The geometry of passivity for quantum systems and a novel elementary derivation of the Gibbs state

Passivity is a fundamental concept that constitutes a necessary condition for any quantum system to attain thermodynamic equilibrium, and for a notion of temperature to emerge. While extensive work has been done that exploits this, the transition from passivity at a single-shot level to the completely passive Gibbs state is technically clear but lacks a good over-arching intuition. Here, we re-formulate passivity for quantum systems in purely geometric terms. This description makes the emergence of the Gibbs state from passive states entirely transparent. Beyond clarifying existing results, it also provides novel analysis for non-equilibrium quantum systems. We show that, to every passive state, one can associate a simple convex shape in a 2-dimensional plane, and that the area of this shape measures the degree to which the system deviates from the manifold of equilibrium states. This provides a novel geometric measure of athermality with relations to both ergotropy and $\beta$-athermality.


I. INTRODUCTION
The Gibbs state is a cornerstone of equilibrium statistical mechanics, and provides a rigorous notion of temperature for any quantum system. There are a range of justifications for this state, and each reveals different facets of thermodynamics. For example, it can be obtained from a micro-canonical ensemble derivation [1], where one posits an equal a priori probability distribution over energies for some large system and then proceeds to analyse the resultant marginal state on a much smaller subsystem. It can also be described from an inferential perspective, where one adopts a maximally unbiased description of the quantum system subject to known constraints -namely a maximum entropy construction [2,3]. One highly active area of research is to determine the conditions under which a dynamically-evolving system tends towards an equilibrium state, and moreover one in which a notion of temperature emerges [4][5][6]. However, there is another route to the Gibbs state that is appealing in its physical simplicity, namely through the concept of passivity of quantum states [7][8][9]. This arose in the study of infinite-dimensional quantum systems, where one must adopt an algebraic description in order to rigorously describe features such as phase transitions [10].
Loosely speaking, passivity captures the inability of a quantum state to be used to 'raise a weight'. The concept of passivity does not tell us when quantum systems dynamically equilibrate, instead it provides a simple characterisation of Gibbs states. This simplicity also makes FIG. 1. The physical and mathematical elements of passivity. Shown on the left in green are the key physical concepts that single out the equilibrium state of a quantum system through the lens of passivity. Each component corresponds to a simple mathematical structure, shown on the right in blue. Together these lead to a purely geometric account of equilibrium states for a quantum system in terms of an asymptotic ensemble V∞(ρ) for a state ρ.
it well-suited to exploring structural aspects of thermodynamics in quantum systems beyond equilibrium, and for elucidating the conceptual ingredients required for a notion of equilibrium to be established.
The original passivity analysis was in terms of C * algebras [7], and so did not lend itself to a widely accessible 2 account. This was in part remedied by Lenard in 1978, who provided an analysis specialised to finite-dimensional quantum systems [8]. There have since been shorter and more compact passivity analyses, for example via the concept of 'virtual temperatures' [9]. While such derivations of the Gibbs state from passivity are technically clear, one might wish for a more intuitive perspective on the structure of passivity that makes the emergence of the Gibbs state entirely inevitable while suggesting natural extensions.
With this aim in mind, we present a novel geometric formulation of passivity. In particular, we show in Section III C that every passive state ρ of a quantum system can be associated to a simple convex shape V ∞ (ρ) in the 2-d plane. This convex shape naturally emerges in the macroscopic regime of many independent copies of the state, and its structure entirely encodes the deviation of the system from the manifold of equilibrium states. With this geometric notion, in Section III D we show how the Gibbs state derivation becomes almost trivial, and that the area of the shape V ∞ (ρ) is a well-defined measure of the degree to which the state deviates from equilibrium. In Section IV B, we show the area is monotonely non-increasing along so-called activation trajectories in the energy-entropy plane, and argue that it can supplement the traditional macroscopic equilibrium variables to capture the macroscopic non-equilibrium properties of a quantum system. We also demonstrate in Section IV C how the area relates to the concept of ergotropy [24], as well as the recently introduced notion of β-athermality [25].

II. THE CORE PHYSICAL PRINCIPLES
We begin by making clear what physical notions we use and why they are either required or appealing conceptually. The two core components involved are the concepts of passivity and extensivity, which we now explain.

