# Quantitative bounds to propagation of quantum correlations in many-body
systems

Davide Girolami<EMAIL_ADDRESS>DISAT, Politecnico di Torino, Corso
Duca degli Abruzzi 24, Torino 10129, Italy Michele Minervini  DISAT,
Politecnico di Torino, Corso Duca degli Abruzzi 24, Torino 10129, Italy

###### Abstract

We investigate how much information about a quantum system can be
simultaneously communicated to independent observers, by establishing
quantitative limits to bipartite quantum correlations in many-body systems. As
recently reported in Phys. Rev. Lett. 129, 010401 (2022), bounds on quantum
discord and entanglement of formation between a single quantum system and its
environment, e.g., a large number of photons, dictate that independent
observers which monitor environment fragments inevitably acquire only
classical information about the system. Here, we corroborate and generalize
those findings. First, we calculate continuity bounds of quantum discord,
which set how much states with a small amount of quantum correlations deviate
from being embeddings of classical probability distributions. Also, we
demonstrate a universally valid upper bound to the bipartite entanglement of
formation between an arbitrary pair of components of a many-body quantum
system. The results confirm that proliferation of classical information in the
Universe suppresses quantum correlations.

## Introduction.

Quantum systems display correlations that cannot be explained by the laws of
classical probability [2, 1, 3]. Such a counterintuitive feature of the
quantum world signals a dramatic departure from what we perceive to be our
macroscopic reality. Also, quantum correlations promise to be the key
resources for quantum technologies, as they allow to overperform classical
devices in computing, communication, and sensing [4, 5, 6, 7, 8]. Indeed,
terms like “Entanglement” are becoming common parlance in many branches of
science.

The co-existence between classical and quantum regimes in our Universe, and
for all practical purposes between our laptops and future quantum computers,
can be explained within quantum theory, in terms of bounds to quantum
correlations. Classical information, i.e., the outcome of a measurement on a
physical system, can be freely communicated to an arbitrary number of
observers. That is, bits of information can be copied and simultaneously
distributed to an arbitrary large network of independent receivers, which can
then reach an agreement about the measured quantity. As a result, a prominent
feature of our description of the world is that properties of physical systems
acquire the status of “objective”.
Yet, fundamental results like the no-cloning theorem [9], and monogamy
relations of entanglement measures [10], suggested limits to broadcasting
quantum information, i.e., the wavefunction of a quantum system. Further
recent works have demonstrated constraints to the concurrent distribution of
quantum information from a single source to a network of observers, formalized
in terms of bounds to quantum correlations [16, 11, 13, 12, 14, 15]. Their
operational meaning is that the very quantum theory dictates that quantum
information cannot be concurrently stored and made available to independent
observers. Consequently, these agents cannot reach consensus on quantum
properties of the source.

These results support the core ideas underpinning Quantum Darwinism, a
genuinely quantum explanation of the emergence of a classical macroscopic
reality [17, 18, 19, 20, 21, 22, 23, 24, 25]. Interactions between physical
systems and their environment select pointer states [26], which encode
effectively classical information that can be copied and redundantly spread
into the environment. That is only kind of knowledge that can be acquired by
many observers at the same time. The reason is that such non-cooperating
observers obtain information about a system by eavesdropping on small,
distinct fractions of the system environment, i.e., scattered photons [27, 28,
29, 26].

In this paper, we review and extend the findings of Ref. [16]. As a
preliminary step, we recall quantitative bounds to the average bipartite
quantum discord [30], the most general kind of quantum correlation, and the
entanglement of formation [31], between a system of interest and fragments of
its environment. In particular, we show the emergence of the bound to the
entanglement of formation with a numerical study of the correlation pattern in
a star-like quantum network.
These bounds are universally valid (they hold for any global pure state of the
system and the environment), confirming that quantum Darwinism is a generic
feature of many-body quantum systems [11]. Further, they are easy to compute:
this is surprising, since the quantification of quantum correlations in
complex, multipartite systems is generally a hard problem [8, 33, 34, 32, 35,
36, 37, 38], and neither quantum discord nor the entanglement of formation are
monogamous [39, 10, 40, 41, 42]. Moreover, these upper limits are physically
meaningful: they are expressed in terms of measures of (dis)-agreement among
observers that eavesdrop on the environment about the received information,
which is inevitably classical. That is, whenever we reach consensus, a
defining feature of classical reality, quantum information is unaccessible.
Only an utopian observer able to intercept large fractions of the environment
(i.e., more than half of the scattered photons carrying relevant information
[16]) could establish non-negligible quantum correlations with the system
under scrutiny.

