Quantum Horn's lemma, finite heat baths, and the third law of thermodynamics

Interactions of quantum systems with their environment play a crucial role in resource-theoretic approaches to thermodynamics in the microscopic regime. Here, we analyze the possible state transitions in the presence of"small"heat baths of bounded dimension and energy. We show that for operations on quantum systems with fully degenerate Hamiltonian (noisy operations), all possible state transitions can be realized exactly with a bath that is of the same size as the system or smaller, which proves a quantum version of Horn's lemma as conjectured by Bengtsson and Zyczkowski. On the other hand, if the system's Hamiltonian is not fully degenerate (thermal operations), we show that some possible transitions can only be performed with a heat bath that is unbounded in size and energy, which is an instance of the third law of thermodynamics. In both cases, we prove that quantum operations yield an advantage over classical ones for any given finite heat bath, by allowing a larger and more physically realistic set of state transitions.


I Introduction
Thermodynamics is traditionally viewed as a statistical theory of a large number of particles. In the last few years, however, there has been a wave of activity in extending the realm of thermodynamics to microscopic quantum systems, motivated by ideas from quantum information theory [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,20,21,22,23,24]. One key idea is to formulate thermodynamics as a resource theory [6,25,26,27,28]: by specifying and restricting to a set of experimental operations that an agent is allowed to implement "for free", one obtains a rigorous framework which allows to address typical thermodynamics questions even in physical scenarios that are far from the usual thermodynamic limit.
heat bath E T 0 > 0 thermal operation Figure 1: Theorem 9 can be seen as a simple instance of the third law of thermodynamics. We start with a qubit in its excited state, which serves on the one hand as a work reservoir, and on the other hand represents the system that we would like to cool. For any finite heat bath, thermal operations can never achieve the ground state exactly; there will always remain finite fluctuations, indicated by the arrow. The final temperature T will always be strictly positive, with a lower bound that depends inversely-extensively on the size and energy of the heat bath. The physical intuition of this will be elaborated in more detail in the conclusions.
One way to formulate a suitable resource theory is by declaring the set of thermal operations as allowed. A thermal operation consists of an arbitrary global energy-preserving unitary transformation on the given quantum system A and an arbitrary heat bath B (at fixed background temperature), and the subsequent disposal of the heat bath. By making A itself consist of several parts (say, a single particle and a work-storage device), we can study the interplay of heat and work in microscopic systems. For example, we can ask how much work can be extracted with high probability from a given quantum state by using only thermal operations. Many of these questions have been answered explicitly, and have been shown to refine -and converge to -the usual thermodynamic results in the limit of a large number of weakly interacting particles.
Even though this approach has been tremendously successful in providing insights into thermodynamics in the microscopic regime, it has also raised questions about the way that thermal operations should be modelled and understood [43,44]. While the quantum system A is allowed to be microscopic (say, a single particle), the heat bath B is often assumed to be macroscopic, or even explicitly assumed to be unbounded. However, in many physical situations, this is not a realistic assumption. In particular, we may think of a situation where a thermodynamic protocol has to be completed in a finite amount of time, allowing A to effectively interact only with a restricted number of degrees of freedom in the bath. The third law of thermodynamics [16] can be seen as an example of this, as we will argue further below. This leads us to study thermal operations with bounded heat baths B in this paper.
The thermodynamical questions notwithstanding, there is also a purely mathematical motivation to study finite baths, which comes from more traditional quantum information problems. For example, consider the special case that all Hamiltonians (of system and bath) are fully degenerate, i.e. proportional to the identity. Then thermal operations are maps of the form with m = dim B, and U an arbitrary unitary on AB. These maps have been called noisy operations [2,32] in quantum information theory. They have been studied first without explicit reference to thermodynamics, often under different names (like "K-unistochastic maps" [35,36,37] or "exactly factorizable maps" [41,42]), formalizing the idea of a map that dilutes quantum systems with random noise. One important question in this context is whether for two given quantum states ρ A and ρ A , there exists a noisy operation that maps ρ A to ρ A . This question has been partially answered in [2]: if the spectrum (vector of eigenvalues) of ρ A majorizes the spectrum of ρ A , denoted then one can obtain ρ A to arbitrary accuracy by noisy operations from ρ A . Since unitary basis changes are allowed operations, we may assume that both ρ A and ρ A are diagonal matrices. Then (2) can be proven by reduction to the case of discrete classical probability distributions, where the unitary U acts as a suitable permutation of basis vectors. This strategy of "proof by reduction to classical" has one important drawback, however: it only works if the heat bath is assumed to be unbounded. To see this, suppose that ρ A is a pure state, and ρ A a mixed state with irrational eigenvalues. In this case, eq. (2) holds. But if U is a classical permutation, then the right-hand side of (1) will always be a density matrix with entries that are all rational numbers, and never be equal to ρ A which has irrational diagonal elements. In other words, to approximate ρ A to arbitrary accuracy, the dimension m of the bath has to become arbitrarily large.
It has been conjectured [36] that one can in fact perform all transitions ρ A → ρ A exactly with a finite bath of dimension dim B = dim A, by using more general unitaries U which are not permutations. In this paper, we give a simple proof of this conjecture (see Lemma 4 below), which can be seen as a quantum version of Horn's lemma. This shows that a small bath is always sufficient to implement any noisy state transition ρ A → ρ A exactly.
A "proof by reduction to classical" has also been the standard approach in the thermodynamic setting, where the Hamiltonians of system and bath are allowed to be non-trivial: the possibility of a transition ρ A → ρ A by thermal operations has standardly been proven by resorting to the classical case, where U is an (energypreserving) permutation. This leads to a setup where one is forced to allow a heat bath B which is unbounded in dimension and energy, in order to approximate all possible state transitions. As mentioned above, this assumption does not always seem to be physically realistic, in particular since it presupposes the fine control to implement arbitrary energy-preserving unitaries on the joint quantum system AB, where B is very large.
One way that this problem has recently been addressed is by considering a small subset of "coarse" operations that are still sufficient to realize all possible thermal state transitions [44]. Similarly, Ref. [45] has studied the role of catalysts (auxiliary systems that ease a state transformation) that are bounded in size or energy. In this paper, we take a different route. We hold on to thermal operations, and ask whether there is an analogue of quantum Horn's lemma for them, in particular in the quasiclassical case where both initial and final state are blockdiagonal in energy. If this was the case, then every thermal state transition between quasiclassical states could be realized exactly with a finite heat bath.
Here we give two results that answer this question in a complementary way. First, we show that "quantum" thermal operations (that is, ones with arbitrary energy-preserving unitaries U ) allow indeed many transitions exactly which could only be realized approximately by "classical" thermal operations (where U is an energy-preserving permutation). For quantum systems with non-degenerate Hamiltonian H A , Figure 4 illustrates this: for a fixed initial state and fixed finite heat bath, the possible final states obtained by classical thermal operations form a discrete finite set. In contrast, the final states obtainable by quantum thermal operations correspond to the convex hull of this set (up to some minor amendments in the case of degeneracies). This removes the unphysical discreteness originating from the finite set of classical permutations, and paints a physically more realistic and also mathematically more convenient picture. It is formalized in our first main result, Theorem 10, which generalizes quantum Horn's lemma to thermal operations. Figure 5 shows a simple consequence of this fact, which is the content of Corollary 12: the set of all states that can be obtained to arbitrary accuracy from an initial state by thermal operations (that is, the states "thermomajorized" by the given initial state) is a compact convex set. All states in its relative interior can be obtained exactly by quantum thermal operation with a suitable finite heat bath.
However, we show that this is not in general true for the states in the relative boundary, which represents our second main result. In particular, for every nondegenerate two-level system A, we show explicitly that there is a state that can be obtained to arbitrary accuracy, but never exactly with any finite heat bath.
This result has an interesting physical interpretation, illustrated in Figure 1. Consider the special case of a two-level system with some energy gap ∆E, which is initially in its (pure) excited state. Suppose we would like to couple this qubit system to a heat bath, in such a way that it is in its ground state after the interaction. Physically, this is arguably the simplest possible instance of cooling: we draw energy from the qubit system itself (coming from the excited state) in order to cool it down to zero temperature, which is less then the ambient (heat bath) temperature. However, we show in Theorem 9 that the ground state can never be obtained exactly with any finite heat bath: it can only be approximated, such that the final temperature T can never be exactly zero. We give a strictly positive lower bound on T which depends on the energy and dimension of the heat bath, which can be seen as an instance of the third law of thermodynamics. While the third law has been examined in more detail elsewhere [16], our result suggests an intuitive picture in terms of the geometry of the state space: states of zero temperature lie on the relative boundary of the set of thermomajorized states (boundary of the light-gray region in Figure 5), and not all of these states can be obtained with a finite heat bath.
Note that our results do not prove that zero temperature can never be attained with any finite heat bath. In the geometric picture of Figure 5, there are states on the boundary which can be obtained exactly (for example, the initial state itself). Thus, we here only consider a very special unattainability result. On the other hand, our formulation suggests a generalization of the third law beyond cooling to zero temperature. For example, in the scenario of Figure 1, we can also look at initial states which are only partially excited, and obtain a potential target temperature T 0 > 0 that can only be achieved approximately, but never exactly, by interactions with any finite heat bath.
Our paper is organized as follows. In Section II, we provide the mathematical framework and notation. We prove our main results for noisy operations in Section III, and for thermal operations in Section IV. In the thermal case, we first consider the case of a twolevel system in Subsection IV.1, and then the general case in Subsection IV.2, before discussing the physical intuition and comparison to other approaches in more detail in the Conclusions, Section V.

