Can multipartite entanglement be characterized by two-point connected correlation functions ?

We discuss under which conditions multipartite entanglement in mixed quantum states can be characterized only in terms of two-point connected correlation functions, as it is the case for pure states. In turn, the latter correlations are defined via a suitable combination of (disconnected) one- and two-point correlation functions. In contrast to the case of pure states, conditions to be satisfied turn out to be rather severe. However, we were able to identify some interesting cases, as when the point-independence is valid of the one-point correlations in each possible decomposition of the density matrix, or when the operators that enter in the correlations are (semi-)positive/negative defined.

Estimating multipartite entanglement for mixed states via the QFI can be difficult in general, and in stronglyinteracting systems it has been performed only in a limited number of cases, mainly at equilibrium [42,45], also establishing a direct link with dynamical susceptibility [33].The main reason is that the QFI for a mixed state cannot be fully expressed in terms of (one-and two-point) correlation functions, instead it requires a sum over matrix elements with the respect to all the states diagonalizing the full density matrix [42] (see the next Section for a precise definition).
In this work, focusing on mixed quantum states, we discuss physical and mathematical conditions that make it possible to bound multipartite entanglement, via oneand two-point correlation functions.The conditions to be satisfied turn out to be rather severe.However, we were able to identity at least two physically relevant situations, such as when the point-independence is valid of the one-point correlations in each possible decomposition of the density matrix or when the operators that enter in the correlations are (semi-)positive/negative defined.

II. MULTIPARTITE ENTANGLEMENT A. Separability and producibility
For a d-dimensional discrete system with N compo-arXiv:2108.03605v3[quant-ph] 5 Oct 2023 nents (e.g., sites), c-partite entanglement, with 1 ≤ c ≤ N , implies that a partition {|ψ i ⟩} exists, where the maximum number of components in a single |ψ i ⟩ is c.The tensor-product state |ψ⟩ = ⊗i |ψ i ⟩ is then said to be c-producible [4], or to have entanglement depth c.In addition, a system is said to host c-partite entanglement if it is c-producible but not (c + 1)-producible.Instead, the number h, with N c ≤ h ≤ N − c + 1, of disentangled subsets is the degree of separability [6,27].The usual separability corresponds to h = N and c = 1.On a lattice, the subsystems are not necessarily adjacent sites.When c = N , |ψ⟩ is said to host genuine ME [58].
For mixed states, c-producibility in h subsets holds if ρ can be decomposed (generally not uniquely) as where p λ > 0 without any lack of generality, and | λ⟩ are c-separable states in h subsets, not necessarily with the same space-partition.If c = N , then Eq. ( 1) is still valid, trivially with a single partition and in every decomposition.In general, the c-producible decomposition | λ⟩ is not orthogonal, thus ρ is not diagonal.Moreover, the producibility of | λ⟩ is generally lost in other decompositions.

B. Bounds for multipartite entanglement
First, we focus on pure states |ψ⟩.We denote by x, y two sites of the lattices and by ô(x), ô(y) local operators based on these sites.In this way, the variance of the Hermitian operator on |ψ⟩ is defined as [4] V being two-point connected correlations [59].V [|ψ⟩, Ô] N coincides with the QFI for pure states [4].In the following, the QFI for pure states will be also denoted as Importantly, the same quantity is a witness of ME for these states.This means that, if (c, h)-entanglement, but not (c + 1, h ′ )-entanglement (with h ′ ≥ N c+1 ), is present, then the inequality holds [27].Indeed, (5) bounds the quantum advantage offered by (c, h)-entanglement in terms of the sensiti-vity, with respect to the shot-noise (separable) limit, More in detail, (5) generates other relevant bounds [27] : for instance, it can be extended choosing h = N c , which yields the bound for c-producibility V [|ψ⟩, Ô] N ≤ 4 k c N [4,26].This implies that its violation signals at least (c + 1)-partite entanglement [8,9].The ultimate limit V [|ψ⟩, Ô] N = 4 k N 2 , when |ψ⟩ hosts genuine ME, is called the Heisenberg limit.Another similar estimator for c, generally supposed to be not an integer, was found in [28], see SM 1.Similarly, one can maximize in c the righthand term of Eq. ( 5), setting c = N −h+1 and obtaining the bound for h : [27,60].
