Energy upper bound for structurally stable N-passive states

Passive states are special configurations of a quantum system which exhibit no energy decrement at the end of an arbitrary cyclic driving of the model Hamiltonian. When applied to an increasing number of copies of the initial density matrix, the requirement of passivity induces a hierarchical ordering which, in the asymptotic limit of infinitely many elements, pinpoints ground states and thermal Gibbs states. In particular, for large values of N the energy content of a N-passive state which is also structurally stable (i.e. capable to maintain its passivity status under small perturbations of the model Hamiltonian), is expected to be close to the corresponding value of the thermal Gibbs state which has the same entropy. In the present paper we provide a quantitative assessment of this fact, by producing an upper bound for the energy of an arbitrary N-passive, structurally stable state which only depends on the spectral properties of the Hamiltonian of the system. We also show the condition under which our inequality can be saturated. A generalization of the bound is finally presented that, for sufficiently large N, applies to states which are N-passive, but not necessarily structurally stable.


I. INTRODUCTION
The passive states of a quantum system A consist of all density matrices that commute with the system Hamiltonian H and have no population inversions [1]- [3]. Originally introduced in Ref. [1] by linking them to the Kubo-Martin-Schwinger thermal stability condition [4]- [6], passive states exhibit zero ergotropy [7], i.e. zero maximum mean energy decrement when forcing the system to undergo through unitary evolutions induced by ciclic external modulations of H. In the Kelvin-Planck formulation of the second law of thermodynamics, ergotropy can be interpreted as the maximum work that can be extracted from a system [7], [8], suggesting the identification of passive states as a primitive form of thermal equilibrium. In view of this property, ergotropy and passive states play a key role in quantum thermodynamics [3], [9], where they help in clarifying several aspects of the theory, spanning from foundational issues at the interplay between physics and information [10]- [19], to more practical issues, such as the characterisation of optimal thermodynamical cycles [3], [20]- [25] and the charging efficiency of quantum batteries models [8], [26]- [30]. Passive states have been also identified as optimisers for several entropic functionals which are relevant in the theory of quantum communication [31]- [33], and as suitable generalizations of the vacuum state for quantum field theory in curved space-time models [34].
A natural generalization of passivity can be obtained by considering multiple copies of the original system [1], [2]. In R. Salvia is with Scuola Normale Superiore and Istituto Nanoscienze-CNR, I-56127 Pisa, Italy.
V. Giovannetti is with NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, I-56126 Pisa, Italy. particular, a density matrix ρ of A is said to be N -passive with respect to the local Hamiltonian H when, given N identical copies of it, one has that ρ ⊗N is passive when considering as joint Hamiltonian of the compound the sum of N copies of H. It turns out that N -passive states are also N -passive for all N ≤ N , the opposite inclusion not be granted in general, inducing a strict hierarchical ordering on the associated sets. In this framework thermal Gibbs states share the exclusive property of being the only density matrices of the system which are completely passive, i.e. passive at all order N , and also being structurally stable [1], [2], [8], [14]. Structural stability ensures that the state under consideration will remain passive even when the system Hamiltonian undergoes small perturbations. This condition is naturally granted to all passive configurations when H has a non-degenerate spectrum, but becomes a non trivial requirement in the presence of degeneracies. A direct consequence of the above mentioned property of Gibbs states is that, for N large enough, the mean energy E(ρ; H) of a structurally-stable, N -passive density matrix ρ must approach the mean energy E β(ρ) (H) of the Gibbs configuration ω β(ρ) that has the same entropy of ρ -the latter being always a lower bound for E(ρ; H), i.e. E β (H) ≤ E(ρ, H). Aim of the present work is to investigate how the gap between E(ρ; H) and E β (H) reduces as N increases. For this purpose we prove an inequality which provides an upper bound for E(ρ; H) in term of E β (H), via a moltiplicative factor which only depends upon the spectral properties of the Hamiltonian, and which converges asymptotically to 1 as N increases. Incidentally, following the same argument presented in Ref. [24], our findings can also be used to give a lower bound for the work that can be extracted from a system, hence providing a practical tool to estimate the usefulness of a given state from the perspective of average work extraction.
We stress that the derivation presented here relies heavily on the structural stability property of the input states; if we lift such condition, the bounds do not apply in general. However, for sufficiently large values of N , we also give a variant of the inequality which remains true for all N -passive states (not necessarily structurally stable) -see Table I for a summary of the results of this paper.
The manuscript is organized as follows: in Sec. II we introduce the notation, set the theoretical framework that will be used in the remaining part of the paper, and present some preliminary observations. Sec. III contains the main result of the work: here we derive our upper bound for the energy of Npassive, structurally stable states and discuss its achievability. Sec. IV presents instead a generalization of the bound for N -   passive states which are not necessarily structurally stables, which applies in the asymptotic limit of sufficiently large N . In Sec. V we present finally some considerations on the case of Hamiltonian characterised by energy levels gaps which are commensurable. Conclusions are drawn in Sec. VI. The manuscript contains also few appendixes which provide technical support for the derivation of the main results (in particular in Appendix B we give a new proof of the fact that Gibbs states and ground states are the only density matrices which are completely passive).

