-l+1}^{\textrm{out}\prime}$
and
$\mathcal{T}\kappa_{l}^{\textrm{out}}=\kappa_{m_{\textrm{in}}\left(T\right)-l+1}^{\textrm{in}\prime}$,
where $\kappa_{k}^{\textrm{in$\prime$}}$ and $\kappa_{k}^{\textrm{out}\prime}$
are some other incoiming and outgoing streams which satisfy the requirement
(i) and suboptimally satisfy theirs corresponding requirements (ii).

###  V.4 Numerical scheme

####  V.4.1 The preparatory stage.

Given $J\left(\omega\right)$ we construct the equivalent semiinfinite chain
representation eq. (2). Then the maximal time interval $\left[0,T\right]$ is
chosen. The one-particle Schrodinger equation eq. (32) is solved for
$\phi_{j}\left(\tau\right)$. Due to the Lieb-Robinson bounds the Schrodinger
equation can be truncated at a finite number of sites. Since this is a single
particle problem, we can keep a huge number of sites and this does not place
practical constraints in numerical computations. The density matrix
$\widehat{\rho}_{+}\left(T\right)$ is computed using eq. (31). The relative
singificance treshold is chosen e.g. $r_{\textrm{cut}}=10^{-4}$. Then the most
significant eigenvalues of $\widehat{\rho}_{+}\left(T\right)$ are found:

$\widehat{\rho}_{+}\left(T\right)\left|\kappa_{l}\right\rangle=\pi_{l}\left|\kappa_{l}\right\rangle,$
(40)

where $\left|\kappa_{l}\right\rangle$ are sorted in the descending order of
$\pi_{l}$ and we keep a finite number $m_{\textrm{in}}\left(T\right)$of modes
which are beyond the significance level, $\pi_{k}/\pi_{1}>r_{\textrm{cut}}$
for $k=1\ldots m_{\textrm{in}}\left(T\right)$. This way we satisfy the
requirement (i) for the incoming and outoing modes. The Fig. 3 presents an
example of such a calculation.

Usually the number $m_{\textrm{in}}\left(T\right)$ is much smaller then the
number of spatial sites we need to keep due to the Lieb-Robinson bounds.
Therefore, here and below we assume that we perform the computations in the
frame of modes
$\left|\kappa_{1}\right\rangle\ldots\left|\kappa_{m_{\textrm{in}}\left(T\right)}\right\rangle$.
Then $\widehat{\rho}_{+}\left(T\right)$ is a
$m_{\textrm{in}}\left(T\right)\times m_{\textrm{in}}\left(T\right)$ matrix.

####  V.4.2 The stream of incoming modes

As we have discussed, the slowest coupling is a time-reversed fastest
decoupling. Therefore, we propagate $\widehat{\rho}_{+}\left(T\right)$
backwards in time, from $t=T$ to $t=0$. Each moment of time we compute its
eigenvalues,

$\widehat{\rho}_{+}\left(t\right)\left|\kappa_{l}^{+}\left(t\right)\right\rangle=\pi_{l}^{+}\left(t\right)\left|\kappa_{l}^{+}\left(t\right)\right\rangle,$
(41)

where $\left|\kappa_{l}^{+}\right\rangle$ are sorted in the descending order
of $\pi_{l}^{+}$ . We remember the times $t_{k}^{\textrm{in}}$ when the lowest
eigenvalue $\pi_{k}^{+}\left(t_{k}^{\textrm{in}}\right)$ falls below the
significance treshold,
$\pi_{k}^{+}\left(t_{k}^{\textrm{in}}\right)/\pi_{1}^{+}\left(t_{1}^{\textrm{in}}\right)<r_{\textrm{cut}}$.
Each time the corresponding mode
$\kappa_{k}^{+}\left(t_{k}^{\textrm{in}}\right)$ is remembered as the incoming
mode,
$\kappa_{k}^{\textrm{in}}\equiv\kappa_{k}^{+}\left(t_{k}^{\textrm{in}}\right)$.
Immediately after that the mode $\kappa_{k}^{\textrm{in}}$ is excluded from
the backpropagation: the dimensions of $\widehat{\rho}_{+}\left(t\right)$
decrease by $1$ at each time moment $t_{k}^{\textrm{in}}$. Recurrently
repeating this step we find the incoming modes satisfying (i) and (ii)

