Axiomatic relation between thermodynamic and information-theoretic entropies

Thermodynamic entropy, as defined by Clausius, characterizes macroscopic observations of a system based on phenomenological quantities such as temperature and heat. In contrast, information-theoretic entropy, introduced by Shannon, is a measure of uncertainty. In this Letter, we connect these two notions of entropy, using an axiomatic framework for thermodynamics [Lieb, Yngvason, Proc. Roy. Soc.(2013)]. In particular, we obtain a direct relation between the Clausius entropy and the Shannon entropy, or its generalisation to quantum systems, the von Neumann entropy. More generally, we find that entropy measures relevant in non-equilibrium thermodynamics correspond to entropies used in one-shot information theory.

Entropy plays a central role both in thermodynamics and in information theory. This is remarkable, as the two theories appear to be fundamentally different. Thermodynamics is a phenomenological theory, concerned with the description of large physical systems, such as steam engines or fridges. It relies on concepts like work or heat, which are defined in terms of macroscopic observables. Information theory, on the other hand, deals with "knowledge" on a rather abstract level. Like statistical mechanics, it refers to the microscopic states of a system, such as the values of the individual bits stored in a memory device.
Accordingly, the notion of entropy is rather different in the two theories. In thermodynamics, entropy is a function of the macroscopic state of a physical system which describes phenomenologically which processes are possible independently of any microscopic model. Following Clausius, it is conventionally defined in terms of the heat that flows into a system at a given temperature, and it lends its operational significance from the second law [1,2]. In information theory, entropy was originally introduced by Shannon to quantify the information content of data or, equivalently, the uncertainty one has about them [3]. Operationally, it characterises properties such as the compression length, i.e., the minimum number of bits needed to store the data. Mathematically, the Shannon entropy is a function of the probability distribution of the random variable that models the data. The von Neumann entropy [4], which is also known from quantum statistical mechanics, provides a generalisation of this concept to the case where information is represented by the state of a quantum-mechanical system.
In this Letter, we take a different, more axiomatic, approach. It relies on a framework proposed by Lieb and Yngvason [22][23][24] for the study of thermodynamic entropy. Within this framework thermodynamic entropy is obtained via a small set of axioms that naturally characterise adiabatic processes. Our work is based on the observation that the framework can as well be used to capture key concepts from information theory. It therefore provides a common ground for the study of entropy measures both in phenomenological thermodynamics and in information theory. This allows us to establish a novel connection between them.
Furthermore, we extend this connection to "singleshot" generalizations of the Shannon and von Neumann entropy. While the latter are only applicable to study data that has a particular structure (e.g., that has been produced by infinitely many independent invocations of a source), single-shot entropies do not suffer from this limitation (see [25,26] for examples of how these entropies are used in applications). These entropy measures, in fact, naturally arise from Lieb and Yngvason's axiomatic framework for nonequilibrium thermodynamics.
The study of entropy has a long tradition both in physics and in information theory. In information theory, several axiomatizations of the Shannon or von Neumann entropy have been proposed [3,[27][28][29][30]. Our approach is different, as it relies on an axiomatization of the thermodynamic entropy. Our work also bears some resemblance to the study of entanglement transformation using similar tools [31][32][33].
The remainder of this Letter is organised as follows. We start with a brief summary of the Lieb-Yngvason framework for thermodynamics and describe how entropy measures are defined within this framework. As a first technical contribution, we show that the framework is as well applicable to an information-theoretic description of thermodynamic systems (Proposition 1). We then show that the corresponding entropy measures defined within the framework coincide with information-theoretic single-shot entropies (Proposition 2). Subsequently, we extend these considerations to various classes of thermo-dynamic processes involving reservoirs and relate them to information-theoretic counterparts.
Lieb and Yngvason's approach.-In their axiomatic framework, Lieb and Yngvason [22][23][24] consider the set Γ of all equilibrium states of a thermodynamic system and equip this space with an order relation, denoted ≺. For X and Y ∈ Γ, X ≺ Y means that Y ∈ Γ is "adiabatically accessible" from the state X ∈ Γ "by means of an interaction with some device consisting of some auxiliary system and a weight in such a way that the auxiliary system returns to its initial state at the end of the process, whereas the weight may have risen or fallen" [22] (see Figure 1). The framework also describes the composition of systems as well as their scaling, corresponding to taking an arbitrary amount of a substance. Within this framework, an entropy measure, S, is a function on Γ that has certain natural properties. In particular, Lieb and Yngvason demand that S should be monotonic with respect to the order relation ≺, additive with respect to the composition operation, and extensive in the scaling (we refer to the Appendix for a more detailed description of these properties). They show that, provided the order ≺ obeys thermodynamically reasonable axioms, S is determined uniquely by these properties, up to an affine transformation that may be fixed by choosing the entropy of two (arbitrary) reference states X 0 and X 1 . Specifically, S is given by where λX denotes the state of a system obtained by scaling a system in state X by a factor λ, and (X, X ) denotes the state of the system obtained by composing systems in state X and X , respectively. Intuitively, if the state X can be reached adiabatically from X 0 , and X 1 can be attained from X, then the entropy S(X) is defined as the optimal λ such that the state X can be created from X 0 and X 1 combined at a ratio (1 − λ) : λ by an adiabatic process. Physically, S corresponds to the usual thermodynamic entropy as defined by Clausius via the reversible heat dissipated δQ rev as S = δQ rev /T . This can be intuitively understood from Clausius' explanation of the second law [1,2] combined with the uniqueness result in [23]; Clausius' physical processes agree with the picture of an adiabatic process in Lieb and Yngvason's framework. The connection is formally established through Lieb and Yngvason's re-derivation of thermodynamics [23]. The framework can be extended to include nonequilibrium states [24]. The states of the corresponding extended state space, Γ ext generally obey weaker axioms than those of Γ. For instance, they may not be scalable. The entropy S can thus not be uniquely extended to Γ ext . Instead, it can be shown that all monotonic extensions S ext of S to the space Γ ext are contained within two bounds, namely These entropy measures can be used to characterise the (im)possibility of state transformations by adiabatic processes (according to the definition above). In particular, they provide a sufficient condition for the possibility to transform X ∈ Γ ext to Y ∈ Γ ext , Similarly, they also provide a necessary condition for such a transformation, (4) We can also use Equations (1a) and (1b) to define two further entropy measures as These have the advantage that their definition does not involve neighbouring states X and X . While in thermodynamics the intervals [S − ,S + ] and [S − , S + ] coincide, the interval [S − ,S + ] is generally -and especially for quantum states -smaller. A detailed analysis is provided in the Appendix. Information-theoretic entropy measures.