A. Passivity
The first core principle in our treatment is passivity, which is physically and operationally clear and fully deterministic. Two examples suffice to illustrate the main idea. First, consider a three-level atom with Hamiltonian H = |1 1|+2 |2 2|+3 |3 3| in the following populationinverted state By applying a unitary evolution to the system that deterministically swaps levels 1 and 3, we can lower the average energy of the system. In particular, the 'passifying' unitary U 1 transforms the state of the system to the new state ρ p : The internal energy of the system has decreased by amount which implicitly corresponds to work extraction. However, given the state ρ P in Eq. (2) the average energy cannot be lowered any further by any unitary. Therefore, there has been a maximal extraction of work from the quantum state ρ with respect to any possible unitary evolution of the system, and the quantity W in Eq. (3) corresponds to the ergotropy of ρ [24]. Another example for the same Hamiltonian H is given by the uniform superposition state Given such an initial state |ψ it is possible to unitarily transform the system precisely to its ground state where ω := e 2πi 3 and U 2 is represented in the {|1 , |2 , |3 } basis. Clearly, we cannot further lower the energy of this atomic system in its ground state |ψ P = |1 via unitary evolution. States such as ρ P and |ψ P , for which no further lowering of energy can occur through deterministic unitary transformations, are called passive. This is formalised by the following definition [7]. Definition 1. A quantum state ρ of a system with Hamiltonian H is passive if tr HU ρU † ≥ tr(Hρ) (6) for all unitaries U acting upon the system.
For simplicity, we restrict our attention to finite dimensional systems with bounded energies, although the notion of passivity extends to infinite dimensional systems [7]. In light of this definition, a natural question is: when exactly is a quantum state passive? The following theorem provides necessary and sufficient conditions for a finite-dimensional state being passive [8]. See Appendix (A) for a majorization-based proof. Thus, passive states are precisely those which are blockdiagonal in the energy eigenbasis with eigenvalues which are 'anti-ordered' with respect to the energies.
The definition of passivity does not invoke the full machinery of thermodynamics. It is a physical statement about the deterministic extraction of energy from a system. However, the notion is embedded within thermodynamics in the Kelvin-Planck formulation of the Second Law [26], which forbids the extraction of work from thermalised systems that are adiabatically isolated.

B. Extensivity
Passivity is defined relative to a Hamiltonian, but one could replace this with another observable [15][16][17][18][19]. This would relate to other types of resources, not just those from which energy can be reversibly extracted. However, even restricting ourselves to energetic considerations, we could define passivity with respect to the second moment H 2 . It is readily seen that this gives the same set of passive states as the original definition 1 .
The expectation value of energy H arises in other derivations of the Gibbs state. The MaxEnt approach singles out the state that maximises the entropy given a fixed average energy [2,3]. Langrange multipliers show that this state takes the Gibbs form, However, if one maximizes the entropy with respect to fixed H 2 this would instead lead to no longer giving the expected Gibbs form. Therefore, the reason why passivity is based on the expectation value of H is motivated by an additional physical property. This second core ingredient is extensivity. H is an extensive observable, unlike H 2 or any other power, and this occurs because energy is additively conserved microscopically. However, if two systems are passive under their respective Hamiltonians, there is no guarantee that the combined system will also be passive under the combined Hamiltonian. This motivates the following extension.
If a state is k-passive for all integers k ≥ 1, it is called completely passive.
As we shall see, a celebrated result is that, for finite dimensional quantum systems, the completely passive states are essentially the Gibbs states.

III. A GEOMETRIC FORMULATION OF PASSIVITY
We now present a novel geometric reformulation of passivity and show how it leads to a transparent derivation of the thermodynamic Gibbs state.
A. The -s Ensemble Consider a quantum system with Hamiltonian H on the finite-dimensional Hilbert space H, and a passive state ρ of the system. Given the necessary and sufficient conditions of Theorem 2, we need only consider (p i ) = eigs(ρ) and ( i ) = eigs(H). For simplicity, we first assume that the spectrum of H is non-degenerate, and discuss the more general case in Appendix B. For this situation, there is an orthonormal basis |e i in which we have ρ = diag(p i ) and H = diag( i ).
In order to analyse passive states, an initial choice of representation might be the pairs {( i , p i ) : 1 ≤ i ≤ d} and the conditions on these that lead to passivity, kpassivity and complete passivity. However, this representation has the disadvantage of treating the two quantities in the pairing differently when systems are combined: energy is additive, whereas probabilities multiply. Additivity suggests that, instead of probabilities p i , a better choice is to use s i := − log p i , which can be viewed as a "single instance" entropy.
The set of points V(ρ) provide an ' -s ensemble' by virtue that if we view ( i , s i ) as defining a two-component random variable with probability distribution (p i ), then the expectation values of the two components have physical interpretations -the average energy and von Neumann entropy of ρ. We explore this further in Section IV.