####  V.4.3 The stream of outgoing modes

This time we propagate the density matrix $\widehat{\rho}_{-}\left(t\right)$
eq. (37) forward in time, from $t=0$ to $t=T$. Initially
$\widehat{\rho}_{-}\left(t\right)$ is a $m_{\textrm{in}}\left(t\right)\times
m_{\textrm{in}}\left(t\right)$ matrix since we project it to the subspace of
incoming modes
$\kappa_{1}^{\textrm{in}}\ldots\kappa_{m_{\textrm{in}}\left(t\right)}^{\textrm{in}}$
which have coupled before the time moment $t$. Each moment of time we compute
its eigenvalues,

$\widehat{\rho}_{-}\left(t\right)\left|\kappa_{l}^{-}\left(t\right)\right\rangle=\pi_{l}^{-}\left(t\right)\left|\kappa_{l}^{-}\left(t\right)\right\rangle,$
(42)

where $\left|\kappa_{l}^{-}\right\rangle$ are sorted in the descending order
of $\pi_{l}^{-}$ . We remember the times $t_{k}^{\textrm{out}}$ when the
lowest eigenvalue $\pi_{k}^{-}\left(t_{k}^{\textrm{out}}\right)$ falls below
the significance treshold,
$\pi_{k}^{-}\left(t_{k}^{\textrm{out}}\right)/\pi_{1}^{-}\left(t_{1}^{\textrm{out}}\right)<r_{\textrm{cut}}$.
Each time the corresponding mode
$\kappa_{k}^{-}\left(t_{k}^{\textrm{out}}\right)$ is remembered as the
outgoing mode,
$\kappa_{k}^{\textrm{out}}\equiv\kappa_{k}^{-}\left(t_{k}^{\textrm{out}}\right)$.
The diagonalizing unitary matrix $U_{k}$ is also remembered and applied to the
modes since it corresponds to the frame rotation eq. (14). Immediately after
that the mode $\kappa_{k}^{\textrm{out}}$ is excluded from the propagation:
the dimensions of $\widehat{\rho}_{-}\left(t\right)$ decrease by $1$ at each
time moment $t_{k}^{\textrm{out}}$. In othe words, we project
$\widehat{\rho}_{-}\left(t\right)$ to the subspace of relevant modes of
dimension $m_{\textrm{in}}\left(t\right)-m_{\textrm{out}}\left(t\right)$.
Recurrently repeating this step we find the outgoing modes satisfying (i) and
(ii).

####  V.4.4 The model parameters

The coupling coefficient $\chi_{l}^{\textrm{rel}}\left(t\right)$ is found as

$\chi_{l}^{\textrm{rel}}\left(t\right)=\left[\widehat{a}_{0}\left(t\right),\widehat{\kappa}_{l}^{\textrm{rel}\dagger}\right]_{\xi}\\\
=\left\langle\phi\left(t\right)\left|\kappa_{l}^{\textrm{rel}}\right.\right\rangle.$
(43)

The generator $D\left(t\right)$ is found as

$D\left(t\right)=i\ln U_{k}/\left(t_{k}^{\textrm{out}}-t_{k}^{*}\right),$ (44)

where $t_{k}^{*}$ is the time of the previous coupling/decoupling event.

Now we write the Schrodinger equation (15)-(16) in a Fock space which is
truncated in maximal occupation of relevant modes, and solve the interaction
quench problem.

##  VI CONCLUSION

In this work we propose the concept of bandlimited quantum noise. On a spin-
boson model we demonstrate that the bandlimited quantum noise is effectively a
discrete time process, which yields simple and intuitive equations of motion.
These equations of motion are ready to be applied to specific problems. A
numerical procedure to find the necessary coupling constants for a given
spectral density $J\left(\omega\right)$ is presented.

The merit of the proposed approach is that it exploits the balance of
complexity which is characteristic of the quantum dissipative motion [43]:
only a bounded number of degrees of freedom are coupled to OQS at any given
instant of time. Moreover, the population of these coupled degrees of freedom
is bounded. This yields a microscopically-derived matrix product state (MPS)
for the quantum dissipative evolution, with a bounded bond dimension.
Interesting enough, we do not use SVD anywhere. In this work we provide an
alternative way of how to obtain the MPS structures.