Then, we present new results. We firstly focus on quantum discord. We prove
continuity bounds nearby the set of states which describe classically
correlated systems. In particular, by employing the relative entropy as
(pseudo)-distance, we demonstrate that quantum discord takes small values for
density matrices that are close to the set of “classical-quantum” states [8].
The result implies that quantum information cannot be communicated via
interactions that can be described, with arbitrarily small error, by classical
physics. A spectacular example is, in fact, the measurements performed by we
humans on macroscopic objects.
Second, we generalize the upper bound to the bipartite entanglement of
formation. By introducing a new measure of (dis)-agreement among observers
about classical information, we derive a limit to the average entanglement
that an arbitrary component of a many-particle quantum system can share with
other parts. The larger the system, the smaller is the amount of entanglement
that can be locally established. Therefore, bipartite quantum correlations are
suppressed, even if the global state displays genuine multipartite
entanglement.

The paper is organized as follows. In Section I, we introduce the information-
theoretic measures of classical and quantum correlations that we will employ
here. In Section II, we review the main results of Ref. [16]. In Section III,
we will demonstrate the continuity bounds to quantum discord and a generalized
bound to the bipartite entanglement of formation in many-body systems. In the
Conclusion, we will outline our findings and suggest further questions that
are worthy of investigation.

##  I Measures of classical and quantum correlations

Consider a quantum Universe that consists of a quantum system ${\cal S}$ and
its $N$-partite environment ${\cal E}:=\cup_{i=1}^{N}\varepsilon_{i}$ (FIG.
1). We define an environment fragment of $k\leq N$ particles ${\cal
F}_{k}:=\cup_{\\#i=k}\varepsilon_{i}$ and its complement ${\cal E}_{/k}:={\cal
E}/{\cal F}_{k}$. In the following, we recall the definitions of widely
employed measures of classical and quantum correlations between ${\cal S}$ and
the fragment ${\cal F}_{k}$.

Being $H(\rho_{{\cal X}}):=-\mathrm{tr}\left\\{\rho_{\cal X}\log_{2}\rho_{\cal
X}\right\\}$ the von Neumann entropy of the state $\rho_{{\cal X}}$ of the
system ${\cal X}$, the statistical dependence between ${\cal S}$ and ${\cal
F}_{k}$ in the state $\rho_{{\cal S}{\cal F}_{k}}$ is given by the mutual
information

$\displaystyle I(\rho_{{\cal S}{\cal F}_{k}}):=H(\rho_{{\cal
S}})+H(\rho_{{\cal F}_{k}})-H(\rho_{{\cal S}{\cal F}_{k}}).$ (1)

The mutual information is the total information shared by two systems.
Remarkably, it splits into classical and quantum components [30, 43].
The classical part is constructed as follows. Suppose one performs a local
positive operator-valued measure (POVM)
$\mathbf{M}_{k}:=\left\\{\mathbf{M}_{\alpha},\sum_{\alpha}\mathbf{M}_{\alpha}^{\dagger}\mathbf{M}_{\alpha}=\mathbb{I}\right\\}$
on ${\cal F}_{k}$. The post-measurement state of the bipartition ${\cal
SF}_{k}$ is

$\rho_{{\cal
SF}_{k,\mathbf{M}_{k}}}=\sum_{\alpha}\left(\mathbb{I}\otimes\mathbf{M}_{\alpha}\right)\,\rho_{{\cal
SF}_{k}}\left(\mathbb{I}\otimes\mathbf{M}_{\alpha}^{\dagger}\right).$ (2)

Then, classical correlations are quantified as the maximal information about
${\cal S}$ an observer can extract by measurements on ${\cal F}_{k}$ [43, 44],
which is given by the maximal mutual information of the post-measurement
state:

$\displaystyle J\left(\rho_{{\cal S}\check{\cal
F}_{k}}\right):=\max\limits_{\mathbf{M}_{k}}I\left(\rho_{{\cal S}{\cal
F}_{k,\mathbf{M}_{k}}}\right).$ (3)

The maximal value of classical correlations, i.e., the maximal classical
information that can flow from ${\cal S}$ to an environment fragment, is
$H(\rho_{{\cal S}})$.
The quantum part of the mutual information, namely _quantum discord_ , is then
defined as the difference between pre- and post-measurement mutual information
[30]:

$D\left(\rho_{{\cal S}\check{\cal F}_{k}}\right):=I\left(\rho_{{\cal S}{\cal
F}_{k}}\right)-J\left(\rho_{{\cal S}\check{\cal F}_{k}}\right).$ (4)

This quantity is the minimal quantum information about ${\cal S}$ that is lost
by ${\cal F}_{k}$ when it is subject to a local measurement $\mathbf{M}_{k}$
[46, 45, 2].
Quantum discord has captured a lot of interest because of its peculiar
properties. It can exist without entanglement and, conversely to entanglement,
can be created by local operations and classical communication (LOCCs).
Therefore, it has been considered for some time an appealing alternative to
entanglement as a resource for quantum information processing [30, 42, 47, 48,
49, 50, 51].
Note that, for pure states of ${\cal SF}_{k}$, quantum discord is equal to the
entanglement entropy: $D\left(\rho_{{\cal S}\check{\cal
F}_{k}}\right)=D\left(\rho_{\check{\cal S}{\cal F}_{k}}\right)=H(\rho_{{\cal
S}})$, taking the maximal value $H(\rho_{{\cal F}_{k}})$ for maximally
entangled states [8]. Yet, for mixed states classical and quantum correlations
are in general not invariant under permutations of a bipartition components:
$J\left(\rho_{{\cal S}\check{\cal F}_{k}}\right)\neq J\left(\rho_{\check{\cal
S}{\cal F}_{k}}\right)$, and $D\left(\rho_{{\cal S}\check{\cal
F}_{k}}\right)\neq D\left(\rho_{\check{\cal S}{\cal F}_{k}}\right)$. Further,
$D\left(\rho_{{\cal S}\check{\cal F}_{k}}\right)=0$ does not imply
$D\left(\rho_{\check{\cal S}{\cal F}_{k}}\right)=0$.

Figure 1: We consider a quantum Universe in which a system ${\cal S}$
interacts with an $N$-particle environment ${\cal E}$. We investigate
fundamental bounds to bipartite quantum correlations, as quantified by quantum
discord and the entanglement of formation, between ${\cal S}$ and an
environment fragment ${\cal F}_{k}$.

Next, we review the definition of entanglement of formation of a state
$\rho_{{\cal SF}_{k}}=\sum_{\alpha}p_{\alpha}\rho_{\alpha,{\cal SF}_{k}}$
[31], with $\rho_{\alpha}=|\alpha\rangle\langle\alpha|$. It is obtained by
convex roof optimization of the entanglement entropy:

$\displaystyle E(\rho_{{\cal S}{\cal
F}_{k}}):=\min\limits_{\left\\{p_{\alpha},\rho_{\alpha}\right\\}}-\sum_{\alpha}p_{\alpha}\text{tr}\left\\{\text{tr}_{{\cal
S}}\left\\{\rho_{\alpha,{\cal SF}_{k}}\right\\}\log_{2}\text{tr}_{{\cal
S}}\left\\{\rho_{\alpha,{\cal SF}_{k}}\right\\}\right\\}.$ (5)

There exists a surprising trade-off relation between the entanglement of
formation and classical correlations in tripartite systems, discovered by
Koashi and Winter [39]:

$E(\rho_{{\cal S}{\cal F}_{k}})\leq\,H(\rho_{{\cal S}})-J\left(\rho_{{\cal
S}\check{\cal E}_{/k}}\right).$ (6)

The inequality is saturated for pure states of the Universe ${\cal SE}$. There
is no loss of generality in this assumption: every mixed state of the Universe
can be purified by dilation. The result has been employed to derive
quantitative relations between quantum discord and entanglement of formation
[52].

