# Canonical typicality under general quantum channels

Pedro Silva Correia  Departamento de Ciências Exatas, Universidade Estadual de
Santa Cruz, Ilhéus, Bahia 45662-900, Brazil<EMAIL_ADDRESS>Gabriel Dias
Carvalho  Escola Politécnica de Pernambuco, Universidade de Pernambuco,
50720-001, Recife, PE, Brazil  Instituto de Física, Universidade Federal
Fluminense, Av. Litoranea s/n, Gragoatá 24210-346, Niterói, RJ, Brazil  Thiago
R. de Oliveira  Instituto de Física, Universidade Federal Fluminense, Av.
Litoranea s/n, Gragoatá 24210-346, Niterói, RJ, Brazil  Raúl O. Vallejos
Fernando de Melo  Centro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier
Sigaud, 150, Rio de Janeiro, RJ, Brazil

###### Abstract

With the control of ever more complex quantum systems becoming a reality, new
scenarios are emerging where generalizations of the most foundational aspects
of statistical quantum mechanics are imperative. In such experimental
scenarios the often natural correspondence between the particles that compose
the system and the relevant degrees-of-freedom might not be observed. In the
present work we employ quantum channels to define generalized subsystems,
which should capture the pertinent degrees-of-freedom, and obtain their
associated canonical state. Moreover, we show that generalized subsystems also
display the phenomena of canonical typicality, i.e., the generalized subsystem
description generated from almost any microscopic pure state of the whole
system will behave similarly as the corresponding canonical state. In
particular we demonstrate that the property regulating the emergence of the
canonical typicality behavior is the entropy of the channel used to define the
generalized subsystem.

Introduction. Ensembles form an integral part of statistical mechanics, being
key ingredients in pillar fields as thermodynamics and statistical mechanics
(both classical and quantum). An ensemble formalizes the idea that for complex
systems one cannot control all its degrees-of-freedom as to prepare it in a
well-defined state. In each run of the experiment only few “macroscopic”
quantities are fixed, and thus any of the “microscopic” states that are
consistent with the macroscopic quantities can be prepared. The collection of
such compatible microscopic states forms the statistical ensemble of a given
experimental scenario.

Two of the most prevailing ensembles in physics are the microcanonical and the
canonical ensembles. For the first we imagine a scenario where the system is
formed by a large number of degrees-of-freedom which are completely isolated
from the rest of the universe, with some macroscopic properties having well-
defined values. For the second, the canonical ensemble can be seen as the
description of a subsystem, with fixed number of degrees-of-freedom, of the
microcanonical one. The canonical ensemble thus characterize the situation
where macroscopic properties are allowed to vary within a given subsystem,
while such properties are fixed when considering the whole system.

While the use of ensembles has been very successful, they introduce a
probabilistic character to the microscopic description, which can be argued
deterministic. Indeed, in a given run of an experiment one might expect that
all properties of the probed system are fixed, and not just the few controlled
macroscopic quantities. Within this reasoning, the microscopic state is well
defined, albeit unknown. To reconcile this deterministic perspective with the
overwhelming success of the probabilistic description is one of the main
discussions in the foundations of statistical mechanics since Boltzmann. The
most common justification, using the microscopic laws of mechanics, invokes
chaos theory and the ergodic hypothesis [1]. Others consider an information
theory approach, exploiting the principle of maximum entropy as championed by
Jaynes [2, 3].

Another possibility is the so-called typicality argument: almost all
microscopic states which are compatible with a given preparation procedure
will behave similarly to the ensemble description for any macroscopic property
[1]. Such a method had been already proposed by Boltzmann, and it was latter
extended to the quantum domain by von Neumann [4]. The latter has remained
forgotten until the recent reanalysis of the typicality argument from the
quantum information perspective [5, 6]. In this modern mindset, the canonical
typicality asserts that any pure state from the whole system, that abides by
the fixed macroscopic quantities defining a preparation scheme, will be
locally, i.e., for the subsystem, close to the canonical state. How close in
this perspective will depend on various ingredients and are somehow a
formalization of the “thermodynamical limit” (see details below).

Crucial in the above discussion is the concept of a subsystem with a fixed
number of degrees-of-freedom. Usually we equate this with the idea of a fixed
number of particles, which is a very natural attitude when dealing, for
instance, with a gas of weakly interacting atoms. However, there are
situations where this identification is not so clear or appropriate. Think for
instance of an optical lattice where individual atoms are loaded onto
potential wells [7, 8], but their reading is done in blocks that contains
multiple wells: In this case a coarse-grained description is more convenient
[9, 10]. Another example is that of a superconducting metal described by the
BCS theory, with Cooper pairs forming individual entities which are
delocalized in space: here the quasi-particle description is more relevant
[11]. In these cases, and many others, a more pertinent description is that of
a generalized subsystem [12]. It is not the presumed degrees-of-freedom or the
real space distribution of them which gives the more fitting account of a
subsystem.

In what follows, we employ the theory of quantum channels to define
generalized subsystems to and their canonical ensemble. Moreover, we show that
such general scenarios also admit a canonical typicality reasoning. In this
way we extend the typicality approach to the foundations of statistical
mechanics over situations where the description of a physical scenario in
terms of its underlying particles is not necessarily the most suitable one.
The “thermodynamics” of a given experimental scenario thus depends on what can
be measured, i.e., on what can be called an effective particle.

Canonical typicality. We start by briefly reviewing the main ingredients and
results of canonical typicality for the case in which the subsystem is just a
subset of the total system [5, 13]. Consider a total system to which we assign
a Hilbert space $\mathcal{H}_{T}$ – such a space can, for instance, be
constructed by the tensor product of $N$ individual particles’ spaces. To
define a subsystem we split the total space in two parts,
$\mathcal{H}_{T}=\mathcal{H}_{S}\otimes\mathcal{H}_{E}$; the first part,
$\mathcal{H}_{S}$, is associated with the subsystem, and $\mathcal{H}_{E}$ is
associated with the rest of the system and it is seen as an environment for
the subsystem. We then define a restricted subspace
$\mathcal{H}_{R}\subseteq\mathcal{H}_{S}\otimes\mathcal{H}_{E}$ in which all
the states abide by the macroscopic constraints.

In such a subspace the total system’s description which represents ignorance
of any other constraint than the already contemplated by the restriction, is
the microcanonical state $\mathcal{E}_{R}=\mathds{1}_{R}/d_{R}$, with
$\mathds{1}_{R}$ the projector into $\mathcal{H}_{R}$ and
$d_{R}=\dim(\mathcal{H}_{R})$. The system’s canonical state $\Omega_{{\rm
tr}_{E}}$ is then obtained by simply discarding the part of the total system
which is not related to the subsystem, i.e., by tracing out the environmental
degrees of freedom: $\Omega_{{\rm tr}_{E}}={\rm tr}_{E}(\mathcal{E}_{R})$.

The canonical typicality approach is based in showing that for almost all
states $\delimiter 69640972\psi\rangle\in\mathcal{H}_{R}$, taken uniformly at
random (from the unitarily invariant measure, the Haar measure [14]), the
local state of the subsystem, ${\varrho}_{{\rm tr}_{E}}^{\psi}={\rm
tr}_{E}(|\psi\rangle\\!\langle\psi|)$, is close to the canonical state
$\Omega_{{\rm tr}_{E}}$ when the effective dimension of the environment,
$d_{E}^{\text{eff}}=1/{\rm tr}(\Omega_{{\rm tr}_{S}}^{2})$ (with $\Omega_{{\rm
tr}_{S}}={\rm tr}_{S}(\mathcal{E}_{R})$), is much bigger than the subsystem
dimension $d_{S}=\dim(\mathcal{H}_{S})$. Concretely, it was shown that the
average distance is bounded as follows:

$\displaystyle\overline{\mathcal{D}({\varrho}_{{\rm
tr}_{E}}^{\psi},\Omega_{{\rm
tr}_{E}})}^{\psi}\leq\frac{1}{2}\sqrt{\frac{d_{S}}{d_{E}^{\text{eff}}}}.$ (1)

In the above, the average is taken over the uniform measure, and
$\mathcal{D}({\varrho},\sigma)=\delimiter 69645069{\varrho}-\sigma\delimiter
86422285_{1}/2$ with the trace-norm defined as $\delimiter 69645069A\delimiter
86422285_{1}={\rm tr}\sqrt{A^{\dagger}A}$. Moreover, employing the so-called
Levy’s lemma [15, 16] (see Appendix A), the authors showed that the
probability for ${\varrho}_{{\rm tr}_{E}}^{\psi}$ being further away from
$\Omega_{{\rm tr}_{E}}$ decays exponentially with $d_{R}$. The canonical state
is then recovered from any pure state in the restricted subspace when
$d_{R}\gg 1$ and $d_{S}\ll d_{E}^{\text{eff}}$, which are then the formal
requirements for equilibration.

The canonical typicality principle can thus be seen as a justification for the
use of ensembles in statistical mechanics: we do not need to assume ignorance
over the whole system to obtain a canonical state for a subsystem, since most
pure states of the whole system appear to be in equilibrium when we look
locally at a subsystem.

Generalized subsystems. In the above scenario, the subsystem is a simple
partition of the pre-established total Hilbert space. Within statistical
mechanics it is commonsensical to split the total system into a system of
interest, the intended subsystem, and its environment, for which no control is
assumed. Such a split is often possible due to the weak interaction between
the two parts. By partial tracing the environmental degrees-of-freedom, we
obtain the subsystem description.

However, to find a physically motivated partition some reshape the total
Hilbert space might be necessary. A text-book example of this is the Hydrogen
atom case. A initial description in terms of an electron and a proton is
rearranged into the degrees-of-freedom of center-of-mass and relative-
particle. In this new split, both effective particles are decoupled and we can
solve Schrödinger’s equation. The idea of quasiparticles, ubiquitous in
condensed matter physics, is a somewhat more recent example of this
reshuffling of Hilbert’ space in order to find meaningful effective
descriptions. Once the more compelling degrees-of-freedom are found, the
remaining ones can be seen, and treated, as an effective environment.

