An interesting feature of Thermal Operations, which follows from the fundamental operations1
and 2, is the fact that coherence over the energy eigenbasis cannot be created. Indeed, as we show in the following, these operations are symmetric with respect to the time translations gen- erated by the system Hamiltonian. The fact that Thermal Operations cannot create coherence implies that coherence itself represents an additional resource in thermodynamics. In order to manipulate coherence in this resource theory, the agent needs to have access to an additional system, known as a coherence reservoir [132]. This coherence reservoir is a large system with non-degenerate Hamiltonian, and is described by a state in a coherent superposition. The agent can then exchange coherence between this reservoir and the main system, while not degradating the reservoir, which can therefore be re-used an arbitrary number of times.
We now show that Thermal Operations cannot create coherence. This was first shown in Refs. [133,134]. Let us introduce the notion of a time-translation covariant map [129,135]. Definition 24 (Time-translation covariant map). Consider an Hilbert space H with Hamilto- nian H, and a quantum operation ε : S (H) → S (H). We say that the map ε is time-translation covariant iff
e−iHtε(ρ) e+iHt= ε(e−iHtρ e+iHt), ∀ ρ ∈ S (H) , ∀ t ∈ R. (2.11) where e−iHt is the unitary evolution generated by the Hamiltonian H at time t.
If a map is time-translation covariant, we can apply it before the evolution, and then evolve the state, or vice versa we can evolve the state and then apply the map. In any case, the final state we obtain is the same. An example of a map which is clearly time-translation covariant is the unitary evolution of the state with respect to the system Hamiltonian, ε(·) = e−iHs · e+iHs, for any s ∈ R. This is not the sole map to be time-translation covariant, and we now show that Thermal Operations satisfy Eq. (2.11), see Ref. [133].
Lemma 5. Consider the resource theory of thermodynamics acting on a finite-dimensional system S with Hamiltonian HS. Then, the maps in the set of allowed operations of the theory, that is, Thermal Operations, are time-translation covariant.
Proof. Let us use the definition of time-translation covariant map, given in Eq. (2.11), and the fact that the most general Thermal Operation is of the form given in Eq. (2.6) – here we map state in S (HS) into states in S (HS), for simplicity. For all ρS∈ S (HS), and for all t ∈ R, we have, εT O(e−iHStρSe+iHSt) = TrB h U e−iHStρ Se+iHSt ⊗ τβU† i = TrB h U e−iHStρ
Se+iHSt ⊗ e−iHBtτβe+iHBt U† i = TrB h Ue−i(HS+HB)tρ S⊗ τβe+i(HS+HB)t U†i = TrB h e−i(HS+HB)tU ρ S⊗ τβU†e+i(HS+HB)t i = e−iHStTr B h U ρS⊗ τβU† i e+iHSt= e−iHStε T O(ρS) e+iHSt, (2.12) where the second equality follows from the fact that τβ = e−βHB/Z, and therefore it com- mutes with HB, while the fourth equality follows from the fundamental operation2composing Thermal Operations, which requires the unitary U ∈ B (HS⊗ HB) to commute with the total Hamiltonian HS+ HB.
Using the result of the above lemma, we can now show that Thermal Operations are unable to create coherence in the energy eigenbasis, unless the Hamiltonian is degenerate.
Proposition 4. Consider the resource theory of thermodynamics acting on a finite-dimensional system S with a non-degenerate Hamiltonian HS. The allowed operations of the theory, Thermal Operations, are unable to create coherence in the eigenbasis of HS.
Proof. Consider a state ρS ∈ S (HS) that commutes with the Hamiltonian HS, and therefore is diagonal in the energy eigenbasis (since HS is non-degenerate). If we use the time-translation invariance of Thermal Operations, Lem. 5, we find that for all εT O, and for all t ∈ R,
e−iHStε
T O(ρS) e+iHSt= εT O(e−iHStρSe+iHSt) = εT O(ρS), (2.13) where the second equality follows from the fact that [ρS, HS] = 0. However, Eq. (2.13) needs to be valid for all t ∈ R, which implies that εT O(ρS) commutes with HS, and therefore is diagonal in the energy eigenbasis.