B. Passivity as a total order on vectors
Passivity implies a certain structure on the state of the system, fully specified by the necessary and sufficient conditions given in Theorem 2. To summarise, a passive state ρ is block-diagonal in the energy eigenbasis, with energies ( i ) and eigenvalues (p i ) that are 'anti-ordered'.
Our introduction of the -s ensemble allows us to recast these conditions for passivity as a simple geometric property in R 2 . The key point here will be the idea of a total ordering on a vector space.
Definition 5. Let r i := (x i , y i ) and r j := (x j , y j ) be two vectors in R 2 . Then we define the relation ≤ such that for any two vectors r i , r j ∈ R 2 , we have r i ≤ r j if and only if x i ≤ x j and y i ≤ y j . Moreover, given a set R = {r 1 , r 2 , . . . } of vectors, we say that R is totally ordered if, given any two r i , r j in R, then either r i ≤ r j or r j ≤ r i .
With this definition in hand, we are now in a position to provide a simple characterisation of passive states in terms of the -s ensemble. Proof. This result follows directly from Theorem 2, since − log p i ≤ − log p j if and only if p i ≥ p j .
As shown in Fig. (2), the total order appearing in Lemma 6 corresponds to a set of geometric constraints on the relative positioning of the elements of the -s ensemble in R 2 . More precisely, if we pick out any element v i of the -s ensemble, all other elements v j , ∀j = i must lie within the upper-right or lower-left quadrant defined by the point v i .

C. k-passivity and the -s ensemble
Passivity is defined for a single system, while complete passivity shifts the focus onto multiple copies. Our introduction of the -s ensemble lends itself well to this change of focus, precisely because it respects additivity under the composition of systems. Before outlining the key result, we first clarify this statement with a few examples. It proves to be useful to 'regularise' the -s ensemble by dividing by the number of copies of the system. For instance, given a qubit system (d = 2) with -s ensemble V(ρ) = {v 1 , v 2 }, the regularised representation for two copies of the state is and for three copies it is where c i ∈ N. These are illustrated for a qutrit example in Fig. (3). The condition of k-passivity for quantum states can now be geometrically restated. Proof. This is simply Lemma 6 stated for the -s ensemble of k copies of the state. To consider complete passivity, we define an asymptotic version of the -s ensemble, taking the limit of V k (ρ) as k → ∞, and making the intuition of Fig. (3) rigorous. The technical definition of this asymptotic ensemble is given in Appendix C, but here we need only consider a more intuitive picture, which views the IID asymptotic ensemble as the convex hull of the single-copy -s ensemble.
Lemma 8 (Asymptotic ensemble). For a d-dimensional quantum system with Hamiltonian H, and state ρ with [ρ, H] = 0 and -s ensemble V(ρ), we have that where conv[S] denotes the convex hull of a set S.
The proof of this is provided in Appendix C. Some examples are pictured in Fig. (4). The significance of this is that in the asymptotic limit the structure of passive states is extremely simple -each passive state in the many-copy limit is described by a convex polygon in the plane that is easily obtained from the single-copy ensemble.

D. The geometric derivation of the Gibbs state
We now state the following elementary planar geometry result, which is needed for our Gibbs state derivation.

Lemma 9.
A convex set C is totally ordered in R 2 if and only if all its points are colinear with a non-negative slope.
Proof. Consider a totally ordered convex set C in R 2 , depicted in Fig. (5). Suppose there exist three points x 1 , x 2 , x 3 in C that are not colinear. Convexity implies that the triangular region formed by their convex hull also lies in C, and in particular any circle in this triangle is also in this set. However, it is impossible for a circle in the plane to be totally ordered, contradicting the initial assumption. The only way to avoid this contradiction is to enforce that all points of C are colinear. Conversely, a convex set of colinear points in R 2 with non-negative slope is, by inspection, clearly totally ordered, which completes the proof.