##  II Quantitative bounds to bipartite quantum correlations in many-body
systems

In this Section, we review the main results of Ref. [16]. We focus on setting
bounds to correlations between ${\cal S}$ and single-site subsystems
$\varepsilon_{i}$. The results apply to fragments of arbitrary size ${\cal
F}_{k}$ straightforwardly.
Classical information about ${\cal S}$ can be freely cloned and simultaneously
distributed to the environment fragments. For instance, consider creation of
classical correlations in a three-bit register by a XOR gate,
$|0\rangle\langle 0|_{{\cal F}_{k}}(|00\rangle\langle 00|+|11\rangle\langle
11|)/2_{{\cal SE}_{/k}}\rightarrow(|000\rangle\langle 000|+|111\rangle\langle
111|)/2_{{\cal F}_{k}{\cal SE}_{/k}}$. More generally, one can saturate the
inequality $\bar{J}\left(\rho_{{\cal
S}\check{\varepsilon}_{i}}\right):=\frac{1}{N}\sum_{i=1}^{N}J\left(\rho_{{\cal
S}\check{\varepsilon}_{i}}\right)\leq H(\rho_{{\cal S}})$.

Figure 2: There exist upper limits to bipartite quantum correlations between a
system ${\cal S}$ and the environment subsystems $\varepsilon_{i}$s of ${\cal
E}$. Proliferation of classical information, as quantified by the amount of
consensus about ${\cal S}$ that can be reached by observers eavesdropping on
$\varepsilon_{i}$s, destroys quantum discord and entanglement.

Conversely, quantum correlations are restricted by the very same quantum laws.
The inner mechanism suppressing quantum information is the creation of
consensus among a large number of observers that access copies of classical
information deposited in different environment fragments (FIG. 2). Let us
quantify the average (dis)-agreement about the classical information on ${\cal
S}$ that an observer tracking a particle $\varepsilon_{i}$ experiences with
another agent that accesses the rest of the environment ${\cal E}_{/i}$: one
can define the parameters

$\displaystyle\delta:=$ $\displaystyle\sum_{i=1}^{N}\delta_{i}/N,$ (7)
$\displaystyle\delta_{i}:=$ $\displaystyle\ \frac{J\left(\rho_{{\cal
S}\check{{\cal E}}}\right)-\min\left\\{J\left(\rho_{{\cal
S}\check{\varepsilon}_{i}}\right),J\left(\rho_{{\cal S}\check{\cal
E}_{/i}}\right)\right\\}}{H(\rho_{{\cal S}})}\,\in\,[0,1].$

We briefly discuss why the index $\delta_{i}$ is a good measure of the (lack
of) consensus between two observers monitoring $\varepsilon_{i}$ and ${\cal
E}_{/i}$, respectively. Assume $J\left(\rho_{{\cal S}\check{{\cal
E}}_{/i}}\right)\geq J\left(\rho_{{\cal S}\check{\varepsilon}_{i}}\right)$. If
$\delta_{i}=0$, then $J\left(\rho_{{\cal S}\check{{\cal
E}}}\right)=J\left(\rho_{{\cal S}\check{{\cal
E}}_{/i}}\right)=J\left(\rho_{{\cal S}\check{\varepsilon}_{i}}\right)$. The
reverse implication is also true. Hence, the parameter $\delta_{i}$ is zero if
and only if the same classical information about ${\cal S}$ is simultaneously
available into $\varepsilon_{i}$ and ${\cal E}_{/i}$. That is, if and only if
observers measuring on the two environment fragments are in perfect agreement.
Further, if $\delta_{i}=1$, then $J\left(\rho_{{\cal
S}\check{\varepsilon}_{i}}\right)=0$, and there is maximal disagreement
between the observers. The reverse statement holds too.