II Mathematical preliminaries
We will assume that all quantum systems (labelled by uppercase letters like A) are finite-dimensional, and carry a Hamiltonian, i.e. a self-adjoint operator H A = H † A . In the context of noisy operations (in particular Section III), we assume that all Hamiltonians are zero. We also assume that every quantum system comes with a choice of orthonormal basis, in such a way that its Hamiltonian is diagonal in that basis.
A classical state is a probability vector p = (p 1 . . . , p n ) ∈ R n , with p i ≥ 0, i p i = 1. We will treat probability vectors as column vectors in matrix calculations, but for notational simplicity, we denote them by row vectors. We identify classical states p ∈ R n with the corresponding quantum states (diagonal density matrices)p on C n , where Given two classical states p, q ∈ R n , we say that p majorizes q [30], and write p q, if and only if where p ↓ = (p ↓ 1 , . . . , p ↓ n ) denotes the entries of p in nonincreasing order, i.e p ↓ 1 ≥ p ↓ 2 ≥ . . . ≥ p ↓ n . Furthermore, we write ρ σ for quantum states ρ, σ if and only if spec(ρ) spec(σ), where spec(ρ) denotes the vector of eigenvalues of ρ. According to the Hardy-Littlewood-Polýa theorem [30], p q if and only if there exists a bistochastic matrix D ∈ R n×n , i.e. a matrix with D ij ≥ 0, i D ij = j D ij = 1, such that q = Dp. The following lemma is easily checked by direct calculation, but will be very useful in the following. It has also been observed in [38]. [38]). Let p ∈ R n be a classical state, and let U ∈ U(n) be a unitary. Then the vector of diagonal elements of UpU † can be written

Lemma 1 (Ruch and Mead
where D is a bistochastic matrix with entries D ij = |U ij | 2 , i.e. D = U • U * for the Hadamard [29] product Given any quantum system A, a noisy operation is a map on the quantum states of A which can be written in the form (1), for some suitable choice of quantum system B of dimension dim B = m and unitary transformation U on AB. (We will not consider more general noisy or thermal operations which map states on one Hilbert space to states on another Hilbert space in this paper.) The known relation between majorization and noisy operations is as follows.
In the case of non-trivial Hamiltonians, all these notions are generalized. We always assume that there is a fixed background inverse temperature β = 1/(k B T ), with k B the Boltzmann constant. The classical Gibbs state of a system B with Hamiltonian H B is γ B = (e −βE1 , . . . , e −βEn )/Z, where E 1 , . . . , E n are the eigenvalues (and thus diagonal elements) of H B , and Z = n i=1 e −βEi is the partition function. The corresponding quantum Gibbs state isγ B = exp(−βH B )/Z. We will sometimes write γ B for γ B if we need space for indices.
A thermal operation on a quantum system A is a map of the form Thermal operations are closely related to a generalized notion of majorization, named thermomajorization (this also corresponds to "majorization relative to d" for d = γ in [30]). For any two classical states p, p on a system A with Gibbs state γ = γ A , we write p γ p and say that "p thermomajorizes p " if and only if there exists a stochastic γ-preserving matrix D such that p = Dp. Stochasticity of D means that D maps states to states, i.e. D ij ≥ 0 and i D ij = 1 for all j, and γpreservation is the property that Dγ = γ.
The following was shown (in different notation) by Janzing et al. [1], see also [39,40]. Lemma 3 (Janzing et al. [1] In the following, we are only interested in transitions between classical states, i.e. instances of (3) where input and output state are both diagonal in the canonical (energy eigen)basis. Note that this does not mean that we restrict to operations Φ that map all diagonal input states to diagonal output states.
We are thus not considering the role of quantum coherence in this paper, which is currently an active area of research [21,24,23]. However, we are going beyond the earlier quasiclassical setups by explicitly considering unitaries U in (3) which are not just classical permutations of basis vectors.