FIG. 5.
Proof that the points of a totally ordered convex set are colinear. Three points in C are selected and a circle is chosen from inside their convex hull. As the righthand circle highlights, the dashed red points are not totally ordered with the black point, demonstrating that a circle cannot be totally ordered in R 2 .
We now present the geometric proof of the following well-known theorem [7,8].
Theorem 10 (Complete passivity of Gibbs states). A state ρ of a d-dimensional quantum system with Hamiltonian H is completely passive if and only if it is a Gibbs state ρ = e −βH /Z β for some β ≥ 0.
The proof is now obvious from the geometry of the situation.
Proof. Suppose a state ρ of a d-dimensional system is completely passive. By definition, it is passive for all k ≥ 1, thus V k (ρ) must be totally ordered for all k ≥ 1. This implies that V ∞ (ρ) must also be a totally ordered set. However, V ∞ (ρ) is convex, therefore Lemma 9 implies it must be a line segment with non-negative slope. Since can be written as ( i , s i ) = ( i , β i − log Z) for some constants Z and β ≥ 0. This implies that p i = e −β i /Z, thus ρ is a Gibbs state at some inverse temperature β ≥ 0.
Conversely, if ρ is a Gibbs state with β ≥ 0 then it is clear that V(ρ) is a set of co-linear points with nonnegative slope, and thus the convex hull V ∞ (ρ) is a totally ordered line segment with non-negative slope. The Gibbs state is therefore completely passive, which completes the proof.
Note that this proof makes it immediately clear that all passive qubit states (d = 2) are also completely passive, because the two points of their -s ensemble have to be colinear. It is only for d ≥ 3 that there is a separation between passivity and complete passivity. We can re-state the condition for complete passivity in a purely geometric form.
Corollary 11. A passive state ρ of a d-dimensional quantum system with Hamiltonian H is completely passive / Gibbsian if and only if V(ρ) is totally ordered and the area of V ∞ (ρ) is zero.
In this geometric perspective, the points of the -s ensemble for a completely passive state are colinear. The gradient of this line has the natural interpretation of inverse temperature β. This connection between temperature and gradient in our geometric picture extends to all passive states, where the virtual temperature is the gradient between the points v i , v j ∈ V(ρ) [9]. While a completely passive state is fully specified by a single inverse temperature β, other passive states are described by a set {β i,j }. This suggests a physical intuition as to why energy cannot be extracted from a completely passive state: if there are different virtual temperatures, a heat engine can be operated between them and work can be extracted [12].
Completely passive states of different systems at the same temperature will remain completely passive when combined. This can be immediately seen in our geometric description -points on the same line are closed under vector addition. This returns the usual thermodynamic temperature. However, the notion arising from complete passivity has the advantage that it applies to individual systems. Even an individual qubit in a completely passive state can be sensibly ascribed an inverse temperature β.
The ensemble V(ρ) provides a fine-grained description of a passive state ρ, while the asymptotic ensemble V ∞ (ρ) gives a fine-grained description of the macroscopic manycopy limit of this state. However, in the macroscopic equilibrium regime, the physics of the system is described by a small number of macroscopic variables [27], such as pressure, volume, entropy, etc. This prompts the following question: Can we supplement the traditional macroscopic equilibrium variables with a well-defined macroscopic non-equilibrium variable that quantifies the degree to which the system deviates from equilibrium?
Two important points arise here. Firstly, such a variable is not considered relative to any one particular equilibrium state, but relative to the complete set of thermal equilibrium states. Secondly, we do not want to exploit the global structure of the manifold of equilibrium states in order to define this variable, but instead wish that it is defined purely from the statistics of the individual state under consideration.
To address this, we focus on the situation in which the two primary macroscopic equilibrium variables for the system are the internal energy E and the equilibrium entropy S. One possible candidate for a non-equilibrium variable is the asymptotic ergotropy [28] of the quantum state -the maximal rate of work that can be extracted from many copies of the passive state. However, this is understood from an optimal work-extraction process on many copies of the state. Here we wish to explore other options that might lend themselves to properties purely at the state level, as opposed to process-dependent concepts.
In the previous analysis, we showed that passive states admit a geometric description in which the nonequilibrium aspect of the system coincides with a nonzero area for the asymptotic ensemble. Therefore it is natural to ask if the area of V ∞ (ρ) can be used as the relevant asymptotic non-equilibrium variable. In this section, we address the degree to which the area fulfils the desired behaviour, and then show how it relates to other measures such as the asymptotic ergotropy.

A. Macroscopic regime and the energy-entropy diagram
To connect with the traditional macroscopic description, we reinterpret V(ρ) as a random variable that takes values ( k , s k ) in R 2 and which has expectation value where E(ρ) := tr[ρH] is the average energy and S(ρ) := −tr[ρ log ρ] is the von Neumann entropy of the state ρ.
Asymptotic thermodynamics describes the limit where the number of non-interacting copies of the system tends to infinity. Roughly speaking, in this limit we call two quantum states ρ and σ asymptotically equivalent, and write ρ σ, if we can transform between the two states via energy preserving unitaries [25]. The fineprint here is that one allows for the addition of some ancilliary system which is sublinear in size. The sublinearity of the ancilla guarantees that, in the asymptotic limit, the amount of energy and entropy that can be transferred between the ancilla and each copy of the system tends to zero, meaning its per-copy contribution can be safely neglected. Asymptotic equivalence has been shown to pick out the two macroscopic variables identified in Eq. (15) as the two relevant quantities to describe the entire state space when the asymptotic limit is taken. This is expressed neatly in the following theorem, which was proven in [25].
Theorem 12. Consider two arbitrary states ρ and σ on a d-dimensional quantum system with fixed Hamiltonian H. Then the following equivalence holds Put simply, in the many-copy limit, any two states can be interconverted via unitaries that conserve energy globally if and only if they have the same average energy The union of all sets D (E,S) spans the entire space of states D. Furthermore, the set of points (E, S) corresponding to physical states ρ ∈ D form a closed convex subset in R 2 , known as the energy-entropy diagram [25]. It follows that, asymptotically, thermodynamics can be restricted to a 2-dimensional planar geometry. Returning to our ensemble description, Eq. (15) shows that the statistical average or 'centre of mass' of all elements of a given ensemble V(ρ) corresponds to a single point (E, S) on the energy-entropy (E-S) diagram.
The key features of the energy-entropy diagram are illustrated in Fig. (6) for a simple finite-dimensional system. The boundaries of the E-S diagram are fully determined by the particular Hamiltonian of the system H. The space of states is bounded from above by the curve of Gibbs states with respect to H, i.e., those which maximise the entropy for a given energy. Quantum states that are not completely passive lie below the thermal curve.