Finally, we notice that, critically, if two lattice sites x and y belong to different partitions and |ψ⟩ is producible, then ⟨ψ|ô(x)ô(y)|ψ⟩ c = 0, for every conceivable local operator ô, see e. g. [61,62].This property can be exploited to demonstrate the bounds of producibility for pure states recalled above, see SM 2. Now, we focus on mixed states.We define as the average variance in a producible decomposition, as in Eq. ( 1), while the average variance in a generic decomposition {p λ , |λ⟩} of ρ (functionally defined as in Eq. ( 6) will be again denoted generically as V [ρ, Ô] N .The corresponding functionals involving the sums of the modula of the connected correlations will be denoted as Starting from Eq. ( 1) and exploiting the bound in Eq. ( 5), we have that, see [27], the inequality in Eq. ( 5) holds also for V [ρ, Ô] N in Eq. ( 6), the average variance in the presence of (c, h)-entanglement for the density matrix ρ in Eq. ( 1), since λ pλ = 1 [63,64].Actually, (c, h)-entanglement is a sufficient but not necessary condition for the validity of Eq. (7).Moreover, maximizing in h the right-hand term in Eq. ( 7), the more common bound [4] V is obtained.Correspondingly, maximizing in c the righthand term in Eq. (7), Eq. ( 7) is obtained exploiting the producibility of all the states | λ⟩, with the same c and h (but not necessarily the same space-partition), so that the bound in Eq. ( 5) holds for all of them.Instead, no orthonormality hypothesis of the | λ⟩ set is required.Mo-reover, note that the average variance for mixed states in a producible decomposition is related to the pure state variance Eq. ( 33) via In this way, V [ρ, Ô] N also saturates the corresponding convexity inequality valid in a generic decomposition [4].As it will be required later on, we introduce the quantity The same bounds in Eqs. ( 7) and ( 8) also hold, in the presence of (c, h)-entanglement for the density matrix ρ in Eq. ( 1), for , both for pure states and mixed states.The proof of the latter statement follows from the derivation presented in SM 2, where the bound V ≤ 4 k c N is shown for pure states.Then, the extension to mixed states is done as from Eq. ( 7) to Eq. ( 6).
as formalized in the so-called "convex roof theorem" [65,66] : for chosen ρ and Ô, the resulting QFI is the minimum average variance between all the possible decompositions {p λ , |λ⟩} of ρ.The same property allows to demonstrate immediately that the bounds in Eqs.(7) and (8) also hold for the QFI, as it is well known in literature [4].We also stress that the convex inequality in Eq. ( 10) is fulfilled also by the QFI, that is saturated for pure states.In a (spectral) decomposition |n⟩, where ρ is diagonal, the QFI, denoted in the following as F [ρ, Ô] N , is written as [4] The described non-trivial relation between the QFI F [ρ, Ô] N and the average variances V [ρ, Ô] N calculated in generic decompositions {p λ , |λ⟩} can be illustrated considering for instance a one-dimensional array of N = 6 spin-1/2, described by the XXZ Heisenberg Hamiltonian in a transverse magnetic field, where J x and J z are exchange interactions, and 2 , is conserved, a fact exploited in [45] to compute the QFI efficiently.Due to this symmetry, the operator Ô = S x has ⟨λ| Ô|λ⟩ = 0, ∀ |λ⟩ with definite S z .To analyze a mixed state scenario, we consider Markovian dissipation, according to a Gorini-Kossakowski-Sudarshan-Lindblad master equation [68][69][70], with local spin-flip and spin-dephasing noise described by jump operators L m = {σ m } respectively.Dissipation rates are denoted by γ Sx and γ Sz .Fig. 1 shows the time evolution of the different functionals F [ρ, Ô] N and V [ρ, Ô] N (here both calculated in the diagonal decomposition), starting from the ground state of H, with S z = 0, in the presence of two forms of dissipation.In Fig. 1 (a), we include dissipation, as As the S z symmetry is conserved throughout the evolution, ⟨λ|σ In contrast, when the dissipation does not preserve the magnetization, as shown in Fig. 1 (b), then the QFI quickly differs from V [ρ, Ô] N , the last quantity being higher in value, as expected from the convex roof theorem.