In the aforementioned cases, the Hilbert space rearrangement was done by a
unitary transformation (like Bogoliubov transformations [17]), and the focus
on a subsystem was done by tracing out the weakly interacting degrees-of-
freedom of the effective environment. In the quantum information lingo, this
definition of subsystem is specified by the quantum channel $\Lambda_{{\rm
tr}_{E}\circ U}:\mathcal{H}_{T}\mapsto\mathcal{H}_{S}$ which acts as
$\Lambda_{{\rm tr}_{E}}^{U}({\varrho})={\rm tr}_{E}(U{\varrho}U^{\dagger})$.

While very useful and successful, this is not the only way to obtain effective
subsystems of quantum systems. As established in [12], and further developed
in [9, 10, 18, 19, 20, 21, 22, 23], generalized subsystems can be achieved by
exploiting the most general form of a quantum channel
$\Lambda:\mathcal{L}(\mathcal{H}_{T})\mapsto\mathcal{L}(\mathcal{H}_{S})$
[24]:

$\Lambda({\varrho})={\rm tr}_{N}(V{\varrho}V^{\dagger}).$ (2)

In the above expression
$V:\mathcal{H}_{T}\mapsto\mathcal{H}_{S}\otimes\mathcal{H}_{N}$ is an
isometry. Like unitary transformations, isometries also preserve the scalar
product as $V^{\dagger}V=\mathds{1}_{T}$. However, as isometries might change
the space dimension, $VV^{\dagger}=\mathds{1}_{SN}$ is a projector onto
$\mathcal{H}_{S}\otimes\mathcal{H}_{N}$. It is this very change of dimensions,
empowered by the isometry, which can be exploited to obtain generalized
subsystems. The total space $\mathcal{H}_{T}$ is reshuffled to
$\mathcal{H}_{S}\otimes\mathcal{H}_{N}$, with the space $\mathcal{H}_{N}$
being associated with an effective (possibly mathematically abstract)
environment for the generalized subsystem acting on $\mathcal{H}_{S}$. Like
before, the now general effective environment is discarded by the partial
trace.

A minimal, though instructive, example of a generalized subsystem is as
follows [19]. Consider a tree-level atom, with ground state $\delimiter
69640972g\rangle$, and quasi-degenerated excited states $\delimiter
69640972e_{1}\rangle$ and $\delimiter 69640972e_{2}\rangle$. In an experiment
where no distinction is made between the excited levels, the atom can be
effectively described by a two-level atom. The generalized subsystem is then
defined by the map $\Lambda:\mathbb{C}_{3}\mapsto\mathbb{C}_{2}$ whose
diagonal entries are $\Lambda(|g\rangle\\!\langle g|)=|0\rangle\\!\langle 0|$
and $\Lambda(|e_{1}\rangle\\!\langle e_{1}|)=\Lambda(|e_{2}\rangle\\!\langle
e_{2}|)=|1\rangle\\!\langle 1|$, with $\delimiter 696409720\rangle$ and
$\delimiter 696409721\rangle$ respectively the ground and excited state of the
effective two-level atom. Clearly this case cannot be expressed by a unitary
transformation followed by a partial trace, as 3 cannot be factorized as 2
times another whole number. It also cannot be seeing as a projection onto a
2-dimensional subspace, for this would not preserve the norm. The full map and
isometry for this case is shown in Appendix B. It is important to notice that
$V:\mathbb{C}_{3}\mapsto\mathbb{C}_{2}\otimes\mathbb{C}_{2}$, that is, to
define this generalized subsystem it was necessary to introduce an artificial,
mathematical, environment with dimension 2. This shows the generality and
versatility of using quantum channels to define generalized subsystems.

Canonical typicality for generalized subsystems. Here we present the main
result of this work, namely that the phenomena of canonical typicality is also
achieved for generalized subsystems. In this way, we extend the typicality
approach to the foundations of statistical mechanics over a wide range of
physical scenarios.

Consider a quantum mechanical system to which we assign a total Hilbert space
$\mathcal{H}_{T}$, and suppose that such a system obeys some arbitrary
restriction $R$, and thus act on $\mathcal{H}_{R}\subseteq\mathcal{H}_{T}$. If
no further constraints are imposed, the description of the system is given by
its microcanonical state:

$\displaystyle\mathcal{E}_{R}=\dfrac{\mathds{1}_{R}}{d_{R}},$

where $\mathds{1}_{R}$ is the identity (projector) operator in
$\mathcal{L}(\mathcal{H}_{R})$, and $d_{R}$ is the dimension of
$\mathcal{H}_{R}$. Our generalized canonical state, i.e., the description of
the generalized subsystem, will then be given by the action of a general
quantum channel
$\Lambda:\mathcal{L}(\mathcal{H}_{R})\mapsto\mathcal{L}(\mathcal{H}_{S})$ on
the microcanonical state: $\Omega_{\Lambda}:=\Lambda(\mathcal{E}_{R})$.

For the generalized canonical typicality principle we are then interested in
showing that for almost any $\delimiter
69640972\psi\rangle\in\mathcal{H}_{R}$, taken uniformly at random, we will
have
${\varrho}_{\Lambda}^{\psi}:=\Lambda(|\psi\rangle\\!\langle\psi|)\sim\Omega_{\Lambda}$.
Following Ref.[5], this is done in two parts.

The first part is to bound the average distance between
${\varrho}_{\Lambda}^{\psi}$ and $\Omega_{\Lambda}$. In Appendix C we show
that

$\overline{\mathcal{D}({\varrho}_{\Lambda}^{\psi},\Omega_{\Lambda})}^{\psi}\leq\dfrac{1}{2}\sqrt{d_{S}{\rm
tr}({J_{\Lambda}}^{2})}=\dfrac{1}{2}\sqrt{d_{S}(1-S_{L}(\Lambda))}.$ (3)

Here
$J_{\Lambda}:=\Lambda\otimes\mathds{1}(|\phi^{+}\rangle\\!\langle\phi^{+}|)$
is the Choi matrix of the quantum channel $\Lambda$, where $\delimiter
69640972\phi^{+}\rangle=\sum_{i}\delimiter
69640972ii\rangle/\sqrt{d_{R}}\in\mathcal{H}_{R}\otimes\mathcal{H}_{R}$. The
Choi matrix holds all the information and properties of the channel –
$J_{\Lambda}$ is isomorphic to $\Lambda$. Using this relationship, in [25, 26]
${\rm tr}(J_{\Lambda}^{2})$ was defined as the channel purity, and
$S_{L}(\Lambda)=1-{\rm tr}(J_{\Lambda}^{2})$ as the channel linear entropy.

As expected, the bound in (3) is the least tight when the effective
environment’s dimension is one, $\dim{\mathcal{H}_{N}}=1$ in Eq.(2). In this
case no information about the microscopic description is discarded; the
subsystem is the full system. The most common example is that of a unitary
channel, as in a closed dynamics, and thus with zero entropy. The tightest
bound is achieved by the channel with maximum entropy, namely the full
depolarization channel $\Lambda_{D}^{1}$: every state
${\varrho}\in\mathcal{L}(\mathcal{H}_{R})$ is substituted by the maximally
mixed state $\mathds{1}/d_{S}\in\mathcal{L}(\mathcal{H}_{S})$. For such a
channel $S_{L}(\Lambda_{D}^{1})=1-1/d_{S}d_{R}$, leading to the upper bound
$\overline{\mathcal{D}({\varrho}_{\Lambda_{D}^{1}}^{\psi},\Omega_{\Lambda_{D}^{1}})}^{\psi}\leq\frac{1}{2}\sqrt{1/d_{R}}$
– the upper bound is the strongest when $d_{R}\gg 1$, i.e., when
$S_{L}(\Lambda_{D}^{1})$ approaches one. As expected, the bound (3) reduces to
(1) in the special case where $\Lambda={\rm tr}_{E}$. See Appendix D for a
detailed proof.

Generally speaking, the average distance between a state
${\varrho}_{\Lambda}^{\psi}=\Lambda(|\psi\rangle\\!\langle\psi|)$, for
$\delimiter 69640972\psi\rangle$ sampled uniformly from
$\mathbb{S}^{2d_{R}-1}$, to the canonical state
$\Omega_{\Lambda}=\Lambda(\mathds{1}_{R}/d_{R})$ will be smaller the more
information about $|\psi\rangle\\!\langle\psi|$ is thrown away by $\Lambda$,
i.e., the higher is the channel entropy.

Another way to understand the result in Eq. (3), is by explicitly using the
expression for $\Lambda$, Eq. (2):

$S_{L}(\Lambda)=1-{\rm tr}\left\\{\left[{\rm
tr}_{N}(V\otimes\mathds{1}|\phi_{+}\rangle\\!\langle\phi_{+}|V^{\dagger}\otimes\mathds{1})\right]^{2}\right\\};$

which can be interpreted in terms of the entanglement in
$V\otimes\mathds{1}\delimiter
69640972\phi_{+}\rangle\in(\mathcal{H}_{S}\otimes\mathcal{H}_{N})\otimes\mathcal{H}_{R}$
with respect to the partition that splits the effective environment against
the rest. If the effective environment is separable from the effective system
and the copy in $\mathcal{H}_{R}$, then no information is lost by tracing it
out. In the other direction, the bound in (3) is tighter the stronger is the
entanglement, generated by the isometry $V$, between the effective system and
its effective environment.

