Quantum majorization and a complete set of entropic conditions for quantum thermodynamics

What does it mean for one quantum process to be more disordered than another? Interestingly, this apparently abstract question arises naturally in a wide range of areas such as information theory, thermodynamics, quantum reference frames, and the resource theory of asymmetry. Here we use a quantum-mechanical generalization of majorization to develop a framework for answering this question, in terms of single-shot entropies, or equivalently, in terms of semi-definite programs. We also investigate some of the applications of this framework, and remarkably find that, in the context of quantum thermodynamics it provides the first complete set of necessary and sufficient conditions for arbitrary quantum state transformations under thermodynamic processes, which rigorously accounts for quantum-mechanical properties, such as coherence. Our framework of generalized thermal processes extends thermal operations, and is based on natural physical principles, namely, energy conservation, the existence of equilibrium states, and the requirement that quantum coherence be accounted for thermodynamically.

For completeness we repeat the statement of theorem 1. Theorem 1 Let ρ AB ∈ B(H A ⊗ H B ) and σ AC ∈ B(H A ⊗ H C ) be two compatible bipartite quantum states. Let {M A j } be an arbitrary, but fixed, informationally complete POVM on system A. Denote the dimension of any system X as d X ∈ N. The following are equivalent: 1. The state ρ AB quantum majorizes σ AC , 2. For any quantum process (CPTP linear map) Φ : 3. Supplementary Equation (2) holds for any measureand-prepare quantum channel Φ : B(H A ) → B(H A ) of the form: while the states {ω A j } can freely vary. 4. g(ρ AB , σ AC ) 1, where the the function g is defined by the following semidefinite programming: In order to prove Theorem 1, we will begin by proving the following lemma. Recall the definition of conditional min-entropy H min (A|B) Ω , of a bipartite state Ω AB , H min (A|B) Ω := − log inf be two sets of n density matrices in B(H B ) and B(H C ), respectively. Let {q i } n i=1 be some arbitrary but fixed probability distribution with q i > 0. For any set of n density matrices define the following tripartite separable matrix: Then, the following are equivalent: 1. There exists a CPTP map E : 2. For any ω A 1 , ..., ω A n ∈ B(H A ): where |φ AC + denotes the maximally entangled state on H A ⊗ H C .
Proof. Consider two families of density matrices, {ρ B i } n i=1 and {σ C i } n i=1 . We want to reformulate, in an equivalent way, the condition By introducing a set of self-adjoint operators {X C j } forming a basis for B(H C ), Supplementary Equation (8) can be written as Let us now consider the set of real vectors obtained by letting E vary over all possible CPTP maps from system B to system C, while the ρ i 's and the X j 's are kept fixed. It is clear that the set is a closed and bounded convex set, as it is the image, under a linear map, of the set of CPTP maps from B to C (that is a closed and bounded convex set). By writing s = (s ij ) when s ij = Tr σ C i X C j , Supplementary Equation (12) becomes At this point, we invoke the separation theorem for convex sets (see, e.g., Ref. [1]), which in particular implies the following: Lemma [Separation theorem] Let S ⊂ R n be a closed and bounded convex set. The vector y ∈ R n belongs to S, i.e. y ∈ S, if and only if, for any vector k ∈ R n , max x∈S k · x k · y.
Applied to our case, it yields that condition (8) is equivalent to namely, Defining self-adjoint operators Z C i = j λ ij X C j , we can reformulate the statement as follows: In this condition we should vary operators Z C i over all self-adjoint operators. However, it turns out that we can only restrict these operators to the set of density operators. In other words, this condition is equivalent to To show this note that for any bounded self-adjoint operator Z C i and positive number is a positive operator, and therefore ω C Consider an arbitrary fixed probability distribution q i , with full support, such that q min ≡ min i q i > 0. Then, it turns out that condition (19) can be reformulated as: To show this we use the fact that for any state ω C i , the convex combination (q min /q i )ω C i + (1 − q min /q i )1 C /d C is also a valid state. Using this together with the fact that Tr E(ρ B i ) = Tr σ C i = 1 we can derive condition (19) from condition (20) and vice versa. The next step is to introduce an auxiliary system A ∼ = C (i.e., d A = d C ), choose two orthonormal bases {|i A } and {|i C }, and define the maximally entangled state |φ AC Noticing that Tr[XY ] = d Tr X ⊗ Y T φ + , where the superscript T denotes the transposition with respect to the basis in (21), and that ω i are density matrices if and only if (ω i ) T are, we arrive at As shown in Ref. [5], the quantity = max for Ω AB ≡ i q i ω A i ⊗ ρ B i , can be written in terms of the conditional min-entropy (6) as We thus proved that statements (1) and (2) of Lemma 1 are indeed equivalent. Moreover, a sufficient condition for (8) is that where now Ω AB and Ω AC are meant as the marginals of the same tripartite extension However, it is easy to verify that the above condition is also necessary: indeed, if (8) holds, due to the data-processing theorem applied to the conditional min-entropy (see, e.g., Ref. [6] and [7]), H min (A|B) Ω H min (A|C) Ω . We thus have that statements (1) and (3) are also logically equivalent, and hence the proof is complete.
We are now ready to prove the main theorem. Proof of Theorem 1: are sub-normalized quantum states (i.e. positive semidefinite matrices). Moreover, since ρ A = σ A we have Tr ρ B j = Tr σ C j ≡ p j . We therefore conclude that that there exists CPTP map E that satisfies Supplementary Equation (1) if and only if there exists a CPTP map E that satisfies where ρ B j ≡ ρ B j /p j and σ C j ≡ σ C j /p j . To apply Lemma 1, we introduce a system A with d A = d C , we fix an arbitrary probability distribution q i > 0, and define where the states ω A i can vary. Then, taking q j = p j , we conclude that Notice that, in case some p i = 0, we can redefine the measurement operators M A j → M A j + δ1 A , in such a way that they still span the set B(H A ) but have non-zero probability everywhere. With Supplementary Eq. (32) at hand, the proof of Theorem 1 follows now from Lemma 1. This completes the proof of theorem 1.