II. DEFINITIONS AND PRELIMINARY OBSERVATIONS
Let A be a quantum system described by a Hilbert space H of finite dimension d and characterised by an assigned Hamiltonian with eigenvectors {| j } j and associated eigenvalues { j } j which we assume to be organized in non-decreasing order, i.e.
Notice that in the writing of (2) we are explicitly allowing for possible degeneracies in the spectrum of H. In particular for future reference we indicate with d 0 the degeneracy of its ground level (meaning that i = 0 for all i ∈ {0, 1, · · · , d 0 − 1}), and use the symbol H G to represent the associated eigenspace, i.e.
Given then an element ρ of the density matrix set S of A we now define its ergotropy as the functional By construction E (1) (ρ; H) is a non-negative quantity which can be explicitly computed by solving the maximization with respect to U (see Appendix A for details on this). In the above theoretical framework the set P H of passive states can now be identified as the subset of S characterised by the property of having zero ergotropy value, It turns out that every passive state ρ must coincide with one of its associated passive counterparts [1], [2] implying that P H is exclusively made of density matrices which verify the following constraints: i) ρ is diagonal in the energy eigenbasis {| j } j , i.e. it can be expressed as ii) ρ has no population inversion, i.e. its eigenvalues {λ j } j fulfil the following ordering A special subsetP H is provided by the passive states which are structurally stable, i.e. passive density matrices which besides obeying i) and ii) also satisfy the extra property which assigns identical population values to energy eigenvalues which are degenerate [1], [2]. One can easily verify that when H has a not-degenerate spectrum (i.e. when (2) holds true with strict inequalities for all j),P can be shown to be closed under convex convolution.
In a similar fashion, given N ≥ 1 integer, we can now introduce the set P (N ) H of N -ordered passive (or simply Npassive) states. Specifically, given the total Hamiltonian of the joint system, H ( ) being the single system Hamiltonian acting on the -th copy we define with E (N ) (ρ; H) the N -order ergotropy functional the maximum being now evaluated with respect to all the (possibly non-local) unitaries of the joint system. Using the same argument that led to the necessary and sufficient conditions i) and ii) given above, it has been shown that ρ ∈ P (N ) H if and only if the following conditions hold true: i) ρ is diagonal in the energy eigenbasis, i.e. must be of the form (7); ii) its eigenvalues fulfil the requirement for all couples of the population sets I N := {n 1 , n 2 , · · · , n d }, J N := {m 1 , m 2 , · · · , m d } formed by d non-negative integers that sum up to N .
Notice that for N = 1 Eq. (13) reduces to the absence of population inversion (8), and that for consistency in the above expression the indeterminate form 0 0 has to be interpreted equal to 1, i.e. explicitly By close inspection of the above definitions, it follows that N -passive states are also N -passive for all N ≤ N . The opposite inclusion however is not necessarily granted implying a specific ordering on the associated sets, i.e.
In a similar fashion of what done in the case of P (1) H , also for P (N ) H we can then introduce the notion structural stability. Specifically, for any given integer k ≤ N we define the subset P (N,k) H of the N -passive, k-structurally stable density matrices as the one formed by the special N -passive elements which, besides the condition i) and ii) detailed above also obey to the extra requirement for all the couples of the population sets I k := {n 1 , n 2 , · · · , n d }, J k := {m 1 , m 2 , · · · , m d } formed by d non-negative integers that sum up to k; (again in writing (16) we assume the convention (14); notice also that for k = N = 1 this expression reduces to (9)).
Hierarchical rules analogous to (15) hold true also for the setsP (N,k) . In this case, one can easily show that and also that while in general for N ≥ N ≥ 1 it is not true that P which, for all N ≥ 1, allows us to expressP (to verify this simply observe that on one hand,P ). In the special case of H which is non-degenerate (i.e. when (2) holds true with strict inequalities for all j) we have already noticed that P We are finally in the position to give the definition of completely passive (CP) states: these are the density matrices of A which are passive at all order N , i.e. which are diagonal in the energy eigen-basis and fulfil the constraint (13) at all order. The set of completely passive can be hence identified with the intersection of all the P (N ) the last identity being a trivial consequence of the ordering (15). In a similar fashion we can also introduce the definition of CP states which are also 1-structurally stable (CP1SS), as the intersection of all the sets P as well as the set of CP states which are structurally stable at all orders (CPCSS) which, thanks to (17) corresponds to the inclusion of all the setsP By construction it is clear thatP H . Most notably however it turn out that irrespectively from H,P both being identified with the set of Gibbs states of the system. This important result will be reviewed in the next subsection, with a complete characterisation of P (∞) H in terms of ground and Gibbs states.

A. Ground states and Gibbs states
Explicit examples of CP states are provided by the ground states density matrix ρ which have their support inside the ground subspace H G [2]. We shall use the symbol S with Π G = d0−1 j=0 |e i e j | being the projector on H G . Notice also that, while S H , for d 0 > 1 its only element that is structurally stable is the uniform ground state mixture Π G /d 0 . As a matter fact, in Ref. [2] it has been shown that these elements of S or more explicitly Closely related to S

(G)
H is the set of Gibbs thermal states G H . They are identified with the collection of density matrices of the form with the parameter β ≥ 0 playing the role of an effective inverse temperature, and the normalization term Z β being the associated partition function. In the infinite temperature regime, ω β (H) converges to the completely mixed state of A, that is ω 0 := lim β→0 ω β = 1/d. On the contrary, in the zero temperature limit, ω β (H) converges to the uniform mixture supported on the ground energy eigenspace (3) of the system, namely which, as we already mentioned, is the special element of S (G) H . Gibbs states play a fundamental role in the study of the thermodynamic properties of A since they embed the very notion of thermal equilibrium [3], [9]. In particular they can be identified by the property of granting the minimal value of the mean energy E(ρ; H) attainable for density matrices ρ with fixed von Neumann entropy S(ρ) := −Tr[ρ log ρ], i.e. with It is worth remembering that both E β (H) and S β are strictly monotonically decreasing functions of β. Thanks to this fact, there is a one-to-one correspondence between these functionals, that ultimately leads to the following relevant expression We also notice that, while S β can saturate the maximum entropy value available for the system A (i.e. S β=0 = log d), due to Eq. (30), the Gibbs state entropy functional is bound to be always larger than or equal to log d 0 which, unless the Hamiltonian has a non-degenerate ground level (i.e. d 0 = 1), is strictly larger than zero. Accordingly, (31) identifies the Gibbs states as minimal energy states only for the subclass of density matrices ρ which have entropy above the log d 0 . For those instead which have S(ρ) < log d 0 , the only possible lower bound on E(ρ; H) is simply provided by elements of S

(G)
H , leading to the following trivial inequality As mentioned in the introductory section, an alternative way of characterising the Gibbs subset G H is by the observation that the density matrices of the form (29) share the exclusive property of being the only density matrices of A which are CP and 1-structurally stable [1], [2]. Furthermore, since the elements of G H are also structurally stable at all order, we can write which explicitly proves the identity (25) anticipated at the end of the previous section. Together with (28), the above expression finally allows us to conclude that In Appendix B we present an alternative proof of these important identities based on a simple geometrical argument.
Here instead we comment on the fact that the they can be simplified in two limiting cases. The first one is for two dimensional systems (d = 2). In this situation all passive states are also structurally stable and completely passive, hence thermal, implying that the hierarchies (17) collapse, so that Eqs. (36) and (37) can be replaced by the ground state set being trivially contained in the Gibbs set, i.e. S

(G)
H ⊂ G H . A similar statement can be extended also for d ≥ 3 for Hamiltonians that have a two-level spectrum, i.e. H which beside the (possibly degenerate) zero, ground energy level are characterised by a unique (possibly degenerate) nonzero eigenvalue. Indeed, in this case one can easily show that P (1,1) H coincides with the Gibbs set, allowing us to replace (36) with while (37) remains the same.