The second part of the canonical typicality result follows from the
application of Levy’s lemma to the distance between
$\Lambda(|\psi\rangle\\!\langle\psi|)$ and $\Omega_{\Lambda}$. As we show in
Appendix A, for any state uniformly sampled from
$\mathbb{S}^{2d_{\text{R}}-1}$ it follows that:

${\rm
Pr}_{\psi}\Big{[}\big{|}\mathcal{D}\left(\Lambda(|\psi\rangle\\!\langle\psi|),\Omega_{\Lambda}\right)-\overline{\mathcal{D}({\varrho}_{\Lambda}^{\chi},\Omega_{\Lambda})}^{\chi}\big{|}>\epsilon\Big{]}\leq
2\textrm{e}^{-C\frac{d_{\text{R}}\epsilon^{2}}{4\eta_{\Lambda}^{2}}}.$ (4)

In the above expression $C$ is a constant that can be taken equal to
$2/9\pi^{3}$, and $\eta_{\Lambda}$ is the channel Lipschitz constant.

From the first part of the result, Eq. (3), we have that if the channel’s
entropy is large, the average distance between
$\Lambda(|\psi\rangle\\!\langle\psi|)$ and $\Omega_{\Lambda}$ is small.
Putting this together with Eq. (4), the probability for
$\mathcal{D}\left(\Lambda(|\psi\rangle\\!\langle\psi|),\Omega_{\Lambda}\right)$
be further from its mean value by an amount $\epsilon$ decays exponentially
with $\epsilon^{2}$ and $d_{R}$. In this generalized scenario, the condition
for the canonical typicality is then $S_{L}(\Lambda)\approx 1$ and $d_{R}\gg
1$.

Application 1: Blurred and saturated detector. Consider an optical lattice
over which hundreds of cold atoms are loaded. Nowadays, such a system can be
very well isolated from external environments for the time of the experiment.
Despite of that, to completely describe the quantum state of the system may be
experimentally unfeasible, and theoretically intractable. It is thus highly
desirable to obtain effective descriptions which are more manageable, while
not loosing connection with the experimental scenario.

To be concrete, take the case where $N$ two-level atoms are loaded into an
optical lattice in a way that there is one atom per potential well, and no
interactions are allowed. The Hamiltonian describing this situation is then
$H=-\hbar\omega\sum_{i=1}^{N}Z_{i}/2$, with $Z_{i}$ the usual $z$ Pauli matrix
for the $i$-th atom. Suppose that due to a energy restriction, a fraction $p$
of the atoms are excited, $\delimiter 696409721\rangle$, and the rest are in
the ground state, $\delimiter 696409720\rangle$. Accordingly,
$\mathcal{H}_{R}\subseteq\mathcal{H}_{T}=(\mathbb{C}_{2})^{\otimes N}$ has
dimension $d_{R}={N\choose Np}$. The microcanonical state for this scenario is
simply:

$\mathcal{E}_{R}=\frac{1}{{N\choose Np}}\sum_{s:|s|=Np}|s\rangle\\!\langle
s|.$ (5)

In the above $s\in\\{0,1\\}^{N}$ are strings with $N$ bits, and $|s|$
represents the number of excited atoms, i.e. the number of 1’s in $s$.

In this type of experiment, the energy measurement of the atoms is frequently
done with a fluorescence technique. A simplified description of this technique
is as follows: a laser is shone over the system, and its frequency is chosen
as to be resonant with a transition of the excited state with a third level
(this level is only used in the measurement process, but not to encode
information). Such third level is broad and the electron quickly decays back
to the $\delimiter 696409721\rangle$ state by emitting a photon in a random
direction, and the process repeats. In this way, if a given atom is in
$\delimiter 696409721\rangle$, it will scatter light. On the contrary, if the
atom is in $\delimiter 696409720\rangle$, the laser is far from resonance and
no light is scattered. The light scattered by the various atoms is collected
by a microscope whose resolution determines if the light scattered by
neighboring atoms can be resolved.

Consider a situation where due to the experimental access one has to the
system, the fluorescence measurement cannot resolve individual wells, but it
takes blocks of $n$ sites. In this way, $n$ atoms behave as a single effective
atom: if at least one underlying atom is excited, light will be scattered; if
no atom is excited, no light is scattered. In such a situation, which
generalized canonical state should be assigned to the system? Although this
situation is not the usual system-environment split found in open quantum
systems, our formalism can be employed in order to construct the generalized
canonical state.

To describe this scenario, inspired by previous works [9, 27, 10, 21], we
define the following CPTP map $\Lambda_{\text{BnS}}^{n\rightarrow
1}:\mathcal{L}(\mathbb{C}_{2}^{\otimes n})\mapsto\mathcal{L}(\mathbb{C}_{2})$:

$\Lambda_{\text{BnS}}^{n\rightarrow 1}[\delimiter
69640972s^{\prime}\rangle\\!\langle s^{\prime\prime}\delimiter
86418188]=\left\\{\begin{aligned} &|0\rangle\\!\langle 0|,\hskip
5.69046pt\text{if }s^{\prime}=s^{\prime\prime}\text{ and }|s^{\prime}|=0,\\\
&|1\rangle\\!\langle 1|,\hskip 5.69046pt\text{if
}s^{\prime}=s^{\prime\prime}\text{ and }|s^{\prime}|\geq 1,\\\
&\frac{1}{\sqrt{2^{n}-1}}\delimiter 696409720\rangle\\!\langle 1\delimiter
86418188,\hskip 5.69046pt\text{if }|s^{\prime}|=0\text{ and
}|s^{\prime\prime}|\geq 1,\\\ &\frac{1}{\sqrt{2^{n}-1}}\delimiter
696409721\rangle\\!\langle 0\delimiter 86418188,\hskip 5.69046pt\text{if
}|s^{\prime}|\geq 1\text{ and }|s^{\prime\prime}|=0,\\\ &0,\hskip
5.69046pt\text{otherwise. }\end{aligned}\right.$

In the above, $s^{\prime},s^{\prime\prime}\in\\{0,1\\}^{n}$ are strings with
$n$ bits. The factor of $1/\sqrt{2^{n}-1}$ for the effective coherence terms
is the largest possible while keeping $\Lambda_{\text{BnS}}^{n\rightarrow 1}$
completely positive. There is no effective coherence generated by underlying
coherence terms as $\delimiter 69640972s^{\prime}\rangle\\!\langle
s^{\prime\prime}\delimiter 86418188$ with $s^{\prime}\neq s^{\prime\prime}$
and $|s^{\prime}|,|s^{\prime\prime}|\geq 1$, because these states cannot be
distinguished by the detection process. It is important to notice that such a
map is not equivalent to the partial trace, as the result $|1\rangle\\!\langle
1|$ is obtained if at least one atom is excited and it is independent of its
position.

Therefore, if we split the $N$ atoms of the total system into $k=N/n$ blocks
of $n$ atoms, we can obtain a generalized effective subsystem description via
the map:

$\Lambda_{\text{BnS}}^{N\rightarrow
k}=\Lambda_{\text{BnS}}^{\frac{N}{k}\rightarrow
1}\otimes\Lambda_{\text{BnS}}^{\frac{N}{k}\rightarrow
1}\otimes\cdots\otimes\Lambda_{\text{BnS}}^{\frac{N}{k}\rightarrow 1}.$ (6)

The effective system is then equivalent to $k$ two-level atoms.

For this scenario, we can evaluate the generalized canonical state,
$\Omega_{\Lambda_{\text{BnS}}}=\Lambda_{\text{BnS}}^{N\rightarrow
k}(\mathcal{E}_{R})$, to be:

$\Omega_{\Lambda_{\text{BnS}}}=\frac{1}{d_{R}}\sum_{|s|=\lceil
kp\rceil}^{\min(Np,k)}\sum_{q=0}^{|s|}{|s|\choose q}{\frac{N}{k}q\choose
Np}(-1)^{|s|-q}\;\Pi_{|s|}.$ (7)

In the above expression $\Pi_{|s|}$ is the projector onto the subspace spanned
by the strings with number of 1’s equal $|s|$. The derivation of this
generalized canonical state is shown in Appendix E.

Although that it is not the traditional statistical mechanics setup, given the
previous results, we expect to observe the canonical typicality result
whenever $d_{R}\gg 1$ and $d_{R}\gg d_{S}=2^{k}$. Within these assumptions,
$\Lambda_{\text{BnS}}^{N\rightarrow k}$ will disregard a considerable amount
of information about the microscopic description, i.e.,
$S(\Lambda_{\text{BnS}}^{N\rightarrow k})$ will be close to one. In this case,
selecting any state at random in $\mathcal{H}_{R}$ and applying
$\Lambda_{\text{BnS}}^{N\rightarrow k}$ to it will lead to a good
approximation of $\Omega_{\Lambda_{\text{BnS}}}$.

Fixed the Hamiltonian as specified above, in Fig.1 we compare two scenarios
where we end up with a subsystem formed by $k$ spins: the first one is the
traditional one where we simply traced out $N-k$ spins, i.e., we apply the map
$\Lambda_{{\rm tr}_{N-k}}$. The second one is the scheme described above, with
the subsystem of $k$ spins being obtained by applying the map
$\Lambda_{\textrm{BnS}}^{N\rightarrow k}$. These two cases are illustrated in
Fig. 1(a).

The first point to be observed is that the canonical states for the two
situations are very different. Noticing that the energy of the system is
proportional to $|s|$, in Fig. 1(b) we plot the energy distribution assuming
$N=10000$, $Np=200$ and $k=100$ for both cases. The distribution more to the
left is the one associated with the partial trace, whereas the one more to the
right is related to the blurred and saturated detection. From this result it
is clear that the appropriate canonical state heavily depends on the
generalized subsystem description, and not exclusively on the underlying
particle structure.