III. DISCUSSION OF CONDITIONS TO ESTIMATE MULTIPARTITE ENTANGLEMENT BY CORRELATION FUNCTIONS A. The general problem
The producible decompositions | λ⟩ and p λ, where V [ρ, Ô] N is defined as in (6), are not generally known a priori.Thus, in order to be a useful witness of ME, V [ρ, Ô] N is required to be calculable without the knowledge of | λ⟩.To focus on this central problem, it is use- dissipation.The observables for all panels are shown in the legend of (a).We also defined (the first term in Eq. (12) , where ρ and ô(x) are meant to be expressed in a generic basis |α⟩ of the Hilbert space, possibly orthonormal, as | λ⟩ = α c λ α |α⟩.This expression does not depend explicitly on | λ⟩, thus it is covariant, and invariant in value, under changes of decomposition.This transformation is realized via a unitary operator U , acting as (see SM 3) : In contrast, the second term is neither invariant nor covariant under the transformation U in general, hence to evaluate it requires the explicit knowledge of | λ⟩.In turn, the non-covariance of V2 [ρ, Ô] N makes it difficult to calculate V [ρ, Ô] N in a generic decomposition, and to use it as an efficient ME estimator.The same problem occurs in calculating the QFI, via sup {p λ ,|λ⟩} λ p λ ⟨λ| Ô|λ⟩ 2 in Eq. ( 12).This is the main reason why the cumbersome expression in Eq. ( 13) must be used, in general.
B. Two conditions for the calculability of Eq. (6) In the present subsection, we discuss two situations when the quantity (18) in Eq. ( 12) can be calculated, in a generic decomposition |λ⟩, in spite of the difficulties mentioned above.
-Let first consider Ô as a semipositive or seminegative defined operator in the considered Hilbert space.We recall that an operator ô is called positive definite on a given Hilbert state if, for any vector |v⟩ on this space, ⟨v|ô|v⟩ > 0. Similarly, it is called semi-positive definite if ⟨v|ô|v⟩ ≥ 0. We mention for instance collective spin operators S (x,y,z) 2 = i s In this condition, if ⟨λ| Ô|λ⟩ = 0, for any state |λ⟩ in a chosen decomposition, then the quantity in Eq. ( 18) vanishes in any other decomposition, since also λ ⟨λ| Ô|λ⟩ = Tr ρ Ô = 0 and Tr ρ Ô is invariant under changes of decompositions.
-Second, let us consider states with the property that is independent of x, for the chosen local operator ô(x) and for every |λ⟩ of a certain decomposition.Translationally invariant states can be included, see more details below.In this condition, the quantity in Eq. ( 18) is evaluated as follows.Defining Tr[ρ ô(x)ô(y)] ≡ c(x, y) (appearing in V1 [ρ, Ô] N , Eq. ( 15), and always depending on x and y), then it is possible to recast the average variance V [ρ, Ô] N , in the present decomposition {p λ , |λ⟩}, as The conditions for the validity of Eq. ( 20) will be discussed in the next subsection.
Although evaluated in a specific decomposition, this functional will be used to determine conditions under which it is possible to bound multipartite entanglement for the mixed state described by ρ.This result will be shown in the next Section, as well as the relation of the same functional with V [ρ, Ô] N in Eq. ( 6) and with the QFI, F [ρ, Ô] N .
C. Derivation of Eq. (19) In this Section, we derive and motivate Eq. (21).In order to perform this task, we have to discuss first the conditions for the validity of Eq. (20).For this purpose, beyond Eq.( 19), the second property that we assume at first is : if |x − y| → ∞ [71][72][73].However, this property will be relaxed in the following subsections, since eventually it will not be strictly required to justify Eq. ( 21) as an entanglement witness.
Notice that the simultaneous validity of Eqs. ( 19) and ( 22) requires strictly that ⟨λ|ô(x)ô(y)|λ⟩ does not depend on the unit vector of x − y, nor on x and y themselves, at least if |x − y| → ∞.Sufficient, but not necessary, conditions for this scenario are rotational invariance and again translational invariance.