-Information theory is concerned with data and their processing. In quantum information theory, which we consider here for generality, data is encoded in quantum systems (which include classical systems as a special case), whose states we describe by the usual density operator formalism. In order to quantify the information content of data, we appeal to a family of information measures, the Rényi entropies [34] (cf. Appendix). Two important members of this family are the min-and the max-entropy. The min-entropy is defined as where ρ ∞ denotes the maximal eigenvalue of ρ. Operationally, it describes the amount of randomness that can be extracted deterministically from data in state ρ [35,36]. The max-entropy is defined as and quantifies the number of (qu)bits needed to store data in state ρ [25]. An Information-Theoretic Description of Adiabatic Processes.-In order to apply Lieb and Yngvason's framework to an information-theoretic description of thermodynamic systems, we need to formally specify the various ingredients (such as the order relation) which the abstract framework requires. First, we identify the set of "equilibrium states" of an information-bearing quantum system. We define these as those quantum states represented by flat density operators (i.e., operators whose non-zero eigenvalues are all equal). This, in turn, allows us to derive the notion of an "adiabatic process," in  Figure 1. (a) Adiabatic process according to Lieb and Yngvason's definition. A system, represented as a cube, interacts with some device and a weight. After the process the device has to be in its initial state again and the weight can only have changed its relative height. The change of the state of the system itself is represented as a deformation of the cube here. (b) The system is connected to a heat bath at temperature T. Adiabatic processes with the aid of an auxiliary system and a weight are applied to this combined system. the sense of Lieb and Yngvason, for microscopic systems. Their notion of an adiabatic process (see Figure 1(a)), here, translates to the following three quantum operations: • addition of an extra ancilla system ("device"), in an equilibrium state; • interaction of the system and the extra device with a weight system with a joint, energy-preserving unitary; • removal of the extra device, restored to its original state.
Consider for simplicity a system with a trivial, i.e. fully degenerate Hamiltonian: energy and thus the weight itself lose their relevance and the interactions reduce to arbitrary unitary transformations on the system and the extra device. In the Appendix, we show that these operations achieve the same state transformations as the set of noisy operations, a class of operations known in information theory [37]. These are processes that can be described as a sequence of the following basic steps: • addition of an ancillary system in a maximally mixed state; • reversible transformation of the system with any joint unitary; • removal of a subsystem.
This can be extended to systems with a non-trivial Hamiltonian, by modeling the weight as a quantum system in a coherent superposition state of many energy levels [38]. Any unitary evolution of the system can then be achieved to good approximation by an appropriately chosen energy-preserving interaction with the weight. (A more detailed explanation of this fact can be found in the Appendix.) Thermodynamics of our Information-Theoretic Model.-We now seek to apply Lieb and Yngvason's framework to quantum information systems. As we have just seen, an adiabatic process, which defines the order relation ≺, corresponds to accessibility by noisy operations, i.e. ρ ≺ ρ if and only if ρ can be reached from ρ via noisy operations.
Noisy operations are characterized by the mathematical notion of majorization: the existence of a noisy operation transforming a state ρ to a state ρ is equivalent to the condition that ρ majorizes ρ [37]. This is defined as follows. Let {p i } i and {p i } i be the eigenvalues of ρ and ρ arranged in decreasing order. Then ρ ma- p i for all k = 1, 2, . . . , dim(H). (Note that we use a different notation than the conventional " " used e.g. in Ref. [39], to avoid confusion with Lieb and Yngvason's adiabatic processes.) We define the composition of states naturally as their tensor product; establishing a reasonable scaling operation is more involved. We assume scaling a quantum system by a natural factor λ ∈ N to mean combining λ such systems. Thus, the scaling coincides with the composition operation. We generalize this concept to noninteger scaling factors by extending the space of equilibrium states to be continuous. For simplicity, consider only states ρ ∈ S(H) which are diagonal in a common eigenbasis, which does not restrict the generality of our considerations, as explained in the Appendix. Represent ρ's eigenvalues p 1 ≥ p 2 ≥ . . . ≥ p dim H as a step function For an equilibrium state σ, for which by assumption all non-zero eigenvalues are equal, the step function is scaled as f λσ (x) = f σ (x 1 λ ) λ , which coincides with the composition for λ ∈ N but allows for a formal continuation to any λ ∈ R ≥0 . This scaled state might be unphysical if λ is not an integer. However, we show in the Appendix that, by considering larger systems, the framework can be reformulated in such a way that only scaling by an integer factor is needed. Main Results.-Consider a Hilbert space H, and consider the order relation ≺ M which corresponds to accessibility by noisy operations, as defined above. We may now state our main results. The axioms can be checked through a straightforward calculation, which we carry out in the Appendix. Proposition 2. For states on H ordered by means of ≺ M , the unique entropy function S for equilibrium states coincides with the von Neumann entropy H(ρ) = − tr (ρ log ρ). Furthermore, the two entropic quantities S − andS + are equal to H min and H max .
We provide the proof in the Appendix. The boundsS − andS + and their relation to S − and S + are visualized in Figure 2. While S + andS + coincide, the discrete quantity S − differs fromS − .
Other thermodynamic processes.-We have, up to this point, considered only isolated systems. It seems however natural to ask for analogues of thermodynamic scenarios describing other types of processes, for instance systems interacting with reservoirs. In the most obvious case of a system connected to a heat bath, depicted in Figure 1(b), we assume our system to be in thermal contact with a heat reservoir at temperature T , which in statistical mechanics would be described by a canonical ensemble. A system in equilibrium is thus in a thermal state with respect to the heat bath.
On the system and the reservoir we still allow for the above adiabatic processes, i.e. we can describe the allowed state transformations by means of noisy operations. As long as the states of the system alone are block diagonal in the energy eigenbasis, our noisy operations on system and reservoir can be reduced to thermal operations on the states of the system, as one can understand from [15]. Thermal operations [13] consist of the following steps: • addition of an ancillary system in a thermal state relative to the heat bath; • unitary transformation of the system and the ancilla commuting with the total Hamiltonian; • removal of any subsystem.