SUPPLEMENTARY NOTE 2: COMPLEXITY OF DECIDING QUANTUM MAJORIZATION
We will show now that the problem of whether there exists a CPTP map E such that E(ρ i ) = σ i (see Lemma 1 above) can be formulated as a semidefinite programming. Following similar lines, also all the other versions of quantum majorization discussed in this paper can be shown to be equivalent to a semidefinite programming.
We start by noting that Supplementary Equation (9) can be written as where In the following we absorb the q i s into ω i s, so that the ω i s become subnormalized, satisfying n i=1 Tr[ω i ] = 1. We get that the above condition is equivalent to the condition α(t) 1 for all t, where with ω i 0. After rescaling τ ≡ 1 t τ and ω i ≡ 1 t ω i we get The condition α(t) 1 for all t is therefore equivalent to one condition, α 1 (more precisely, α = 1 since it can be shown that α can never exceed 1), where We now show that the above minimization problem is an SDP. To see it, we define the following vector space, which is a direct sum of n + 2 Hilbert spaces: The vector space V 1 is consisting of matrices ζ ∈ V 1 of the form: , and X i ∈ B(H A ) for each i = 1, ..., n. In addition, we define the vector space , and a linear transformation Γ : V 1 → V 2 given by: Clearly, the map above is linear.
We also denote C ≡ (0, I, 0, ..., 0). With these notations: To bring the above optimization problem to a canonical SDP form, we denote and E j is a basis of B(H A ⊗H B ). We also denote H 0 ≡ σ.
With this notations we get α ≡ min Tr [Cζ] subject to ζ 0 It is interesting to note that the dual problem is given by where the dual map Γ * is given by: Therefore, the dual problem can be expressed as β = max y subject to τ AB 0 ; τ B I ; and ∀ i = 1, ..., n Note that α (or β) can be commuted efficiently using standard SDP algorithms. As mentioned in Remark 3 of the main text, and as shown in Note 2 above, all the instances of quantum majorization considered in this paper can be formulated as semidefinite programs. These are well known for being efficiently solvable. One may be left wondering, then, about the role and relevance of the alternative formulation of quantum majorization that we provide in Theorem 1, in terms of an infinite set of inequalities between state monotones. Clearly, if the problem is just to decide whether quantum majorization holds or not, one should run the corresponding semi-definite program. However, the SDP formulation does not provide us any further insight about why a solution exists or not, nor does it tell us anything about the resources at stake and the way to quantify them. In other words, it does not tell us much about the physics behind quantum majorization.
Ideally, a resource theory should not only provide an efficient way to check whether a free transformation exists between two states, but also a way to measure resources as state functions. While the SDP formulation fulfills the former requirement, the formulation in terms of a complete set of monotones fulfills the latter.
The fact that here we find a complete set of monotones comprising infinitely many such functions is not an artifact of the present approach, but it is something that appears in many other contexts too. For example, already in classical statistics, the majorization relation with catalytic transformations (i.e., the "trumping" relation) is known to be equivalent to an inequality that must hold for all (uncountably many) Rényi entropies [30], and no discrete set of equivalent conditions is known. Again in the classical case, catalytic thermal operations have also been characterized in terms of an infinite set of "second laws" involving free energy functionals [2]. In the quantum theory of entanglement, when the local dimension is four or higher, it is known that an infinite number of entanglement monotones is not only sufficient but also necessary, in order to determine state conversion [3].
The characterization of quantum majorization in terms of a complete set of monotones, as we show in what follows, is also able to completely capture the notion of quantum thermal processes, with respect to both energy and coherence. This is non-trivial (and perhaps even surprising), given that it was shown that no direct analogue, in terms of "simple" free energy functionals, would ever be able to capture the subtle interplay between energy and coherence appearing in genuinely quantum thermal processes between non-commuting states [4]. It is hence a merit of our approach to circumvent this obstacle providing, at the same time, a novel insight into the theory. Indeed, the monotones constructed here are able to go be-yond free energy functionals, by explicitly bringing into the picture an external reference system, with respect to which information about energy and time (i.e., coherence) is measured. Such an insight, that suggests also an intriguing physical picture behind quantum majorization, cannot be gained by looking at the SDP formulation alone.
Finally, the characterization in terms of monotones has, with respect to the SDP formulation, another advantage, which is due to the fact that our monotones can be expressed as min-conditional entropies [5,6]. This allows us to apply, in principle, the powerful tools developed for single-shot quantum information theory [7] in order to study their behavior in the asymptotic scenario, something that we leave open for future investigations.