We also stress that, in order to guarantee that the onepoint correlations are not point-dependent, here translational invariance must be understood for every translation a i , connecting two sites of the lattice and for every state |λ⟩.For regular lattices, these translations decompose into sums of translations inside a unit cell and primary lattice vectors, see e. g. [74].Finally, if the condition in Eq. ( 22) holds for every local operator ô(x), this is called "cluster decomposition" in the literature.
Importantly, Eq. ( 22) does not imply at all that Tr ρ ô(x)ô(y) tends, even for large space separations, to the product Tr ρ ô(x)) Tr ρ ô(y) : this can be seen also as an effect of the classical weights p λ , not encoding entanglement.Moreover, the fact that o λ is not x-dependent implies the same fact for Tr ρ ô(x)) = λ p λ o λ .The opposite implication does not hold in general, requiring further assumptions, as translational invariance.
We also comment that the latter property sets a tight limitation to the degree of producibility c, even for a single pure state |ψ⟩ : indeed, consider two adjacent points x and y : as argued in Section II B, if they belong to the same entangled subset |ψ i ⟩, then ⟨λ|ô(x)ô(y)|λ⟩ c is nonzero in general, otherwise the same quantity is forced to vanish.Translational invariance, for every translation a i , connecting two sites of the lattice, implies From the discussion in Section II B, the latter equality implies immediately that c = 1 or c = N .More involved scenarios would be possible instead if translational invariance was allowed only for a subset of the entire set of lattice-translation {a i }.
Under the two conditions in Eqs. ( 19) and ( 22), the quantity in Eq. ( 18) can be evaluated straightforwardly, leading to Eqs. (20) and (21) (see more details in SM 5).We stress that the same derivation exploits that all the states of the decomposition |λ⟩ fulfill Eq. (22).However, the same property ceases to hold in general, as the decomposition is changed along Eq. ( 16).Consequently, one can verify that Eq. ( 21) is not equal to V [ρ, Ô] N nor to F [ρ, Ô] N , in general.Nevertheless, the same functional (21) will reveal still useful to bound multipartite entanglement in any decomposition.

D. Comments on the space independence of the one-point correlations
Here, having in mind the point ii) of Section III B, we discuss under which conditions the independence of the one-point correlations on the point itself can hold in every decomposition.
At first, we consider, in a given decomposition |λ⟩, the matrix elements ⟨λ|ô(x)|λ⟩ and ⟨λ|ô(y)|λ⟩, y ̸ = x, and we assume the stronger condition of translational invariance.Note that we do not a priori require that these conditions hold in every decomposition.We can interpolate between the two matrix elements above, writing : where Ty−x = e i P (y−x) is a unitary translation operator, generated by the momentum operator P .Here y − x is a multiple number of the lattice steps, possibly even in different primary directions for regular lattices.If |λ⟩ in translationally invariant, then Ty−x |λ⟩ = e i k (y−x) |λ⟩, k being the momentum quantum number.The space-dependent phases cancel out in the matrix element ⟨λ|ô(y)|λ⟩, so that, in the end : as expected.Notice that, critically, the same cancellation does not occur in general for the off-diagonal matrix elements ⟨λ ′ |ô(y)|λ⟩, |λ ′ ⟩ ̸ = |λ⟩.Consider now a second decomposition |η⟩ : this is obtained from |λ⟩ as in Eq. ( 16), so that : Importantly, the involved unitary matrix U η λ is not space dependent, as well as the numerical factors √ p η and √ p λ .Therefore, the only difference between the elements ⟨η|ô(y)|η⟩ and ⟨η|ô(x)|η⟩ can come from the phases e i k (y−x) , e i k ′ (y−x) , not canceling out in the off-diagonal matrix elements ⟨λ ′ |ô(y)|λ⟩, |λ ′ ⟩ ̸ = |λ⟩.We conclude that, if the states of the decomposition |λ⟩ are such that ⟨λ|ô(x)|λ⟩ does not depend on x, the same property holds for another related decomposition |η⟩ (at least) if ⟨λ ′ |ô(x)|λ⟩ = 0 when |λ ′ ⟩ ̸ = |λ⟩ and ∀ x.A required, but not sufficient, condition, is clearly that, in the same condition, also ⟨λ ′ | Ô|λ⟩ = 0. Due to Eq. ( 26), the same conclusion is reached in the more general case when the independence of ⟨λ|ô(x)|λ⟩ is realized without the strongest assumption of translational invariance (still provided the vanishing of the off-diagonal matrix elements, assumed x-dependent).