For states which are diagonal in the energy eigenbasis, the existence of a state transformation by a thermal operation is expressed in terms of thermo-majorization [15]. Let ρ = i p i |E i E i | ∈ S(H) be a density matrix diagonal in the energy eigenbasis and let d = dim H. Represent its spectrum as a step function with ordered eigenvalues p 1 ≥ p 2 ≥ . . . ≥ p d . The Gibbsrescaled version of this step function is defined as  [15,16]. For states ρ and σ ∈ S(H) which are block diagonal in the energy eigenbasis, the order relation of thermo-majorization ≺ T can be defined as (12) It obeys Lieb and Yngvason's axioms (shown in the Appendix), which enables calculating the unique potential for thermal states as well as bounds on the extension of the potential function to arbitrary block diagonal states. In fact, we recover the "single-shot free energies" F min and F max introduced in [15,20] to describe the work needed for the formation of a state as well as the extractable work.
Proposition 3. For states ordered by means of thermal operations through the relation ≺ T the unique potential S for thermal states coincides with the Helmholtz free energy F , and the two quantitiesS − andS + correspond to F max and F min .
A detailed calculation can be found in the Appendix. Scenarios including other types of reservoirs [21], such as a particle or an angular momentum reservoir, yield analogous results. Various settings, along with their corresponding order relation, equilibrium states and resulting state functions are summarized in Table I.
Conclusions.-We have shown that, with minor adaptations, Lieb and Yngvason's approach is directly applicable to information theory, allowing us to put thermodynamic and information theoretic entropy on the same footing. Compared to other works based on Landauer's

Fmin, Fmax
Interaction with a heat bath and a particle reservoir Grand potential Ω

Ωmin, Ωmax
Interaction with an angular momentum reservoir Table I. An overview on the application of Lieb and Yngvason's framework to various scenarios. In the first line we describe Lieb and Yngvason's original scenario in the case of quantum states. The rest of the table contains adapted scenarios, where the systems have additional interactions with reservoirs, as denoted in the first column. In the second column, we present the corresponding resource theoretic scenarios, the selected class of processes as well as their associated order relation. The remaining columns detail the equilibrium states and the entropic quantities S,S− andS+ corresponding to each setting.

Information Theory
Statistical Mechanics Thermodynamics identification of relevant quantities Landauer's principle this work Figure 3. The contribution of this work is to draw an explicit connection between thermodynamic and information theoretic entropy without the need for microscopic notions of heat and work. In contrast, those usually need to be specified for proofs of Landauer's principle, which thus rely on an underlying model of (quantum) statistical mechanics. Finally, on a macroscopic scale, the connection between thermodynamics and statistical mechanics is traditionally drawn by identifying the physical quantities from both frameworks.
principle [5,6,12,17], which relies on an underlying microscopic model of work, our axiomatic approach directly relates information theory to phenomenological thermodynamics without the need for statistical mechanics (see diagram in Figure 3). We thus avoid the conceptual difficulties of defining microscopic notions of heat and of appealing to particular models of work storage. Instead, our approach immediately points out the formal and conceptual parallels of phenomenological thermodynamics and information theory.
Furthermore, we can apply our approach to models of different possible interactions of a system with a reservoir, represented by an appropriate mathematical order relation. For each of the considered scenarios we find that the unique "entropy function" for equilibrium states predicted by Lieb and Yngvason coincides with a wellknown corresponding resource monotone such as the von Neumann entropy for isolated systems and the free energy for a system connected to a heat bath. Moreover, we may directly apply Lieb and Yngvason's results for thermodynamic non-equilibrium states. Possible extensions of the unique entropy function to non-equilibrium states are bounded, for an isolated system, by the information theoretic min-and max-entropy. Note also that, in addition to modeling microscopic adiabatic processes, the majorization relation is tightly related to information processing: an encoding operation can be expressed as a noisy operation, while the inverse of a randomness extraction process is such an operation as well.
We expect that this approach can be extended further; by slightly changing the order relation to a "smoothed majorization relation", we presume it to yield the corresponding smoothed entropy measures. Also, for a nonisolated system, our approach is limited to states that are block diagonal in a corresponding eigenbasis, e.g. in the particular case of a heat bath, in the energy eigenbasis [15]. We leave the question of generalizing our results for interacting systems to states with nonzero off-diagonal entries open for further investigation. In addition, we have not allowed the agent carrying out the processes to be assisted by side information about the system, which could be useful for performing thermodynamic operations [12]. We might expect that an appropriate extension of Lieb and Yngvason's framework would provide an axiomatic and operationally well-justified definition of the conditional entropy.
Acknowledgements.-This project has been supported by the Swiss National Science Foundation (SNSF) via the NCCR "QSIT", by the European Research Council (ERC) via project No. 258932, and by the COST Action MP1209.

APPENDIX Appendix A: Revisiting Lieb and Yngvason's axiomatic approach
Lieb and Yngvason have contributed an axiomatic approach to derive an entropy function for thermodynamic equilibrium states [22,23]. Recently, they have extended their approach to non-equilibrium states [24], which also enables them to make predictions relevant for non-equilibrium thermodynamics.
Lieb and Yngvason consider a preorder ≺ on a set Γ; physically Γ is the space of all equilibrium states of a system. A preorder is reflexive, transitive, but in contrast to a partial order not antisymmetric. This means that if two elements denoted by X and Y ∈ Γ satisfy X ≺ Y as well as Y ≺ X, this does not imply that they are the same element of the set Γ. In line with Lieb and Yngvason's terminology, we will call ≺ an "order relation" or just "order" in the following. Whenever both relations Elements X 1 ∈ Γ 1 and X 2 ∈ Γ 2 of possibly different sets Γ 1 and Γ 2 can be composed, physically meaning that they can be set next to each other and considered as one combined system. The arising composed state is symbolically denoted as (X 1 , X 2 ) ∈ Γ 1 × Γ 2 . Note that the cartesian product Γ 1 × Γ 2 denotes the space of composed systems (X 1 , X 2 ), where the composition operation is associative and commutative.
In addition, any element X ∈ Γ can be scaled, i.e. for any λ > 0 one can define a scaled element denoted as λX ∈ λΓ. The scaling is required to obey 1X = X as well as λ 1 (λ 2 X) = (λ 1 λ 2 )X. For the sets Γ, the required properties are 1Γ = Γ and λ 1 (λ 2 Γ) = (λ 1 λ 2 )Γ, where λΓ symbolically denotes the space of scaled elements λX. Scaling a system by a factor λ means taking λ times the amount of substance contained in the original system.