The second point concerns the canonical typicality for the blurred and
saturated detection, given that this aspect for the scenario of partial trace
has been already explored in [5]. In Fig. 1(c) the solid symbols refer to the
mean and variance for the distribution of
$\mathcal{D}({\varrho}_{\Lambda_{\textrm{BnS}}}^{\psi},\Omega_{\Lambda_{\textrm{BnS}}})$
with $\psi$ taken at random from the Haar measure, while the hollow markers
show the bound $\sqrt{d_{S}(1-S_{L}(\Lambda_{\textrm{BnS}}))}/2$. In this
plot, the underlying system is assumed to be formed by 8 spins, $N=8$, and the
case where $k=4$ is shown in blue (circles), while the case of $k=2$ is shown
in red (squares). All the quantities are shown as a function of the number of
excitations, $Np$, which is directly linked to the restriction dimension
$d_{R}$. One immediately observes that as $k$ decreases, the smaller are the
mean distances and the bound. Also it is clear that the bound is tighter the
bigger is $d_{R}$. Both behaviors are can be understood by noticing that, for
fixed $N$, the smaller $k$ and the larger $d_{R}$ are, the more information
about the microscopic state is being thrown away, i.e., the larger is the
channel entropy. Lastly, from the error bars, it is clear that the
concentration around the mean value gets stronger as the fraction of excited
spins increases. In Eq. (4) this expressed by the Lipschitz constant of
$\Lambda_{\text{BnS}}$, which is smaller than one and decreases as $Np$ grows.
For instance, for $Np=7$ no block of 4 (or 2) spins will have all the spins in
the ground state, and thus the only possible state for the $k$ effective spins
is to have all of them in the excited state. Therefore, all pure states in the
subspace with $Np=7$ are mapped to $k$ excited effective spins and the
canonical state is also given by $k$ excited state, and as such their distance
and $\eta_{\Lambda_{\textrm{BnS}}}$ are both zero.

(a)
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|  |  |
---|---|---
$\Lambda_{{\rm tr}_{N-k}}$ |  | $\Lambda_{\textrm{BnS}}^{N\rightarrow k}$
(b)
(c)
Figure 1:  (a) Subsystems scenarios. On the left panel it is shown the usual
link between particles and degrees-of-freedom. In this case, a subsystem is
simply a subset of the constituent particles: from a system of $N$ spins, a
subsystem of $k$ spins is obtained by tracing out $N-k$ spins, i.e., by
applying the map $\Lambda_{{\rm tr}_{N-k}}$. On the right is an example of a
generalized subsystem, which is suggested by a blurred and saturated
fluorescence detection. In this case, blocks of $n$ spins give rise to one
effective spin. The description of a effective subsystem of $k$ spins coming
from an underlying system of $N$ spins is obtained by the action of the
channel $\Lambda_{\textrm{BnS}}^{N\rightarrow k}$. (b) Canonical
Distributions. Considering the Hamiltonian
$H=-\hbar\omega\sum_{i=1}^{N}Z_{i}/2$ for both scenarios shown in (a), it is
clear that the number of excited states $|s|$ is directly related to the
energy of the system. In this panel we plot the probability distribution of
$|s|$ for both cases: the points on the left (circles) represent the energy
distribution for the partial trace case, while the points on the right
(squares) are related to the blurred and saturated situation. For this plot we
used $N=10000$, $Np=200$ and $k=1000$. (c) Canonical typicality for the
blurred and saturated generalized subsystem. For this plot the filled markers
represent the average distance to the generalized canonical state, and the
hollow markers show the bound in Eq. (3). The underlying system here is
composed by $N=8$ spins, with the red (squares) points representing the case
where $k=2$ and the blue (circles) points the case $k=4$.

Conclusions. This work is a first step towards a statistical mechanics of
generalized subsystems. We introduced the canonical state for generalized
subsystems and showed that there are situations where it is drastically
different from the canonical state one would obtain by associating the
relevant degrees-of-freedom of a physical scenario with the particles that
constitute the system. Moreover, we also showed that the phenomena of
canonical typicality is also present for generalized subsystems, and thus all
the foundational ingredients of traditional statistical mechanics are
inherited by this new picture. Given our all-encompassing description of
physical systems, it became explicit that the quantity which controls the
emergence of canonical typicality is the entropy of the channel used to define
the generalized subsystem.

One last remark is in order. In the formalism shown above we employed quantum
channels exclusively to define the generalized subsystems. However, given the
formalism’s flexibility, we could simply change $\Lambda$ by a composition of
quantum channels as $\Lambda\circ\Gamma$, with
$\Gamma:\mathcal{L}(\mathcal{H}_{R})\mapsto\mathcal{L}(\mathcal{H}_{R})$,
without altering any of the presented results. In this situation $\Gamma$ can
be thought as a pre-processing of the underlying description, with the pure
states of $\mathcal{H}_{R}$ generically being mapped into mixed ones. Such a
possibility is interesting in at least two ways. First it shows that the
phenomena of canonical typicality is robust to noise, which might substantiate
the success of ensembles. Second, this might alleviate the unreasonable
requirement for canonical typicality that the underlying states whose
effective description behave similarly to the canonical state, are states
taken uniformly from the Haar measure. For complex systems we do not expect
Nature to be able to efficiently produce such states [28], and thus this
requirement puts in doubt the relevance of the typicality approach. By pre-
processing the states from the Haar measure, we are effectively inducing a new
measure, which might lead to more physically feasible samples. For instance,
it is expected that the average amount of entanglement to drop after the pre-
processing, possibly making the production of the output states a much easier
task.

Altogether, the presented formalism not only will give a more suitable
description of the statistical mechanics of highly isolated and strongly
interacting complex systems, but it might also shed light on some conceptual
issues of traditional statistical mechanics.

Acknowledgements. This work is supported in part by the National Council for
Scientific and Technological Development, CNPq Brazil (projects: Universal
Grant No. 406499/2021-7, and 409611/2022-0), and it is part of the Brazilian
National Institute for Quantum Information. TRO acknowledges funding from the
Air Force Office of Scientific Research under Grant No. FA9550-23-1-0092.

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##  Appendix A Appendix A: Applying Levy’s Lemma to
$\mathcal{D}({\varrho}_{\Lambda}^{\psi},\Omega_{\Lambda})$

Lemma (Levy’s Lemma): Given a Lipschitz continuous function
$f:\mathbb{S}^{d}\rightarrow\mathbb{R}$, and a point $\psi\in\mathbb{S}^{d}$
chosen uniformly at random,

$\displaystyle{\rm
Pr}_{\psi}\big{[}|f(\psi)-\overline{f}|\geq\epsilon\big{]}\leq
2\exp\bigg{(}\dfrac{-2G(d+1)\epsilon^{2}}{\eta^{2}}\bigg{)},$ (8)

where $\eta$ is the Lipschitz constant of $f$ with respect to the Euclidean
norm (see below), $G$ is a positive constant (which can be taken to be
$G=(18\pi^{3})^{-1}$), and $\overline{f}$ is the mean value of $f$ over the
uniform measure in $\mathbb{S}^{d}$.

After noticing that the space of pure quantum states in a $d$-dimensional
Hilbert space ($\mathcal{H}_{d}$) can be described as a hyper-sphere embedded
in a $2d$ real space, $\mathbb{S}^{2d-1}$, we can apply Levy’s Lemma to
Lipschitz continuous properties of pure states by sampling them from the Haar
measure.

In our case, we are interest in applying Levy’s Lemma to
$f(\psi)=\mathcal{D}({\varrho}_{\Lambda}^{\psi},\Omega_{\Lambda})$ with
$\psi\in\mathbb{S}^{2d_{R}-1}$. It remains to show that such a function is
Lipschitz continuous with respect to the Euclidean norm.

Generally, a function $f:X\mapsto Y$ between metric spaces is Lipschitz
continuous if there exists a constant $\eta\in\mathds{R}^{+}$ such that for
all $a,b\in X$ we have $\delimiter 69645069f(a)-f(b)\delimiter
86422285_{Y}\leq\eta\delimiter 69645069a-b\delimiter 86422285_{X}$. Here
$\delimiter 69645069\cdot\delimiter 86422285_{Z}$ is a norm associated with
space $Z$. Lispschitz continuity is then a strong form of continuity.

For our case, we must upper bound
$|\mathcal{D}({\varrho}_{\Lambda}^{\psi},\Omega_{\Lambda})-\mathcal{D}({\varrho}_{\Lambda}^{\phi},\Omega_{\Lambda})|$
for any $\psi$ and $\phi$ in $\mathbb{S}^{2d_{R}-1}$. It goes as follows [16]:

$\displaystyle|\mathcal{D}({\varrho}_{\Lambda}^{\psi},\Omega_{\Lambda})-\mathcal{D}({\varrho}_{\Lambda}^{\phi},\Omega_{\Lambda})|^{2}$
$\displaystyle=|\delimiter
69645069\Lambda(\psi)-\Lambda(\mathcal{E}_{R})\delimiter
86422285_{1}-\delimiter
69645069\Lambda(\phi)-\Lambda(\mathcal{E}_{R})\delimiter 86422285_{1}|^{2}/4;$
$\displaystyle\leq\delimiter 69645069\Lambda(\psi)-\Lambda(\phi)\delimiter
86422285_{1}^{2}/4;$ $\displaystyle\leq\eta_{\Lambda}^{2}\delimiter
69645069|\psi\rangle\\!\langle\psi|-|\phi\rangle\\!\langle\phi|\delimiter
86422285_{1}^{2}/4;$
$\displaystyle=\eta_{\Lambda}^{2}(1-|\langle\psi|\phi\rangle|^{2});$
$\displaystyle\leq\eta_{\Lambda}^{2}\delimiter 69645069\delimiter
69640972\psi\rangle-\delimiter 69640972\phi\rangle\delimiter 86422285^{2}.$

This proves that
$f(\psi)=\mathcal{D}({\varrho}_{\Lambda}^{\psi},\Omega_{\Lambda})$ is
Lipschitz continuous with constant $\eta_{\Lambda}$.