Introducing a measure of (lack of) objectivity about classical information is
instrumental in proving a bound to bipartite quantum discord in many-body
systems for any pure state of the universe $|\psi\rangle_{{\cal SE}}$:

$\displaystyle\bar{D}\left(\rho_{{\cal S}\check{\varepsilon}_{i}}\right):=$
$\displaystyle\frac{1}{N}\sum_{i=1}^{N}D\left(\rho_{{\cal
S}\check{\varepsilon}_{i}}\right),$ $\displaystyle\bar{D}\left(\rho_{{\cal
S}\check{\varepsilon}_{i}}\right)\leq$ $\displaystyle\,\delta\,H(\rho_{{\cal
S}}).$ (8)

Therefore, consensus about classical information, i.e., the emergence of
classical objectivity about properties of ${\cal S}$ by indirect observation
(intercepting fragments of the environment), suppresses quantum correlations.
An equivalent bound holds for the entanglement of formation. By employing the
Koashi-Winter inequality in Eq. (6), a few algebra steps show that

$\displaystyle E(\rho_{{\cal S}\varepsilon_{i}})\leq$
$\displaystyle\,\delta_{i}\,H(\rho_{{\cal S}}),$ (9)
$\displaystyle\bar{E}\left(\rho_{{\cal S}\varepsilon_{i}}\right):=$
$\displaystyle\frac{1}{N}\sum_{i=1}^{N}E\left(\rho_{{\cal
S}\varepsilon_{i}}\right)\leq\,\delta\,H(\rho_{{\cal S}}).$

Figure 3:  We show the bound to the average entanglement of formation (Eq.
(9)) in action. The following quantities are computed in the final state ${\bf
U}_{{\cal SE}}(a)|+\rangle_{\cal S}|0\rangle_{\cal E}^{\otimes N}$, for
different values of $N$ (see the main text for full details): the average
entanglement of formation, $\bar{E}\left(\rho_{{\cal
S}\varepsilon_{i}}\right)$ (blue line —); the upper bound in Eq. (9) (dashed
blue line
$\mathbin{\vbox{\hbox{\rule{1.55pt}{1.55pt}}}}\mathbin{\vbox{\hbox{\rule{1.55pt}{1.55pt}}}}\mathbin{\vbox{\hbox{\rule{1.55pt}{1.55pt}}}}$);
average classical correlations $\bar{J}\left(\rho_{{\cal
S}\check{\varepsilon}_{i}}\right)$ (black line —). The average values of
classical and quantum correlations are the ones computed for an arbitrary pair
${\cal S}\varepsilon_{i}$, because of the symmetry under environment subsystem
permutations of the final state of the Universe. The newfound bound is
informative in the limit $a\rightarrow 0$, when the global state is highly
entangled. Proliferation of a larger amount of classical correlations
suppresses quantum correlations. Comparing these results with Fig. 3 of
Ref.[16], we note that the entanglement of formation declines by increasing
$N$ much faster than quantum discord.

We elucidate the bound with a numerical study. We consider the quantum
Universe to be in the initial uncorrelated state $|+\rangle_{{\cal
S}}|0\rangle^{\otimes N}_{{\cal E}}$. Then, one applies the unitary ${\bf
U}_{{\cal SE}}(a)\equiv\Pi_{i=1}^{N}{\bf U}_{{\cal S}\varepsilon_{i}}(a)$,
where the two-site transformation ${\bf U}_{{\cal S}\varepsilon_{i}}(a)$ is
the “c-maybe” gate
$\mathbb{I}_{2}\oplus\left(\begin{smallmatrix}a&\sqrt{1-a^{2}}\\\
\sqrt{1-a^{2}}&-a\end{smallmatrix}\right),a\in[0,1],$ on ${\cal
S}\varepsilon_{i}$ [14]. This dynamics models the interaction of a quantum
system ${\cal S}$ with a large photonic environment [53, 18].
We calculate bipartite classical correlations and the entanglement of
formation in the marginal density matrix $\rho_{{\cal S}\varepsilon_{i}}$ of
the final state. Their values can be computed analytically [54, 14]. The
results, which we plot in FIG. 3, display how the entanglement of formation
obeys a “weak monogamy relation” dictated by the abundance of classical
information about ${\cal S}$ simultaneously available throughout the
environment, as defined by the bound in Eq. (9). For $a\rightarrow 0$, the
universe comes close to be in a (generalized) GHZ state and such a behaviour
is magnified: quantum correlations vanish, while classical information
proliferation is maximized.