The order relation ≺ satisfies by assumption the following six axioms E1 to E6 as well as the Comparison Hypothesis.
Note that Lieb and Yngvason do not regard the Comparison Hypothesis as an axiom but rather derive it from additional reasonable axioms about thermodynamical systems. Lieb and Yngvason's contribution concerns possible "entropy functions" on Γ. More precisely, they show that there is a (essentially) unique function S on Γ which satisfies the following properties: • Additivity: For any two states X ∈ Γ and X ∈ Γ , S((X, X )) = S(X) + S(X ) holds.
• Monotonicity: If two states X andX are comparable with ≺, then X ≺X ⇔ S(X) ≤ S(X).
Lieb and Yngvason's second law is restated in the following theorem.
Theorem 4 (Lieb & Yngvason). Provided that the six axioms E1 to E6 as well as the Comparison Hypothesis are fulfilled, there exists a function S that is additive under composition, extensive in the scaling and monotonic with respect to ≺. Furthermore, this function S is unique up to affine transformations.
For a state X ∈ Γ, the unique function S is given as where arbitrary elements X 0 ≺≺ X 1 ∈ Γ can be chosen, defining a gauge. Since the scaling is continuous, the supremum and the infimum are attained, and the function S can be conveniently expressed as To describe non-equilibrium states of a thermodynamic system, additional elements, not satisfying the scaling property, are introduced. Adding such non-scalable elements to the set Γ we obtain an extended set Γ ext for which we require the following.
The first requirement ensures that the non-scalable elements are comparable to at least two elements X 0 and X 1 of the set Γ, and hence via transitivity to all of them. Furthermore, it assures that the non-scalable elements are not at the boundary of the extended set Γ ext with respect to the preorder ≺. For such non-equilibrium states, Lieb and Yngvason [24] come to the following conclusion.
Proposition 5. On condition that N1 and N2 hold for any non-equilibrium state X ∈ Γ ext , the two functions S − and S + defined as bound all possible extensions S ext of S to the set Γ ext which are monotonic with respect to the relation ≺.
This implies that for a state X ∈ Γ ext , the attained value S ext (X) of such an extension always lies in between the values S(X ) and S(X ) of its neighboring scalable elements according to the order relation ≺: We will prefer to work with the following alternative quantities, which only rely on the state X and not on any "neighbouring" equilibrium states X and X : rather than using Lieb and Yngvason's bounds (A4) and (A5). Operationally,S − specifies the portion of the system that can maximally be in state X 1 if one wants to create the state X by composing subsystems in states X 0 and X 1 . The minimal portion of X 1 one can obtain by transforming X into a composition of two smaller systems in states X 0 and X 1 is characterized byS + . Note that in thermodynamics the two sets of bounding quantities {S − , S + } and {S − ,S + } coincide, as due to the continuity of the thermodynamic quantities an equilibrium state X ∈ Γ with X ∼ ((1 − λ)X 0 , λX 1 ) exists for any λ. However, this does not directly follow from the axioms and the interval [S − ,S + ] is generally smaller than [S − , S + ] and may not contain all monotonic extensions S ext of the entropy function S. We will see in Appendix C that our quantitiesS − andS + are advantageous in the quantum case.
Appendix B: A short introduction to selected resource theoretic aspects In general, a resource theory deals with an agent that is only allowed to execute operations of a predefined class, and investigates which tasks can be accomplished.
For quantum resource theories, the state space on which these operations act consists of density operators ρ, i.e. positive semidefinite operators of unit trace on a Hilbert space H. We denote the set of all density operators on H as S(H).
In a resource theory, the value of a state as a means to achieve a certain task can be quantified by a monotone, a measure that is monotonic under the selected class of operations. A state is said to be a resource if it can not be prepared with operations of the allowed class only.

The resource theory of noisy operations
The resource theory of noisy operations [37,43,44] can be defined by allowing the following operations: • Addition of an ancillary system in a maximally mixed state.
• Unitary transformation of the system.
• Removal of any subsystem (by taking the partial trace).
Horodecki and Oppenheim [37] have shown that, for finite dimensional systems, a state ρ ∈ S(H) can be transformed into a stateρ ∈ S(H) by noisy operations if and only if the spectrum of ρ majorizes the spectrum ofρ.
Note that to avoid confusion with Lieb and Yngvason's order relation later, we use a non-standard notation for the majorization ≺ M , which in particular differs from the notational convention from Bahtia [39].
Majorization can just as well be expressed by means of the step functions f ρ introduced in the main text: (B2) As f ρ (x) is monotonically decreasing in x and due to the normalization ∞ 0 f ρ (x)dx = 1, the condition k ∈ {1, 2, . . . , d} from Definition 6 is equivalently replaced with k ∈ R ≥0 .
• There exists a unital quantum operation achieving the transition ρ →ρ.
A proof can be found for instance in [44,45]. The resource theory of noisy operations has numerous monotonic functions [44]. One of the most popular families of monotones under these operations are the so-called Rényi entropies [34].
Definition 9. The α-Rényi entropy of a density operator ρ ∈ S(H) is defined as For α → ∞ and α = 0 we recover two quantities from the smooth entropy framework, the min-entropy H min and the max-entropy H max [25,47]. Note that taking the limit α → 1 leads to the von Neumann entropy H = − tr(ρ log ρ).
Definition 10. For a density operator ρ ∈ S(H) the min-and max-entropies are defined as where ρ ∞ denotes the maximal eigenvalue of ρ.
Note that our definition does not coincide with the usual terminology [47], instead we resume the notation from [25], where H max is defined as H 0 . Our nonstandard choice is well justified, as we regard the maxentropy as a physical concept, which can be quantified by different entropy measures all attaining very similar values.

The resource theory of thermal operations
The resource theory of thermal operations describes quantum systems interacting with a heat bath at a temperature T [13,15,48]. The allowed operations in this framework are: • Addition of an ancillary system in a thermal state relative to the heat bath.
• Application of any unitary operation which commutes with the total Hamiltonian on system and ancilla.
• Removal of any subsystem (by taking the partial trace).
Thermal states, also called Gibbs states, are of the form where Z is the partition function and the E i denote the energy eigenstates; the constant β = 1 kBT is inversely proportional to the temperature T and k B denotes the Boltzmann constant. These states are preserved under thermal operations, while all athermal states are resource states [13,15]. Note that for a system with a trivial Hamiltonian, i.e. a system where all energy levels are degenerate, the resource theory of thermal operations is equivalent to the resource theory of noisy operations.