B. Area as a non-equilibrium variable
Asymptotic equivalence does not on its own describe a notion of thermal equilibrium. Therefore, we want to supplement asymptotic equivalence with a notion of thermal equilibrium in a natural way. In the asymptotic limit, the ensemble V ∞ (ρ) is given by the convex hull of the single-copy ensemble V(ρ), and has an associated area A(ρ). Having shown that ρ is completely passive if and only if V ∞ (ρ) has zero area and non-negative slope, we might wonder if the area variable A can be added to (E, S) to replace the statistical description of the system with non-equilibrium thermodynamic variables (A, E, S). Moreover, we ask whether the area A has a straightforward physical interpretation. First, we need an explicit expression for the area variable. The area of V ∞ (ρ) is fully determined by the vertices of the convex hull.
Definition 14. For a d-dimensional quantum system in passive state ρ, the set of n ≤ d vertices of the convex set V ∞ (ρ) is defined as where ext[C] denotes the set of extremal points of the convex set C and the vertices v 1 , . . . , v n are labelled clockwise modulo n around the convex hull V ∞ (ρ).
By the trapezium rule [29], the area of V ∞ (ρ) is where the sum runs over the n elements of V vert (ρ), and ∆ i,j := i − j is the energy level spacing between the corresponding elements. For a general d-dimensional system, where d > 3, the area A(ρ) can vary as we range over all states ρ in the 8 equivalence class D (E,S) . However, we now exploit the asymptotic equivalence given in Theorem 12 to single out a canonical area A for a given (E, S) pair, which we suggestively call the geometric athermality.
Definition 15. Consider a d-dimensional quantum system with fixed Hamiltonian H. The geometric athermality A for each (E, S) pair is where A(ρ) is the area of V ∞ (ρ).
If the only information we have about a system is its average energy and entropy (E, S), the geometric athermality A is a function of E and S but provides new information about the underlying state of the system. More precisely, it tells us whether or not the system is in thermal equilibrium with respect to any temperature. In other words, A is a witness of athermality. This motivates the introduction of geometric athermality A as our additional thermodynamic variable to supplement the asymptotic description (E, S), forming the triple of numbers (A, E, S).
Does this additional thermodynamic variable admit a further operational interpretation beyond being a witness of athermality? In the remainder of this section, we answer this question in the affirmative by upgrading the statement that A is a witness of athermality to the statement that A is a measure of athermality. More precisely, we show that A is monotonically non-increasing under a set of physically motivated trajectories in the E-S plane that bring the state of the system closer to the manifold of Gibbs states.
We first clarify the set of physical trajectories in E-S space that concern us here. We restrict our attention to transformations of passive states that lead to work extraction and require that no ordered energy is injected into the system. Such transformations must therefore never increase the average energy or decrease the entropy of the system, since trajectories without these restrictions could involve the implicit smuggling-in of work resources. Hence, any infinitesimal evolution of the state is restricted in the direction of the shaded region in Fig. (6). Inspired by the activation maps introduced in [25], we call any state trajectory constructed from such infinitesimal transformations an activation trajectory, because it extracts ergotropy from the asymptotic collection of passive states ρ ⊗n , with n 1.
Definition 16. Any trajectory on the E-S diagram is called an activation trajectory if and only if its tangent unit vector u := (u E , u S ) satisfies u E ≤ 0 and u S ≥ 0 at all points along the trajectory directed towards the manifold of equilibrium states.
Equipped with the above definition, we now present the following theorem that justifies the use of A as a genuine athermality measure.
Theorem 17 (Monotonicity of A). For any ddimensional quantum system with non-degenerate Hamiltonian H, the geometric athermality A of a passive state ρ is monotonically non-increasing along all activation trajectories.
A full proof is given in Appendix D, but for illustrative purposes, we here sketch the proof for a qutrit system. The extension to higher dimensions involves a protocol that generalises the simple qutrit case.
Consider a passive, but not completely passive, state ρ of a qutrit system with non-degenerate Hamiltonian H. For a qutrit system, there is a unique state 3 corresponding to each pair (E, S), i.e., |D (E,S) | = 1. Therefore, the geometric athermality A can be simply computed using Eq. (19) without performing the optimization in Definition 15. Due to the non-linearity of entropy, we do not have an analytic expression for A in terms of E and S. However we can find a general, explicit expression for the differential of the geometric athermality A for any passive qutrit state ρ: which is derived in Appendix D. The partial derivatives of this expression satisfy ∂ E A > 0 and ∂ S A < 0, therefore the geometric athermality is monotonically decreasing along any activation trajectory, because the directional derivative at any point is Since the geometric athermality is monotonically decreasing along activation trajectories for qutrits, we have shown it constitutes a non-trivial measure of athermality for such systems.
C. Relation of geometric athermality to β-athermality, ergotropy and maximal heat extraction The manifold of thermal equilibrium states is the boundary of quantum states as depicted in Fig. (6) and correspond to zero geometric athermality. However, A is not the only measure of non-equilibrium behaviour that attains a zero value on the equilibrium curve. For example, completely passive states are the only states from FIG. 7. Contour plot for A. Shown is the contour plot for the geometric athermality within the passive region for a qutrit with energy spectrum (0, 1, 2). The thick black contour in the light blue region corresponds to the set of thermal equilibrium states of the system. We rescale the value of A to f (A) in the colour scaling, where f (x) := 1 − e −x is a monotonically increasing function of x, so as to highlight the structure of the contours. It can be seen on the plot that geometric athermality is monotonically non-increasing towards the equilibrium curve.
which no work can be extracted in the IID limit, as well as the only states that can attain zero β-athermality [25]. In this section, we consider how these quantities relate to the geometric athermality and, in particular, how the available work of a system in the asymptotic limit varies with A.
The β-athermality of a state ρ measures the "distance" on the E-S plane between the state and a specific Gibbs state γ β at the point E(γ β ), S(γ β ) on the equilibrium curve. It is given by the relative entropy between the two states, where F (σ) := E(σ) − β −1 S(σ) is the free energy of state σ. This measure explicitly depends on the particular choice of Gibbs state γ β . In particular, any Gibbs state γ β with β = β has non-zero β-athermality. This measure is relevant for the usual resource-theoretic approach to thermodynamics, where systems can freely equilibrate with a thermal reservoir at inverse temperature β [30]. However, in more general contexts, where all states on the equilibrium curve are considered equilibrium states, it is desirable to have a measure of athermality, like the geometric athermality, that does not have such a temperature dependence. In fact, Theorem 18 shows that the quantity inf β∈[0,∞) a β (ρ) is simply the entropy difference of state ρ along the activation trajectory that thermalises it with zero work output. It can therefore be considered as a modified measure of athermality that no longer depends on temperature and attains a zero value for all Gibbs states. We can therefore turn our attention to investigating the relation of the geometric athermality with the available work per system in the asymptotic limit. This is done by investigating certain simple activation trajectories on the E-S plane.
An isentropic trajectory that brings the state closer to the equilibrium curve is equivalent to work extraction with no entropic losses, so that the maximal extractable work W max of a state ρ is where β max is chosen so that the entropy remains constant along the trajectory, S(γ βmax ) = S(ρ). This upper bound on the extractable work, which one could call the asymptotic ergotropy [24], has been shown to be attainable in [28] for the asymptotic regime of many identical copies ρ ⊗n of a state ρ.
Correspondingly, an isoenergetic trajectory that brings the state on the equilibrium curve is equivalent to no work extraction. We can rephrase this thermodynamically as the activation trajectory that outputs the maximal entropic gain. It is given by an expression that is dual to Eq. (24) in the thermodynamic variables (E, S), where β min is now chosen so that the average energy remains constant, E(γ βmin ) = E(ρ). This bound is attainable simply by thermalising the system at inversetemperature β min . This bound can in fact be thought of as the minimal β-athermality of state ρ.
Theorem 18. The maximal entropic gain for a system in a state ρ along all possible activation trajectories is given by its minimal β-athermality, Proof. By the defining relation Eq. (25) for ∆S max and β min , we always have To prove the second equality, we consider the the following two possible cases for any given state ρ.
If ρ = |e 1 e 1 |, it coincides with γ βmin ≡ γ ∞ and If ρ = |e 1 e 1 |, we expand Eq. (23) to get so that a β (ρ) attains a unique extremum when E(γ) = E(ρ). Since a β (ρ) Geometrically, Theorem 18 says that the Gibbs state closest to ρ with respect to the family of athermality functions a β is the one lying directly above ρ on the E-S diagram.
To illustrate the relation between the geometric athermality and asymptotic ergotropy (or minimal βathermality), we consider in Fig. (8) a maximally energetic passive qutrit state ρ 0 [11] under a Hamiltonian with equal energy spacing. The plot shows that the area of a 3-dimensional system and its asymptotic ergotropy are positively correlated along any activation trajectory. Similar results follow for the minimal β-athermality.
In fact, we can show directly from their definitions that the two quantities W max , ∆S max are also measures of athermality. The differentials of asymptotic ergotropy and maximal entropic gain are where the temperatures are calculated at the point where the derivative is considered.
Theorem 19. The asymptotic ergotropy W max and minimal β-athermality ∆S max are both monotonically nondecreasing functions of the geometric athermality A along any differentiable activation trajectory.
Proof. Let the activation trajectory be labelled S(E), with non-positive slope dS/dE ≤ 0 along the entire trajectory due to Definition 16. Then, substituting dS = (dS/dE)dE in the differential forms of asymptotic er-gotropy W max , minimal β-athermality ∆S max and geometric athermality A, we find that, along the entire trajectory, the derivatives dW max /dE, d∆S max /dE are positive. Therefore, dA dW max ≥ 0 and dA d∆S max ≥ 0, (33) with equality whenever dA = 0.
The asymptotic ergotropy W max and minimimal βathermality ∆S max are associated with the two extremal activation trajectories that a passive state ρ can take towards the equilibrium curve. Therefore, one can calculate them by constructing the E-S diagram that corresponds to the -s ensemble V(ρ) and then compute temperatures β −1 max and β −1 min respectively. The two measures are in a sense "dual". On the contrary, the geometric athermality measure A is not associated with any specific activation trajectory and solely depends on the -s ensemble V(ρ). This means that the geometric athermality does not require the whole structure of the E-S diagram and treats all points on the equilibrium curve equally, in the sense that there arises no special temperature like β min or β max .
A second point of comparison worth noting is the domain in which the three measures remain monotonic. The measures W max and ∆S max are clearly monotonic at any point on the E-S plane that corresponds to a valid quantum state, while the monotonicity of the geometric athermality A is guaranteed over the passive region of the E-S diagram. Although this reasoning seems to suggest that the geometric athermality is rather restricted, we can in fact easily extend its validity to all quantum states. Any state can be mapped to a unique passive state, associated to it via sorting its spectrum, while, conversely, one can reach any quantum state by unitarily transforming a passive state. Therefore, the entire E-S diagram can be divided into equivalence classes, each represented by a passive state ρ P , such that the value of the geometric athermality of a passive state ρ P can be associated to all quantum states in its equivalence class.