III. UPPER BOUNDS FOR THE MEAN ENERGY OF
Equation (31) establishes that Gibbs states provide a natural lower bound for the mean energy value of the density matrices which have the same entropy. In particular, given a N -passive state ρ ∈ P (N ) H of entropy larger than or equal to log d 0 , we can write Notice that, thanks to the uniform population constraint (9), the requirement of having entropy larger than or equal to log d 0 is naturally fulfilled by the elements of P , accordingly for such states we need not to worry about such condition.
For two dimensional systems (d = 2), thanks to (38) the inequality in (40) is replaced by an identity. By exploiting (39) a similar statement can be extended also for d ≥ 3 for those H which have a simple spectrum: in this case however the equality in (137) holds true only for the passive states which are also explicitly structurally stable, i.e. ρ ∈P On the contrary, for d ≥ 3 and Hamiltonian H that possesses at least two distinct non-zero energy eigenvalues, the gap between the right-hand-side and left-hand-side of Eq. (137) is typically finite. Yet, as a consequence of Eq. (36), we expect that such gap should reduce as N increases, irrespectively from the spectral properties of H. Aim of the present section is to provide a quantitative estimation of this fact at least for the elements of P (N ) H which are at least 1-structurally stable. In particular we shall see that the following upper bound hold true, with max (= d ) being the maximum energy eigenvalue of H and with R(H) being a non-negative constant that only depends upon the spectral properties of the system Hamiltonian. Explicitly for H not having a simple (two-level) spectrum one has the maximum being computed among all possible triples of ordered energy levels c > b > a -for H being two-level we can just put R(H) = 0 and recover (42) via (137). Notice that for H with non-simple spectrum R(H) is always greater than or equal to one and that the optimization over c can be explicitly carried out leading to Interestingly enough, when β(ρ) max > 1 (low temperature regime), the upper bound of Eq. (43) can be improved by means of the following inequality (47) A direct comparison between (43) and (47) reveals that when β(ρ) max > 1 and N is sufficiently large, the former is tighter than the latter (otherwise (43) always wins).