In this work we also propose a Monte Carlo simulation technique which samples
the pure-state evolution interrupted by random quantum jumps. This has a
benefit with respect to the density matrix methods due to a quadratically
smaller dimension of the state vector. Importantly, these quantum jumps are
irreversible and completed. There is no need to simulate the reversed quantum
jumps [46, 47].

The model of bandlimited quantum noise also yields some conceptual insights.
In particular, it shows that the classical records are stored in the
asymptotically invariant entanglement structures. Given a pure state, we can
compute the entropy of the ensemble of classical records which this pure state
contains. It also presents a novel discrete-time input-output formalism.

The concept of bandlimited quantum noise is expected to be applicable to any
kind of open system and environment. The work is in progress to cover more
situations.

###### Acknowledgements.

The work was carried out in the framework of the Roadmap for Quantum computing
in Russia

##  Appendix A CLASSICAL KOTELNIKOV SAMPLING: VARIATIONAL FORMULATION

Here we reformulate the classical Kotelnikov sampling theorem so that it could
be easily generalized to the case of quantum noise. We begin by noting that
the ensemble of random classical signal samples $z\left(t\right)$ on a finite
time interval $\left[0,T\right]$ is analogous to a many body state since it
generates an inifinite hierarchy of correlators

$C_{\textrm{cl}}\left(t_{1},t_{2},\ldots
t_{p}\left|t_{1}^{\prime},t_{2}^{\prime},\ldots
t_{q}^{\prime}\right.\right)\\\
=\overline{z^{*}\left(t_{1}\right)z^{*}\left(t_{2}\right)\ldots
z^{*}\left(t_{p}\right)z\left(t_{1}^{\prime}\right)z\left(t_{2}^{\prime}\right)\ldots
z\left(t_{q}^{\prime}\right)}.$ (45)

Then the time correlator
$C_{\textrm{cl}}\left(t\left|t^{\prime}\right.\right)=\overline{z^{*}\left(t\right)z\left(t^{\prime}\right)}$
can be interpreted as a one-body reduced density matrix
$\widehat{\rho}_{\textrm{cl}}$ with matrix elements

$\left\langle
t\left|\widehat{\rho}_{\textrm{cl}}\right|t^{\prime}\right\rangle=\overline{z^{*}\left(t\right)z\left(t^{\prime}\right)}.$
(46)

Suppose we are given some normalized temporal function $\phi\left(t\right)$
denoted as $\left|\phi\right\rangle$. We may consider it as a DoF and ask a
question: is it significant for $z\left(t\right)$ or we may discard it? In
other words, we seek for a minimal number $m$ of DoFs
$\left|\kappa_{k}\right\rangle$, $k=1\ldots m$, which play the most
significant role in the representation of the signal, so that it effectively
becomes $m$-dimensional,
$z\left(t\right)\approx\sum_{k=1}^{m}z_{m}\left\langle
t\left|\kappa_{k}\right.\right\rangle$. To answer this question it is enough
to consider the variance

$\mathcal{K}\left[\phi\right]=\left\langle\phi\left|\widehat{\rho}_{\textrm{cl}}\right|\phi\right\rangle.$
(47)

If $\overline{z\left(t\right)}=0$, then a negligible
$\mathcal{K}\left[\phi\right]$ indicates that the signal is effectively zero
along the direction of $\left|\phi\right\rangle$, and it is irrelevant for the
representation of $z\left(t\right)$. Then the most significant DoF
$\left|\kappa_{1}\right\rangle$ is found variationally

$\left|\kappa_{1}\right\rangle=\textrm{arg}\underset{\kappa_{1}}{\max}\frac{\left\langle\kappa_{1}\right|\widehat{\rho}_{\textrm{cl}}\left|\kappa_{1}\right\rangle}{\left\langle\kappa_{1}\left|\kappa_{1}\right.\right\rangle}.$
(48)