Various examples are plausible with the feature described above.Consider for instance a density matrix with an orthogonal decomposition (as a spectral one) |λ⟩, such that its elements are eigenstates of Ô = S z = i σ (z) i (i labelling the sites, now), as in the example of Section II C, with periodic boundary conditions.However, now all these states are characterized by different eigenvalues of Ô.Since clearly S z , σ In the following, we analyze some consequences of Eq. ( 22), relevant for our purposes.In particular, the same equation immediately implies, and is implied by, that, for a certain pure state |ψ⟩, the scaling of V [|ψ⟩, Ô] N with N is below the Heisenberg one.Similarly, if the average variance V [ρ, Ô] N in a certain decomposition {p λ , |λ⟩} can be written in the invariant form (21) (then it is possible to set λ p λ o 2 λ = c ∞ ), its scaling with N is below the Heisenberg one, and vice versa.
These facts hold because, if then |V |[|ψ⟩, Ô] N , scales with N → ∞ as [33,41,75] : Notice that, as required by the simultaneous validity of Eqs. ( 19) and ( 22) (see Section III E), in Eq. ( 27 of the sum in x and y in its definition, as in Eq. (11).
Finally, we notice that a lower bound for the scaling of c can be obtained further from the failure of Eq. ( 22) and without exploiting the scaling of |V |[|ψ⟩, Ô] N : c ∼ N γ , γ ≥ (d − 1)/d, see SM 4. This bound reflects the entanglement between a bulk point and the boundary of the lattice implied by the failure of Eq. ( 22).Moreover, it demonstrates c ∼ N in the mean-field limit d → ∞.

IV. USE OF THE FUNCTIONAL IN EQ. (21) TO BOUND MULTIPARTITE ENTANGLEMENT
We are not able to provide constraints on a density matrix ρ, sufficient to assure that the property in Eq. ( 22) holds in any decomposition.However, this task will turn out not to be strictly required for our purposes.
Indeed, in the present Section, we show how the functional in Eq. ( 21) can be exploited to bound multipartite entanglement, provided that the property in Eq. ( 19) holds for each decomposition -even without invoking the cluster decompistion property (22).
Two main situations can occur : -c < N : in this case, the property in Eq. ( 22) holds for the decomposition {p λ, | λ⟩} in Eq. ( 1), from the discussion of the previous subsection.Therefore, the average variance V [ρ, Ô] N can be expressed in the form of Eq. (21).The same expression is covariant and invariant under changes of decomposition (therefore, it can be equivalently evaluated in {p λ, | λ⟩} or in any other decomposition), and it is also equal to the QFI in the present conditions : Indeed, referring to Eq. ( 12) : Therefore, Eq. ( 21) can be exploited to bound, via the bounds in the subsections II B, the actual value of c.
Clearly, this value must be lower than N , as implied also by the construction of Eq. ( 21)).
c = N : in this case, if the property in Eq. ( 22) holds, everything works as in the previous case, apart from the fact that now the estimated value of c by the functional V [ρ, Ô] N in Eq. ( 21) can saturate to N .
Instead, if Eq. ( 22) does not hold, the same functional (not sharing any clear general relation with F [ρ, Ô] N , at the best of our understanding), still calculated in a chosen working decomposition, can yield a lower value c < N (again by construction of Eq. ( 21)).Therefore, a lower bound for c is still established.

V. CONCLUSIONS
We discussed some physical and mathematical conditions that make it possible to bound multipartite entanglement for mixed quantum states, via one-and twopoint correlation functions.
Our analysis holds for discrete systems, but it could be extended -although not straightforwardly -to continuous systems [28,76].Further investigation is required to identify other physically interesting cases, where our approach can be useful.