V. OUTLOOK
In this paper, we have shed light on the structure of passive states and provided a clear and simple derivation of the Gibbs state. The key tool for this was a geometric reformulation of passivity as a total ordering. Equipped with the area of the -s ensemble in this geometric framework, we then introduced the geometric athermality as a measure of athermality for quantum states that does not depend on a particular temperature. We also found that geometric athermality supplements the concept of asymptotic equivalence in a way that in a sense provides a non-equilibrium 'equation of state' A(E, S). It might be fruitful to explore this line further to determine its scope and application.
We finish with a range of open questions that arise from the framework introduced in this paper: 1. The main question that we have not explored is the following. For a d-dimensional quantum system with Hamiltonian H, and fixed values of (E, S) what is the shape of the asymptotic ensemble that has minimal area? In other words, what is the general form of the quantum state ρ whose asymptotic ensemble has area equal to the geometric athermality?
2. It turns out that the accumulation of 'face points' (points in V(ρ) that lie on the boundary faces of V ∞ (ρ)) appear to be significant (see the proof of the monotonicity of A). These lie on boundary line segments and so we can interpret each line segment as defining a Gibbs state at some inverse temperature β f . Can we describe the approach to equilibrium purely in terms of these boundary 'equilibrium' parameters?
3. It would be of interest to provide a more unified geometric account of this structure and extend to other scenarios such as multiple non-commuting conserved charges (where V ∞ will now consist of a volume in R n ).
4. Can we extend the geometric description to include non-passive states, and states with coherences between energy eigenspaces?
5. Can our geometric framework be extended to infinite-dimensional systems to enable the reformulation of the minimal area as a calculus of variations problem? What shape does the minimal surface V ∞ assume in this continuum setting?
6. We chose the Euclidean metric to define area. Is there a metric structure that provides a more natural form for the geometric athermality? Perhaps using ideas from information geometry?