Next we find the DoF $\left|\kappa_{2}\right\rangle$ which is the most
singificant after $\left|\kappa_{1}\right\rangle$. It satisfies the same
variational principle under the constraint
$\left|\kappa_{1}\right\rangle\perp\left|\kappa_{2}\right\rangle$ since it
should be a new independent DoF. Repeating this recursively, we recongnize
this as the Rayleigh-Ritz formulation of the eigenvalue problem,

$\widehat{\rho}_{\textrm{cl}}\left|\kappa_{l}\right\rangle=\pi_{l}\left|\kappa_{l}\right\rangle,$
(49)

where the eigenvectors $\left|\kappa_{l}\right\rangle$ are sorted in the
descending order of $\pi_{l}$. By choosing some relative significance treshold
$r_{\textrm{cut}}$ (e.g. $r_{\textrm{cut}}=10^{-7}$), we keep only the first
$m$ modes which surpass this treshold, $\pi_{k}/\pi_{1}>r_{\textrm{cut}}$ for
$k=1\ldots m$. I

It can be shown by a numerical computation that indeed
$\left|\kappa_{l}\right\rangle$ span the same subspace as the sinc basis in
eq. (5), and $m=2BT$. We call the modes $\left|\kappa_{l}\right\rangle$
surpassing the significance treshold the Kotelnikov modes.

We conclude that the message of the Kotelnikov sampling theorem is that a
bandlimited signal has a finite number $m$ of statistically significant DoFs
on a finite interval $T$, and $m$ scales as $\propto BT$. These DoFs are found
as eigenvectors of the largest eigenvalues of the one body reduced density
matrix.

##  Appendix B SOLUTION FOR $\widehat{\rho}_{+}\left(t\right)$

We interpret the equation (29) as a stochastic equation in the Stratonovich
sense

$id\left|\Psi_{\xi+}\right\rangle=\left\\{dz^{*}\widehat{a}_{0}\left(t\right)+dz\widehat{a}_{0}^{\dagger}\left(t\right)\right\\}\left|\Psi_{\xi+}\right\rangle.$
(50)

where $dz$ is increment of complex white noise (Wiener increment). The
corresponding Ito form is

$d\left|\Psi_{\xi+}\right\rangle\\\
=\left\\{-idz^{*}\widehat{a}_{0}\left(t\right)-idz\widehat{a}_{0}^{\dagger}\left(t\right)-\left(\widehat{a}_{0}^{\dagger}\left(t\right)\widehat{a}_{0}\left(t\right)+\frac{1}{2}\right)dt\right\\}\left|\Psi_{\xi+}\right\rangle.$
(51)

The one-body density matrix is

$\rho_{+kl}\left(t\right)=\overline{\left\langle\Psi_{\xi+}\left(t\right)\left|\widehat{a}_{k}^{\dagger}\widehat{a}_{l}\right|\Psi_{\xi+}\left(t\right)\right\rangle},$
(52)

where $\overline{\left(\cdot\right)}$ is stochastic average (over noise $z$
realizations). Using the identity

$\widehat{a}_{0}^{\dagger}\left(t\right)=\sum_{j}\widehat{a}_{j}^{\dagger}\phi_{j}\left(t\right),$
(53)

where $\phi_{j}\left(t\right)$ satisfies eq. (32), we apply the Ito formula
for the increment $d\rho_{+kl}\left(t\right)$. After some algebra we obtain:

$d\rho_{+kl}\left(t\right)=dt\phi_{k}^{*}\left(t\right)\phi_{l}\left(t\right).$
(54)

Therefore, we get the equation (31).