A. Derivation of the bounds
In order to derive Eqs. (43) and (46) first of all, we need to understand in which way the eigenvalues of the state ρ are constrained by each other by the requirement of being an element of the N -passive set P (N ) H . Since the problem is non trivial only for the case where the spectrum of H is not twolevel, in what follows we shall focus on these cases where H admits at least 3 different energy levels.
Preposition 1. Given an Hamiltonian H admitting three nondegenerate energy levels a < b < c , and ρ ∈ P the following inequality holds then we must have Proof. The result follows directly from the N -passivity condition Eq. (13) and results in the geometrical constraint depicted in Fig. 1. In particular Eq. (49) corresponds to impose (13) for the special case of the population sets I N contains as only non-zero term n b = N , while J N contains as only non-zero terms n c = m and n a = N − m. The inequality (51) instead follows by applying Eq. (13) to the special case in which I N contains as only non-zero terms n a = m+1, n c = N −m−1, while J N contains as only non-zero term n b = N .
We can now proceed to estimate how much the populations of a N -passive, 1-structurally stable state ρ can differ from those of a thermal equilibrium matrix. Preposition 2. Let k < j be two distinct eigenvalues of H and ρ ∈ P the following implication must hold Instead if exist real numbers β, Z ∈ [0, ∞) fulfilling at least one of the two following inequalities: or then we must have To prove instead Eq. (63) set a = k , b = j , and c = i and select m ∈ [0, N [ such that Eq. (50) of Preposition 1 applies. Then we have In the case in which inequalities (55) hold, exploiting the fact that 1 − N/(m + 1) ≤ 0 we can bound the RHS of (58) by Also, if the inequalities (54) are true, we can write the same bound for λ i : Therefore, in either one of the two cases (54) and (55) we have where in the last inequality we used The results of Preposition 2 apply to density matrices ρ which are just N -passive, but not necessarily 1-structurally stable. In order to treat the degenerate cases i = j and i = k , henceforth in this section we will assume as hypothesis the 1-structural stability of ρ. Corollary 1. Let k < j be two distinct eigenvalues of H and ρ ∈P (N,1) H a N -passive, 1-structurally stable state. Then under the condition (52) holds the implication and under either one of the conditions (54) or (55) it is valid the implication Proof. In the cases in which i is neither equal to j or to k , (62) reduces to (53), which is implied by condition (52). Likewise, when i = j , (63) is equivalent to (56), which is implied by (54) or by (55). Consider now the case where i = j : under this circumstance Eqs. (62) trivially follow from (52) and by the fact that since ρ is 1-structurally stable, we must have λ i = λ j , see e.g. Eq. (9). The same reasoning can be applied also for (63) by exploiting either (54) or (55).
Finally if i = k , we must have λ i = λ k , and Eq. (62) follows again from (52). See Fig. 2 for a graphical representation of these relations.
be a N -passive, 1-structurally stable state. Then under the condition (54) of Preposition 2 we must have where R(H) is the quantity defined in Eq. (44). Furthermore, if k is the ground state energy of the system (i.e. k = 0 = 0), then the following inequalities hold Proof. For i = k , Eq. (64) is a trivial consequence of the first inequality of assumed hypothesis (54). For k < i ≤ j instead we notice that (54) implies (52) and hence (62), i.e.
The thesis then follows from the observation that For i > j instead we use the fact that (54) implies (63), i.e.
and the inequality Suppose next that k is the ground energy level of the system: in this case the bound (64) on λ i applies for all energy levels, i.e.
therefore we can write On the contrary replacing i and hence H be a N -passive state. Then under the condition (55) of Preposition 2 we must have Proof. The assumed hypothesis (55) implies (56). The thesis follows from (56), with the observation (70).
The inequalities (62) and (63) are the key ingredient to arrive at our main result. To complete our derivation of equation (46), we find it useful to first state a set of hypotheses under which, exploiting Preposition 2, one can guarantee that Eq. (46) is true (see Preposition 3 below). Then we show that these conditions are always satisfied by all the elements ofP be a N -passive, 1-structurally stable state of a Hamiltonian H characterised by at least three distinct eigenvalues. Then its mean energy energy E(ρ; H) is bounded by the inequalitity (46) if at least one of the two following conditions holds true for the spectrum of ρ: with Z β(ρ) is the partition function of the isoentropic Gibbs state ω β(ρ) , or Condition 2: the population of the ground state is lower bounded by Z −1 β(ρ) , i.e.
Proof. When Condition 1 applies we notice that the hypothesis (52) of Preposition 2 are fulfilled by identifying k and j with 0 = 0 and c respectively, and taking Z = Z β(ρ) , β = β(ρ). Accordingly we can then invoke (62) to establish that for i such that 0 ≤ i ≤ c the following inequality must hold which in particular implies (for 0 < i ≤ c this follows by the fact that R(H) ≥ c / i , for i = 0 instead (79) is a trivial consequence of the first inequality of (76)). Now we can apply (78) for i = b to establish that Notice also that from (76) we have Therefore we notice that the condition (54) of Preposition 2 are fulfilled one taking k and j with b and c respectively, and taking Z = Z and β = β(ρ). Hence invoking (63) we can claim that for all i ≥ c , we must have which finally leads to (46) by using (40) and (33) to enforce the identity Consider next the case where Condition 2 holds. Under this circumstance let us introduce the Gibbs state ω β with inverse temperature β ≥ β(ρ), such that Because Tr ρ = Tr ω β = 1, there exists at least one eigenvalue c > 0 such that λ c being the associated population of the Gibbs state ω β . Accordingly the hypothesis of Corollary 2 are fulfilled with k = 0 = 0, j = c , Z = Z β , and β = β . Hence invoking (65) we can write 88) where in the left-hand-side we used (85) to express S(ρ) in terms of E β(ρ) (H). In case β = β(ρ) Eq. (88) is just (46) and the proof ends. On the contrary if β > β(ρ), exploiting the fact that function ln Z β is concave we can write with ∆β := β − β(ρ) > 0. Using the inequality (89) into (88), and rearranging the terms, we hence arrive to which leads to (46) by the strict positivity of β . Proof. If ρ is such that λ 0 ≥ Z −1 β(ρ) then the thesis follows by application of Condition 2 of Preposition 3. Consider next the case λ 0 < Z −1 β(ρ) . Here we observe that since the von Neumann entropies of ρ and ω β(ρ) coincide, i.e. S(ρ) = S(ω β(ρ) ), the spectrum of ρ does not majorize, nor is majorized by the spectrum of ω β(ρ) [35], [36]. As shown in Appendix C we can then claim that there must exist c > b > 0 such that λ b ≥λ b and λ c ≤λ c , wherê are the eigenvalues of the Gibbs state ω β(ρ) . Therefore this time the thesis derives as a consequence of Condition 1 of Preposition 3. Furthermore, the inequality (43) also holds if ρ ∈ P (N ) H is a generic N -passive state, provided that the populations of ρ satisfy the condition Proof. Consider first the case in which the population of the ground state of ρ ∈P . In this case, the majorization argument of Appendix C implies that there exist c > b > 0 such that λ b ≥λ b and λ c ≤λ c (again we use the convention (91) to indicate the eigenvalues of the Gibbs state ω β(ρ) ). This, as seen during the proof of Preposition 3, implies the inequality (83). We now observe that replacing i R(H) N in the right-hand-side of (83) with max and hence Next we consider the case in which ρ ∈ P  (9) the condition (92) is implied by λ 0 ≤ Z −1 β(ρ) ; therefore, this case will also complete the proof of (43) for all the N -passive, 1structurally stable states. Since ρ and ω β(ρ) have both trace one, there must exist some other eigenvalue λ b of ρ such that The assumption (92) ensures that b > ln d G , i.e., that b > 0. Chooseλ b to be the first eigenvalue of ρ which is smaller than the corresponding eigenvalue of ω β(ρ) , i.e. λ i ≥λ i for every i < b . This implies The hypotheses (55) of Preposition 2 are fulfilled by identifying k and j respectively with 0 = 0 and b , and taking Z = Z β(ρ) , β = β(ρ). Therefore, for any i > b the inequality (75) holds: Given the non-majorization condition, we can infer from (95) that there must exist some other eigenvalue λ c , with c > b, such that which in turns implies that Using (99) and then (96), we finally conclude that which proves the thesis.