We then obtain the desired result:

${\rm
Pr}_{\psi}\Big{[}\big{|}\mathcal{D}\left(\Lambda(|\psi\rangle\\!\langle\psi|),\Omega_{\Lambda}\right)-\overline{\mathcal{D}({\varrho}_{\Lambda}^{\chi},\Omega_{\Lambda})}^{\chi}\big{|}\geq\epsilon\Big{]}\leq
2\textrm{e}^{-C\frac{d_{\text{R}}\epsilon^{2}}{\eta_{\Lambda}^{2}}}.$ (9)

where $C$ is a constant that can be taken equal to $2/9\pi^{3}$, and
$\eta_{\Lambda}$ is the channel Lipschitz constant.

##  Appendix B APPENDIX B: Full map
$\Lambda:\mathbb{C}_{3}\mapsto\mathbb{C}_{2}$

For completeness, here we present the full map that singles out an effective
2-dimensional subsystem from a tree-level system whose excited level cannot be
distinguished. The map action in a generic linear operator
${\varrho}=[{\varrho}]_{ij}$ with $i,j\in\\{0,1,2\\}$ is as follows:

$\Lambda({\varrho})=\begin{pmatrix}{\varrho}_{00}&\frac{{\varrho}_{01}+{\varrho}_{02}}{\sqrt{2}}\\\
\frac{{\varrho}_{10}+{\varrho}_{20}}{\sqrt{2}}&{\varrho}_{11}+{\varrho}_{22}\end{pmatrix}.$
(10)

The isometry $V:\mathbb{C}_{3}\mapsto\mathbb{C}_{2}\otimes\mathbb{C}_{2}$
defining this channel is given by:

$V=\begin{pmatrix}1&0&0\\\ 0&0&0\\\ 0&\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\\
0&\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}\end{pmatrix}.$ (11)

##  Appendix C APPENDIX C: PROVING
$\mathcal{D}({\varrho}_{\Lambda}^{\psi},\Omega_{\Lambda})\leq\frac{1}{2}\sqrt{d_{B}(1-S_{L}(\Lambda))}$

Given that the definitions of the trace-norm $\parallel A\parallel_{1}={\rm
tr}\big{[}\sqrt{A^{\dagger}A}\big{]}$ and Hilbert-Schmidt norm $\parallel
A\parallel_{2}=\sqrt{{\rm tr}[A^{\dagger}A]}$. For any $n\times n$ matrix $A$
the following inequality is satisfied:

$\displaystyle\parallel A\parallel_{1}\leq\sqrt{n}\parallel A\parallel_{2}.$
(12)

This relation can be easily proved, considering that $A$ has eigenvalues
$\lambda_{i}$, and the convexity of the square function:

$\displaystyle\parallel
A\parallel_{1}^{2}=n^{2}\bigg{(}\dfrac{1}{n}\sum_{i}|\lambda_{i}|\bigg{)}^{2}\leq
n^{2}\dfrac{1}{n}\sum_{i}|\lambda_{i}|^{2}=n\parallel A\parallel_{2}^{2}.$

Let us derive another useful relation. Doing
$A={\varrho}_{\Lambda}^{\psi}-\Omega_{\Lambda}$ and taking the average in the
Hilbert-Schmidt norm:

$\displaystyle\overline{\parallel{\varrho}_{\Lambda}^{\psi}-\Omega_{\Lambda}\parallel_{2}}^{\psi}$
$\displaystyle\leq\sqrt{\overline{\parallel{\varrho}_{\Lambda}^{\psi}-\Omega_{\Lambda}\parallel_{2}^{2}}^{\psi}}$
$\displaystyle=\sqrt{\overline{{\rm
tr}[({\varrho}_{\Lambda}^{\psi}-\Omega_{\Lambda})^{2}]}^{\psi}}=\sqrt{\overline{{\rm
tr}[({\varrho}_{\Lambda}^{\psi})^{2}]}^{\psi}-2\overline{{\rm
tr}[{\varrho}_{\Lambda}^{\psi}\Omega_{\Lambda}]}^{\psi}+\overline{{\rm
tr}[\Omega_{\Lambda}^{2}]}^{\psi}}=\sqrt{\overline{{\rm
tr}[({\varrho}_{\Lambda}^{\psi})^{2}]}^{\psi}-2{\rm
tr}[\overline{{\varrho}_{\Lambda}^{\psi}}^{\psi}\Omega_{\Lambda}]+{\rm
tr}[\Omega_{\Lambda}^{2}]}$ $\displaystyle=\sqrt{\overline{{\rm
tr}[({\varrho}_{\Lambda}^{\psi})^{2}]}^{\psi}-{\rm
tr}[\Omega_{\Lambda}^{2}]},$ (13)

in the first line we used the Jensen’s inequality
$f(\overline{M})\leq\overline{f(M)}$ with $f(\cdot)=(\cdot)^{2}$. In the third
line we used
$\overline{\Omega_{\Lambda}}^{\psi}=\overline{{\varrho}_{\Lambda}^{\psi}}^{\psi}=\Omega_{\Lambda}$.

Then, from (12) and (13):

$\displaystyle\overline{\mathcal{D}({\varrho}_{\Lambda}^{\psi},\Omega_{\Lambda})}^{\psi}=\dfrac{1}{2}\overline{\parallel{\varrho}_{\Lambda}^{\psi}-\Omega_{\Lambda}\parallel_{1}}^{\psi}\leq\dfrac{1}{2}\sqrt{d_{B}\big{(}\overline{{\rm
tr}[({\varrho}_{\Lambda}^{\psi})^{2}]}^{\psi}-{\rm
tr}[\Omega_{\Lambda}^{2}]\big{)}}$ (14)

###  C.1 Calculating $\overline{{\rm
tr}[({\varrho}_{\Lambda}^{\psi})^{2}]}^{\psi}$

To calculate $\overline{{\rm tr}[({\varrho}_{\Lambda}^{\psi})^{2}]}^{\psi}$,
it is helpful to duplicate the original Hilbert space, extending the problem
from $\mathcal{H}_{R}$ to $\mathcal{H}_{R}\otimes\mathcal{H}_{R^{\prime}}$. We
start by written ${\rm tr}[({\varrho}_{\Lambda}^{\psi})^{2}]$ as:

$\displaystyle{\rm tr}[({\varrho}_{\Lambda}^{\psi})^{2}]$
$\displaystyle=\sum_{i}({\varrho}_{ii})^{2}$
$\displaystyle=\sum_{i,j,i^{\prime},j^{\prime}}({\varrho}_{ij})({\varrho}_{i^{\prime}j^{\prime}})\langle
ii^{\prime}|jj^{\prime}\rangle\langle i^{\prime}i|jj^{\prime}\rangle$
$\displaystyle={\rm
tr}_{SS^{\prime}}[({\varrho}_{\Lambda}^{\psi}\otimes{\varrho}_{\Lambda^{\prime}}^{\psi})F_{SS^{\prime}}],$
(15)

where the $F_{SS^{\prime}}$ is the swap operation $S\leftrightarrow
S^{\prime}$:

$\displaystyle F_{SS^{\prime}}=\sum_{ss^{\prime}}\delimiter
69640972s^{\prime}\rangle\langle s\delimiter 86418188\otimes\delimiter
69640972s\rangle\langle s^{\prime}\delimiter 86418188$ (16)

and then

$\displaystyle\overline{{\rm tr}[({\varrho}_{\Lambda}^{\psi})^{2}]}^{\psi}$
$\displaystyle=\overline{{\rm
tr}_{SS^{\prime}}[({\varrho}_{\Lambda}^{\psi}\otimes{\varrho}_{\Lambda^{\prime}}^{\psi})F_{SS^{\prime}}]}^{\psi}$
$\displaystyle=\overline{{\rm tr}_{SS^{\prime}}[(\Lambda[\delimiter
69640972\psi\rangle\langle\psi\delimiter 86418188]\otimes\Lambda[\delimiter
69640972\psi\rangle\langle\psi\delimiter 86418188])F_{SS^{\prime}}]}^{\psi}$
$\displaystyle=\overline{{\rm
tr}_{SS^{\prime}}[(\Lambda\otimes\Lambda)[\delimiter
69640972\psi\rangle\langle\psi\delimiter 86418188\otimes\delimiter
69640972\psi\rangle\langle\psi\delimiter 86418188]F_{SS^{\prime}}]}^{\psi}$
$\displaystyle={\rm
tr}_{SS^{\prime}}[(\Lambda\otimes\Lambda)[\overline{\delimiter
69640972\psi\rangle\langle\psi\delimiter 86418188\otimes\delimiter
69640972\psi\rangle\langle\psi\delimiter 86418188}^{\psi}]F_{SS^{\prime}}]$
(17)

Using the result [bibid]

$\displaystyle\overline{\delimiter 69640972\psi\rangle\langle\psi\delimiter
86418188\otimes\delimiter 69640972\psi\rangle\langle\psi\delimiter
86418188}^{\psi}=\dfrac{\mathds{1}_{RR^{\prime}}+F_{RR^{\prime}}}{d_{R}(d_{R}+1)}$
(18)

where $\mathds{1}_{RR^{\prime}}=\mathds{1}_{R}\otimes\mathds{1}_{R^{\prime}}$.
Using above equation in (17):

$\displaystyle\overline{{\rm tr}[({\varrho}_{\Lambda}^{\psi})^{2}]}^{\psi}$
$\displaystyle={\rm
tr}_{SS^{\prime}}\big{[}(\Lambda\otimes\Lambda)\big{[}\dfrac{\mathds{1}_{RR^{\prime}}}{d_{R}(d_{R}+1)}\big{]}F_{SS^{\prime}}\big{]}+{\rm
tr}_{SS^{\prime}}\big{[}(\Lambda\otimes\Lambda)\big{[}\dfrac{\mathds{F}_{RR^{\prime}}}{d_{R}(d_{R}+1)}\big{]}F_{SS^{\prime}}\big{]}$
$\displaystyle\leq{\rm tr}_{S}[\Omega_{\Lambda}^{2}]+{\rm
tr}_{SS^{\prime}}\big{[}(\Lambda\otimes\Lambda)\big{[}\dfrac{F_{RR^{\prime}}}{d_{R}^{2}}\big{]}F_{SS^{\prime}}\big{]}$
(19)

in the last line we used $d_{R}(d_{R}+1)\geq d_{R}^{2}$.