## References

  * Dowling and Milburn [2003] J. P. Dowling and G. J. Milburn, Quantum technology: the second quantum revolution, Phil. Trans. R. Soc. Lond. A 361, 1655 (2003).
  * Bruzewicz  _et al._ [2019] C. D. Bruzewicz, J. Chiaverini, R. McConnell, and J. M. Sage, Trapped-ion quantum computing: Progress and challenges, Appl. Phys. Rev. 6, 021314 (2019).
  * Scarlino  _et al._ [2019] P. Scarlino, D. J. v. Woerkom, A. Stockklauser, J. V. Koski, M. C. Collodo, S. Gasparinetti, C. Reichl, W. Wegscheider, T. Ihn, K. Ensslin, and A. Wallraff, All-microwave control and dispersive readout of gate-defined quantum dot qubits in circuit quantum electrodynamics, Phys. Rev. Lett. 122, 206802 (2019).
  * Kjaergaard  _et al._ [2020] M. Kjaergaard, M. E. Schwartz, J. Braumuller, P. Krantz, J. I.-J. Wang, S. Gustavsson, and W. D. Oliver, Superconducting qubits: Current state of play, Annual Review of Condensed Matter Physics 11, 369 (2020).
  * Childress and Hanson [2013] L. I. Childress and R. Hanson, Diamond nv centers for quantum computing and quantum networks, Mrs Bulletin 38, 134 (2013).
  * Smith  _et al._ [2020] T. B. Smith, M. C. Cassidy, D. J. Reilly, S. D. Bartlett, and A. L. Grimsmo, Dispersive readout of majorana qubits, PRX Quantum 1, 020313 (2020).
  * Breuer and Petruccione [2007] H.-P. Breuer and F. Petruccione,  _The Theory of Open Quantum Systems_ (Oxford University Press, Oxford, 2007).
  * Blais  _et al._ [2004] A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation, Phys. Rev. A 69, 062320 (2004).
  * Krantz  _et al._ [2019] P. Krantz, M. Kjaergaard, F. Yan, T. P. Orlando, S. Gustavsson, and W. D. Oliver, A quantum engineer’s guide to superconducting qubits, Appl. Phys. Rev. 6, 021318 (2019).
  * Hewson [1997] A. C. Hewson,  _The Kodno Problem to Heavy Fermions_ (Cambridge University Press, Cambridge UK, 1997).
  * Pustilnik and Glazman [2004] M. Pustilnik and L. Glazman, Kondo effect in quantum dots, J. Phys.: Condens. Matter 16, R513 (2004).
  * Thoenniss  _et al._ [2022] J. Thoenniss, A. Lerose, and D. A. Abanin, Non-equilibrium quantum impurity problems via matrix-product states in the temporal domain (2022), arXiv:2205.04995 [cond-mat] .
  * Sonner  _et al._ [2021] M. Sonner, A. Lerose, and D. A. Abanin, Influence functional of many-body systems: Temporal entanglement and matrix-product state representation, Ann. Phys. 431, 168552 (2021).
  * Lerose  _et al._ [2021a] A. Lerose, M. Sonner, and D. A. Abanin, Influence matrix approach to many-body floquet dynamics, Phys. Rev. X 11, 021040 (2021a).
  * Campisi and Hanggi [2011] M. Campisi and P. Hanggi, Fluctuation, dissipation and the arrow of time, Entropy 13, 2024 (2011).
  * Andrieux and Gaspard [2008] D. Andrieux and P. Gaspard, Quantum work relations and responce theory, Phys. Rev. Lett. 100, 230404 (2008).
  * Talkner and Hanggi [2020] P. Talkner and P. Hanggi, Colloquium: Statistical mechanics and thermodynamics at strong coupling: Quantum and classical, Rev. Mod. Phys. 92, 041002 (2020).
  * Batalhao  _et al._ [2015] T. B. Batalhao, A. M. Souza, R. S. Sarthour, I. S. Oliveira, M. Paternosto, E. Lutz, and R. M. Serra, Irreversibility and the arrow of time in a quenched quantum system, Phys. Rev. Lett. 115, 190601 (2015).
  * Batalhao  _et al._ [2018] T. B. Batalhao, S. Gherardini, J. P. Santos, G. T. Landi, and M. Paternostro, Characterizing irreversibility in open quantum systems, in  _Thermodynamics in the Quantum Regime: Fundamental Aspects and New Directions_, edited by F. Binder, L. A. Correa, C. Gogolin, J. Anders, and G. Adesso (Springer International Publishing, Cham, 2018) pp. 395–410.