III Noisy operations with small auxiliary systems
As mentioned in the introduction, in most of the literature on single-shot quantum thermodynamics (e.g. [2,5,6,11,32]), noisy operations are constructed by implementing a classical permutation in a high-dimensional degenerate eigenspace of system and bath. To approximate target states to arbitrary accuracy, this needs in general arbitrarily large auxiliary systems. (We also use the word "bath" for these auxiliary systems for brevity, even though they do not carry a non-trivial Hamiltonian and are thus not heat baths in a strict sense.) We will now show that this is not necessary if we consider more general unitaries. In this case, all transitions can be implemented exactly with a bath that is of the same dimension as the system.

Lemma 4.
If p, p ∈ R n are states such that p p , then the transition p → p can be accomplished by a noisy operation with an auxiliary system of size n. That is, there is a unitary U on AB, A = B = C n , such that Proof. We will apply the Schur-Horn Theorem [33], which tells us that there is a unitary V on A such that VpV † has diagonal elements p 1 , . . . , p n . There- It remains to show that Φ can be implemented as a noisy operation with a bath of size n. To this end, choose an arbitrary orthonormal basis |1 , . . . , |n on B = C n , and let π be the cyclic permutation with π|i = |i + 1 if i < n and π|n = |1 . Define the unitary W by Then, if ρ = Tr B W ρ ⊗ 1 n W † , we have i|ρ |j = 1 n i|ρ|j tr(π i−j ), which is zero if i = j and i|ρ|j otherwise. In other words, the unitary W implements Φ as a noisy operation, and we can set U : This result confirms a conjecture by Bengtsson anḋ Zyczkowski [36], who however use slightly different terminology. They call a noisy operation on a system of size n, mediated by a bath of size n, "unistochastic", regarded as a map between quantum states (note that this conflicts with our use of this term in Lemma 13 in the appendix). For two quantum states ρ and σ, write ρ σ if and only if the spectrum of ρ majorizes the spectrum of σ. Then, in the terminology of [35,36,37]), Lemma 4 implies "Quantum Horn's Lemma": If ρ σ, then there exists a unistochastic quantum operation that maps ρ to σ exactly. Clearly, if p and p denote the probability vectors of eigenvalues of ρ and σ, then there is a unitary U such that (4) holds true, with a bath of size n. Since there are unitaries V and W such thatp = V ρV † andp = W σW † , we can set U := (W † ⊗ 1)U (V ⊗ 1), and then U will generate a "unistochastic quantum operation" that maps ρ to σ.
It is interesting to see that Lemma 4 above is closely related to (and, in fact, has an alternative proof in terms of) a variant of the quantum marginal problem as formulated, for example, in [46]: given the spectrum of some bipartite quantum state ρ AB , then what are the possible spectra of the local reduced state ρ A ?
The following lemma gives a complete answer to this question in the case dim A ≤ dim B. In a nutshell, it says that the possible spectra of ρ A are those that are majorized by the partial trace of the (non-increasingly ordered) spectrum of ρ AB .

Lemma 5. Let ρ be a quantum state on AB, and σ a quantum state on
where λ(ρ) is the vector of eigenvalues of ρ, ordered in non-increasing order, andλ(ρ) is the corresponding diagonal density matrix.
This lemma was conjectured in [46] in 2005, where it was also shown that (5) is necessary for the existence of a suitable unitary U which satisfies the statement of the lemma. After Lemma 5 was proven by the first author in his Master thesis [47], it turned out (cf. Acknowledgments) that the statement had just recently been proven independently in [48]. The lemma can also be inferred from the results in [49]. We give our proof in the appendix for completeness.
Lemma 4 follows as a corollary from Lemma 5 by setting dim A = dim B = n, ρ = ρ A ⊗ 1/n, σ = ρ A , and noting that where λ 1 ≥ λ 2 ≥ . . . are the eigenvalues of ρ A . The partial trace corresponds to summing over n-blocks, thus These are the eigenvalues of ρ A , which must therefore majorize the eigenvalues of ρ A . Given that all possible transitions on a system A of size n can be achieved with a bath B of size n, it is natural to ask whether one can more generally classify the set of all maps on the classical probability distributions that can be implemented in this setting. We give a partial answer in Lemma 13 in the appendix, where we show that the corresponding set of maps lies somewhere in between the unistochastic and the bistochastic maps.
The relation to the quantum marginal problem allows us to explore what happens in the case that the heat bath dimension m = dim B is smaller than the system dimension n = dim A. The case m = 2, n = 3 has been considered in [46] in Sec. 7.3, where the authors have worked out the full set of inequalities relating the eigenvalues of ρ AB to the eigenvalues of possible marginals ρ A . Their result implies the following: Lemma 6 (Case n = 3, m = 2). If p, p ∈ R 3 are states such that p p , then the transition p → p can be accomplished by a noisy operation with an auxiliary That is, in this case, even a bath that is smaller than the system is sufficient to implement all noisy state transitions.
We do not currently know the smallest possible value of m that would allow all noisy state transitions on a system of dimension n. However, we have the following elementary bound:

Lemma 7. If all noisy state transitions on a system of size n can be implemented with a bath of size m, then
Proof. If all noisy state transitions can be implemented, then in particular a pure state can be mapped to a maximally mixed state. That is, there is a pure state |ψ on A = C n and a unitary U on AB such that where {|i m i=1 is any orthonormal basis of B = C m . The Schmidt rank of the pure state U |ψ A |i B is upperbounded by m, and so the rank of every density operator Tr B [. . .] on the right-hand side is upper-bounded by m. Due to the subadditivity of the rank, this means that the rank of the right-hand side is less than or equal to m 2 . Hence rank(1/n) = n ≤ m 2 .
We do not know whether this bound is tight. If it was, then this would say that any noisy operation on k qubits can be achieved with a bath of k/2 qubits only (rounding up accordingly). This would be a significant tightening of quantum Horn's lemma.