Since coherence is a resource in thermodynamics, efforts have been spent to study how this quantity evolves under Thermal Operations [136], and whether it can be traded for another resource, for example, for work [137].
An additional question naturally arises, namely, how the agent can create a state with non- zero coherence over the energy eigenbasis within the formalism of Thermal Operations. This problem is equivalent to that considered in the last paragraph of Sec. 2.2.3, on the implemen- tation of unitary operations that do not commute with the system’s Hamiltonian. Coherence manipulation with Thermal Operations was first considered in Ref. [132], where it is shown that an additional system, referred to as a coherence reservoir, is needed in order to modify the coherence of the main system. In its simplest form, this ancillary system is infinite-dimensional, with a Hamiltonian which is unbounded both from below and above, and the state describing this system is in a uniform superposition over a large subset of energy eigenstates. In the fol- lowing, we consider the easiest case in which the main system S is a qubit with Hamiltonian HS = E0|0i h0| + E1|1i h1|, with energy gap ∆E = E1− E0, the coherence reservoir C has Hamiltonian HC =P`∈Z` ∆E |`i h`|, and the state describing this system is |Ψi =PL` √1L|`i, where L >> 1, see Fig.2.2.
With the help of this coherence reservoir, the agent can implement a unitary operation over the main system S which creates coherence. Suppose, for instance, that the agent wants to implement an Hadamard UH ∈ B (HS) over the main system S, mapping |0i → |+i and |1i → |−i. This transformation can be realised, using Thermal Operations, by applying the
Figure 2.2: In order to create coherence over the energy eigenbasis of the main system, we need to use a coherence reservoir. This is an ancillary system whose Hamiltonian is a double- infinite ladder – modification to this Hamiltonian can be made so as to obtain a more physical system – described by the state |ΨiC =PL
` 1 √
L|`i, which is in a large (L 1) superposition of its energy eigenstates (represented by the blue ellipse on the left-side ladder). In order to create coherence on the main system, and to map its state from |0iS into |+iS, we can use the energy-preserving unitary operation VH described in Eq. (2.14). The effect of this unitary over the reference frame is to create a superposition between the original state |ΨiC and the same state slightly displaced (the green ellipse on the right-side ladder). Since these two states significantly overlap, the final state is approximately equal to |ΨiC⊗ |+iS.
following global operation over the system S and the coherence reservoir C,
VH = X `∈Z 1 X n,m=0
|ni hn| UH|mi hm|S⊗ |` − (n − m)i h`|C. (2.14)
It is easy to show that this operation is energy preserving, since it commutes with the global Hamiltonian of system and coherence reservoir HS+ HC. Furthermore, when VH is applied to the initial global state |0iS⊗ |ΨiC, we get
VH|0iS⊗ |ΨiC = 1 √ 2|0iS⊗ L X ` 1 √ L|`iC ! +√1 2|1iS⊗ L X ` 1 √ L|` − 1iC ! ≈ |+iS⊗ |ΨiC, (2.15)
where the last approximate equality follows from the fact that, for L → ∞, the displaced state of the coherence reservoir,PL
` √1L|` − 1iC, almost completely overlap with the state |ΨiC, see Fig.2.2. Thus, coherence can be created with Thermal Operations if the agent has access to a coherence reservoir, and if we consider approximate transformations along with exact ones.
Notice that, at a first glance, the above coherence reservoir might seem unphysical, and a too-strong resource that cannot be accessed in a laboratory. However, coherence manipulation is possible even in the case in which the reservoir has a Hamiltonian that is not unbounded from below, making it a physically meaningful system [137]. This system allows for the same power an unbounded reservoir provides, although it gets degraded with time and needs energy to be kept functional. Furthermore, a coherence reservoir of this kind can be realised in the laboratory, since the state of the radiation produced by a laser is a good approximation of the state |ΨiC used in the above protocol. Finally, it is interesting to notice that the coherence reservoir we have introduced here plays a very similar role to the one reference frames play in asymmetry theory [138,139]. Indeed, reference frames are systems that can be used to lift the super-selection rules imposed by some conservation laws on the main system, which is the same function the coherence reservoir fulfils in the context of thermodynamics.