B. Saturation of the inequality (46)
The inequality (46) implies that for N > R(H), the ratio between the mean energy of a N -passive, 1-structurally stable state ρ and the energy of the Gibbs state ω β(ρ) that has the same entropy of ρ, can be at most In this section we shall exhibit an explicit example of Npassive, 1-structurally stable states whose energy are arbitrarily close to the limit (102) -see also Fig. 3. Although this example is contrived (it requires very small temperatures and very large degeneracies), it works for all N ≥ 2, for some specific value of α max . In particular our example requires that for some integer 1 ≤ m < N one has or equivalently α max m+1 m . Given N ≥ 2, consider the set of N -passive, 1-structurally stable state ρ ∈P (N,1) H associated with a Hamiltonian H characterised by three energy levels: with r > 1 being the parameter that provides the R(H) of the model, i.e.
We also assume 1 and 2 to have degeneracies g 1 and g 2 respectively, whose values will be specified later on. Under the above premise, in what follows we shall use 1 , g 1 , and g 2 as free parameters over which we optimize to enforce the saturation of the bound (102). First of all, given a the associated iso-entropic Gibbs counterpart ω β(ρ) of ρ ∈P (N,1) H we exploit 1 to force it into the low temperature regime imposing the constraint β(ρ) 1 1 .
For a reason that will soon become apparent, we impose an additional condition on 1 , namely that This condition is clearly compatible with (106). Now we fix g 1 and g 2 to ensure that despite the fact that λ 1 λ 2 , the total population in the 2 energy level will still be bigger than the total population in the 1 , i.e. we assume Condition (109) ensures that the energy and the entropy of the thermal equilibrium states ω β (ρ) are dominated by the contribution from the higher energy level of the system, i.e.
Take now m < N integer such that Note that such m can be identified with the same m of (103): accordingly to fully comply with such constraint we requires 1/r to be very close to the upper bound of (112), so that we can also write m where in the last passage we used equation (113) and the fact that λ 0 1. Now we focus on the special subset of the density matrices ρ ∈P (N,1) H that have λ 0 1 and λ 1 1, and which saturate the limit posed by Eq. (114). We parametrize the populations of ρ as with η ≤ 1 and ξ 1 which in particular we assume to fulfil the inequality ξ e −k0 (r−1)β(ρ) 1 with k 0 1 being some large fixed constant (notice that thanks to (106), Eq. (116) is in perfect agreement with the request of having ξ 1, indeed the larger is β(ρ) 1 the smaller is ξ). We now impose the energy and the entropy of the state (115) to be dominated by λ 1 . Accordingly we set a new condition for g 1 and g 2 , requiring that which thanks to our choice (116) is perfectly compatible with our previous assumption (109).
Equations (117) and (115) lead to the following approximated expressions for the mean energy and entropy of ρ: On one hand, together with (110), the first allows us to write the ratio (101) in terms of ξ as On the other hand, from the second expression of Eq. (118) we get the extra condition that follows from the request that ρ and ω β(ρ) have the same entropy (once more it is worth noticing that no conflict arises with our previous assumptions, since the large values of β(ρ) 1 imposed by (106) are in agreement with small values of ξ). Replacing (119) for ξ into (120) leads to a transcendental equation for the ratio (101) of the model: We now claim that it is possible to set the parameters of the model (i.e. the quantities r, 1 , g 1 , g 2 , and k 0 ) in such a way that the bound (46) saturates, by forcing the solution α α max from Eq. (121) while fulfilling all the constraints we invoked in the derivation, i.e. the inequalities (103), (106), (108), (109), and (116). To see this let first observe that from (109) it follows that α −1 < 1 − ln(rα) β(ρ) 1 which simply says that α is a quantity greater than 1. On the contrary a lower bound for α −1 can be obtained via the constraint (116) which via (119) rewrites as ln(g2/g1) Replacing this into (121) this implies α −1 r−1 rαmax−1 , whose right-hand-side is strictly smaller than α −1 max due to the fact that α max ≥ 1 by construction. Therefore, as far as it concerns to (109) and (116), α max is inside of the domain of the allowed values of α obtainable when solving (121). To check the compatibility of such result with (103) and (106) let us solve (121) for ln(g2/g1) when α is taken to be equal to α max , i.e.
which to be in agreement with (109) would require a condition which is equivalent to (108).

IV. NON STRUCTURALLY STABLE, N -PASSIVE STATES
The bounds derived in the previous section in general do not apply to states which are just N -passive. Indeed the conditions of Preposition 3 may not be fulfilled by a passive state which is not structurally stable: one can always find a couple of eigenstates λ b > λ a of ρ such that λ a < Z −1 β(ρ) e −β(ρ) a and λ b > Z −1 β(ρ) e −β(ρ) b , but their energies could be equal ( a = b ), and in this case (46) or (43) needs not to be valid.
Of course this problem may arise only if the spectrum of H is degenerate since, due to Eq. (21), for non-degenerate Hamiltonians all N -passive states are also N -passive and 1structurally stable and the bounds we have derived trivially hold true. At least for the bound (43) a similar conclusion can be drawn in the presence of degeneracies of the spectrum of H, for all N -passive state ρ ∈ P (N ) H whose ground state populations are larger than or equal to the ground state population of their associated isoentropic Gibbs states, i.e. when Eq. (92) is true: under such condition, by proposition Preposition 5 a generic ρ ∈ P (N ) H will still respect the bound (43) -see Table I.
In summary, the only cases which are left uncovered by at least one of our two bounds, is when H is degenerate, ρ ∈ P (N ) H is not 1-structurally stable and violate the condition (92). Aim of the present section is to deal with these special configurations. To being with, it is worth remarking that this case includes both the situation where ρ admits has a sufficiently large entropy which allows us to identify an isoentropic Gibbs counterpart ω β(ρ) , as well as the more pathological cases where S(ρ) < log d 0 for which ω β(ρ) does not even exists. Still, in both scenarios we can associate to ρ a N -passive, 1-structurally stable density matrixρ ∈P (N,1) H obtained by replacing the populations λ j s of ρ with their mean values computed by averaging them over all the energy levels with the same energy eigenvalues, i.e.
where d i is the degeneracy of the energy level i . One can easily verify that the spectrum ofρ majorizes the one of ρ [35], [36]. Therefore, while by constructionρ has the same energy of ρ, its entropy is certainly not smaller than S(ρ), i.e.
Furthermore, sinceρ is N -passive and 1-structurally stable it respects the inequalities (43) and (46). This means that, given a Hamiltonian with at least three distinct eigenvalues, for any N > R(H) we can write where as usual β(ρ) indicates the inverse temperature of the Gibbs state ω β(ρ) that has the same entropy ofρ. By expanding Eq. (126) at large N we can finally arrive to the following compact expression where now u(ρ) := min{1, β(ρ) max }. Assume next that S(ρ) ≥ log d 0 so that ω β(ρ) does exist. Notice that by the monotonocity relation that connects the Gibbs functionals (32) and (33), from (125) we have β(ρ) ≤ β(ρ) and also that E β(ρ) (H) cannot be smaller than E β(ρ) (H), i.e.
(128) In order to convert (126) or (127) into a bound that links the energy of ρ with the energy of its Gibbs counterpart we need to find a way to reverse the inequality (128), constructing a lower bound for E β(ρ) (H) in terms of E β(ρ) (H). For this purpose in the next paragraphs we determine an upper bound of the quantity which using again the monotonocity connection between (32) and (33) will then be converted into the inequality we are looking for. Our final result will be that, whenever the condition (92) is false and S(ρ) ≥ log d 0 , we can write In the case of a Hamiltonian with a non-degenerate ground state (d 0 = 1) the above expression can be further simplified reducing to The above expressions refers to all the cases where H has at least three independent eigenvalues. The only non-trivial configuration which is left unsolved is the one where H is a two-level Hamiltonian and the system has dimension d ≥ 3.
In this case, we show that (130) is replaced by Finally consider the situation where S(ρ) < log d 0 which even prevents us the possibility of identifying a Gibbs counterpart of ρ. Here -as shown in Sec. IV-C-Eq. (130) can be replaced by A. Derivation of the asymptotic bound (130) In order to calculate how much the entropy of the system increases when passing from ρ ∈ P (N ) H to its 1-structurally stable counterpartρ ∈P (N,1) H defined in (124), we need to know how much the eigenvalues λ j of ρ can be "spread out" around their mean valueλ j . To tackle this issue, for all eigenvalues of H, we find it useful to introduce the corresponding minimal and maximal populations of ρ, i.e. the quantities which clearly fulfil the inequality In view of the previous discussion we shall then assume the condition namely the negation of condition (92). Preposition 6. Given N ≥ 2 and ρ ∈ P H a N -passive state with entropy larger than or equal to log d 0 and satisfying the condition (137), the following inequality hold for all the populations λ j and λ i of ρ associated with a nonzero energy level > 0 of H (i.e. i = j = ).
Proof. If the energy level is not degenerate (i.e. d = 1) the inequality (139) is trivial (in the this case the left-hand-side term is null, while the right-hand-side is non-negative due to (137). On the contrary, let if is degenerate, let λ j and λ i two different populations of ρ that are associated to it, i.e.
Apply hence the N -passivity condition (13), choosing a population set I N which contains as only non-zero term n j = N , and a population set J N which contains as only non-zero terms n i = N − 1 and n = 1 with ≤ d 0 − 1 referring to one of the eigenvalues of the ground state energy level. Simple algebra allows us to recast this result into the inequality which leads to (140) when taking λ = λ min (0), and enforcing (137).