IV Thermal operations with small heat baths
We now turn to the more general case of thermal operations. We will start by analyzing two-level systems in Subsection IV.1, which illustrates most of the features of arbitrary-dimensional systems that will be proven in Subsection IV.2.

IV.1 The qubit case with a finite heat bath
Consider a two-dimensional Hilbert space with orthonormal basis {|1 , |2 } and Hamiltonian and assume that ∆E > 0. The corresponding Gibbs state at inverse temperature β > 0 is given bŷ Now we would like to see which (classical) states p are thermomajorized by a given state p. According to Lemma 3, we need a stochastic matrix D which satisfies In two dimensions the stochastic matrices that preserve γ are parametrized by a single parameter α: The set of states thermomajorized by p is thus also parametrized by α, and we can use the notation p α := D α p. Three special cases are as follows: • α = 0 gives D α = 1 and p α = p.
, which maps all states p to the thermal state p α = γ.
The set of states thermomajorized by p, including these special cases, is shown in Figure 2. In the case ∆E = 0, the D α are bistochastic, and we recover the case of standard majorization from Section III. In order to implement the D α as thermal operations, it is necessary to choose H B in a way that H A + H B exhibits some degeneracies, since the unitaries in (3) have to commute with the total Hamiltonian. So the spectrum of H B has to contain the energy gap ∆E, and furthermore it seems plausible that carrying the energy gap between many different energy levels should allow for a greater variety of transitions. Let us therefore consider a truncated harmonic oscillator that is, an m-level system that has ∆E as energy difference between all neighbouring eigenstates. In the case m = 2, we have one copy of the original system, and for m → ∞ we obtain a harmonic oscillator. The joint Hamiltonian of system and bath All energy levels of H AB except for the lowest and highest one are doubly degenerate. As explained above, this degeneracy is necessary to implement non-trivial thermal operations. In the case m = 2, we can implement one of the special cases above (apart from the trivial case), namely D α for α = γ 2 . This is achieved by the energy-preserving permutation π with π |12 = |21 and π |21 = |12 which swaps the two systems, leading top ⊗γ →γ ⊗p. Tracing out the second system, we end up with the Gibbs state of the qubit, independent of the state we started with.
By implementing other unitaries U 2×2 on the subspace span{|12 , |21 }, we can in fact implement all maps D α with 0 ≤ α ≤ γ 2 , and thus obtain all corresponding target state p α . This follows from the simple observation that SU (2) is connected, and thus by varying over all U 2×2 ∈ SU(2), we obtain target states p that vary continuously between p = p and p = γ.
Can we obtain more target states than this? To answer this in the negative, consider the parametrization with κ, λ ∈ C such that |κ| 2 + |λ| 2 = 1. Implementing this unitary on the degenerate subspace, we obtain a target state p (U 2×2 ). A simple calculation shows that where π is the swap introduced above. This shows that the extremal cases are generated by classical permutations (swap or identity on the degenerate subspace) and choosing general energy-preserving unitaries give us all states that are in the convex hull or 'in between' these two states, but not more. Now we would like to see which states we can achieve for m > 2. Consider the permutation which swaps the basis vectors of all two-dimensional energy eigenspaces of H AB . Intuitively, π max is the classical permutation for which we expect to obtain the D α with the largest possible value of α (for this specific heat bath). Calculating the thermal operation for p = (1, 0) and reading off 1 − α from p = (1 − α, α) shows that π max yields which is indeed the minimal value for 1 − α that is achievable by choosing any energy-preserving permutation. This becomes clear from the the fact that all energy-preserving permutations can be written as π J , where J ⊂ {2, . . . , m}, and That is, J indicates in which subspaces we choose to swap the two basis vectors, and π max = π {2,...,m} . Choosing some permutation π J leads a corresponding value α of 1 − α = j∈{1,...,m}\J γ B j which is indeed minimal for J = {2, . . . , m}. We thus define which is the maximal value of the parameter α that we can achieve with this kind of heat bath of dimension m, even if we choose not just a permutation but a more general unitary for the thermal operation. This can be seen similarly as in the case m = 2. The energy spectrum of H AB is 0, ∆E, . . . , m∆E, and all energies except for the highest and lowest are two-fold degenerate. Thus, all energy-preserving unitaries consist of (m − 1) independently chosen unitaries from SU(2) acting in the respective energy eigenspaces. For every single one of these eigenspaces, choosing a general unitary instead of a swap or the identity (as in π J ) will give a state that is a mixture of the two permutation cases, as above. Thus we can conclude that all target states p obtainable by energy preserving unitaries are in the convex hull of those that we obtain by energy preserving classical permutations.
So the set of states p that we can obtain with a truncated harmonic oscillator as a heat bath is This means that the set of achievable states p is growing with increasing m, but never exhausts the set of all thermomajorized states: p * , corresponding to α = e −β∆E , can never be achieved exactly for finite m. However, we can get arbitrarily close to p * by increasing the dimension m since lim m→∞ α (m) max = e −β∆E . In particular, all states p = p * which are thermomajorized by p can be obtained exactly with a finite heat bath of suitable size. This is also sketched in Figure 3.
(1, 0) (0, 1) What if we choose any other (finite) heat bath Hamiltonian H B which is not of the truncated harmonic oscillator form -could it be that we achieve the state p * exactly? The following result answers this question in the negative, and gives a bound on how close we can get to p * . It also says that truncated harmonic oscillators are in a certain sense optimal.