For d-dimensional N -component discrete systems, the QFI for pure states equals the variance V [|ψ⟩, Ô] N in Eq. ( 3) of the main text, and it can be cast in terms of two-point connected correlations (the θ dependence being neglected) [59] as For two mixed states, described by density matrices ρ and σ, the fidelity, similar to that for pure states, is defined as [4,78] : The last expression is considered applied to two infinitesimally-separated density matrices and ρ(θ + dθ) = e −i dθ Ô ρ(θ) e i dθ Ô , Ô labelling a generic Hermitian operator.There, 0 ≤ p λ ≤ 1, and the basis |λ⟩ is chosen orthonormal, then ρ(θ) ≡ ρ D is in its diagonal form.The resulting QFI, −2 d 2 F (θ,dθ) dθ 2 , reads : that for a pure state clearly reduces to the variance V [|ψ⟩, Ô] N in Eq. ( 3) of the main text, since in that case p λ = 1 just for a certain state |λ⟩, vanishing for the other ones.The QFI in Eq. ( 36) fulfills the inequality [4] p λ ≥ 0 and the bound being saturated by pure states.This convexity inequality is strictly required by an entanglement witness, since physically this property reflects the fact that mixing quantum states cannot increase the entanglement content, as well as the related achievable estimation sensitivity [4].The QFI cannot be expressed entirely in terms of two-point connected correlation functions, indeed the following relation holds [42,79] Part of the recognized importance of the QFI (33) is that it witnesses ME, both for pure and mixed states [8,9,26].Indeed, the violation of the inequality where N c ≤ h ≤ N − c + 1 is the number of disentangled subsets, signals at least (c + 1)−partite entanglement between the N components of the considered system, with 1 ≤ c ≤ N [27].In (39) Actually, c can also diverge with N c ∼ N l .The ultimate limit V [|ψ⟩, Ô] N = N 2 is called the Heisenberg limit and |ψ⟩ is then said to host genuine ME.Eq. ( 39) generates other relevant bounds [27] : for instance, it can be increased choosing h = N c , which yields the more common bound [4,26] useful if h is unknown.A direct derivation of this bound for pure states will be given in SM 2.
SM 2 : derivation of the bound in Eq. ( 8) of the main text for pure states In this Section, we discuss the bounds for pure states in the main text, presenting a treatment different from the original one in [26].In particular, the present discussion is valid for the quantum Fisher information (QFI) F [|ψ⟩, Ô] N only on pure states, and proceeds by the direct analysis of the two-point connected correlation function.We recall that, for a pure state |ψ⟩, V [|ψ⟩, Ô] N and V [|ψ⟩, Ô] N being defined as in the main text.We start from the verified [61,62] assumption that, if entanglement does not hold between two different sets, all the connected correlation between them (the discrete components, e.g. the sites) must vanish, and one can rewrite the quantum Fisher information on a generic pure state |ψ⟩ as follows : where D a are the h unentangled subsets of |ψ⟩.If the decomposition in Eq. ( 42) holds, exploiting the triangular inequality, Eq. ( 42) can be bounded as follows : where n a ≤ c are the number of components in the domain D a .We also have that where by i, j ∈ |ψ⟩ we denote that i, j belong to the support of |ψ⟩ and the number k ψ is defined (if |ψ⟩ is assumed normalized to 1) as the maximum difference between the squares of the eigenvalues of ôi and ôj , denoted by λ m and λ n (and m, n labelling the eigenvalues of ôi ) : Therefore, from Eq. ( 43), and defining with the same notation of the main text the last equality holding for the same reason as for Eq. ( 42) , we finally obtain : The finite number k ψ defined in Eq. ( 45) coincides with that for k in the main text (see after Eq. ( 5) ) at least for operators such that n = −m (notation of the main text), as spin operators.
The bound in Eq. ( 47) can be improved further, choosing an element D a * (existing by hypothesis) such that n a * = c : this assumption leads to that is the bound in Eq. ( 8) of the main text.From Eq. ( 48), it is clear that the Heisenberg scaling, F [|ψ⟩, Ô] N ∼ N 2 , can strictly hold only if the parameter k ψ in the expressions above does not scale with N .This is the case of spin-1 2 operators, for which k ψ = 1 4 .Importantly, also for our purposes in the Section III E of the main text, the same bound in Eq. ( 8) turns out to hold for the quantity |V |[|ψ⟩, Ô] N ≥ F [|ψ⟩, Ô] N , defined in Eq. ( 46) for a pure state |ψ⟩, and involving the sum of modula of the connected correlations.
The expression in Eq. ( 47) is also the starting point to prove the bound in Eq. ( 8) of the main text, as shown in Appendix A of [27].Therefore, the same bound turns out also valid again for |V |[|ψ⟩, Ô] N .