Corollary 4.
Given N ≥ 2 and ρ ∈ P (N ) H a N -passive state with entropy larger than or equal to log d 0 and satisfying the condition (137), the following inequalities hold for all the energy level > 0 of H.
Inequalities (138) and (140) are valid only for energy levels which are not the ground state. In the case i = 0, we can enforce only a looser upper bound: Preposition 7. Given N ≥ 2 and ρ ∈ P (N ) H a N -passive state with entropy larger than or equal to log d 0 and satisfying the condition (137), there exists an eigenvalue a of H such that Proof. Since ρ and ω β(ρ) have the same mean energy, there must exists at least one eigenvalue of ρ (say λ a ) associated with an energy level a > 0 for which (indeed if by contradiction such level would not exist then E β(ρ) (H) will be strictly larger than E(ρ; H)). Let then λ i and λ j two populations associated with the ground state energy level of the system (i.e. i = j = 0). Apply the N -passivity equation (13), when selecting a population set I N which contains as only non-zero term n 0 = N , and a population set J N which contains as only non-zero terms n j = N − 1 and n a = 1 to obtain the inequality Identifying λ j and λ i with λ max (0) and λ min (0) respectively, Eq. (143) leads to the last passage following from (137) and (142).
We are now ready to estimate the entropy gain ∆S(ρ) for each degenerate energy level of H. We treat separately the three cases of the ground state, of the excited states with a population λ j higher than the corresponding populationλ j = Z −1 β(ρ) e −β j of the Gibbs state ω β(ρ) , and of the excited states with populations λ j smaller thanλ j .
then following inequality holds true, withλ j the eigenvalues ofρ defined in (124).
Proof. Given j = the following chain of inequality can be written where in the first passage we used (136), in the second we used Corollary 4, and in the last one we used (145). Equation (146) then follows by multiplying the above expression by λ j and summing over all possible energy levels of energy equal to .
Preposition 9. Given N ≥ 2 and ρ ∈ P H a N -passive state with entropy larger than or equal to log d 0 and satisfying the condition (137), such that there exist a strictly positive energy level > 0 of H for which then following inequality holds true, withλ j the eigenvalues ofρ defined in (124) and d the degeneracy of .
Proof. Given j = , we can write where the first inequality follows from (136) and the second from Preposition 6 setting λ i = λ min ( ) in Eq. (138). Multiplying then by λ j and summing over all possible choices of j = we heve that where in the second inequality we used the fact thatλ( ) ≤ 1/2 (a property which applies to all non-ground state populations of all Gibbs states), the fact that the function f (λ) := −λ ln λ is monotonically increasing for 0 < λ < 1 2 , and the inequality (148).
152) withλ j the eigenvalues ofρ defined in (124), d 0 the degeneracy of the ground state, and max the greatest eigenvalues of H.
Proof. Expunging from the sum the negative terms we can write where the last sum contains at most d 0 − 1 terms, because there is at least one λ j smaller than the mean. Invoking hence Preposition 7 twice and Eq. (137) we arrive to which replaced into (153) yields the thesis.
We have now all the ingredients to estimate the maximum amount of entropy that we can gain converting ρ ∈ P with d 0 and max the degeneracy of the ground state and the maximum eigenvalue of H respectively.
Proof. Observe that where in the second line we used the fact that for j = i one hasλ j =λ i and that j = λ j = j = λ j . Combining Prepositions 8 and 9 we hence notice that the part of the sum in Eq. (156) that involves all the energy levels above the ground state can be bounded as follows where in the last line we used the definitions of E(ρ; H) and E β(ρ) (H). On the contrary the part of the sum in Eq. (156) that instead involves only degenerate ground states can be instead bounded as in Eq. (152) of Preposition 10.
Equation (130) can be finally derived by using the identity (34) which links the energy and the entropy of Gibbs states. Accordingly, at first order in ∆S(ρ) we get where we used also e β max N −1 The bound (130) is hence obtained by first using the fact that thanks to the property β(ρ) ≤ β(ρ) we have u(ρ) ≤ u(ρ), and then replacing (158) into the inequality (127).