SUPPLEMENTARY NOTE 4: RE-DERIVATION OF THERMO-MAJORIZATION AS THE CLASSICAL CASE
Thermo-majorization generalizes ordinary majorization in a natural way [8][9][10][11][12][13]. Given two probability distributions p = (p i ) and q = (q i ) together with the Gibbs distribution γ = (γ i ) = ( 1 Z e −βEi ) at temperature T = (kβ) −1 , we say that p thermo-majorizes q and write p T q exactly when the following holds for all t 0. This can be shown to be equivalent [8-11, 13, 22] to the existence of a stochastic map S such that Sp = q and Sγ = γ. In what follows, we show that quantum majorization reduces to Thermo-majorization in the classical case. In particular, we will show that the conditions in Theorem 1 (specifically, Supplementary Equation (10) of Lemma 1) reduces to Supplementary Eq. (46). We first start with the semi-classical case.

The semi-classical case
In this case, we assume that the n states, {σ C i }, in Lemma 1 commute with each other. Therefore, we can assume that they are all diagonal with respect to a fixed basis. We show now that this immediately implies that the n states {ω i } in Lemma 1 can also be taken to be diagonal in the same basis. In fact, in the following lemma we show that if {σ C i } are all symmetric with respect to some group, then the states {ω i } also have the same symmetry. Lemma 2 Using the same notations as in Lemma 1, let ∆ : B(H C ) → B(H C ) be a CPTP map, and suppose ∆(σ T i ) = σ T i for all i = 1, ..., n. Then, in all the statements of Lemma 1 we can replace the set Remark 1. The lemma above is particularly interesting if the map ∆ corresponds to some symmetry. That is, where {U g } is some unitary representation of a compact group G. In this case, one can take ∆ to be the G-twirling, and thereby assume that all the ω A i s of Lemma 1 are also symmetric with respect to the same representation of G.
Proof. The proof follows from the two sides of Supplementary Eq. (9). On one hand, where ∆ † is the dual (adjoint) unital map of ∆. On the other hand, if for some non-normalized state τ , then since ∆ † is a unital CP map we get That is, Combining (47) and (50) with (9) we conclude that if (9) holds for all states of the form {∆ † (ω A i )} then it holds for any set of n states {ω A i }. This completes the proof of lemma 2.
The case that we are interested here is the one in which all the σ i s are diagonal with respect to some fixed basis. This is the case considered in Corollary 1 of Ref. [21]. In this case, we can take ∆ to be the completely decohering map with respect to the fix basis. Since the set{∆(ω i )} consists of diagonal matrices, we can assume w.l.o.g. that all the ω i s in Lemma 1 are diagonal. We can therefore write so that is a classical quantum state. It is well known that for classical quantum states, the conditional min-entropy can be expressed in terms of a guessing probability [5]. In the case that d A = 2 the conditional-min entropy of Ω AB can be further simplified and we get However, even if the σ i s all commute, it is not enough in general to restrict the comparison only to twodimensional auxiliary states ω i , if the goal is that of showing the existence of a CPTP map achieving ρ i → σ i . If such a restriction is made, what one can show is the existence of a weaker map, namely, a 2-statistical morphism [14,21], but counterexamples have been shown for which neither a CPTP nor a PTP map exists [23]. There are two very important exceptions to this. The first is the case in which there are only two commuting states {ρ 1 , ρ 2 } and two commuting states {σ 1 , σ 2 }, namely, the case of two classical dichotomies. In this case, already Blackwell showed that two-dimensional commuting states ω i suffice [11].