Lemma 8. Let
A be a qubit system with energy gap ∆E > 0, and B a finite-dimensional heat bath with arbitrary Hamiltonian H B . Then the stochastic maps D α of the form (6) which can be implemented by thermal operations on this system and heat bath satisfy where  (7)) can never be created exactly from p with any finite heat bath. However, all other states that are thermomajorized by p can be obtained exactly with a finite heat bath.
Proof. Denote the orthonormal basis vectors of B by |i and the corresponding energies by E i (there may be i = j with E i = E j ). Furthermore, denote the set of "matching" energies of H B by M B , where an energy E is matching if there is some i such that E i = E, and if there is some j such that E j = E − ∆E. That is, the matching energies E ∈ M B are those energy values which have a suitable energy gap downwards in order to allow for thermal operation in conjunction with the qubit on A.
To every E ∈ M B , there is a corresponding global energy eigenspace Let P E be the orthogonal projector onto S E , and P c := 1 − E∈M B P E the projector onto the orthogonal complement S c of the span of all these (matching) energy eigenspaces.
Any energy-preserving unitary must be a direct sum of unitaries U E acting on S E and U c acting on S c . Define (p ⊗γ) E := P E (p ⊗γ)P E and similarly (p⊗γ) c . Any thermal operation D α acting on a probability vector p which can be implemented with the given heat bath can thus be written In particular, since thermal operations do not generate off-diagonal elements [5], the right-hand side will yield a diagonal density matrix on A. Due to (6), we can determine the corresponding parameter α by setting p = (1, 0) and computing α = 2| (D α (p)) ∧ |2 . Since with γ B i := exp(−βE i )/Z and Z = tr e −βH B , the energy-preserving unitary U c cannot map any |1i in this sum to any state of non-zero overlap with |2j for any j. Thus, Tr B U c (p ⊗γ) c U † c = |1 1|, and so Lemma 8 has a compelling physical interpretation in terms of the third law of thermodynamics. Think of the task to cool a physical system. In general, this needs two ingredients: first, a heat bath, and second, a work reservoir. We can model both by considering a qubit system in a state p = (0, 1), that is, an excited eigenstate of energy ∆E. Since p thermomajorizes all other classical qubit states p , it is clear that this state contains enough free energy to allow a transition to any other state p to arbitrary accuracy by coupling it to a suitable heat bath.
Suppose we would like to obtain another state of inverse temperature β = 1/(k B T ). The third law of thermodynamics tells us that we cannot achieve temperature T = 0 in finite time. More in detail, Nernst's formulation [16], first stated in 1912 [17], can be phrased as follows [18,19]: It is impossible by any procedure, no matter how idealized, to reduce any assembly to the absolute zero in a finite number of operations. The thermal state of temperature zero is the ground state p = (1, 0) which is nothing but the state p * from (7) (for our choice of initial state p). And, in fact, Lemma 8 says that we cannot obtain p = p * exactly with any finite heat bath. But we can say more.

Theorem 9 (Third law of thermodynamics for qubits).
Let A be a qubit system with energy gap ∆E which is in its excited energy eigenstate, and B an n-dimensional heat bath with smallest and largest energy eigenvalues E B min and E B max , which is in a thermal state at temperature T . If we apply any thermal operation that yields a new qubit state p with non-negative temperature T ≥ 0 (i.e. without population inversion), then is the free energy of the heat bath.
Proof. If we apply a map D α of the form (6) to the excited state p = (0, 1), then D α p = p with p as in (11) yields and the argument inside the log is strictly positive due to α < e −β∆E . Furthermore, β ∈ [0, +∞] implies that α ≥ 1 2 e −β∆E . From eq. (10), it follows in particular that α ≤ e −β∆E 1 − γ B max (since g B max ≥ 1), and so where Z B = tr(exp(−βH B )), with β = 1/(k B T ) the inverse temperature. Substituting the definition of F B proves the first claimed inequality. The second one follows from the additional estimate i denotes the energy eigenvalues of H B . In the special case of the truncated harmonic oscillator, with H B as in (8) and thus E B max = (m − 1)∆E, we get for large m (to first order in 1/m). For weakly interacting multi-particle baths, the denominators in the lower bounds for T in Theorem 9 will be extensive, that is, proportional to the particle number N . In physically realistic scenarios, only a polynomial number N = poly(t) of particles can interact with the qubit system within a finite time interval [0, t]. Supplemented by these plausible physical assumptions, Theorem 9 tells us that where t is the time in which the given thermodynamical protocol is supposed to be completed, and r > 0 is some model-dependent exponent. This result is compatible with the findings in [16], where the authors specify the exponent r in more detail, dependent on the spectrum of the heat bath. For example, they argue that r = 7 for a radiation bath in three spatial dimensions, which is in some sense the largest physically plausible r. However, the approach of [16] is somewhat different from the one in this paper. We will give a more detailed comparison, and some more physical intuition on our result, in the conclusions. It is also interesting to see that the specific form of our lower bound in Theorem 9 is very close to the temperature obtained via a cooling method described in [16]. It relies on a different scenario that has been explored, for example, in [34], which is a variation of a traditional bit reset protocol. Given any two-level system with energy gap ∆E, raise the energy of the higher level isothermally while in contact with a heat bath of temperature T . Thermalization will bring the two-level system closer to its ground state. Traditionally, one would raise the energy level up to infinity, but suppose we stop at some E max , and subsequently reduce the higher energy level to its initial value. The resulting temperature of the qubit turns out to be (T ∆E)/(∆E + E max ), which is comparable to the bound in Theorem 9 (and in particular (12)).

IV.2 General systems with a finite heat bath
In the previous subsection, we have seen by example of a qubit that the classical target states which can be obtained with a fixed finite heat bath (in particular, a truncated harmonic oscillator) are the convex hull of those that can be achieved by classical permutation of energy levels on the same heat bath. We will now prove that this is true in general, up to some minor amendments.
To this end, let us introduce some notation. For any given initial classical state p (with corresponding diagonal quantum statep) and Hamiltonians H A , H B , we denote by T Q (p, H A , H B ) (p, H A , H B ), we denote the set of final states p which can be obtained by a suitable choice of energy-preserving permutation π on AB such thatp = Tr B π(p ⊗γ B )π † (14) (which will always result in diagonal states). In other words, T C (p, H A , H B ) denotes the set of states that can be created from p classically (which is a finite set), whereas T Q (p, H A , H B ) denotes those states that can be obtained quantum-mechanically. (These sets also depend on the inverse temperature β, which we regard as fixed in this section.) We have the following result.
Theorem 10. We have If furthermore H A is non-degenerate, then we have