B. Derivation of the asymptotic bound (133)
We now consider the case of an Hamiltonian H whose spectrum has only two distinct eigenvalues, 0 and max > 0. Here (126) can be replaced by Assume once more the entropy of ρ is larger than or equal to log d 0 so that E β(ρ) is well defined. In order to have E(ρ; H) > E β(ρ) , the populations of ρ must necessarily satisfy the condition (137), and also the additional condition The validity of (137) allows us to use Preposition 10, from which it follows equation (152). On the other hand, the condition (160) can be identified with the condition (145) in Preposition 8, implying that inequality (146) holds for the energy level max . Combining equations (152) and (146), we deduce that for a two-level hamiltonian The bound (161) is similar to the bound (156), but it lacks the term proportional to E β(ρ) . Using (34) and e β max N −1 = 1 + O 1 N , we can convert the bound (161) on ∆S(ρ) in an asymptotic bound on E β(ρ) , which is equation (158) without the term proportional to E β(ρ) , i.e., Replacing (162) into the inequality (159), we therefore obtain the bound (133).

C. Derivation of Eq. (134)
Here we focus on the case where we have a too small entropy to even identify a Gibbs isoentropic counterpart, i.e.
Let ∆S 0 (ρ) denote the cost of levelling up only the ground state populations of ρ, i.e. constructing a density matrixρ 0 obtained by replacing the ground state populations of ρ with λ 0 = 1 d0 j =0 λ j while leaving all the other populations untouched, By majorization it is easy to verify that the entropy ofρ 0 is not smaller than the one of the Gibbs ground state ω ∞ , therefore we can write Furthermore reasoning along the same lines of Preposition 10, we also have that for every λ i such that i > 0, which implies (167) There are d − d 0 levels above the ground state, and their contribution to the total entropy is bounded by  (168) and (169) we deduce the inequality which in conclusion gives us V. SOME CONSIDERATIONS ABOUT COMMENSURABLE SPECTRA In Sec. II we commented about the fact that for twodimensional systems (d = 2) the hierarchy (15) trivialises (the structurally stable passive states being also N passive for all N ) due to the fact that all density matrices which are diagonal in the energy basis can be cast in the Gibbs form for some proper choice of β and Z. On the contrary, as the dimensionality increases, Eq. (37) implies that the exponential connection which according to Eq. (29) links the energy levels and the associated populations, is recovered only with the hypotheses of complete passivity and structural stability. This general rule admits some notable exceptions when the spectrum of the system Hamiltonian exhibits special properties. In particular, it is possible to show that if a subset of the energy levels of H are commensurable, then the associated populations of a state ρ which is structurally stable and N -passive (with N sufficiently large but finite), must be expressed as in Eq. (172) for some proper choice of β and Z. More specifically Preposition 12. If a < b < c are three energy levels of H such that for some integers p and q, and if N ≥ q, then in any Npassive, N -structurally stable state ρ ∈P and | c ⊗q ⊗ | a ⊗(p−q) ⊗ | 0 ⊗(N −p) of ρ ⊗N have the same energy (notice that we are using here that since ρ ∈P it is diagonal in the energy eigenbasis). Thence according to structurally stable condition (16) they must have the same populations, i.e.

Corollary 5.
For an hamiltonian with equally spaced energy levels ( n = n 1 ), there are no nontrivial N -passive, Nstructurally stable states for N ≥ 2.
Corollary 6. For a generic discrete hamiltonian H whose energy levels are commensurable, there are no nontrivial Npassive, N -structurally stable states for any N ≥ N * , where The last statement leads us to the following observation which holds for continuous variable systems -the definition of N -passivity being easily generalized in this case.

Corollary 7.
For an hamiltonian H with a purely continuous energy spectrum, there are no nontrivial, N -passive, Nstructurally stable states for N ≥ 2.
Proof. Take any two energies a < c . Since the spectrum is continuous, there exist eigenstates with any possible energy between a and c ; then we can always find a suitable b to satisfy the condition of Preposition 12.

VI. CONCLUSIONS
We derived upper bounds for the mean energy of N -passive, structurally stable configurations ρ. We also give inequalities that apply for N -passive states which are not necessarily structurally stable, in the asymptotic limit of large N . Our inequalities depend on the spectral quantity R(H); the latter will typically be larger for larger values of the Hilbert space dimension d, resulting in looser upper bounds. On the contrary, we expect that the ratio between the maximal energy of an N -passive state and the energy of the isoentropic Gibbs state will, in general, be smaller for larger dimensions d, because the eigenvalues of ρ will be constrained by more conditions. In the continuum limit, as we have seen, the set of N -passive, N -structurally stable collapses on the set of Gibbs states.
Possible future development of the present approach could be the study the connection between higher momenta of the energy distribution of ρ and those of its Gibbs isoentropic counterpart ω β(ρ) . More generally one could also employ the technique we present here for estimating how the distance between ρ and ω β(ρ) drops when N increases.
ACKNOWLEDGMENT We thank G. M. Andolina for comments.