Closely connected to the resource theory of thermal operations is the order relation of thermo-majorization, which is defined as a Gibbs-rescaled majorization [15,16].
be a density matrix block diagonal in the energy eigenbasis and let d = dim H. Represent its spectrum as a step function The Gibbsrescaled version of this step function is given as Thermo-majorization can be defined in terms of Gibbsrescaled step functions, analogous to the formulation of majorization ≺ M in Equation (B2).
Definition 12. Let ρ and σ ∈ S(H) be two quantum states which are block diagonal in the energy eigenbasis. We define the order relation of thermo-majorization ≺ T as Note that this definition is different but equivalent to the one in [15] Horodecki and Oppenheim [15] have shown that thermal operations can be characterized in terms of this order relation of thermo-majorization ≺ T , at least for states that are block diagonal in the energy eigenbasis. 1 Proposition 13. Let ρ and σ ∈ S(H) be two states which are (block) diagonal in the energy eigenbasis. Then there exists a thermal operation achieving the transition ρ → σ if and only if the state ρ thermo-majorizes the state σ, denoted as ρ ≺ T σ.
As shown by Brandao et al. [20], a family of measures which are monotonic under thermal operations for block diagonal states ρ and for all α ≥ 0 is given as where τ is the thermal state of the system and F (τ ) = −k B T ln Z τ its free energy. The Rényi divergences D α (ρ||τ ), α ≥ 0 for states ρ and τ which commute are where p i are the eigenvalues of ρ and t i are the eigenvalues of τ . These measures contain for α = 0 the F min and in the limit α → ∞ the F max , given as where D 0 (ρ||τ ) = − log tr Π ρ τ and D ∞ (ρ||τ ) = log min {λ : ρ ≤ λτ } correspond to the relative entropies, introduced in [36,49], with respect to the thermal state of the system. Note that for a thermal state τ the equality F min (τ ) = F (τ ) = F max (τ ) holds. Horodecki and Oppenheim [15] originally introduced these two quantities to describe the extractable work as well as the work needed to form a state. Assuming to have no errors for these two processes and that the states ρ ∈ S(H) are diagonal in the energy eigenbasis, the extractable work under thermal operations is given as whereas the work of formation is In the thermodynamical limit Horodecki and Oppenheim recover the extractable work of a state ρ ∈ S(H) to be W (ρ) = F (ρ) − F (τ ). In this limit, the same quantity is used to describe the work of formation.

Appendix C: Lieb and Yngvason's entropy for quantum states
Lieb and Yngvason's framework is based on abstract axioms, which admits its application to other physical contexts. We apply their approach to quantum states, by considering the space S(H) of all density operators on a Hilbert space H as Γ ext . The equilibrium states, forming the set Γ, are defined as all those states characterized by a spectrum for which all non-zero eigenvalues are equal.
We show in the following how the noisy operations, characterized by the majorization relation ≺ M , arise as the information-theoretic counterpart to adiabatic accessibility in phenomenological thermodynamics.
We express the adiabatic processes including a device and a weight as introduced in Appendix A, denoted by A →, by concrete physical operations consisting of the steps: • Adding a device to the system, formally expressed as taking the tensor product with an ancillary system in an equilibrium state.
• Letting the system and the device interact with the weight in an adiabatic, i.e. energy conserving, process.
• Removing the device in its initial state, which means taking the partial trace of the ancillary system in the state it has been added initially.
To describe the weight, a system which according to Lieb and Yngvason's framework should only change its height, i.e. provide or remove energy, we rely on a particular model byÅberg [38] describing a coherent weight W with a Hamiltonian corresponding to an energy ladder, where the {|w } w are orthonormal states and the constant s ∈ R ≥0 defines the energy level spacing of the Hamiltonian. The weight, assumed to be in a state σ = |η L,l0 η L,l0 | with |η L,l0 = 1 √ L L−1 w=0 |w + l 0 , is connected to a quantum system S with Hamiltonian H S , in our case consisting of the system and the ancilla. We allow operations on the system and the weight which commute with translations of the weight, i.e. do not depend on the absolute energetic state of the weight, and which at the same time conserve the total energy, i.e. commute with H S ⊗ 1 W + 1 S ⊗ H W .
This realizes the idea that only the relative change in energy of the weight, being the energy that is added to or removed from system S, influences the system, giving the weight the role of a work reservoir. For large L, these operations on the system and the weight allow the implementation of any unitary operation on the system S alone, catalytically. Thus, the interaction of system and ancilla with the weight reduces to the application of arbitrary unitaries to the system and the ancilla. Intuitively, this weight can be imagined as eliminating the role of energy from the framework or just as allowing to change the Hamiltonian at will.
Compared to other models of weights [19] this particular model allows us to obtain superpositions of energy eigenstates of a system which was initially in such an eigenstate. It thus allows us to include the treatment of states which are not block diagonal in the energy eigenbasis, which could not be created otherwise.
The processes A → resemble intuitively the noisy operations introduced in Appendix B 1. However, the processes A → allow the addition of any equilibrium ancilla yet require us to return the ancillas in their initial state.
The following proposition shows that these processes are also characterized by the majorization relation ≺ M , and are thus equivalent to noisy operations.

Proposition 14.
For two states ρ and σ ∈ S(H S ) the following are equivalent: • (A): The spectrum of ρ majorizes the spectrum of σ, i.e. ρ ≺ M σ.
Proof. (A) ⇒ (B): Horodecki and Oppenheim [37] show that if ρ ≺ M σ, then there exists a noisy operation bringing ρ to σ, i.e. there exists a unitary U SA acting on an additional ancillary system A such that where 1 A ∈ S(H A ) is a maximally mixed ancilla and U † SA denotes the adjoint of U SA . 2 In the noisy operations that Horodecki and Oppenheim construct explicitly, one can see that the unitary U SA does not change the reduced state on the ancillary system, i.e.