  * Manzano  _et al._ [2019] G. Manzano, R. Fazio, and E. Roldan, Quantum martingale theory and entropy production, Phys. Rev. Lett. 122, 220602 (2019).
  * Elouard and Mohammady [2018] C. Elouard and M. H. Mohammady, Work, heat and entropy production along quantum trajectories, in  _Thermodynamics in the Quantum Regime: Fundamental Aspects and New Directions_, edited by F. Binder, L. A. Correa, C. Gogolin, J. Anders, and G. Adesso (Springer International Publishing, Cham, 2018) pp. 363–393.
  * Deffner and Lutz [2011] S. Deffner and E. Lutz, Nonequilibrium entropy production for open quantum systems, Phys. Rev. Lett. 107, 140404 (2011).
  * Esposito  _et al._ [2010] M. Esposito, K. Lindenberg, and C. V. d. Broeck, Entropy production as correlation between system and reservoir, New J. Phys. 12, 013013 (2010).
  * Breuer [2003] H.-P. Breuer, Quantum jumps and entropy production, Phys. Rev. A 68, 032105 (2003).
  * Vu and Saito [2022] T. V. Vu and K. Saito, Thermodynamics of precision in markovian open quantum dynamics, Phys. Rev. Lett. 128, 140602 (2022).
  * Hasegawa [2021] Y. Hasegawa, Irreversibility, loschmidt echo, and thermodynamics uncertainty, Phys. Rev. Lett. 127, 240602 (2021).
  * Horowitz and Parrondo [2013] J. M. Horowitz and J. M. R. Parrondo, Entropy production along nonequilibrium quantum jump trajectories, New. J. Phys. 15, 085028 (2013).
  * Menczel  _et al._ [2020] P. Menczel, C. Flindt, and K. Brandner, Quantum jump approach to microscopic heat engines, Phys. Rev. Res. 2, 033449 (2020).
  * Popovic  _et al._ [2021] M. Popovic, M. T. Mitchison, A. Strathearn, B. W. Lovett, J. Goold, and P. R. Eastham, Quantum heat statistics with time-evolving matrix product operators, PRX Quantum 2, 020338 (2021).
  * Peterson  _et al._ [2019] J. P. S. Peterson, T. B. Batalhao, M. Herrera, and A. M. Souza, Experimental characterization of a spin quantum heat engine, Phys. Rev. Lett. 123, 240601 (2019).
  * Jacobs [2014] K. Jacobs,  _Quantum Measurement Theory and its Applications_ (Cambridge University Press, 2014).
  * Glazman and Catelani [2021] L. I. Glazman and G. Catelani, Bogoliubov quasiparticles in superconducting qubits, SciPost Phys. Lect. Notes , 31 (2021).
  * Campisi  _et al._ [2015] M. Campisi, J. Pekola, and R. Fazio, Nonequilibrium fluctuations in quantum heat engines: theory, example, and possible solid state experiments, New J. Phys. 17, 035012 (2015).
  * Rivas [2020] A. Rivas, Strong coupling thermodynamics of open quantum systems, Phys. Rev. Lett. 124, 160601 (2020).
  * Plenio and Knight [1998] M. B. Plenio and P. L. Knight, The quantum-jump approach to dissipative dynamics in quantum optics, Rev. Mod. Phys. 70, 101 (1998).
  * Gardiner and Zoller [2004] C. W. Gardiner and P. Zoller,  _Quantum Noise_ , 3rd ed. (Springer-Verlag Berlin Heidelberg, 2004).
  * Alkauskas  _et al._ [2014] A. Alkauskas, B. B. Buckley, D. D. Awschalom, and C. G. V. d. Walle, First-principles theory of the luminescence lineshape for the triplet transition in diamond nv centers, New J. Phys. 16, 073026 (2014).
  * Alkauskas  _et al._ [2016] A. Alkauskas, M. McCluskey, and C. G. V. d. Walle, Tutorial: Defects in semiconductors-combining experiment and theory, J. App. Phys. 119, 181101 (2016).
  * Yablonovich [1987] E. Yablonovich, Inhibited spontaneous emission in solid-state physics and electronics, Phys. Rev. Lett. 58, 2059 (1987).
  * Ferreira  _et al._ [2021] V. S. Ferreira, J. Banker, A. Sipaigil, M. H. Matheny, A. J. Keller, E. Kim, M. Mirhosseini, and O. Painter, Collapse and revival of an artifical atom coupled to a structured photonic reservoir, Phys. Rev. X 11, 041043 (2021).