##  III Extending and clarifying limits to quantum information propagation

###  III.1 Behavior of quantum discord in the proximity of classical states

In this Section, we derive new results that show how bipartite quantum
correlations are restricted in many-body systems. We observe that the results
outlined in the previous section imply that, if $\delta=0$, and therefore
$J\left(\rho_{{\cal S}\check{\varepsilon}_{i}}\right)=H(\rho_{\cal
S}),\,\forall\,i$, then there is no environment fragment that can share
quantum discord with ${\cal S}$. We prove a statement about the degenerate
case of this scenario: all the subsystems store the very same amount of
classical information about ${\cal S}$, but its value is zero, i.e., no
classical correlations exist.
Remark: There are not quantum correlations without classical correlations:

$\displaystyle J\left(\rho_{{\cal
S}\check{\varepsilon}_{i}}\right)=0\Rightarrow D\left(\rho_{{\cal
S}\check{\varepsilon}_{i}}\right)=0.$ (10)

Proof – This claim can be proved in several ways. For example, from the
Koashi-Winter relation, it follows that $J\left(\rho_{{\cal
S}\check{\varepsilon}_{i}}\right)=0\Rightarrow E(\rho_{{\cal
SE}_{/i}})=D\left(\rho_{{\cal S}\check{\cal E}_{/i}}\right)=H({\cal S})$.
Since $E(\rho_{{\cal SE}_{/i}})+E\left(\rho_{{\cal
S}{\varepsilon}_{i}}\right)=D\left(\rho_{{\cal
S}\check{\varepsilon}_{i}}\right)+D(\rho_{{\cal S}\check{\cal E}_{/i}})$ [52],
one has $D\left(\rho_{{\cal S}\check{\varepsilon}_{i}}\right)=0$.

Next, we explore more nuanced aspects of the transition from quantum to
classical regimes. We ask whether quantum discord is “continuous”, in the
sense of taking small values for states that are geometrically close (and
physically similar) to classically correlated density matrices. The bound in
Eq. (II) establishes that simultaneous maximal classical correlations between
${\cal S}$ and each $\varepsilon_{i}$ destroy quantum discord throughout the
Universe. Hence, quantum information about ${\cal S}$ is not accessible to
independent observers that monitor different $\varepsilon_{i}$. Proving that
quantum discord is subject to sharp continuity bounds at the frontier with
classical states would mean that, whenever a classical description of the
correlation pattern is sufficiently precise, quantum correlations are
inevitably negligible. That is, classical objectivity and a significant amount
of quantum correlations cannot co-exist.
It is known that $D\left(\rho_{{\cal S}\check{\cal F}_{k}}\right)=0$ if and
only if there exists a measurement $\mathbf{M}_{k}$ such that $\rho_{{\cal
SF}_{k}}=\rho_{{\cal SF}_{k,\mathbf{M}_{k}}}$. We here prove continuity bounds
to quantum discord about the zero value. First we show that if a state of a
partition ${\cal SF}_{k}$ (which we assume to be a full rank density matrix)
is close to the set of post-measurement states $\rho_{{\cal
SF}_{k,\mathbf{M}_{k}}}$, then its discord is small. Given the subset of the
projective measurements $\\{\mathbf{P}_{k}\\}\subset\\{\mathbf{M}_{k}\\}$
which can be performed on ${\cal F}_{k}$, recalling the definition of relative
entropy $H\left(\rho_{\cal X}||\rho_{{\cal Y}}\right):=\text{Tr}\\{\rho_{\cal
X}\log_{2}\rho_{\cal X}\\}-\text{Tr}\\{\rho_{\cal X}\log_{2}\rho_{\cal Y}\\}$,
one has