Thus, the ancilla is removed in the maximally mixed state in which it was added and the process is an adiabatic process ρ A → σ according to our definition. (B) ⇒ (A): As we know from Proposition 8, ρ ≺ M σ is equivalent to the existence of a unital map from ρ to σ. In the following, we show that the processes ρ A → σ are unital and thus also imply the ordering ρ ≺ M σ. Now let χ ∈ S(H A ) be a state with a flat spectrum, i.e. it can be written as χ = The inequality follows by subadditivity of h. We know Since the ancillary system has to be in state χ at the end of the process, (C7) As this inequality can only be satisfied if Recall that for two states ρ andρ ∈ S(H) the majorization condition ρ ≺ Mρ can be equivalently expressed in terms of the spectral step functions f ρ and fρ as introduced in (B2). Even though the functions f ρ are not in one-to-one correspondence with the states ρ but rather represent all states having a certain spectrum, this description is enough for this purpose. As the corresponding processes A → include the application of an arbitrary unitary, it is perspicuous that the order relation ≺ M is independent of the eigenbases of the compared states.
For an equilibrium state ρ, which has a flat spectrum, the step function f ρ has the simple form Thus, for two equilibrium states ρ andρ the rank alone determines which one majorizes the other, as We define the composition of two states ρ ∈ S(H) and ρ ∈ S(H) as their tensor product ρ ⊗ρ ∈ S(H ⊗H). The scaling of an equilibrium state ρ is assumed to coincide with its composition for scaling factors λ ∈ N. The step function of a scaled state λρ ∈ λS(H), which is thus defined as ρ ⊗λ ∈ S(H ⊗λ ), is To obtain a continuous scaling operation, this way of scaling the step function is applied for any λ ∈ R >0 . For most values of λ the scaled copies λρ do not represent physical states and no actual space λS(H) exists. However, any normalized, but possibly unphysical, function f (x), can be turned into a physical state by actually considering the function 1 n f (x/n) for a large enough n: the step now coincides to a good approximation with an integer abcissa and the function now represents to a good approximation a physical state on a larger system. Now, notice that we can always combine states with a fully mixed state of a given rank, and that the following rules apply: These rules are easily seen with the representation in terms of the step function. They allow now to give a precise signification to any relative statements between states which would be required to be scaled in an unphysical way: for example, we have (λρ, µρ) ≺ σ if and Thus, if λρ does not actually correspond to a physical state, then the second expression should in fact be considered; indeed for large enough n the state λ ρ, 1n n is actually physical to a good approximation.
The entropy function (A1) can thus be equivalently rewritten as The last expression, for n large enough, only involves physical states. We now proceed to show that this order relation along with its scaling operation fulfills Lieb and Yngvason's axioms.
Proposition 15. Consider the majorization relation ≺ M between the ordered spectra of states ρ ∈ S(H). and define the composition operation as well as the scaling of non-equilibrium states as introduced above. Then, for equilibrium states, i.e. states with a flat spectrum, the six axioms E1 to E6 as well as the Comparison Hypothesis hold. Moreover, for non-equilibrium states axioms N1 and N2 are satisfied.
Proof. As axiom N2 requires that axioms E1 to E3 as well as E6 also hold for non-equilibrium states, we directly show that they hold for non-equilibrium states as well.
Stability (E6): Note that χ 0 and χ 1 are necessarily equilibrium states, as in Lieb and Yngvason's framework only those can be scaled. Let ρ ∈ S(H) have eigenvalues p 1 ≥ p 2 ≥ . . . ≥ p dim(H) . Assume that (ρ, εχ 0 ) ≺ M (σ, εχ 1 ) for a sequence of ε's tending to zero. Let this sequence be denoted by (ε i ) i . Then for all ε i we have and thus which follows by dominated convergence. Note that and similar for f (σ,εiχ1) . Thus, taking the limit leads to which is equivalent to ρ ≺ M σ.

Corollary 16. For a system with a trivial Hamiltonian on which processes
A → can be performed, there exists a unique entropy function S for equilibrium states, as well as bounds S − and S + on the entropy of non-equilibrium states, all up to an affine transformation.
Proof. Proposition 15 allows the application of Lieb and Yngvason's Theorem 4 and Proposition 5 to quantum states compared by majorization. This directly implies the corollary. Proof. For λ = S(ρ) we have Exploiting that ρ 0 , ρ and ρ 1 are equilibrium states, for which the ordering ≺ M only depends on their rank, this is equivalent to which can be reformulated as The parameters a and b are defined as a = In particular, S(ρ) = H(ρ) is obtained with the choice rank ρ 0 = 1 and rank ρ 1 = 2. As Lieb and Yngvason [22,23] have defined the handling of negative scaling factors, this choice of ρ 0 and ρ 1 is not problematic in order to reasonably define the entropy for an equilibrium state ρ obeying ρ 1 ≺ M ρ; such a state ρ is associated with a negative scaling factor (1 − λ) and thus characterized by an entropy larger than 1. Proof. For any state ρ ∈ S(H) with eigenvalues p 1 ≥ p 2 ≥ . . . ≥ p dim(H) and for any equilibrium states ρ 0 ≺≺ M ρ 1 , let λ be such that ((1 − λ)ρ 0 , λρ 1 ) ≺ M ρ. Then, (C29) Letk = (rank ρ 0 ) 1−λ (rank ρ 1 ) λ . As ρ 0 and ρ 1 are equilibrium states, we know that for 0 ≤ k ≤k, Thus, by considering k ≤ 1, Equation (C29) directly implies which can be rewritten as with a = for all k ∈ R ≥0 , i.e. implies (C29); the second inequality follows as the step function f ρ (x) is monotonously decreasing and normalised. Thus, taking the supremum over λ in (C32) concludes the proof forS − , as H min (ρ) = − log ρ ∞ , where ρ ∞ denotes the maximal eigenvalue of the state ρ, i.e. equals p 1 .
In the case of quantum states considered here, the bounds {S − ,S + } do not equal Lieb and Yngvason's original {S − , S + }. This can be most easily shown with the following example.
The intuitive explanation is that in the case of quantum states -contrary to thermodynamics -there does not exist an equilibrium state for each value of λ, but for a quantum system of a particular size, the equilibrium states are discretely distributed and give only discrete entropy values. Note that forS + and S + no such difference is observed.
The quantitiesS − andS + are reasonable, as for two states ρ, σ ∈ S(H) they have the desirable properties Thus, if σ is an equilibrium state, we observẽ which for S − is generally not the case. Moreover,S − enables us to distinguish for instance a state ρ = 2 3 |0 0|+ 1 3 |1 1| from the state ρ considered in the previous example, while with S − this is impossible.