APPENDIX A MORE ON THE ERGOTROPY FUNCTIONAL
The erogotropy functional (4) can be casted in a more compact formula by explicitly solving the optimization over U . For this purpose let us write ρ as with eigenvectors {|φ j } j and associated eigenvalues {λ j } j , which, without loss of generality, we shall assume to be organized in non-increasing order, i.e.
A passive counterpart ρ p of ρ is now identified as an element of S which is diagonal with respect to the energy eigeinbasis {| j } j and which can be expressed as where {λ j } j is a relabelling of {λ j } j that fulfils the following ordering In other words ρ p is an element of P H that is iso-spectral to ρ, i.e. which admits the {λ j } j has eigenvalues. Accordingly there exists always a unitary transformation U p such that connects them, i.e. ρ p = U p ρU † p . It should also be noticed that due to the special ordering we fixed in (176) and (2) an examples of passive state (177) is given by the density matrix obtained from ρ by simply replacing |φ j with | j for all j. If the Hamiltonian H is explicitly not degenerate (i.e. if in Eq. (2) is verified with strict inequalities),ρ p is the unique passive counterpart of ρ. However, if H instead admits some degree of degeneracy then this is not true and ρ may admits other passive counterparts others than (179) which can be obtained from the latter by means of arbitrary unitary rotations that do not mix up eigenspaces associated with different eigenvalues (this freedom in the definition of ρ p is associated with the fact that indeed if i = j for some i = j, then Eq. (178) does not fix any relative ordering between the associated populations).
In any case all passive counterparts of ρ will have the same mean energy, i.e.
Most importantly one can verify that the unitaries U which leads to the maximum in the right-hand-side of Eq. (4) are exactly the one that maps ρ into one of it passive counterparts, accordingly we can write [1], [2] with δ j,j being the Kronecker delta.
The identity (37) establishes that Gibbs and ground states are the only CP configurations of the system A, while (36) specifies that the Gibbs are also the only CPSS density matrices. Explicit proofs of these statements can be found in Refs. [1], [2], [8], [14]. In what follows however we give a simple, alternative demonstration of this fact based on some simple geometric considerations. H , are diagonal in the energy eigenbasis, we can restrict the analysis to this special case assuming that our ρ has the form (7), i.e.
Consider then condition ii) that enforces N -order passivity. Introducing the positive quantities b i := − ln λ i from Eq. (13) it follows that ρ is CP if and only if, for all N , and for all allowed choices of the sets I N := {n 1 , n 2 , · · · , n d }, J N := {m 1 , m 2 , · · · , m d }, we have d i=0 where the regularization (14) translates into (notice however that we do not need to enforce an analogous regularization for opposite situation for the product (n = ∞)( = 0) which we leave explicitly indeterminate). If we interpret b i and i as component of vectors in R d , Eq. (184) can be reframed as with I N , J N ∈ R d obtained by promoting the elements of I N and J N into vectorial components respectively, i.e. I N := (n 0 , n 1 , · · · , n d−1 ) and J N := (m 0 , m 1 , · · · , m d−1 ). Calling then 1 the vector (1, 1, . . . , 1) of R d , by construction we have that I N · 1 = J N · 1 = N , implying that the vector I N − J N is orthogonal to 1, i.e. ( I N − J N ) · 1 = 0. Accordingly Eq. (186) rewrites where V N := is the subset of R d of the allowed (normalized) vectors. For N → ∞, V N tends to a limit subset V ∞ := N ≥1 j≥N V j , and the CP requirement can be expressed as Since V ∞ is dense in the subspace of the unitary sphere which is orthogonal to 1, Eq. (188) is possible only if, once projected into that subspace, the vectors and b are linearly dependent by a non-negative proportionality constant β ≥ 0. Projecting in the subspace perpendicular to 1 is equivalent to add Z1 for some real constant Z. Therefore we must have which expanded in components leads to which formally coincides with the request to have ρ in the Gibbs form (29) (the value of Z being forced to coincide with Z β by normalization). The only exception to (190) occurs in the limiting case where the identity (189) is fulfilled with an infinite value of β. Under this circumstance for all i which are strictly larger than zero (i.e. for all i ≥ d 0 ) we have that b i diverges forcing the associated λ i to be exactly equal to zero. On the other hand when i = 0 (i.e. for all i ∈ {0, 1, · · · , d 0 }) the form (β = ∞)( = 0) is indeterminate -see comment below Eq. (185) -and the constraint (190) needs not to be applied leaving us the freedom to chose the associated values of λ i as we wish. This leads us to identify the ground state elements as the only other possible choices for being CP, concluding the proof of Eq. (37). Proof. According to Preposition 13 the only CP states are the Gibbs and the ground states elements. For d 0 > 1 however ground states need not to fulfil the constraint (9) required for being 1-structural stable, on the contrary Gibbs density matrices have λ i = e −β i /Z which naturally implement such requirement.
More generally the Gibbs states verify also the stronger requirement (16) for all value of k, hence they are also kstationary stable at all order:

APPENDIX C MAJORIZATION ARGUMENT
Here we present an explicit proof of the majorization argument used in the proof of Preposition 4, i.e. we show that if then there must exist must exist c > b > 0 such that whereλ j = Z −1 β(ρ) e −β(ρ) j are the eigenvalues of the Gibbs state ω β(ρ) .
For the sake of completeness we briefly recall that given two probability sets P := {p j } j=1,··· ,d and Q := {q j } j=1,··· ,d whose elements are labelled in non-decreasing order, i.e. p j ≥ p j+1 , q j ≥ q j+1 for all j ∈ {1, · · · , d − 1}, one say that Q majorizes P when [35], [36] the inequality being always saturated with an identity for k = d due to normalization conditions. Furthermore if there exists at least one value k ≤ d − 1, for which (193) is fulfilled with a strict inequality we say that Q strictly majorizes P . It turns out that majorization induces an ordering for the entropy of the two sets, so that whenever Q majorizes P , then the entropy of the former is always smaller than or equal to the entropy of the latter, the inequality being strict if the strict majorization condition applies. It is hence clear that if the two probability sets have identitical entropy then there neither Q can majorize P , nor Q can majorize by P . Taking into accounts the above facts let now go back to the proof of the property (192). The existence of b > 0 fulfilling (192) can be established from (191) and from the fact that ρ and ω β(ρ) have both trace one. We can further observe that one can select as b a level of H which has not maximum energy value; indeed, if by contradiction for all j smaller than the maximum energy value of H, from (191) it would follow that ρ would be strictly majorized by ω(ρ), which is impossible as the two states have the same entropy. Now take as b the one which has the smallest energy. Accordingly for all i < b we have λ i ≤λ i and hence, the strict inequality being a consequence of (191). Therefore there must exist c ∈ {b, · · · , d − 1} such that otherwise {λ i } i would strictly majorize {λ i } i and the two could not have the same entropy. Observe then that the normalization conditions impose that which can only be satisfied if there exist c ≥ c +1 > b such that λ c ≤λ c , hence proving the thesis.