, and d i denotes the dimensions of the (energy) eigenspaces of H A .
An illustration of Theorem 10 (the non-degenerate case) is given in Figure 4. Note that the dimension of C could be chosen even smaller if it turned out that a generalization of quantum Horn's lemma along the lines of Lemma 6 was possible in general (for example, if Lemma 7 turned out to be tight).
Proof. Let p ∈ T Q (p, H A , H B ), and let U be a unitary such that (13) where Ω i are matrices with all diagonal elements zero, and D i := U i • U * i are bistochastic matrices. Let Ω := i Ω i , then all diagonal elements of Ω are zero, i.e. jk|Ω|jk = 0 for all j, k, and thus j| Tr B Ω|j = 0 for all j. Every D i is a bistochastic matrix acting on S i , and thus, according to Birkhoff's Theorem [29], a convex combination of permutations π (j) i acting on S i . Therefore, according to Lemma 14 in the appendix, D := i D i is a convex combinations of direct sums of permutations π i , which is an energy-preserving permutation due to the block matrix form. We have and sincep is diagonal, this leads tô which is a convex combination of outputs of maps of the form (14). Thus p ∈ conv T C (p, H A , H B ), which proves (15). Now we would like to show the second half of the theorem, that every p ∈ conv T C (p, H A , H B ) can be obtained quantum-mechanically. So suppose there are energy-preserving permutations π (k) and coefficients µ k such that (18) holds. Energy preservation implies that act on the global energy eigenspaces S i . Let us definê which givesp = Tr Bλ . As before, we can decompose λ = i λ i and λ = i λ i into blocks corresponding to the global energy eigenspaces, and it follows that i λ i , and so λ i λ i , i.e. λ i majorizes λ i by a general result on majorization [30]. Thus, according to [30,Thm. B.6], there are unistochastic maps D i such that λ i = D i λ i (note that D i need not be equal to i ), and therefore unitary matrices U i such that i . We set U := i U i and by Lemma 1, as in (17), we will have where Ω = Ω i is a matrix with all diagonal elements zero. It follows that For thermal operations, we know that Tr B UλU † must be block-diagonal [5] in the energy eigenbasis due to energy conservation. Thus, if H A is non-degenerate, it follows that this matrix is diagonal in the unique energy eigenbasis, and we havê p = Tr B U (p ⊗γ)U † , which proves (16). However, if H A is degenerate, we do not necessarily obtain a diagonal matrix. We deal with this by introducing an auxiliary system used to implement a decoherence map that sets the off-diagonal elements of ρ A := Tr B (UλU † ) to zero. Since ρ A is blockdiagonal in the energy eigenbasis, with block sizes (at most) d i , Lemma 11 below shows that we can implement the decoherence map by adding a fully degenerate auxiliary system of dimension 1+ i (d i −1) and performing a suitable thermal operation.
Note that this theorem contains Lemma 4 from Section III as a special case: in the case of a trivial n × n Hamiltonian H A = 0, we have a single degeneracy d 1 = n. Due to Birkhoff's Theorem [29], all bistochastic maps can be implemented as convex combinations of classical permutations on system A alone. Therefore, denoting a trivial heat bath B of dimension dim B = 1 (and arbitrary Hamiltonian H B ∈ R) simply by a bullet, we have Then Theorem 10 tells us that we can implement all these transitions exactly by noisy quantum operations with a heat bath C of size The following lemma has been used in the proof of Theorem 10.

Lemma 11 (Thermal decoherence). Let
In other words: small heat baths allow to decohere block-diagonal quantum states. If H A is nondegenerate, i.e. all d i = 1, then ρ A is already diagonal and we do not need any heat bath at all, i.e. dim B = 1.
Proof. We construct a thermal operation Φ on A which sets suitable matrix elements of the input density matrix ρ to zero. Suppose we are given any subset S ⊂ {1, 2, . . . , n}, then we will construct Φ such that it sets all elements in rows and columns appearing in S to zero (except for the diagonal elements) and leaves all other matrix elements invariant. For example, suppose that n = 4, and that S = {2, 4}. Then we will construct a thermal operation Φ such that    ρ 11 . . . ρ 14 . . . . . .
Clearly, if ρ is already block-diagonal with block sizes d i (e.g. in this case d 1 = 2 and d 2 = 2), then this map fully decoheres ρ, as long as we decohere (d i − 1) rows and columns in every i-th block. In other words, we will choose the subset S such that and if Φ acts as stated, we achieve the decoherence as claimed in the statement of the lemma. Let B = C d , d = |S| + 1, and |1 , . . . , |d some orthonormal basis on B. Define the permutation matrix π by π|i = |i + 1 if 1 ≤ i ≤ d − 1 and π|d = |1 . We have π T = π † = π −1 , and, for j ∈ Z, Label the elements of S (in increasing order) by S = {s 1 , s 2 , . . . , s d−1 }, and define the unitary U on AB by Clearly [U, H A ⊗ 1] = 0, so if we regard B as a quantum system with trivial Hamiltonian H B = 0, the following map Φ : ρ → ρ is a thermal operation: It is easy to check by direct calculation that where ρ ij = i|ρ|j , and similary for ρ ij . For example, if i ∈ S and j ∈ S, then ik|U = ik|, and where m is such that s m = j. Thus, we get All other cases of i and j can be checked similarly.
Theorem 10 implies in particular the following result, which is illustrated in Figure 5. Proof. We use Lemma 3 which shows that all states p ∈ T (p, H A ) can be obtained to arbitrary accuracy by classical thermal operations with suitable heat baths. Formally, if we denote the set of all states that can be  T (p, HA), is a compact convex set (light gray). If HA is non-degenerate, then the set of states TQ(p, HA, HB) that one can obtain with a fixed, finite heat bath, by quantum thermal operations, is a compact convex subset (dark gray, second row). With increasing resources (for example, by having several copies of the heat bath), this subset will grow (third column), and finally cover every point in the relative interior of T (p, HA). However, some points on the boundary will never be covered (we have proven this for the qubit in Lemma 8, and conjecture that it is also true for higher-level systems). This can be interpreted as an instance of the third law of thermodynamics. On the other hand, states in the relative boundary cannot always be obtained exactly. This is shown by example in Lemma 8.