$\displaystyle D\left(\rho_{{\cal S}\check{\cal F}_{k}}\right)$
$\displaystyle\leq\min_{\mathbf{P}_{k}}\left\\{I\left(\rho_{{\cal S}{\cal
F}_{k}}\right)-J\left(\rho_{{\cal S}\check{\cal F}_{k}}\right)\right\\}$ (11)
$\displaystyle=\min_{\mathbf{P}_{k}}\left\\{H\left(\rho_{{\cal S}{\cal
F}_{k}}||\rho_{{\cal S}}\otimes\rho_{{\cal F}_{k}}\right)-H\left(\rho_{{\cal
S}{\cal F}_{k,\mathbf{P}_{k}}}||\rho_{{\cal S}}\otimes\rho_{{\cal
F}_{k,\mathbf{P}_{k}}}\right)\right\\}$
$\displaystyle=\min_{\mathbf{P}_{k}}\left\\{H\left(\rho_{{\cal S}{\cal
F}_{k}}||\rho_{{\cal S}{\cal F}_{k,\mathbf{P}_{k}}}\right)-H\left(\rho_{{\cal
F}_{k}}||\rho_{{\cal F}_{k,\mathbf{P}_{k}}}\right)\right\\}$
$\displaystyle\leq\min_{\mathbf{P}_{k}}\left\\{H\left(\rho_{{\cal S}{\cal
F}_{k}}||\rho_{{\cal SF}_{k,\mathbf{P}_{k}}}\right)\right\\}.$

Finally, we obtain

$\min_{\mathbf{P}_{k}}\left\\{H\left(\rho_{{\cal S}{\cal F}_{k}}||\rho_{{\cal
SF}_{k,\mathbf{P}_{k}}}\right)\right\\}\leq\epsilon\Rightarrow
D\left(\rho_{{\cal S}\check{\cal
F}_{k}}\right)\leq\epsilon,\forall\,\epsilon.$ (12)

Therefore, states that are geometrically close ($\epsilon\rightarrow 0$) to be
embeddings of classical probability distributions (classical-quantum states)
display small values of quantum discord.
For the sake of completeness, we calculate the maximal relative entropy
between a state and the closest classically correlated state when an upper
bound to quantum discord, which we obtain by maximizing in Eq. (3) over
projective measurements rather than POVMs, takes arbitrary small values. As a
preliminary step, we recall an upper limit to the relative entropy between two
arbitrary states [55]:

$\displaystyle H\left(\rho_{{\cal X}}||\rho_{{\cal Y}}\right)$
$\displaystyle\leq(\lambda_{min}(\rho_{{\cal Y}})+d_{{\cal X},{\cal
Y}})\,\log\left(1+\frac{d_{{\cal X},{\cal Y}}}{\lambda_{min}(\rho_{{\cal
Y}})}\right)$ (13) $\displaystyle-\lambda_{min}(\rho_{{\cal
X}})\,\log\left(1+\frac{d_{{\cal X},{\cal Y}}}{\lambda_{min}(\rho_{{\cal
X}})}\right),$ $\displaystyle d_{{\cal X},{\cal Y}}$
$\displaystyle\equiv||\rho_{{\cal X}}-\rho_{{\cal Y}}||_{1}/2,$

in which $\lambda_{min}(\rho_{{\cal X}})$ is the smallest eigenvalue of
$\rho_{{\cal X}}$. Then, calling $\tilde{\mathbf{P}}_{k}$ the projective
measurement performed on ${\cal F}_{k}$ that maximizes the post-measurement
mutual information (see Eq. (3)), we obtain

$\displaystyle H\left(\rho_{{\cal S}{\cal F}_{k}}||\rho_{{\cal S}{\cal
F}_{k,\tilde{\mathbf{P}}_{k}}}\right)-H\left(\rho_{{\cal F}_{k}}||\rho_{{\cal
F}_{k,\tilde{\mathbf{P}}_{k}}}\right)$ $\displaystyle\leq\epsilon\Rightarrow$
$\displaystyle H\left(\rho_{{\cal S}{\cal F}_{k}}||\rho_{{\cal S}{\cal
F}_{k,\tilde{\mathbf{P}}_{k}}}\right)$
$\displaystyle\leq\epsilon+H\left(\rho_{{\cal F}_{k}}||\rho_{{\cal
F}_{k,\tilde{\mathbf{P}}_{k}}}\right)$ $\displaystyle H\left(\rho_{{\cal
S}{\cal F}_{k}}||\rho_{{\cal S}{\cal F}_{k,\tilde{\mathbf{P}}_{k}}}\right)$
$\displaystyle\leq\epsilon+f\left(\rho_{{\cal
F}_{k}},\tilde{\mathbf{P}}_{k}\right),$ (14)