Thus, (19) in (14):

$\displaystyle\overline{\parallel{\varrho}_{\Lambda}^{\psi}-\Omega_{\Lambda}\parallel_{1}}^{\psi}\leq\sqrt{\dfrac{d_{S}}{d_{R}^{2}}\;{\rm
tr}_{SS^{\prime}}[(\Lambda\otimes\Lambda)[F_{RR^{\prime}}]F_{SS^{\prime}}]}.$
(20)

Using (16) in ${\rm
tr}_{SS^{\prime}}[(\Lambda\otimes\Lambda)[F_{RR^{\prime}}]F_{SS^{\prime}}]$:

$\displaystyle{\rm
tr}_{SS^{\prime}}[(\Lambda\otimes\Lambda)[F_{RR^{\prime}}]F_{SS^{\prime}}]$
$\displaystyle={\rm
tr}_{SS^{\prime}}\big{[}(\Lambda\otimes\Lambda)\big{[}\sum_{r,r^{\prime}=0}^{d_{R}-1}\delimiter
69640972r^{\prime}\rangle\langle r\delimiter 86418188\otimes\delimiter
69640972r\rangle\langle r^{\prime}\delimiter
86418188\big{]}F_{SS^{\prime}}\big{]}$ $\displaystyle={\rm
tr}_{SS^{\prime}}\big{[}\big{(}\sum_{r,r^{\prime}=0}^{d_{R}-1}\Lambda\big{[}\delimiter
69640972r^{\prime}\rangle\langle r\delimiter
86418188\big{]}\otimes\Lambda\big{[}\delimiter 69640972r\rangle\langle
r^{\prime}\delimiter 86418188\big{]}\big{)}F_{SS^{\prime}}\big{]}$ (21)

using the Kraus representation (22) of $\Lambda$. Any quantum channel
$\Lambda:\mathcal{L}(\mathcal{H}_{R})\rightarrow\mathcal{L}(\mathcal{H}_{S})$,
there exist a representation

$\displaystyle\Lambda[\psi]=\sum_{m=1}^{\tau}K_{m}\psi K_{m}^{\dagger}$ (22)

for all $\psi\in\mathcal{L}(\mathcal{H}_{R})$,
$K_{m}\in\mathcal{L}(\mathcal{H}_{R},\mathcal{H}_{S})$ for all
$m\in\\{1,\dots,\tau\\}$ and $\tau$ need not be any larger than $d_{R}d_{S}$.
The expression (22) is a Kraus representation of the map $\Lambda$, unlike the
Choi representation, Kraus representations are not unique.

Using (16) and (22) in (21)

$\displaystyle{\rm
tr}_{SS^{\prime}}[(\Lambda\otimes\Lambda)[F_{RR^{\prime}}]F_{SS^{\prime}}]$
$\displaystyle={\rm
tr}_{SS^{\prime}}\big{[}\big{(}\sum_{r,r^{\prime}=0}^{d_{R}-1}\sum_{m,n=1}^{\tau}K_{m}\delimiter
69640972r^{\prime}\rangle\langle r\delimiter 86418188K_{m}^{\dagger}\otimes
K_{n}\delimiter 69640972r\rangle\langle r^{\prime}\delimiter
86418188K_{n}^{\dagger}\big{)}\sum_{s,s^{\prime}=0}^{d_{S}-1}\delimiter
69640972s^{\prime}\rangle\langle s\delimiter 86418188\otimes\delimiter
69640972s\rangle\langle s^{\prime}\delimiter 86418188\big{]}$
$\displaystyle={\rm
tr}_{SS^{\prime}}\big{[}\sum_{r,r^{\prime}=0}^{d_{R}-1}\sum_{m,n=1}^{\tau}\sum_{s,s^{\prime}=0}^{d_{S}-1}K_{m}\delimiter
69640972r^{\prime}\rangle\langle r\delimiter 86418188K_{m}^{\dagger}\delimiter
69640972s^{\prime}\rangle\langle s\delimiter 86418188\otimes K_{n}\delimiter
69640972r\rangle\langle r^{\prime}\delimiter 86418188K_{n}^{\dagger}\delimiter
69640972s\rangle\langle s^{\prime}\delimiter 86418188\big{]}$
$\displaystyle=\sum_{t,t^{\prime}=0}^{d_{S}-1}\sum_{r,r^{\prime}=0}^{d_{R}-1}\sum_{m,n=1}^{\tau}\sum_{s,s^{\prime}=0}^{d_{S}-1}\langle
t\delimiter 86418188K_{m}\delimiter 69640972r^{\prime}\rangle\langle
r\delimiter 86418188K_{m}^{\dagger}\delimiter
69640972s^{\prime}\rangle\overbrace{\langle s\delimiter
86418188t}^{\delta_{st}}\rangle\langle t^{\prime}\delimiter
86418188K_{n}\mathds{1}_{R^{\prime}}\delimiter 69640972r\rangle\langle
r^{\prime}\delimiter 86418188K_{n}^{\dagger}\delimiter
69640972s\rangle\overbrace{\langle s^{\prime}\delimiter
86418188t^{\prime}}^{\delta_{s^{\prime}t^{\prime}}}\rangle$
$\displaystyle=\sum_{r,r^{\prime}=0}^{d_{R}-1}\sum_{m,n=1}^{\tau}\sum_{s,s^{\prime}=0}^{d_{S}-1}\langle
s\delimiter 86418188K_{m}\delimiter 69640972r^{\prime}\rangle\langle
r\delimiter 86418188K_{m}^{\dagger}\delimiter 69640972s^{\prime}\rangle\langle
s^{\prime}\delimiter 86418188K_{n}\delimiter 69640972r\rangle\langle
r^{\prime}\delimiter 86418188K_{n}^{\dagger}\delimiter 69640972s\rangle$
$\displaystyle=\sum_{m,n=1}^{\tau}\sum_{s,s^{\prime}=0}^{d_{S}-1}\langle
s\delimiter
86418188K_{m}\underbrace{\big{(}\sum_{r^{\prime}=0}^{d_{R}-1}\delimiter
69640972r^{\prime}\rangle\langle r^{\prime}\delimiter
86418188\big{)}}_{=\mathds{1}_{R}}K_{n}^{\dagger}\delimiter
69640972s\rangle\langle s^{\prime}\delimiter
86418188K_{n}\underbrace{\big{(}\sum_{r=0}^{d_{R}-1}\delimiter
69640972r\rangle\langle r\delimiter
86418188\big{)}}_{=\mathds{1}_{R}}K_{m}^{\dagger}\delimiter
69640972s^{\prime}\rangle$
$\displaystyle=\sum_{m,n=1}^{\tau}\sum_{s,s^{\prime}=0}^{d_{S}-1}\langle
s\delimiter 86418188K_{m}K_{n}^{\dagger}\delimiter 69640972s\rangle\langle
s^{\prime}\delimiter 86418188K_{n}K_{m}^{\dagger}\delimiter
69640972s^{\prime}\rangle$ $\displaystyle=\sum_{m,n=1}^{\tau}{\rm
tr}[K_{m}K_{n}^{\dagger}]{\rm
tr}[K_{n}K_{m}^{\dagger}]=\sum_{m,n=1}^{\tau}|{\rm
tr}[K_{m}K_{n}^{\dagger}]|^{2}$ (23)

above result in (20), we have:

$\displaystyle\mathcal{D}({\varrho}_{\Lambda}^{\psi},\Omega_{\Lambda})\leq\dfrac{1}{2}\sqrt{\dfrac{d_{S}}{d_{R}^{2}}\sum_{m,n=i}^{\tau}\big{|}{\rm
tr}\big{[}K_{m}K_{n}^{\dagger}\big{]}\big{|}^{2}}.$ (24)

Rewriting the above equation in terms of Choi-Jamiołkowski state,
$J_{\Lambda}:=\Lambda\otimes\mathds{1}(|\phi^{+}\rangle\\!\langle\phi^{+}|)$
of the quantum channel $\Lambda$, where $\delimiter
69640972\phi^{+}\rangle=\sum_{i}\delimiter
69640972ii\rangle/\sqrt{d_{R}}\in\mathcal{H}_{R}\otimes\mathcal{H}_{R}$. Using
the Krauss form:

$\displaystyle J_{\Lambda}$
$\displaystyle=(\Lambda\otimes\mathds{1}_{R})[\delimiter
69640972\phi_{+}\rangle\langle\phi_{+}\delimiter 86418188]$
$\displaystyle=\dfrac{1}{d_{R}}\sum_{i,j=0}^{d_{R}-1}\Lambda[\delimiter
69640972i\rangle\langle j\delimiter 86418188]\otimes\delimiter
69640972i\rangle\langle j\delimiter
86418188=\dfrac{1}{d_{R}}\sum_{i,j=0}^{d_{R}-1}\sum_{m=1}^{\tau}K_{m}\delimiter
69640972i\rangle\langle j\delimiter 86418188K_{m}^{\dagger}\otimes\delimiter
69640972i\rangle\langle j\delimiter 86418188$ (25)

Calculating ${d_{R}}^{2}{\rm tr}[{J_{\Lambda}}^{2}]$:

$\displaystyle{d_{R}}^{2}{\rm tr}[{J_{\Lambda}}^{2}]$ $\displaystyle={\rm
tr}_{SR}\bigg{[}\big{(}\sum_{i,j=0}^{d_{R}-1}\sum_{m=1}^{\tau}K_{m}\delimiter
69640972i\rangle\langle j\delimiter 86418188K_{m}^{\dagger}\otimes\delimiter
69640972i\rangle\langle j\delimiter
86418188\big{)}\big{(}\sum_{i^{\prime},j^{\prime}=0}^{d_{R}-1}\sum_{n=1}^{\tau}K_{n}\delimiter
69640972i^{\prime}\rangle\langle j^{\prime}\delimiter
86418188K_{n}^{\dagger}\otimes\delimiter 69640972i^{\prime}\rangle\langle
j^{\prime}\delimiter 86418188\big{)}\bigg{]}$ $\displaystyle={\rm
tr}_{SR}\bigg{[}\sum_{i,j,i^{\prime},j^{\prime}}\sum_{m,n}K_{m}\delimiter
69640972i\rangle\langle j\delimiter 86418188K_{m}^{\dagger}K_{n}\delimiter
69640972i^{\prime}\rangle\langle j^{\prime}\delimiter
86418188K_{n}^{\dagger}\otimes\delimiter 69640972i\rangle\overbrace{\langle
j\delimiter 86418188i^{\prime}\rangle}^{\delta_{ji^{\prime}}}\langle
j^{\prime}\delimiter 86418188\bigg{]}$ $\displaystyle={\rm
tr}_{SR}\bigg{[}\sum_{i,j,j^{\prime}}\sum_{m,n}K_{m}\delimiter
69640972i\rangle\langle j\delimiter 86418188K_{m}^{\dagger}K_{n}\delimiter
69640972j\rangle\langle j^{\prime}\delimiter
86418188K_{n}^{\dagger}\otimes\delimiter 69640972i\rangle\langle
j^{\prime}\delimiter 86418188\bigg{]}$
$\displaystyle=\sum_{s=0}^{d_{S}-1}\sum_{i,j,j^{\prime}}\sum_{m,n}\langle
s\delimiter 86418188K_{m}\delimiter 69640972i\rangle\langle j\delimiter
86418188K_{m}^{\dagger}K_{n}\delimiter 69640972j\rangle\langle
j^{\prime}\delimiter 86418188K_{n}^{\dagger}\delimiter
69640972s\rangle\overbrace{\sum_{r=0}^{d_{R}-1}\langle r\delimiter
69640972i\rangle\langle j^{\prime}\delimiter
86418188r\rangle}^{\delta_{j^{\prime}i}}$
$\displaystyle=\sum_{s}\sum_{i,j}\sum_{m,n}\langle s\delimiter
86418188K_{m}\delimiter 69640972i\rangle\langle j\delimiter
86418188K_{m}^{\dagger}K_{n}\delimiter 69640972j\rangle\langle i\delimiter
86418188K_{n}^{\dagger}\delimiter 69640972s\rangle$
$\displaystyle=\sum_{s}\sum_{j}\sum_{m,n}\langle s\delimiter
86418188K_{m}\bigg{(}\sum_{i}\delimiter 69640972i\rangle\langle i\delimiter
86418188\bigg{)}K_{n}^{\dagger}\delimiter 69640972s\rangle\langle j\delimiter
86418188K_{m}^{\dagger}K_{n}\delimiter 69640972j\rangle$
$\displaystyle=\sum_{s}\sum_{j}\sum_{m,n}\langle s\delimiter
86418188K_{m}K_{n}^{\dagger}\delimiter 69640972s\rangle\langle j\delimiter
86418188K_{m}^{\dagger}K_{n}\delimiter 69640972j\rangle$
$\displaystyle=\sum_{m,n}\bigg{(}\sum_{s}\langle s\delimiter
86418188K_{m}K_{n}^{\dagger}\delimiter
69640972s\rangle\bigg{)}\bigg{(}\sum_{j}\langle j\delimiter
86418188K_{m}^{\dagger}K_{n}\delimiter 69640972j\rangle\bigg{)}$
$\displaystyle=\sum_{m,n}{\rm
tr}\big{[}K_{m}K_{n}^{\dagger}\big{]}\overbrace{{\rm
tr}\big{[}K_{m}^{\dagger}K_{n}\big{]}}^{{\rm
tr}\big{[}K_{n}K_{m}^{\dagger}\big{]}}=\sum_{m,n=1}^{\tau}|{\rm
tr}[K_{m}K_{n}^{\dagger}]|^{2}$ (26)

So (24) can be written in terms of the purity of the Choi-Jamiołkowski state

$\displaystyle\mathcal{D}({\varrho}_{\Lambda}^{\psi},\Omega_{\Lambda})\leq\dfrac{1}{2}\sqrt{d_{S}{\rm
tr}\big{[}{J_{\Lambda}}^{2}\big{]}},$ (27)

or in terms of the linear entropy of the map:

$\displaystyle\mathcal{D}({\varrho}_{\Lambda}^{\psi},\Omega_{\Lambda})\leq\dfrac{1}{2}\sqrt{d_{S}(1-S_{L}(\Lambda))}.,$
(28)

since $S_{L}(\Lambda)=1-{\rm tr}\big{[}{J_{\Lambda}}^{2}\big{]}$.

##  Appendix D Appendix D: Demonstrating that $\sqrt{d_{S}{\rm tr}(J_{{\rm
tr}_{E}}^{2})}=\sqrt{\frac{d_{S}}{d_{E}^{eff}}}$

This section aims to demonstrate that (1) is a particular case of (3) when
$\Lambda={\rm tr}_{E}$. So we need to show that ${\rm tr}(J_{{\rm
tr}_{E}}^{2})=1/{d_{E}^{\text{eff}}}$.

The partial trace act on the entire global space ${\rm
tr}_{E}:\mathcal{L}(\mathcal{H}_{S}\otimes\mathcal{H}_{E})\rightarrow\mathcal{L}(\mathcal{H}_{S})$
and ${\rm
tr}_{S}:\mathcal{L}(\mathcal{H}_{S}\otimes\mathcal{H}_{E})\rightarrow\mathcal{L}(\mathcal{H}_{E})$.
However, note that the microcanonical state $\mathds{1}_{R}/d_{R}$ is defined
in the restricted space
$\mathcal{H}_{R}\subseteq\mathcal{H}_{S}\otimes\mathcal{H}_{E}$, so it is
convenient we choose a basis $\\{\delimiter
69640972R_{r}\rangle\\}\in\mathcal{H}_{R}$ in which elements can be written
as:

$\displaystyle\delimiter 69640972R_{r}\rangle=\sum_{ij}c^{r}_{ij}\delimiter
69640972i\rangle\otimes\delimiter 69640972j\rangle$ (29)

with $\delimiter 69640972i\rangle$ and $\delimiter 69640972j\rangle$
orthonormal basis for $\mathcal{H}_{S}$ and $\mathcal{H}_{E}$, respectively.

In order to find the Choi state, let us first consider the maximally entangles
state $\delimiter
69640972phi_{+}\rangle\in\mathcal{H}_{R}\otimes\mathcal{H}_{R}$, in which its
density matrix can be conveniently written as:

$\displaystyle\delimiter 69640972\phi_{+}\rangle\langle\phi_{+}\delimiter
86418188$ $\displaystyle=\dfrac{1}{d_{R}}\sum_{r,t}\delimiter
69640972R_{r}\rangle\langle R_{t}\delimiter 86418188\otimes\delimiter
69640972R_{r}\rangle\langle R_{t}\delimiter 86418188$
$\displaystyle=\dfrac{1}{d_{R}}\sum_{r,t}\left(\sum_{i,j,k,l}c^{r}_{ij}c^{t\ast}_{k,l}\delimiter
69640972ij\rangle\langle kl\delimiter 86418188\right)\otimes\delimiter
69640972R_{r}\rangle\langle R_{t}\delimiter 86418188$

So the Choi state related to the map ${\rm tr}_{E}$ is given by

$\displaystyle J_{{\rm tr}_{E}}$ $\displaystyle=({\rm
tr}_{E}\otimes\mathds{1}_{R})(\delimiter
69640972\phi_{+}\rangle\langle\phi_{+}\delimiter
86418188)=\dfrac{1}{d_{R}}\sum_{r,t}\left(\sum_{i,j,k,l}c^{r}_{ij}c^{t\ast}_{k,l}{\rm
tr}_{E}(\delimiter 69640972ij\rangle\langle kl\delimiter
86418188)\right)\otimes\delimiter 69640972R_{r}\rangle\langle R_{t}\delimiter
86418188$
$\displaystyle=\dfrac{1}{d_{R}}\sum_{i,k,r,t}\left(\sum_{j}c^{r}_{ij}c^{t\ast}_{k,j}\right)\delimiter
69640972i\rangle\langle k\delimiter 86418188\otimes\delimiter
69640972R_{r}\rangle\langle R_{t}\delimiter 86418188$ (31)

Calculating the purity

$\displaystyle{\rm tr}(J_{{\rm tr}_{E}}^{2})$ $\displaystyle={\rm
tr}\left[\dfrac{1}{d_{R}^{2}}\sum_{i,k,r,t,m,n,p,q}\left(\left(\sum_{j}c^{r}_{ij}c^{t\ast}_{kj}\right)\delimiter
69640972i\rangle\langle k\delimiter 86418188\otimes\delimiter
69640972R_{r}\rangle\langle R_{t}\delimiter
86418188\right)\left(\left(\sum_{z}c^{p}_{mz}c^{q\ast}_{nz}\right)\delimiter
69640972m\rangle\langle n\delimiter 86418188\otimes\delimiter
69640972R_{p}\rangle\langle R_{q}\delimiter 86418188\right)\right]$
$\displaystyle=\dfrac{1}{d_{R}^{2}}\sum_{i,k,r,t,m,n,p,q}\left(\sum_{j}c^{r}_{ij}c^{t\ast}_{kj}\right)\left(\sum_{z}c^{p}_{mz}c^{q\ast}_{nz}\right)\delta_{km}\delta_{tp}\delta_{in}\delta_{rq}$
$\displaystyle=\dfrac{1}{d_{R}^{2}}\sum_{j,z}\left(\sum_{j}c^{r}_{ij}c^{t\ast}_{kj}\right)\left(\sum_{z}c^{t}_{kz}c^{r\ast}_{iz}\right)=\dfrac{1}{d_{R}^{2}}\sum_{j,z}\left(\sum_{r,i}c^{r}_{ij}c^{r\ast}_{iz}\right)\left(\sum_{t,k}c^{t}_{kz}c^{t\ast}_{kj}\right)$
(32)