Thermo-majorization
In the completely classical case, in addition to the ω i s, also the set {ρ B i } consists of diagonal matrices. Denoting we get that (55) Now, in this case, the conditional min-entropy is given by where for each x and y, r x is the n-dimensional vector whose components are {q i r x|i } n i=1 , and s y is the n-dimensional probability vector whose components are {s y|i } n i=1 . Similarly, denoting by we conclude that where t z is the probability vector whose components are t z|i . Therefore, in the classical case, the condition in (10) is equivalent to for any sub-linear functional f of the form f (s) = max x r x · s. Note that y s y = z t z = (1, 1, ..., 1) T . Finally, to obtain themo-majorization, we consider the case n = 2. That is, we have two input states ρ 1 and ρ 2 , and two output states σ 1 and σ 2 . We can think of ρ 2 and σ 2 as Gibbs states. Note that all the vectors r x , s y , and t z are two-dimensional since n = 2. Therefore, in this case, it is sufficient to consider in (59) only sublinear functionals with two elements; that is, of the form f (s) = max{r 1 · s, r 2 · s} (see [20] for more details). We therefore conclude that the condition in (10) is equivalent to y max{r 1 · s y , r 2 · s y } z max{r 1 · t z , r 2 · t z } (60) for all r 1 , r 2 ∈ R 2 + . Using the relation max{a, b} = a+b 2 + |a−b| 2 for any two real numbers a and b, the equation above becomes equivalent to where we used the fact that y s z = z t z = (1, 1, ..., 1) T . Denoting by r 1 − r 2 ≡ a b ∈ R 2 , the above equation becomes Dividing by a and denoting r ≡ −b/a we conclude that our condition in (10) reduces in the classical case to the thermo-majorization condition: Note that there is an equality above if r < 0 so we assume w.l.o.g. that r 0.

Proof of Corollary 2
The proof of Corollary 2 can now be established. Suppose we are interested in the conversion of ρ A into σ A under TPs. Moreover suppose that [ρ A , H A ] = 0, as explained in the main text one may restrict without loss of generality to η 1 and η 2 being incoherent in energy. Therefore the state Ω RA is a classical state. Since TPs are covariant, and ρ A is incoherent in energy it implies that the states accessible under this class must also be incoherent in energy and so [σ A , H A ] = 0 is a necessary condition. Since both input and output states are incoherent the problem reduces to the interconversion of the distributions over energy under stochastic maps that preserve the Gibbs state. This coincides with the conditions for thermo-majorization as stated above.
On the other hand, suppose [σ A , In what follows, we consider three unitary representations g → U g of the same compact group G on systems A, B, and C. We use the following notations: with obvious meaning of symbols. We also introduce the bipartite twirling operation be two sets of n density matrices in B(H B ) and B(H C ), respectively. Let {q i } n i=1 be some arbitrary but fixed probability distribution with q i > 0. For any set of n density matrices define the following tripartite separable matrix: and its twirled version Then, the following are equivalent: 1. There exists a covariant CPTP map E : 2. For any ω A 1 , ..., ω A n ∈ B(H A ): 3. For any ω A 1 , ..., ω A n ∈ B(H A ): Proof. The proof of Lemma 1 goes through unchanged, with the only difference being that we want to find a CPTP map E that is covariant, i.e., that satisfies the following property: Hence, we can start from Supplementary Eq. (22), which in the covariant case becomes Using the covariance of the channel Supplementary Eq. (70), and the so-called "ricochet property" of the maximally entangled state, that is, (1 A ⊗ X C )|φ AC + = (X T A ⊗ 1 C )|φ AC + ), we can rewrite the left-hand side of the above inequality as follows: where, we recall, the channel E is assumed to be covariant.
Le us now consider the quantity where the maximization now is allowed to run over all possible CPTP maps, not only covariant ones. However, since both Ω AB and φ AC + are invariant for the action U g ⊗ U g , we immediately have that and hence, using the conditional min-entropy, Hence, statement (1) is equivalent to (Remember that d A = d C .) Following the same arguments used in the proof of Lemma 1, we also obtain the equivalence between statement (1) and statement (3). This completes the proof of lemma 3.
Remark 2. The existence of a covariant CPTP map achieving the transformation ρ i → σ i is of course a stronger requirement than the existence of a general CPTP map doing the same. Indeed, once we rewrite Supplementary Eq. (9) of Lemma 1 as and since, as a consequence of the data-processing inequality, 2 −Hmin(A|B)Ω 2 −Hmin(A|B) Ω , it is clear that it is in principle harder to satisfy condition (2) of the covariant Lemma 3, than its non-covariant counterpart (9).