  * Bylicka  _et al._ [2017] B. Bylicka, M. Johansson, and A. Acin, Constructive method for detecting the information backflow of non-markovian dynamics, Phys. Rev. Lett 118, 120501 (2017).
  * Reich  _et al._ [2015] D. M. Reich, N. Katz, and C. P. Koch, Exploiting non-markovianity for quantum control, Sci. Rep. 5, 12430 (2015).
  * Polyakov [2022] E. A. Polyakov, Real-time motion of open quantum systems: Structure of entanglement, renormalization group, and trajectories, Phys. Rev. B 105, 054306 (2022).
  * Polyakov and Rubtsov [2019] E. A. Polyakov and A. N. Rubtsov, Dressed quantum trajectories: novel approach to the non-markovian dynamics of open quantum systems on a wide time scale, New. J. Phys. 21, 063004 (2019).
  * Polyakov and Rubtsov [2018a] E. A. Polyakov and A. N. Rubtsov, Information loss pathways in a nuremically exact simulation of a non-markovian open quantum system (2018a), to be published.
  * Piilo  _et al._ [2008] J. Piilo, S. Maniscalco, K. Harkonen, and K.-A. Suominen, Non-markovian quantum jumps, Phys. Rev. Lett. 100, 180402 (2008).
  * Piilo  _et al._ [2009] J. Piilo, K. Harkonen, S. Maniscalco, and K.-A. Suominen, Open system dynamics with non-markovian quantum jumps, Phys. Rev. A 79, 062112 (2009).
  * Lerose  _et al._ [2021b] A. Lerose, M. Sonner, and D. A. Abanin, Scaling of temporal entanglement in proximity to integrability, Phys. Rev. B 104, 035137 (2021b).
  * Frias-Perez and Banuls [2022] M. Frias-Perez and M. C. Banuls, Light cone tensor network and time evolution (2022), arXiv:2201.08402 [quant-ph] .
  * Lerose  _et al._ [2022] A. Lerose, M. Sonner, and D. A. Abanin, Overcoming the entanglement barrier in quantum many-body dynamics via space-time duality (2022), arXiv:2201.04150 [quant-ph] .
  * Feynman and Vernon [1963] R. Feynman and F. Vernon, The theory of a general quantum system interacting with a linear dissipative system, Annals of Physics 24, 118 (1963).
  * Strathearn  _et al._ [2018] A. Strathearn, P. Kirton, D. Kilda, J. Keeling, and B. W. Lovett, Efficient non-markovian quantum dynamics using time-evolving matrix product operators, Nat. Commun. 9, 3322 (2018).
  * Luchnikov  _et al._ [2019] I. A. Luchnikov, S. V. Vintskevich, H. Querdane, and S. N. Filippov, Simulation complexity of open quantum dynamics: Connection with tensor networks, Phys. Rev. Lett. 122, 160401 (2019).
  * Somoza  _et al._ [2019] A. D. Somoza, O. Marty, J. Lin, S. F. Huelga, and M. B. Plenio, Dissipation-assisted matrix product factorization, Phys. Rev. Lett. 123, 100502 (2019).
  * Jorgensen and Pollock [2019] M. R. Jorgensen and F. A. Pollock, Exploiting the causal tensor network structure of quantum processes to efficiently simulate non-markovian path integrals, Phys. Rev. Lett. 123, 240602 (2019).
  * Polyakov [2020] E. A. Polyakov, Structure of entanglement and renormalization in non-equilibrium open quantum systems (2020), arXiv:2012.04065v1 [quant-ph] .
  * Kotelnikov [1933] V. A. Kotelnikov, On the capacity of the ’ether’ and of the cables in electrical communication, in  _Procs. of the first All-Union Conference on the technological reconstruction of thecommunications sector and low-current engineering_ (Moscow, 1933).
  * Polyakov and Rubtsov [2018b] E. A. Polyakov and A. N. Rubtsov, Stochastic wave-function simulation of irreversible emission processes for open quantum systems in a non-markovian environment, AIP Conf. Proc. 1936, 020028 (2018b).
  * Neumann [1996] J. Neumann,  _Mathematische Grundlagen der Quantenmechanik_, 2nd ed. (Springer-Verlag, 1996).