In the light of our analysis above, we stress again the deep connection between entanglement and connected twopoint correlations.Importantly, it emerges quite straightforwardly that c-partite can hold even if not signaled by F [|ψ⟩, Ô] N , for some or any choice of Ô, as found in previous works (see e.g.[36,41]).
Finally, let us discuss a bit more in detail the paradigmatic family of cases involving spin-1 2 operators.Having in mind an array of N 1 2 -spins, consider for instance ôi = s If a pure state |ψ⟩ of this system is fully producible, then c = 1 and Eq. ( 42) reduces to Therefore, Eq. ( 43) becomes : on the N sites), then also : Clearly max i |⟨ψ|s In this appendix, we illustrate the derivation of the general law for a change of decomposition of a density matrix ρ, written in Eq. ( 16) of the main text.
Let us consider a generic N ×N density matrix ρ, expressed in a certain basis (complete set) {|v i ⟩} on the considered M = V U −1 still being a unitary matrix, since it is the product of the unitary matrices V and U −1 .Therefore, Eq. ( 60) rules a general change of decomposition.Notice that, in Eq. ( 16) of the main text, we set M ≡ U .
SM 4 : on Eq. ( 22) on the main text and on the cluster-decomposition theorem for pure quantum states In this Section, we provide additional details on the role of Eq. ( 22) of the main text, for a pure state |λ⟩ on a lattice and for the specific operator ô(x), as well as that of the related stronger property, called cluster decomposition theorem.In particular, the latter theorem states that Eq. ( 61) holds for every local operator, and it encodes the locality principle [71][72][73] for Hamiltonian systems.The same theorem can be valid at least for all the physical systems where the area law for the entanglement entropy is not violated more than logarithmically, like in systems with volume-law violation.Eq. ( 61) is equivalent to the limit ⟨λ|ô(x)ô(y)|λ⟩ c → 0 as |x − y| → ∞.
The condition ⟨λ ′ |ô(x)|λ⟩ → 0 is violated by highly-entangled states |λ⟩, |λ ′ ⟩, e. g. cat-states.More general, Eq. ( 61) implies that at least if |a λ,λ ′ | < ∞, a generally fulfilled condition in physical systems.The latter argument suggests that Eq. ( 61) and the cluster-decomposition theorem, often referred to non-degenerate ground-state(s) of local Hamiltonians, must hold also for excited states, although they are more entangled in general (for instance fulfilling a volume law for the Von Neumann entropy, see e. g. [13,81,82]).Indeed, Eq. ( 63) means that a local operator ô(x) applied on (one point of) the infinite boundary of the states |λ⟩ or |λ ′ ⟩ (no matter their energies), cannot change the same states in a way to induce a nonvanishing overlap with other states of the same orthogonal basis.
Expressed in a alternative manner, Eq. ( 61) and the cluster decomposition theorem must be valid for all the states of the considered systems at least if c < ∞, and therefore if some producibility holds.Otherwise, the entanglement between the infinite boundary and a point in the bulk, or between two points of the boundary, would immediately set c → ∞.More in detail, if N ∼ L d , then in the latter case, c ∼ L δ , with δ ≥ d − 1, d − 1 being the dimension of the boundary, then also c ∼ N γ , γ ≥ (d − 1)/d.The bound for δ can be further improved by noticing that, if Eq. ( 61) fails and translational invariance is assumed (at least asymptotically), then : Merging Eq. ( 64) and the scaling property f (see more details in [33] for critical lattice systems and in [41] for one-dimensional gapped lattices), we obtain δ = d + 2β ≥ d.Therefore, the failure of Eq. ( 61) and implies the reach of genuine multipartite entanglement.Thus, no producibility of any kind is allowed.The same result can be also inferred directly from the bound in Eq. (39).For this purpose, it is crucial that, in the original demonstration of the bound [4,26], Eq. (61) and has not been used.For the same reason, we point out that Eq. ( 61) turns out to be strictly required to prove the bound in SM 2, even though in those proofs no explicit mention to it is made.Finally, the validity of Eq. ( 61) and of the cluster decomposition theorem does not require translational invariance, avoiding any prescription on the limit operations in Eqs. ( 63) and (64).