G-covariant version of Theorem 1
As before we used Lemma 1 to prove Theorem 1, here we use Lemma 3 to prove Theorem 2. Theorem 2 Let ρ AB ∈ B(H A ⊗ H B ) and σ AC ∈ B(H A ⊗ H C ) be two compatible bipartite quantum states. Denote the dimension of any system X as d X ∈ N. The following are equivalent: 1. There exists a G-covariant CPTP map E : 2. For any quantum process (CPTP linear map) Φ :

Supplementary Eq. (88) holds for any
where {M A j } is an arbitrary, but fixed, informationally complete POVM on system A, while the states {ω A j } can freely vary. (90)

For any Φ : B(H
where |φ A C + is the maximally entangled state between systems A and C.
We are now ready to prove the main theorem.

Proof of Theorem 2: Let {Q
are sub-normalized quantum states (i.e. positive semidefinite matrices). Moreover, since ρ A = σ A we have Tr ρ B j = Tr σ C j ≡ p j . We therefore conclude that that there exists a covariant CPTP map E that satisfies (87) if and only if there exists a covariant CPTP map E that satisfies where ρ B j ≡ ρ B j /p j and σ C j ≡ σ C j /p j . To apply Lemma 3, we introduce a system A with d A = d C , we fix an arbitrary probability distribution q i > 0, and define where the states ω A i can vary. The corresponding twirled state is Then, taking q j = p j , we conclude that , Notice that, in case some p i = 0, we can redefine the measurement operators M A j → M A j + δ1 A , in such a way that they still span the set B(H A ) but have non-zero probability everywhere. With Supplementary Eq. (96) at hand, the proof of Theorem 2 follows now from Lemma 3.

Covariant Stinespring dilations
Given systems A and A , with Hilbert spaces H A and H A , we assume that each carry a unitary representation of a compact group G given by U : G → B(H A ) and U : G → B(H A ) respectively. A quantum process E : The following lemma was proved in [28], and we provide the proof here for convenience. Lemma 4 [28] Given a covariant quantum process E : with Kraus operators K λ,m,k : H A → H A that transform as where (v λ (g) jk ) are the matrix elements of the λ-irrep of G and m is a multiplicity label.
Proof Let {K i } be a set of linearly independent Kraus operators for E. Since E is covariant we have that U A g • E • (U A g ) † = E for any g ∈ G, and so it follows that {U A (g)K i U A (g) † } i forms another set of Kraus operators for E for any fixed g ∈ G. Since the Kraus representation is unique up to unitary mixing this implies that U A (g)K i U A (g) † = j V (g) ij K j . Moreover, since the Kraus operators are linearly independent it follows that this unitary V (g) is unique for any fixed g and so the matrices V (g) form a non-projective unitary representation of G. Using the unitary freedom to choose the basis {K i } we can choose a basis for which V (g) is block diagonal in terms of a sum of unitary irreps of G. We denote this basis {K λ,m,k }, with {K λ,m,k } transforming as a λ irrep under G for each multiplicity m as in Equation (98), and k labels the basis vector of the irrep. This completes the proof.
Such Kraus operators are said to transform irreducibly under the group action, and are irreducible tensor operators. Theorem 3 [Covariant Stinespring [29]] For any covariant quantum process E : where |σ B ∈ H B is a symmetric state under the unitary representation U B of G on system B, system C carries a unitary representation U C of G, and V : for all g ∈ G.
Remark 3. This theorem was proved in [29] for the case H A = H A . The proof of the general case is essentially identical, and we provide the proof below for convenience. Proof. From the previous lemma, a covariant quantum process E : B(H A ) → B(H A ) always has a Kraus decomposition {K λ,m,k } such that where λ labels an irrep of G, m is a multiplicity label and k is the basis vector label of the irrep. Let B be a system with Hilbert space H B = span{|σ }, with the state |σ being symmetric under the action of G. For any pair (λ, m) appearing in the Kraus decomposition of E, let W (λ * ,m) be a Hilbert space isomorphic to the λ * -irrep of G and for which we choose a basis {|λ * , m, k } k . We define H C := H B (λ,m) W (λ * ,m) , where the direct sum ranges over all (λ, m) occurring in the Kraus decomposition of E. The space H C carries the unitary group action where (v λ (g) jk ) are the unitary matrix components of the irrep λ of G.
We define the operator V : Using that the {K λ,m,k } k transform irreducibly under the action of G, together with the fact that (v λ (g) jk ) is a unitary matrix, it is readily verified that Equation (100) holds for all g ∈ G, and so V is covariant under the action of G. Moreover since λ,m,k K † λ,m,k K λ,m,k = 1 A , and {|λ * , m, k } λ,m,k is an orthonormal set of states we have that V † V = 1 A ⊗ |σ σ| B and so V is an isometry and so have constructed the required dilation for the covariant quantum process E. This completes the proof of theorem.
The following lemma clarifies that any mixed symmetric state can always be purified to a pure quantum state that is also symmetric under the group action. Lemma 5 Consider a quantum system A, carrying a unitary representation U A : G → B(H A ), and a mixed quantum state σ A for which U A g (σ A ) = σ A for all g ∈ G. Then, there exists a purification |ψ AB of σ A onto a composite system AB, and a unitary representation Since any density operator on H A can be thought as a quantum process from a 1 dimensional input Hilbert space to B(H A ), this lemma follows immediately from theorem 3 on Covariant Steinespring dilation. Here, we present a more direct proof.
Let {|j A } R j=1 be an orthonormal basis of the support subspace of σ A , where r is the rank of σ A . Let (105) be a purification of σ A . Then, Therefore, this completes the proof by taking V g ≡ U B g .