where

$\displaystyle f\left(\rho_{{\cal F}_{k}},\tilde{\mathbf{P}}_{k}\right)=$
$\displaystyle\left\\{\lambda_{min}\left(\rho_{{\cal
F}_{k,\tilde{\mathbf{P}}_{k}}}\right)+d_{{\cal F}_{k},{\cal
F}_{k,\tilde{\mathbf{P}}_{k}}}\right\\}\,\log\left\\{1+\frac{d_{{\cal
F}_{k},{\cal F}_{k,\tilde{\mathbf{P}}_{k}}}}{\lambda_{min}\left(\rho_{{\cal
F}_{k,\tilde{\mathbf{P}}_{k}}}\right)}\right\\}$
$\displaystyle-\lambda_{min}\left(\rho_{{\cal
F}_{k}}\right)\,\log\left(1+\frac{d_{{\cal F}_{k},{\cal
F}_{k,\tilde{\mathbf{P}}_{k}}}}{\lambda_{min}(\rho_{{\cal F}_{k}})}\right).$

This constraint is certainly less neat than Eq. (12) for generic mixed states.
We leave to future studies to shape this claim, as we conjecture that a
cleaner continuity bound may exist.

###  III.2 Generalized bound to the entanglement of formation

We now investigate how the bound to the bipartite entanglement in a star-like
configuration (Eq. (9)) can be generalized. We focus on the correlation
structure of the environment ${\cal E}$, which is a generic $N$-partite
quantum system. We show that there is an upper bound to the bipartite
entanglement of formation between two components of the environment, in terms
of how much classical information is shared by the environment parts.
We define a new disagreement quantifier:

$\displaystyle\delta^{\varepsilon}_{i}:=1-\frac{\min\limits_{\varepsilon_{j}}J\left(\rho_{\varepsilon_{i}\check{\varepsilon}_{j}}\right)}{H(\rho_{\varepsilon_{i}})}\,\in\,[0,1].$
(15)

The quantity manifestly enjoys the same properties of the parameter introduced
in Eq. (7). Then, the entanglement of formation between an environment
subsystem $\varepsilon_{i}$ and any other subsystem is limited by the (lack
of) consensus about measurement outcomes (classical information) on
$\varepsilon_{i}$ across the environment. By employing again the Koashi-Winter
relation, one has

$\displaystyle
J\left(\rho_{\varepsilon_{i}\check{\varepsilon}_{j}}\right)\geq$
$\displaystyle\,(1-\delta^{\varepsilon}_{i})\,H(\rho_{\varepsilon_{i}}),\,\forall\,j\Rightarrow$
$\displaystyle E\left(\rho_{\varepsilon_{i}{\cal E}_{/i}}\right)\leq$
$\displaystyle\,\delta^{\varepsilon}_{i}\,H(\rho_{\varepsilon_{i}}),\,\forall\,i\Rightarrow$
$\displaystyle E\left(\rho_{\varepsilon_{i}\varepsilon_{j}}\right)\leq$
$\displaystyle\,\delta^{\epsilon}_{i}\,H(\rho_{\varepsilon_{i}}),\,\forall
i,j.$ (16)

The bound is clearly saturated, for example, for the GHZ state.

## Conclusion

We have investigated quantitative limits to the propagation of quantum
information in many-body systems. Specifically, we have extended the results
of [16], calculating a continuity bound to quantum discord nearby classical
states (Eq. (12)), and proving an upper bound to the entanglement of formation
(Eq. (III.2)) between two arbitrary components of a multipartite system.
Classical correlations are not subject to any limitations. Consequently,
classical information can be freely broadcast from a source to an arbitrary
number of receivers. Yet, the very same possibility that observers can reach
consensus on such classical information of target physical systems dictates
bounds to quantum information, which are here formulated in terms of limits to
quantum discord and the entanglement of formation.
The results further corroborate the key ideas of Quantum Darwinism, a
theoretical framework that explain the emergence of classical reality within
quantum mechanics. We hope these findings will propel further studies on the
subtleties of the transition between the quantum and classical regimes, which
may lead to derive stronger bounds than the one here demonstrated. Also,
quantitative limits to genuinely multipartite quantum correlations may exist
[56, 57].

###### Acknowledgements.

This research was supported by the Italian Ministry of Research, grant number
MUR-PINR2022, Contract Number NEThEQS (2022B9P8LN).

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