Above, we claimed that Eq. ( 61) and the cluster-decomposition can hold if the ground state is unique.Instead, if the ground-state is degenerate, the same properties do not hold in a generic basis for the ground-states.However, at least in the presence of a finite number of degenerate states, the same properties can be recovered, provided one chooses the combination of them properly.This fact can be illustrated via a paradigmatic example, i.
However, it is fulfilled for the states | ↑⟩ and | ↓⟩.This means that, adopting the basis formed by | ↑⟩, | ↓⟩, and by the excited states above them, Eq. ( 61) can be recovered, and locality made explicit.This fact parallels the possibility, described in the main text, to hidden producibility in the presence of degeneracies.In a sense, in the presence of a finite degenerate ground state, Eq. ( 61) still holds, up to global unitary transformations, as that linking the states |±⟩ and | ↑⟩ and | ↓⟩ in the example above.In Eq. ( 63), they transform the states |λ⟩ and |λ⟩ ′ but, critically, not the operators ô(x) and ô(y), set at the beginning to act on them.Therefore, the value of the sums in the same equation can change.Clearly, the bounds after Eq. ( 64) still hold ; in particular not-validity of Eq. ( 61) in any basis implies the simultaneous absence of any sort of producibility.
In real experiments, the states corresponding to | ↑⟩ and | ↓⟩ are generically selected by the fluctuations (also classical) or by perturbations suitably added, as well as in simulations : for instance, in the example above, by an additional term h ′ N i=1 σ(z) i , h ′ → 0, customary e.g. in DMRG calculations.The discussion above does not cover the important cases of continuous degeneracies, as for spontaneously broken continuous symmetries and for genuine topologically ordered matter [83].Moreover, it is not clear whether a volume-law dependence for the Von Neumann entropy is enough to guarantee the violation of Eq. ( 30) and of cluster decomposition.These issues are beyond the scope of the present work.
in this Section, we obtain Eq. ( 21) again of the main text.This step is strictly required to operatively evaluate V [ρ, Ô] N , since in general the producible basis | λ⟩, fulfilling Eq. ( 1) of the main text, is not known.For this purpose, it is useful to consider how Eq. ( 67) evolves under a transformation to an orthonormal, and generally entangled, basis, i.e. via the Gram-Schmidt procedure corresponding to a not unitary transformation.Setting this transformation as where |n⟩ are all orthonormal, and forgetting for the moment the asymptotic terms involved in the connected correlations, we can write : Exploiting again the orthonormality of the |n⟩ basis, it is now easy to convince ourselves that the latter expression is equal to x,y Tr ρ ô(x)ô(y) , Notably, Eqs. ( 76) and ( 77) become particularly manageable in the orthonormal basis |α⟩ where ρ is diagonal.Indeed : V [ρ, Ô] N = 4 x,y α p α,α ⟨α|ô(x)ô(y)|α⟩ − α p α,α lim |x−y|→∞ ⟨α|ô(x)ô(y)|α⟩ .
The limits in Eqs. ( 76), (77), and (78) can be evaluated in a number of physically interesting cases, making these expressions manageable even at an operative level.

Figure 1 :
Figure 1: Dissipative evolution of a XXZ chain of N = 6 site, using exact diagonalization, with a time-step of Jxdt/ℏ = 0.01, chosen to ensure numerical convergence.We begin the evolution from the ground state of H, with Sz = 0, in the gapless phase for Jz/Jx = 0.8, and study the evolution with (a) symmetry-respecting σ (z) i dissipation, and (b) symmetry-breaking σ (x) i

j
since the domain D a ≡ D i contain now a single site (the i-th one by convention).Actually, since |ψ⟩ the average of s (z) j correctly establishing the shot-noise limit bound for F [|ψ⟩, Ô] N , as well for the sum of modula |V |[|ψ⟩, Ŝ(z) ] N = 4 i |⟨ψ|s (z) 2 i |ψ⟩ c | (equal to F [|ψ⟩, Ŝ(z) ] N in the particular case of a fully producible state |ψ⟩).SM 3 : change of decomposition in Eq. (16) of the main text e. the ferromagnetic quantum Ising model in a transverse field, governed by the Hamiltonian ĤIS = h