SUPPLEMENTARY NOTE 6: GENERALIZED THERMAL PROCESSES
We prove the general result in the presence of thermodynamic observables {H A , X A 1 , . . . , X A n }, which may have non-trivial commutation relations between them. The case on the Hamiltonian being the only thermodynamic observable follows as a special case of this result.
Assumptions (A1) and (A2), together with the requirement that the resource theory be non-trivial in these observables implies that the free state must take the form of the generalized Gibbs ensemble γ A , for constants β, µ 1 , . . . , µ n . This is picked out in several different ways, for example perhaps the simplest to interpret is within the theory of equilibration. An alternative route is through a complete passivity argument in which one has additional access to an ordered macroscopic 'bath' for each observable that can give or take arbitrary amounts of that observable. Given an unbounded number of copies of the free state one wishes to know if one can trivialise the theory in terms of providing an arbitrary displacement for any of these observables. However in the presence of thermodynamic constraints, these are coupled in such a way that one must only consider an "effective" energy bath with Hamilto-nianH = H− k µ k X k . Complete passivity with respect to this observable implies the above generalized Gibbs state through standard arguments. We now give a precise statement of assumption (A3) in the context of thermodynamic observables {H S , X S 1 , . . . , X S n } for any quantum system S. We first note there are two components to any TP process E at the microscopic level: the particular interactions between A and an auxiliary system B, and the state σ B of the auxiliary system. Under assumption (A1) there are no couplings present between eigenspaces of different eigenvalues of the additively conserved observables, however this does not mean that coherence cannot be injected into A. Assumption (A3) places a minimal constraint on the use of coherence sources outside of A. The key idea is that while E may be realised through some specific interaction V between A and its environment B, and this environment may even contain quantum coherences in its state σ B , we can guarantee that E is not exploiting any of these coherences if it is the case that if we were to remove the coherences present in σ B then the transformation E would still be possible through interactions with B. This motivates the following condition.
(A3) (Incoherence) Given a thermodynamically free process E : B(H A ) → B(H A ) there exists an interaction isometry W that obeys (133) and a quantum state η B such that and with η B = G(η B ) where and where U B (g) is the group representation on B generated by the observables {H B , X B 1 , . . . , X B n }. Below in Lemma 7 we show that this assumption captures the demand that no coherence is being exploited from the environment, but before this we establish that the set TP has a compact formulation in terms of covariance. Lemma 6 Given a set of thermodynamic observables {H S , X S 1 , . . . , X S n } for any quantum system S, the set TP of quantum processes from A into A defined by (A1-A3) coincides with the set GPC of Gibbs-preserving processes on A that are covariant under the group G generated by the thermodynamic observables on A and A . Proof We first show that T P ⊂ GP C. Assumption (A2) ensures that the image of the Gibbs state γ A under T P is the fixed point γ A , so it suffices to establish covariance. For any system S we define X S 0 := H S so as to make notation compact. Given a process E ∈ T P , assumption (A1) implies that for some V that obeys the conservation laws given by Equation (133). In particular, this implies that for all k = 0, . . . d and for all θ k ∈ R. Therefore the observables {X k } generate a representation of a group G, with elements g indexed by (θ 0 , . . . , θ d ), and U A C (g)V = V U AB (g) for all g ∈ G. Therefore the process sending any χ AB → V χ AB V † is G-covariant. As discussed, assumption (A3) says that the above σ B can be taken to be symmetric under this group action: U B g (σ B ) = σ B . Since discarding systems is G-covariant, and also composing of G-covariant processes results in a G-covariant process, we see that is a G-covariant process for any E of the form (110). Therefore T P ⊂ GP C.
Conversely, let E ∈ GP C. Since E(γ A ) = γ A , assumption (A2) holds automatically. Since E is G-covariant with respect to the group generated by {X A k } as shown there exists a Stinespring dilation of the process E of the form where V is a G-invariant isometry and |ψ B is invariant under the group action on B. The invariance of V implies that assumption (A1) holds, while the symmetry of |ψ implies that there are no coherences between eigenspaces of the distinguished observables and so (A3) holds. Therefore E ∈ T P , and so the two sets of processes coincide as claimed. This completes the proof of lemma.
To summarize, the state interconversion under TPs is equivalent to the following requirement: where E is required to be a G-covariant process.
We can now show that no coherences are exploited from the environment for any E in TP. Lemma 7 Suppose E is in TP and realised as by some isometry V obeying (133 and interacting with a system B in a state σ B . Then (117) also holds with σ B replaced by G(σ B ).
Proof Since E is a covariant map, we have that Expressing E in terms of (V, σ B ) and exploiting the fact that for any g ∈ G. Integrating over all g and using linearity we deduce that as required.
Remark 4. Note that Lemma 5 shows that replacing any auxiliary σ B with its dephased version G(σ B ) as discussed in the main text is consistent with the existence of a Stinespring form in which the auxiliary system is taken to be in a pure symmetric quantum state. Also note that that we implicitly assume that the group G generated by the thermodynamic observables on the input system coincides with the group generated by those on the output system, which is a basic physical requirement.