Appendix D: System interacting with a heat bath In typical laboratory experiments, the systems of interest interact with a thermal environment. Considering systems connected to a heat reservoir is thus a natural and relevant extension of the scenario of an isolated system, which was described in Appendix C.
Assume that a heat bath at a temperature T is connected to an isolated system S, shown in Figure 1 of the main text. As the system S now thermalizes with the reservoir, the equilibrium states are thermal states, as introduced in (B6), which can also be constrained to a subsystem; this means that the eigenvalues {p i } i of such an equilibrium state τ = i p i |E i E i | are either e −βE i Z or zero, where Z is the normalisation.
System S and reservoir R together still form an isolated system SR, which we assume to undergo adiabatic processes with the aid of an ancillary system and a weight. For an isolated system such a process between two states of the system SR is expressed by means of the majorization relation ≺ M . We follow Horodecki and Oppenheim's treatment [15], that majorization for states of the system SR can be expressed as thermo-majorization ≺ T on the corresponding states on S, at least as long as the final state on S is block diagonal in the energy eigenbasis. Further, the system and the reservoir are assumed to be initially in a product state. The operations that correspond to this order relation ≺ T are the thermal operations introduced in Appendix B 2.
As this treatment is only fully applicable as long as the compared states on S are block diagonal in the energy eigenbasis, the relation ≺ T captures whether equilibrium states -which are always diagonal in the energy eigenbasis -can be interconverted. For those nonequilibrium states which are not diagonal in the energy eigenbasis, however, the order relation ≺ T is not sufficient to express whether two states can be interconverted by a thermal operation 4 . In the following, we therefore restrict our treatment to equilibrium states as well as to non-equilibrium states which are block diagonal in the energy eigenbasis.
The following proposition ensures that we can apply Lieb and Yngvason's results to quantum states ordered with the relation ≺ T .
Proposition 19. Consider the order relation of thermomajorization ≺ T . Then, for equilibrium states τ the six axioms E1 to E6 as well as the Comparison Hypothesis hold, whereas all block diagonal athermal states satisfy axioms N1 and N2.
Proof. By Definition 12, block diagonal states ρ ∈ S(H) can be represented by their Gibbs-rescaled step functions f T ρ . For equilibrium states τ the functions f T τ assume the simple form where Z is the partition function considering all occupied states. We can define the composition of two arbitrary states as their tensor product, like before. For scaling factors λ ∈ N the scaling of an equilibrium state is again assumed to coincide with its composition. This scaling operation can be formally extended to any scaling factor λ ∈ R >0 on the level of the functions f T again; the Gibbs-rescaled step function of a thermal state λτ has the form Note that because these functions are flat, we can again, as we did in Appendix C, give a meaning to scaling with a non-integer factor λ if we consider these states on a larger system.
The Gibbs-rescaled functions f T ρ are normalized monotonically decreasing step functions like the f ρ . Furthermore, they are flat for equilibrium states, where the partition function Z alone determines which state thermomajorizes the other; for two equilibrium states τ andτ Thus for the f T ρ the partition function takes the role of the rank in f ρ .
Substituting the f ρ with the f T ρ and the rank with the partition function Z, we can apply the proof of Proposition 15 for the majorization relation ≺ M to prove the axioms E1 to E6 as well as the Comparison Hypothesis for the order relation of thermo-majorization ≺ T . Note that when adapting the proof of E3, the choice of the {m i } i and the {n i } i is not problematic, as all energies are positive and thus the factors e −βEi are always smaller or equal to one. Axiom N1 holds, as a state ρ = i p i |E i E i | ∈ S(H) always thermo-majorizes the equilibrium state τ 1 = i e −βE i Z |E i E i | and is thermo-majorized by the equilibrium state τ 0 = |E 1 E 1 |.
As all axioms are satisfied by the order relation ≺ T , Theorem 4 implies that there is a unique additive and extensive function which is monotonic under thermal operations.
Proposition 20. The unique function S T (τ ) for equilibrium states τ ∈ S(H) is given by S T (τ ) = ln Z τ up to affine transformations, where Z τ is the partition function.
Proof. For λ = S T (τ ) we have Recalling that for equilibrium states τ ≺ Tτ ⇔ Zτ ≥ Z τ according to (D3), this is equivalent to and can be rewritten as where a T = 1 ln Zτ 1 Zτ 0 and b T = −a T · ln Z τ0 . This concludes the proof.
The function S T (τ ) obtained for a system in contact with a heat bath is therefore equal to the Helmholtz free energy up to affine transformations. For a choice of two states with partition functions Z τ0 = 1 and Z τ1 = e −β one has exactly S T (τ ) = −k B T ln Z τ = F (τ ). Note that this choice of parameters obeys τ 1 ≺≺ T τ 0 instead of the usual τ 0 ≺≺ T τ 1 . With this gauge the sign of S T is thus reversed and the quantity F decreases under thermal operations.
The quantitiesS T− andS T+ , calculated for block diagonal non-equilibrium states, are not only bounds for the potential F , but are also directly related to the work of formation F max as well as to the extractable work F min introduced by Horodecki and Oppenheim [15] and described in Appendix B 2.
Proposition 21. For block diagonal states ρ ∈ S(H) the quantitiesS T− andS T+ correspond to the quantities F max and F min defined in Equations (B13) and (B12) up to affine transformations.
For a state ρ that is block diagonal in the energy eigenbasis where p res max is the maximal rescaled eigenvalue of the state ρ. Thus, taking the supremum over λ in (D12) implies thatS T− (ρ) = F max (ρ) up to affine transformations.
ForS T+ the proof works similarly. Let ρ ∈ S(H) and let τ 0 ≺≺ T τ 1 be two equilibrium states. Now let λ be such that ρ ≺ T ((1 − λ)τ 0 , λτ 1 ) and thus (D15) First we show by contradiction that This contradicts Equation (D15) as Thus we havek ≥ Z ρ , which can be rewritten as with a T and b T defined as above. Moreover, (D16) implies for all k ∈ R ≥0 , i.e. implies (D15); the second inequality holds as f T ρ (x) is monotonously decreasing and normalized.
For F min we find where Π ρ is the projector onto the support of ρ. Taking the infimum over λ in (D19) implies thatS T+ (ρ) = F min (ρ) up to affine transformations and concludes the proof.
In particular, for constants a T = −k B T and b T = 0 corresponding to states with partition functions Z τ0 = 1 and Z τ1 = e −β as above, we obtain preciselỹ 1. System connected to a heat and a particle reservoir Considering a heat and a particle reservoir is a common practice in statistical physics. Our framework can be extended to cover this scenario.