V Conclusions
The study of quantum operations with finite heat baths reveals some interesting mathematical structure and physical insights. In this paper, we have analyzed basic properties of noisy and thermal operations with finite heat baths. In order to study whether some state transitions are unattainable with finite baths (resembling the third law of thermodynamics), we first had to remove an unphysical apparent"unattainability" that is simply a mathematical artefact of a semiclassical treatment: the fact that infinite auxiliary systems are needed for some simple state transitions if one restricts to classical permutations in the energy eigenbasis.
We did this by proving Corollary 12, which is also sketched in Figure 5: the target states that can be obtained from a given (classical) initial state with unbounded heat baths is a compact convex set. All the states in its interior can be obtained exactly via thermal quantum operations with finite heat baths. It is then (some of) the states in the boundary of this set that become the subject of potential unattainability results.
In the case of a two-level quantum system, this boundary consists of two states (cf. Figure 3): the initial state p itself, and another state p * that is in some sense "thermodynamically antipodal" to p. In fact, we have shown that p * can never be obtained exactly from p with any finite heat bath. In the case of a pure excited initial state, this led us to an instance of the third law of thermodynamics (Theorem 9), giving a (physically meaningful) lower bound on the temperature to which the system can be cooled by using any finite-dimensional heat bath. While we have formally proven this result only for qubits, it seems plausible that an analogous statement holds for higher-dimensional quantum systems, too. This gives an intuitive structural picture of unattainability in terms of the geometry of the state space (cf. Figure 5), and it suggests a generalization of the third law to situations beyond cooling to absolute zero (since p * is not in general the ground state).
We can elucidate the relation to other recent unattainability results by elaborating on the physical intuition of our result. Consider cooling with thermal operations, as in the scenario of Figure 1. Denoting the ground state of the qubit by |1 and the excited state by |2 , we would like to apply an energy-preserving unitary to the global initial state |2 2| ⊗γ B in a way that moves as much probability weight as possible (in fact, all of it) into the local ground state, i.e. the subspace spanned by the global eigenstates |1i . However, if |m denotes the maximal energy level of the bath (assuming for the moment that it is unique), then the (globally maximal) energy level |2m is "trapped": energy preservation forbids to move its probability weight into any other subspace. And since the bath starts in a thermal state, this "trapped" population will be non-zero, and it will not be transferred into the ground state. We expect that this simple intuition will remain robust also in more complicated cases beyond the simple qubit scenario, for example in cases where the quantum system A is itself composite, A = A 1 A 2 . This encompasses in particular situations in which A 1 is the system of interest (say, a quantum system to be cooled), while A 2 is a work storage system, such as a weight [54] or work bit [5].
This "inability to move probability weight around" in high-lying energy levels of the heat bath is also a major ingredient of the version of the third law proven in [16]. In our setting, however, this inability comes from an assumption of finite-dimensionality of the bath, while in [16], it is the bath's spectral density together with the assumption of a strict upper bound on the available work w max that limits the available energy levels. Our approach should not be understood as claiming that physical heat baths are generically finite-dimensional. Instead, the motivation is to give a very general formulation of the idea that we can only engineer interactions with a finite number of energy levels of the bath in a controlled way within a thermodynamic protocol. If understood in this way, the approach of [16] is conceptually related to ours (but [16] goes into much more physical detail).
Our unattainability statement (and the one in [16]) is also different from the scenario studied by Janzing et al. in [1], where they consider the amount of non-thermal resource states needed to cool a system (close) to absolute zero. In contrast, our formulation assumes that all non-thermal resources are in principle there (e.g. the energy of the excited state in Figure 1 is enough for cooling to zero), but they can only be harvested by thermodynamic protocols in the presence of an unbounded heat bath.
Several further recent publications have considered limitations for cooling that arise from finite heat baths, cf. [50] and [51] (see also [52] and [53]), but within approaches that are not formulated within the resourcetheoretic framework of thermal operations. Instead, arbitrary (in general energy-non-preserving) unitaries are allowed, and the energy cost or yield is taken into account in retrospect, by bookkeeping the system's average energy content. That is, the fundamental microscopic reversibility of the physical laws is taken into account directly by restricting to unitary transformations, but the principle of total energy conservation is only considered indirectly.
In the context of the third law, this leads to a subtle restriction that comes from the resulting focus on entropic properties. Typically, in approaches that allow arbitrary unitaries, the impossibility of cooling to absolute zero is ultimately attributed to a simple property of linear algebra: namely, that unitary operations cannot create zero eigenvalues from non-zero eigenvalues. In contrast, our setup yields more severe restrictions that rely on microscopic reversibility in conjunction with total energy preservation. The simplest example is the one depicted in Figure 1: even though we start with a pure state of the system, the pure ground state is still unattainable.
Our paper leaves several interesting open questions. First, we have shown by example in Lemma 6 that quantum Horn's lemma is true (Lemma 4), but not optimal: for example, all noisy state transitions on a threedimensional quantum system can be accomplished by means of a two-dimensional auxiliary system. What is the smallest possible bath dimension in general? Is the bound in Lemma 7 tight? If so, then this would also improve our more general result for non-trivial Hamiltonians, Theorem 10. As we have shown, this question is closely related to a version of the quantum marginal problem.
From the point of view of the physical interpretation, the most interesting open problem is to analyze in more detail whether, and if so in what way, our third law for qubits (Theorem 9) generalizes to higher-dimensional quantum systems and scenarios with a work storage device in a physically meaningful way. Restricting the dimension of the heat bath, as we have done here, is only one of several possibilities to study unattainability results in thermodynamics. But the picture of some states, not necessarily of temperature zero, that lie "on the edge" of the set of attainable states (as in Figure 5) is a quite suggestive hint towards possible generalizations beyond perfect cooling. Lemma 13. Let Uni(n) and Bi(n) be the sets of unistochastic resp. bistochastic maps on R n , i.e. Uni(n) := D ∈ R n×n | ∃U : Moreover, let Noisy(n, n) be the set of maps p → p on R n which can be written in the form with some unitary U on C n ⊗ C n . Then we have Uni(2) = Noisy(2, 2) = Bi (2), Moreover, the inclusion Noisy(n, n) ⊆ Bi(n) is strict if and only if Noisy(n, n) is not convex.
Remark. We do not currently know whether Noisy(n, n) is convex for n ≥ 3, which would imply that Noisy(n, n) = Bi(n).
Proof. If D ∈ Uni(n) then there is a unitary matrix V such that D ij = |V ij | 2 . Thus, for any state p ∈ R n , we have in other words, VpV † has the vector Dp on its diagonal. We can compose V with the unitary W from the proof of Lemma 4 which implements the decoherence map to remove all off-diagonal elements with an n-dimensional bath. Then U = W (V ⊗ 1 B ) will realize the map D as a noisy operation. This proves that Uni(n) ⊆ Noisy(n, n). On the other hand, for any given operation in Noisy(n, n) with associated unitary U , define then it is straightforward to check that i D ik = k D ik = 1 and D ik ≥ 0. Moreover, ifp is the output of the noisy operation on inputp, then p = Dp. This shows that Noisy(n, n) ⊆ Bi(n). For n = 2, we have Uni(2) = Bi (2), which proves the first statement of the lemma. For n ≥ 3, we construct a matrix D ∈ Bi(n) \ Uni(n) which can be implemented by a noisy operation. Let π be the cyclic permutation on C n from the proof of Lemma 4, and define Obviously D is bistochastic. To see that it is not unistochastic, suppose U is a unitary matrix with D ij = |U ij | 2 , then orthogonality of the first two rows of U is n j=1 U * 1j U 2j = 0. But for every j = 1, either D 1j = 0 or D 2j = 0, hence this sum equals U * 11 U 21 = e iθ √ D 11 D 21 = e iθ 1 − 1 n 1 n for some θ ∈ R, and this is not zero. Define the unitary A simple calculation shows that this unitary generates a noisy operation which mapsp to (Dp) ∧ . This shows that D ∈ Noisy(n, n), but D ∈ Uni(n).
Finally, since Bi(n) is convex, non-convexity of Noisy(n, n) would imply that Noisy(n, n) Bi(n). Conversely, suppose that this strict inclusion was true (note that we do not know whether this is the case). Since every permutation is unistochastic, and the bistochastic maps are the convex hull of the permutations due to Birkhoff's Theorem [29], Noisy(n, n) could then not be convex, because otherwise the permutations that it contains would generate all bistochastic maps.
We conjecture that Noisy(n, n) is not convex (and thus a strict subset of the bistochastic maps), but we were not able to show this. This contrasts the simple behavior of the set of transitions that can be realized with an n-dimensional bath, characterized by majorization.
The following lemma has been used in the proof of Theorem 10.
Proof. The case n = 1 is trivial, and the cases n ≥ 3 follow from repeated application of the statement for n = 2, so it is sufficient to prove the lemma for n = 2. Suppose that Then Note that this statement can easily be illustrated by writing down the block matrices concretely.
The following lemma concerns the relative interior of a convex set [31]. It has been used in the proof of Corollary 12.