Since we want to prove ${\rm tr}(J_{{\rm tr}_{E}}^{2})=1/d_{E}^{\text{eff}}$
and the effective dimension of the environment is defined as
$d_{E}^{\text{eff}}={\rm tr}(\Omega_{{\rm tr}_{S}}^{2})$, let us explicitly
describe the state $\Omega_{{\rm tr}_{S}}$:

$\displaystyle\Omega_{{\rm tr}_{S}}=$ $\displaystyle{\rm
tr}_{S}\left(\dfrac{\mathds{1}_{R}}{d_{R}}\right)=\dfrac{1}{d_{R}}{\rm
tr}_{S}\left(\sum_{r}\delimiter 69640972R_{r}\rangle\langle R_{r}\delimiter
86418188\right)$ $\displaystyle=\dfrac{1}{d_{R}}{\rm
tr}_{S}\left(\sum_{r}\sum_{i,j,k,l}c^{r}_{ij}c^{r\ast}_{kl}\delimiter
69640972ij\rangle\langle kl\delimiter 86418188\right)$
$\displaystyle=\dfrac{1}{d_{R}}\sum_{j,l}\left(\sum_{r,i}c^{r}_{ij}c^{r\ast}_{il}\right)\delimiter
69640972j\rangle\langle l\delimiter 86418188$ (33)

Now, lt us calculate the purity,

$\displaystyle{\rm tr}\left(\Omega_{{\rm tr}_{S}}^{2}\right)$
$\displaystyle={\rm
tr}\left[\dfrac{1}{d_{R}^{2}}\sum_{j,l,m,n}\left(\left(\sum_{r,i}c^{r}_{ij}c^{r\ast}_{il}\right)\delimiter
69640972j\rangle\langle l\delimiter
86418188\right)\left(\left(\sum_{t,k}c^{t}_{km}c^{t\ast}_{kn}\right)\delimiter
69640972m\rangle\langle n\delimiter 86418188\right)\right]$
$\displaystyle=\dfrac{1}{d_{R}^{2}}\sum_{j,l,m,n}\left(\sum_{r,i}c^{r}_{ij}c^{r\ast}_{il}\right)\left(\sum_{t,k}c^{t}_{km}c^{t\ast}_{kn}\right)\delta_{l,m}\delta_{j,n}$
$\displaystyle=\dfrac{1}{d_{R}^{2}}\sum_{j,l}\left(\sum_{r,i}c^{r}_{ij}c^{r\ast}_{il}\right)\left(\sum_{t,k}c^{t}_{kl}c^{t\ast}_{kj}\right)$
(34)

We can easily check that above equation is equal to (32) if we change the
indices labels $l\rightarrow z$, so we finish the demonstration.

##  Appendix E Appendix E: Blurred and Saturated channel
$\Lambda_{\text{BnS}}^{n\rightarrow 1}$ and derivation of
$\Omega_{\Lambda}^{\Lambda_{\text{BnS}}^{N\rightarrow k}}$

Considering that we split the $N$ atoms of the total system into
$k=\frac{N}{n}$ blocks of $n$ atoms, the effective system will be equivalent
to $k$ two-level atoms. Our starting point is the expression

$\Omega_{\Lambda_{\textrm{BnS}}}=\frac{1}{{N\choose
Np}}\sum_{s\in\\{0,1\\}^{k}:|s|\in\\{\lceil
kp\rceil,...,\min(Np,k)\\}}\sum_{n_{1}+n_{2}+...+n_{|s|}=Np-|s|}{\frac{N}{k}\choose
n_{1}+1}{\frac{N}{k}\choose n_{2}+1}...{\frac{N}{k}\choose
n_{|s|}+1}|s\rangle\\!\langle s|.$ (35)

In the expression above, $|s|$ is the number of 1’s in the string $s$ and
$n_{i}<\frac{N}{k}$, $\forall i\in[\,|s|\,]$. Simpliflying the second
summations,

$S=\sum_{n_{1}+n_{2}+...n_{|s|}=Np-|s|}{\frac{N}{k}\choose
n_{1}+1}{\frac{N}{k}\choose n_{2}+1}...{\frac{N}{k}\choose n_{|s|}+1},$ (36)

with $n_{i}<\frac{N}{k}$. Therefore, $n_{i}\leqslant\frac{N}{k}-1$ and

$S=\sum_{n_{1}=0}^{\frac{N}{k}-1}\sum_{n_{2}=0}^{\frac{N}{k}-1}...\sum_{n_{|s|}=0}^{\frac{N}{k}-1}{\frac{N}{k}\choose
n_{1}+1}{\frac{N}{k}\choose n_{2}+1}...{\frac{N}{k}\choose
n_{|s|}+1}\delta(n_{1}+n_{2}+...+n_{|s|}-(Np-|s|)).$ (37)

Noticing that

$\delta(n_{1}+n_{2}+...+n_{|s|}-(Np-|s|))=\frac{1}{2\pi}\int_{0}^{2\pi}d\theta\,\,e^{i\theta
n_{1}+n_{2}+...+n_{|s|}-Np-|s|},$ (38)
$S=\frac{1}{2\pi}\int_{0}^{2\pi}d\theta\,\,e^{-i\theta(Np-|s|)}\bigg{(}\sum_{n=0}^{\frac{N}{k}-1}{\frac{N}{k}\choose
n+1}e^{i\theta n}\bigg{)}^{|s|}.$ (39)

Making $m=n+1$,

$\begin{split}S&=\frac{1}{2\pi}\int_{0}^{2\pi}d\theta\,\,e^{-i\theta(Np-|s|)}\bigg{(}\sum_{m=1}^{\frac{N}{k}}{\frac{N}{k}\choose
m}e^{i\theta(m-1)}\bigg{)}^{|s|}\\\
&=\frac{1}{2\pi}\int_{0}^{2\pi}d\theta\,\,e^{-i\theta(Np-|s|)}\bigg{(}\sum_{m=0}^{\frac{N}{k}}{\frac{N}{k}\choose
m}e^{i\theta(m-1)}-e^{-i\theta}\bigg{)}^{|s|}\\\
&=\frac{1}{2\pi}\int_{0}^{2\pi}d\theta\,\,e^{-i\theta
Np}\bigg{(}\sum_{m=0}^{\frac{N}{k}}{\frac{N}{k}\choose m}e^{i\theta
m}-1\bigg{)}^{|s|}.\\\ \Rightarrow
S&=\frac{1}{2\pi}\int_{0}^{2\pi}d\theta\,\,e^{-i\theta
Np}\sum_{q=0}^{|s|}{|s|\choose
q}\bigg{[}\sum_{m=0}^{\frac{N}{k}}{\frac{N}{k}\choose m}e^{i\theta
m}\bigg{]}^{q}(-1)^{|s|-q}.\end{split}$ (40)

Therefore,

$\begin{split}S&=\sum_{q=0}^{|s|}{|s|\choose
q}(-1)^{|s|-q}\frac{1}{2\pi}\int_{0}^{2\pi}d\theta e^{-i\theta
Np}\sum_{m_{1}=0}^{\frac{N}{k}}{\frac{N}{k}\choose m_{1}}e^{i\theta
m_{1}}...\sum_{m_{q}=0}^{\frac{N}{k}}{\frac{N}{k}\choose m_{q}}e^{i\theta
m+q}\\\ &=\sum_{q=0}^{|s|}{|s|\choose
q}(-1)^{|s|-q}\sum_{m_{1}=0}^{\frac{N}{k}}\sum_{m_{2}=0}^{\frac{N}{k}}...\sum_{m_{q}=0}^{\frac{N}{k}}{\frac{N}{k}\choose
m_{1}}{\frac{N}{k}\choose m_{2}}...{\frac{N}{k}\choose
m_{q}}\delta(m_{1}+m_{2}+...+m_{q}-Np)\\\ &=\sum_{q=0}^{|s|}{|s|\choose
q}(-1)^{|s|-q}\sum_{m_{1}+m_{2}+...+m_{q}=Np}{\frac{N}{k}\choose
m_{1}}{\frac{N}{k}\choose m_{2}}...{\frac{N}{k}\choose m_{q}}\\\ \Rightarrow
S&=\sum_{q=0}^{|s|}{|s|\choose q}(-1)^{|s|-q}{\frac{Nq}{k}\choose
Np}.\end{split}$ (41)

The canonical state is given by

$\Omega_{\Lambda_{\textrm{BnS}}}=\frac{1}{d_{R}}\sum_{s\in\\{0,1\\}^{k}:|s|\in\\{\lceil
kp\rceil,...,\min(Np,k)\\}}\sum_{q=0}^{|s|}{|s|\choose q}{\frac{Nq}{k}\choose
Np}(-1)^{|s|-q}\,\,\Pi_{|s|},$ (42)

with $d_{R}={N\choose Np}$ and $\Pi_{|s|}$ the projector onto the subspace
spanned by the strings with number os 1’s equal $|s|$. Finally,

$\Omega_{\Lambda_{\textrm{BnS}}}=\frac{1}{d_{R}}\sum_{|s|=\lceil
kp\rceil}^{\min(Np,k)}\sum_{q=0}^{|s|}{|s|\choose q}{\frac{Nq}{k}\choose
Np}(-1)^{|s|-q}\,\,\Pi_{|s|}.$ (43)