SDP solution for thermomajorization with coherence
Here we illustrate how to use SDP to solve the decision problem of determining if ρ ∈ B(H) can be converted to σ ∈ B(H) by a generalized thermal processes. For simplicity of the exposition, we consider the case of no charges with the same Hamiltonian for the input and output spaces. We want to know if there exists a thermal process, that is, a CPTP map E : B(H) → B(H) that is both Gibbs preserving and symmetric under time translation, such that σ = E(ρ). Denoting the Hamiltonian by H and the Gibbs state by γ = 1 Z e −βH , from (45) it follows that there exists such a Gibbs preserving symmetric map E if and only if f (ρ, σ) 1, where f (ρ, σ) is given by the SDP problem Here we added the condition G(τ AB ) = τ AB to (45) to ensure that E is symmetric under time translation with respect to the Hamiltonian H. The G-twirling is given as in (64) with g replaced by the time parameter t, and the group element U t (·) = e iHt (·)e −iHt . The dual to this problem is very similar to (37) and the only difference is that one has to take the G-twirling on the term n i=1 X i ⊗ ρ i . For the more specific case we consider here, we have: As before, the above minimization problem is an SDP. To see it, we follow now the same steps that led to Supplementary Eq. (42). We define the vector space V 1 , consisting of all Hermitian matrices ζ ∈ V 1 of the form: where η ∈ B(H ⊗ H), and X, Y, Z ∈ B(H) are all Hermitian. In addition, we define the vector space V 2 of all Hermitian matrices in B(H ⊗ H), and a linear transformation Γ : V 1 → V 2 given by: Clearly, the map above is linear. Set τ ≡ (0, 0, σ T , γ T ) so that Tr[σζ] = Tr σ T X + Tr γ T Y . We also denote C ≡ (0, I, 0, 0). With these notations: subject to ζ 0 , Tr (ζH j ) = δ 0,j ∀j ∈ {0, 1, ..., d 4 } .
The above optimization form is written in a canonical form and can be solved with SDP packages such as CVX.