VI. ACKNOWLEDGEMENTS
We would like to thank Carlo Sparaciari for many useful discussions. NK, RA and TH are supported by the EPSRC Centre for Doctoral Training in Controlled Quantum Dynamics. DJ is supported by the Royal Society and also a University Academic Fellowship.
Structural stability follows from complete passivity unless the state is a ground state (where the only non-zero eigenvalues of ρ reside in the lowest energy subspace). Complete passivity imposes that the eigenvalues of ρ within a degenerate energy subspace must be equal (assuming ρ is not a ground state). For suppose they were different, then V ∞ (ρ) would not be totally ordered, and hence no longer completely passive. This is handled straightforwardly in our geometric framework, as seen in Fig. (9). The only caveat to this arises for ground states. All ground states are completely passive -it is clearly impossible to lower their energy. However, complete passivity does not imply structural stability for ground states. Complete passivity permits non-uniform eigenvalues in the ground subspace if no other energy subspaces are occupied, because the corresponding -s ensemble remains totally ordered for arbitrary numbers of copies. However, should a small perturbation break the degeneracy of H, there is no guarantee that a small perturbation to the ground state will make it once again completely passive.
Our physical framework should be robust to small perturbations, therefore structural stability is a desirable feature. This comes for free with the requirement of complete passivity, however it must be imposed additionally when considering ground states.