Lemma 15.
Let C ⊂ R n be convex, and X ⊂ C a dense subset of C. Then relint C ⊆ conv X.
Proof. Denote the ε-ball around some point c ∈ R n by B ε (c) = {x ∈ R n | x − c ≤ ε}. Let c ∈ relint C, then there is δ > 0 small enough such that B δ (c) ∩ aff(C) ⊂ C. Since B δ (c) ∩ aff(C) is an m-dimensional Euclidean unit ball, with m = dim C, there is a regular m-simplex with vertices v 1 , . . . , v m+1 on the surface of that ball, i.e. v i − c = δ, and center c = 1 m+1 m+1 i=1 v i . Since X is dense in C, we can find v 1 , . . . , v m+1 ∈ X that are arbitrarily close to the vertices v 1 , . . . , v m+1 . If they are close enough, it is geometrically clear that their convex hull must contain c. Figure 6: Illustration of the proof of Lemma 15. As mentioned in the main text, Lemma 5 has been proven independently in [48], and it also follows from the results in [49]. Nevertheless, we give a self-contained proof for completeness.
where λ(ρ) is the vector of eigenvalues of ρ, ordered in non-increasing order, andλ(ρ) is the corresponding diagonal density matrix.
Proof. Suppose there is some unitary U such that Tr B (U ρU † ) = σ. Let us assume without loss of generality that ρ and σ are diagonal and their diagonal elements are non-increasingly ordered, soλ(ρ) = ρ andλ(σ) = σ.
With Lemma 1 we can write Uλ(ρ) U † = (Dλ(ρ)) ∧ + Ω where D is a stochastic matrix with D ij = |U ij | 2 and Ω some matrix with only zeros on the diagonal. Thus, it holds λ(ρ) Dλ(ρ), and Tr B Ω has also only zeros on its diagonal. Since σ is diagonal, the off-diagonal elements of Tr B Ω must vanish identically, and we get Tr B Ω = 0 and λ(σ) = Tr B (Dλ(ρ)) ∧ . If p, q are probability vectors on a joint system AB and p is non-increasingly ordered, then from p q, it follows that Σ B p Σ B q where Σ B is a the marginal map with respect to the B system which satisfies Tr Bλ = (Σ B λ) ∧ . So we have Tr Bλ (ρ) Tr B (Dλ(ρ)) ∧ =λ(σ) which proves the "only if" direction of the statement (and is independent of the dimensions of A and B).
To prove the converse direction, set n = dim A, m = dim B. Lemma 1 that we have just used is basically the one half of the Schur-Horn Theorem that states that the vector of diagonal elements of a Hermitian matrix is always majorized by the vector its eigenvalues. For the second half of the proof one main ingredient will be the other half of the theorem which states: if λ is the vector of eigenvalues of a Hermitian matrix then for each vector p with λ p there exists a basis in which p is the vector of diagonal elements of this Hermitian matrix.
So assume that Tr Bλ (ρ) σ which means that Σ B λ(ρ) λ(σ). Without loss of generality, we may assume that σ is diagonal (a local diagonalizing unitary can always be implemented via the global unitary U that we are constructing now), σ =λ(σ). By the Schur-Horn theorem we know there exist a unitaryŪ on the A system such thatŪ Tr Bλ (ρ) Ū † has diagonal elements λ(σ). We will use this to define a new unitary on the whole AB system such that Tr B U ρU † = σ is satisfied. So define where u ij are the components ofŪ , and π j−i is a (later to be specified) permutation matrix π on the B system to the power (j − i). Since π † = π −1 andŪ itself is unitary, this is a unitary matrix as well. Now the rest of the proof is merely showing by direct calculation that Tr B Uλ(ρ)U † =λ(σ).
For this we write the diagonal matrixλ(ρ) aŝ and with Lemma 1 we write again Uλ(ρ)U † = (Dλ(ρ)) ∧ + Ω where D is the unistochastic matrix D = U • U * and Ω some matrix with only zeroes on the diagonal. If we definē D =Ū •Ū * it is simple to show by direct calculation that where the second equality sign holds by assumption. So we know that Tr B (Dλ(ρ)) ∧ =λ(σ) and it remains to show that Tr B Ω = 0.
By direct calculation, starting with the form ofλ(ρ) and U as just given, we get the form where Ω (ii) = i| Ω |i is a matrix on the B system with all diagonal elements equal to zero. Since Tr B Ω (ii) = 0 the second term vanishes when calculating Tr B Ω. However we still have to look at the first term where we calculate tr π i−k Λ k π k−j = tr π i−j Λ k and set s = i − j, with s ∈ {−n + 1, . . . , −1, 1, . . . , n − 1} since i = j. Now it is time to specify the permutation π. We choose π such that π(i) = i + 1 for i < m and π(m) = 1. This is a permutation with no fixed point in the set {1, . . . , m} and thus tr(π) = 0 since the permutation matrix has only zeros on the diagonal. This is as well valid for all powers π s of the matrix as long as s = m and s = 0. This is true due to the restriction on s above and since we assumed dim A ≤ dim B i.e. n ≤ m. Now, since π s has no fixed point in {1, . . . , m}, it follows that Tr π s Λ k = m µ=1 λ iµ µ|π s |µ = 0, finally showing that Tr B Ω = 0.