Connecting a particle reservoir to the system and the heat bath allows for particle exchange with this reservoir. Systems in contact with both, a heat and a particle reservoir, have equilibrium states of the form where Z is the grand canonical partition function. For simplicity we assume that there are only particles of one kind.
The system and the two reservoirs together form an isolated system exposed to adiabatic processes assisted by an ancillary system and a weight, expressed by the order relation of majorization ≺ M . As in the case of a heat bath these processes can be described as processes on the system alone. For quantum states ρ = i p i |E i , N i E i , N i |, which are (block) diagonal in the energy-particle eigenbasis, the possibility of such processes can be expressed by an order relation ≺ N,T , which consists again of a rescaling followed by majorization. Analogous to the Gibbs-rescaling, we can define a N,Trescaled step function f N,T ρ (x) as follows.
be a density matrix. Denote its spectrum by a step function The N,T-rescaled version of this step function is given as For an equilibrium state ρ this can be written as Definition 23. Let ρ, σ ∈ S(H) be two states which are block diagonal in the basis {|E i , N i } i . The order of N,T-rescaled majorization ≺ N,T is defined as This is analogous to Definition 12 for a heat bath, just using an adapted rescaling operation. Proceeding as in the case of only a heat bath, the operations corresponding to this order relation ≺ N,T consist of the following steps: • Addition of ancillary systems in any equilibrium state.
• Unitary transformation conserving energy and particle number of system and ancilla together.
• Removal of any subsystem.
For equilibrium states -which are always diagonal in the basis {|E i , N i } i -this treatment suffices, whereas for those non-equilibrium states which contain coherent superpositions of such eigenstates it does not apply.
As for ≺ T , the relation ≺ N,T fulfills Lieb and Yngvason's axioms. According to Theorem 4 there is thus a unique potential Proof. Let ρ ∈ S(H) be an equilibrium state and let the equilibrium states ρ 0 ≺≺ N,T ρ 1 ∈ S(H) define a gauge. Then for λ = S N,T (ρ), As for equilibrium states ρ ≺ N,Tρ ⇔ Zρ ≥ Z ρ this is equivalent to and can be written as where a N,T = 1 ln Zρ 1 Zρ 0 and b N,T = −a N,T · ln Z ρ0 .
The "entropy function", i.e. the potential for a system in contact with a heat bath and a particle reservoir is thus related to the grand potential Ω, since Ω = −k B T ln Z. (E9) For non-equilibrium states, the bounding functions S N,T− andS N,T+ can be calculated analogously to the scenario including only a heat bath. They define bounds Ω max and Ω min on the potential Ω for non-equilibrium states. These are operationally related to the formation and destruction of a state in this scenario, again analogous to the case of a heat bath.

System in contact with an angular momentum reservoir
In the study of Landauer's principle [5], the question whether energy should obtain a special role among the conserved quantities or whether processes such as erasure could as well be realized at an angular momentum instead of an energy cost was brought up by Barnett and Vaccaro [50,51] and answered in the affirmative. We investigate here whether Lieb and Yngvason's framework is applicable to systems connected to angular momentum reservoirs, i.e. whether it does not attribute a special role to energy.
As the consideration of an angular momentum reservoir is not common practice, Barnett and Vaccaro's concrete model of a spin reservoir is presented here. The reservoir consists of N mobile spin-1 2 particles, for which the possible spin states are denoted by |0 and |1 . The spin states are assumed to be degenerate in energy and thus decoupled from the spatial degrees of freedom, which are in equilibrium with a heat bath. The equilibrium probability for the reservoir to be in a particular state with n particles in state |1 and N − n particles in state |0 is where γ is an appropriate parameter analogous to the inverse temperature β for a heat bath. As for each value n there are N n such reservoir states, the normalization is given as Z res J = (1 + e −hγ ) N and has the form of a partition function for the angular momentum reservoir. This construction allows us to consider an angular momentum reservoir of arbitrary size N . In the following, we consider a reservoir in the limit N → ∞.
The state of a system S in contact with a spin angular momentum reservoir can be described by a density operator. As for systems in contact with a heat reservoir, we restrict ourselves to the treatment of states ρ = i p i |J i J i | that are block diagonal in the eigenbasis of the z-component of the spin operator, denoted as {|J i } i . To ensure that energy does not affect our considerations, we assume all spin-levels |J i to be degenerate in energy. A system in equilibrium with the reservoir is described by a density operator of the form ρ = i e −J ih γ ZJ |J i J i | with partition function Z J = i e −Jihγ .
On system S and reservoir R we consider again adiabatic processes assisted by an auxiliary system in an equilibrium state and an "angular momentum weight", which we describe again according toÅberg's model, just considering the z-component of the spin angular momentum instead of energy. As for energy, the existence of such processes is expressed by the majorization relation ≺ M . The processes occurring on system S alone are formally determined as in the case of a heat bath, but by exchanging the energy with the z-component of the spin angular momentum. Following again Horodecki and Oppenheim [15], the processes on the system S for block diagonal states can be expressed with a rescaled order relation ≺ J .
Definition 25. Let ρ = i p i |J i J i | be a density matrix diagonal in the eigenbasis of the z-component of the spin operator. Denote its spectrum as The J-rescaled step function of ρ is defined as For an equilibrium state ρ this simplifies to Definition 26. Let ρ, σ ∈ S(H) be two states which are block diagonal in the basis {|J i } i . The relation of J-rescaled majorization ≺ J is defined as We can describe the processes on system S corresponding to the order relation ≺ J as: • Addition of ancillary systems in an equilibrium state.
• Unitary transformation of system and ancilla conserving angular momentum.
• Removal of any subsystem.
As in the case of a heat bath, the rescaled majorization relation ≺ J fulfills Lieb and Yngvason's axioms, which gives rise to a unique potential for equilibrium states. For non-equilibrium states, the two bounds on the potential denoted asS J− andS J+ are calculated analogously as in the case of a heat bath, but do not correspond to known quantities. We have thus found an angular momentum based resource theory corresponding to the order relation ≺ J , which has S J ,S J− andS J+ as monotones. In agreement with Barnett's and Vaccaro's conclusion [50,51] that erasure can be achieved at an angular momentum instead of an energy cost, energy can be substituted with angular momentum in our resource theoretic picture and does not obtain a special role among the conserved quantities.