Monotonically non-increasing deformations
Definition 27. (Monotonically non-increasing deformation). Consider a d-dimensional quantum system with d ≥ 3 and passive, full rank, non-Gibbsian state ρ of the system with non-degenerate Hamiltonian H. Then we define a monotonically non-increasing deformation to be some infinitesimal change to the area of V ∞ (ρ), A(ρ) → A(ρ) + dA = A(ρ ), generated by varying any number of occupation probabilities p ki → p ki + dp ki and ki dp ki = 0, for i ∈ {1, . . . , m}, where m ≤ d, in such a way that we obtain Then for dE < 0 or dS > 0 in Eq. (D10) we have dA ≤ 0.
Definition 28. (Upper branch). We define the upper branch V upper (ρ) ⊆ V vert (ρ) such that where the k labels are ordered clockwise mod n around V ∞ (ρ).
We can then define the lower branch as follows.
Lemma 31. Let ρ and H be defined as in Definition 27. Then for any such initial state ρ ∈ D (E,S) , ∀E, S there exists at least one monotonically non-increasing deformation.
Proof. Consider an initial passive, non-Gibbsian state ρ ∈ D (E,S) of a d-dimensional quantum system with d ≥ 3 and non-degenerate Hamiltonian H. We show that via a careful selection of points V qut (ρ) := {v ki } i∈{1,2,3} we can always find a qutrit deformation of ρ such that we obtain ∂ E A ≥ 0 and ∂ S A ≤ 0, for infinitesimal transformations. Inspection of Eq. (D4) reveals that we obtain a monotonically non-increasing deformation if we choose our virtual qutrit such that ∆ ki+1,ki−1 (v k (i+1) − v k (i−1) ) ≥ 0 for each i ∈ {1, 2, 3}. Since ρ is passive, this reduces to the following set of energetic constraints: ∆ ki+1,ki−1 mod n ∆ k (i+1) ,k (i−1) mod 3 ≥ 0, for i ∈ {1, 2, 3}. (D14) To show that it is always possible to find such a set from any initial passive configuration, we can therefore project our ensemble V vert (ρ) onto the -axis and consider the relative signs of these energy differences. Importantly, since our Hamiltonian is assumed to be non-degenerate the mapping v k → k , ∀v k ∈ V vert is a bijection. We now state the protocol which is shown graphically in Fig. (10). We first consider the case where V upper (ρ) is not empty. The extremal points are always in the upper branch (white), so purely in terms of combinatorics there are four cases we need to consider, shown in Fig. (10) Complications can arise if there are face points associated with one or more of the vertices of our chosen virtual qutrit. Then our assumption that our function for computing area Eq. (19) is continuous over the domain on which the transformation takes place may break down. To combat this issue, in such cases, we can always appropriately replace the vertices of our virtual qutrit with face points or interior points in such a way that all k labels in Eq. (19) have a fixed correspondence to points before and after the transformation. If all points get replaced with non-vertex points the qutrit deformation will yield dA = 0 from Eq. (D4), which is indeed monotonically non-increasing.
For any initial passive state ρ, the protocol described above returns a set of points V qut (ρ) that give rise to ∂ E A ≥ 0 and ∂ S A ≤ 0 under a qutrit deformation. Moreover, from Lemma 25 we have that this deformation allows for finite dE < 0 or dS > 0. Therefore, we can always find a qutrit deformation that is a monotonically non-increasing deformation concluding the proof of Lemma 31. and so ∂ S A(E, S) ≤ 0. Since the above argument applies to any passive non-Gibbsian initial state ρ ∈ D (E,S) , we thus have that ∂ E A(E, S) ≥ 0 and ∂ S A(E, S) ≤ 0, for all (E, S) in the region of passive states "off the thermal curve". In accordance with Eq. (22), we therefore find for d ≥ 3 that ∇ u A(E, S) ≤ 0, whenever u has components which satisfy u E ≤ 0 and u S ≥ 0, in this region. Finally, as a consequence of Theorem 10, if the initial state ρ is completely passive then the area A(ρ) assumes its global minimum value (zero). Since the thermal curve defines the set of states at the boundary of the state space, they can only be the end point of an activation trajectory. Therefore the geometric athermality is monotonically non-increasing over activation trajectories.