SUPPLEMENTARY NOTE 7: PROOF OF THEOREMS 2 AND 3
Theorem 2 is a special case of theorem 3, where the only additive conserved observable is the Hamiltonian (See remark 5). Therefore, it suffices to prove theorem 3.
The proof is basically a corollary of lemma 3. For any system S = R, A, A , let {U S (g)} be the symmetry group generated by the additively conserved observables {H S , X S k ; k = 1, . . . , n}, where for system R we define and the superscript T denotes the transpose. Note that this definition implies that for any group element g, We are interested to determine if there exists a CPTP map E : and (ii) is covariant, i.e. for all group elements g, where U S g [·] = U S (g)[·]U † (g) for any system S = A, A , R. Therefore, to apply lemma 3, we assume systems A, B and C in the statement of lemma 3 correspond, respectively, to systems R, A and A in the statement of theorem 3. Furthermore, to apply the lemma we assume the set of input states {ρ i } has two elements, namely {ρ A , γ A }, and the corresponding output states are {σ i } = {σ A , γ A }. Finally, in the statement of lemma 3 states {ω i } denote possible states of the reference system. Here, we denote them by {η R 1 , η R 2 }. Under these assignments state Ω ABC in the statement of lemma becomes Then, the equivalence of statements (1) with n = 0, 1, . . . N − 1. Therefore the dynamics of any single quantum system can always be approximated by such a discrete, finite action for some N ∈ N and sufficiently large.
In the case that we have multiple systems A 1 , A 2 , . . . , A M with periods τ 1 , τ 2 , . . . , τ M respectively, we may choose τ = M k=1 τ k as the time-scale for the composite system. Therefore for multiple systems, there will always exist an N ∈ N, sufficiently large so that the mapping n → exp[−in H A k ] is a unitary representation of Z N on each H A k , and which approximates the unitary dynamics of each A k under its Hamiltonian to the specified level of precision. Given this, condition (A3) for incoherence of thermal processes can be replaced with the following.
(A3 ) Approximate incoherence. Consider the case of the Hamiltonian being the only thermodynamic observable, and assume the finite precision approximations described above. If the thermodynamically free process E : B(H A ) → B(H A ) is realized microscopically as with V obeying Equation then we also have E(ρ A ) = Tr C W (ρ A ⊗ G (σ B ))W † .
with G (σ B ) being the group average over Z N of the state σ B given by with U B (n) := exp[−in H B ] is the finite precision time evolution on B, and we interact this state with A through some potentially different isometry W that also obeys (133). This implies that the constraint of time-translation covariance is replaced with Z N -covariance to this level of precision. Given this, the analysis for state interconversion may be repeated under (A3 ) and results in the replacement of 1 τ τ 0 dt(·) with 1 N N −1 k=0 (·) and U R (t) ⊗ U A (t) by the discrete approximation U R (n) ⊗ U A (n).

SUPPLEMENTARY NOTE 10: CLOCK TIMES AND GUESSING PROBABILITIES
As in the previous section, we may restrict our attention to a fully discrete setting with quantum systems of finite dimension and finite level of precision for time resolution. Covariance of the dynamics is now described with respect to the discrete group Z N for some sufficiently large N ∈ N.
For q → 1 we obtain the Z N covariance constraint alone, and the corresponding state Ω RA takes the form (136) For a sufficiently large reference frame R there exists a Hamiltonian H R such that R allows a perfect encoding of the group elements of G = Z N . In particular for dim(H R ) = N with orthonormal basis {|E k R }, we can choose where ω := e 2πi N is an N th root of unity. We then have that (U R (1)) n = U R (n) for any n = 1, 2, . . . and U R (N ) = U R (0) = 1 R as required.
Defining |k R := F |E k R , with F being the discrete Fourier transform operator it is readily seen that and n|m R = 0 for n = m and equal to 1 for n = m. Therefore the reference system R provides a perfect classical encoding of the group elements of Z N in the pure states {|k R }. Setting η R 1 = |0 0| R in equation 136 gives the classicalquantum state where we define ρ A (n) := U A (n)ρ A (U A (n)) † for the state of A at time t = n . These states fully encode the set of clock times t = 0, , . . . , n , . . . , (N − 1) for the joint system.
Since Ω RA is a classical-quantum state, we have that [5] H min (R|A) Ω = − log p guess , where p guess is the optimal Helstrom guessing probability for the ensemble of states {( 1 N , ρ A (n))} N −1 n=0 on A. This implies that 2 −Hmin(R|A)Ω is the optimal guessing probability of the clock time t = n for the joint system, given the single copy of ρ A . Monotonicity of H min (R|A) Ω under the thermal processes implies monotonicity of the clock time guessing probability for the system.