Clock-Driven Quantum Thermal Engines

We consider an isolated autonomous quantum machine, where an explicit quantum clock is responsible for performing all transformations on an arbitrary quantum system (the engine), via a time-independent Hamiltonian. In a general context, we show that this model can exactly implement any energy-conserving unitary on the engine, without degrading the clock. Furthermore, we show that when the engine includes a quantum work storage device we can approximately perform completely general unitaries on the remainder of the engine. This framework can be used in quantum thermodynamics to carry out arbitrary transformations of a system, with accuracy and extracted work as close to optimal as desired, whilst obeying the first and second laws of thermodynamics. We thus show that autonomous thermal machines suffer no intrinsic thermodynamic cost compared to externally controlled ones.

of the system. Furthermore, after an optimal protocol neither the clock nor the weight are degraded relative to their initial states in their power to carry out subsequent transformations. We thus show that clock-driven thermal engines suffer no intrinsic thermodynamic cost compared to externally controlled ones.
We end with a discussion on the viability of alternative clocks, average energy conservation, and the use of the clock also to measure time in the system.

The clock
We first consider a Hilbert space divided into two parts, the clock and the engine,  = ⨂ e c. The engine e is an arbitrary quantum system that can start in any initial state ρ e and be subject to any Hamiltonian H e . As an example, it could be a simple qubit system or it could be a sophisticated thermal machine composed of several high-dimensional subsystems.
The clock, c, is a way of controlling the evolution of the engine, without having to provide external input. In our idealized framework, it has the continuous-spectrum Hamiltonian = H vP c c , where P c is the momentum operator in c and, for convenience, we take = − v 1 m s 1 . Under its free evolution, the clock state, ρ c , thus moves to the right with constant unit velocity, and one may interpret its position X c as reflecting time.
The engine and the clock interact through a static Hamiltonian H int by means of which the control is implemented. The total Hamiltonian thus takes the form Note that all of these Hamiltonians are time-independent. For conciseness, we omit the identities here on out.
In order to accurately and repeatably implement unitary operations on the engine, we only need a simple assumption on the initial state.
Assumption. The initial state is a product state e c and the support of ρ (0) c in position is contained inside a known finite interval. We define K as the size of this interval.
Knowing that the clock is located inside some region is all we require of it, which is in stark contrast to the stronger assumption of the clock being initially in a very narrow position state (corresponding to a well-defined 'time') which one could have expected here. This is a mathematically convenient assumption, which simplifies the calculations at very little cost. Any normalizable state is always close in trace distance to a state with finite support. Since the steps involved in the proofs below never increase the trace distance, all results will hold up to arbitrary precision for any clock state, even if it has infinite tails-say, a Gaussian distribution.

Energy-conserving unitaries
We now constructively show that it is possible within our framework to exactly implement any energyconserving unitary U e on the engine via interactions with the clock. This succeeds for any initial state satisfying equation (2), and the clock and engine are always unentangled at the end.
We start by choosing an interaction Hamiltonian of the form where | 〉 x c are the clockʼs position eigenstates. Since the clockʼs position increases linearly with time, this means the clock is driving the engine by applying on it an effective time-dependent Hamiltonian. We thus guaranteeing that the interaction will never transfer energy between the clock and the engine. This also prevents the clock and engine from becoming entangled.
Given this commutation relation, H int described in the interaction picture takes the form Thus, shifting → + x x t , one can see that the total evolution operator between times 0 and t is given by is the time ordering operation. Note that the clock moves forward at constant speed, despite the interaction. We now choose H e int to have support inside an interval of size L immediately to the right of the clock stateʼs support (the interaction region), define τ = + K Land choose τ > t so the clock has time to completely 'cross-over' the support of H e int (see figure 1). This causes the time integral in equation (6) to go over the entire support of H e int , becoming independent of x.
In other words, denoting as f x supp[ ( )]the support of f on x, we have τ And so, for all τ > t , the time evolution acts separately on the clock and on the engine in order to implement any desired energy-conserving U e . One possibility is a fixed Hamiltonian which switches on and off, i.e.
, where f is a normalized function with support inside the interaction region. However, note that our approach applies to any form of time dependence, which incorporates a broader range of experimental procedures.
Of course, the clock and engine can become entangled during the procedure, but they will always be in a product state at the end ( τ > t ). In fact, the state of the clock doesn't change other than being translated. This means the operation never degrades the clock.

The weight
Here, we show that one can extend the above considerations to implement general, not necessarily energyconserving, unitaries on a subsystem of the engine, by compensating on the rest of it. For that, consider the engine to be composed of two parts, = ⨂ e s w.
the clock state will be entirely to the right of the interaction region.
The system, s, is the part which we wish to transform. It is finite-dimensional and has arbitrary Hamiltonian H s and initial state ρ (0) s . Our objective is that, after applying U e on ρ ρ where V s is a general unitary on s. The weight, w, acts as an energy-storage device, which can be raised or lowered to extract or supply energy [2]. It has the continuous-spectrum Hamiltonian The primary purpose of the weight is to compensate the systemʼs energy change, since the final state of s can have a different energy than the initial state. However, as we show below, the initial state ρ (0) w of the weight may limit how optimally we can transform a state. In particular, to transform non-diagonal states optimally we need the initial state of the weight to be narrow in momentum and be centered around a known p 0 so it can also act as a resource of coherence. Åberg [3] pointed out that this resource can be used catalytically, and we show that this holds in our framework.

Arbitrary transformations
Here, we constructively prove that for any unitary V s on s there is a choice of H int which approximately implements it. First, we choose U e to have the form where P w is the weightʼs momentum operator. This serves two purposes: (i) it guarantees the protocol works regardless of the initial height of the weight (or how much energy is stored in it), (ii) it enables the catalytic use of the coherences in the weight to transform the system [3]. By the proof above, in order to implement this U e , we may choose an H int which also satisfies Alternatively, equation (8) can be written as where | 〉 p w are the momentum eigenstates of the weight. In this form, one sees that, upon applying U e on the engine state, the reduced state of the system becomes is the initial momentum distribution of the weight. As shown in the appendix, this can be made arbitrarily close to ρ is to a delta function, the closer this operation is to the exact V s (which is U p ( ) s 0 ). Therefore, for any ϵ > 0, there is always a good enough ρ (0) w such that is conserved by the operations. Also, note that the full evolution includes both the operations and the free evolution, but the weightʼs free evolution only shifts p 0 without affecting the shape of μ p  (11) simplifies to 3 The continuous-spectrum Hamiltonian was chosen for convenience. It has been shown in [3,4] that discrete weights also work arbitrarily well.

Work cost of transformations
The results above are of very general application in the field of quantum thermodynamics, more specifically in quantum resource theories. We exemplify this by answering a question posed in [1,2]: 'Should unitary operations pose a thermodynamic cost during work extraction?' As such, we further divide the system into two parts, ⨂ u b, so that the total Hilbert space is figure 2). The working subsystem u has arbitrary Hamiltonian H u and finite dimension d u .
The thermal bath, b, is composed of an arbitrary number of finite dimensional systems with arbitrary Hamiltonians in a thermal state at temperature T. Any protocol must specify which bath systems it will use, with their combined Hamiltonian given by H b , and the initial state being ρ Hb kT Hb kT . The weight, w, follows the same rules as above, but gains a new importance. Its change in average energy now also represents the thermodynamic work cost or gain.
Using this structure, we can explicitly define the thermodynamic quantities such as internal energy change, extracted work and emitted heat up to time t, where = t 0 is the time when the protocol starts: Following [2], we also define the free energy of a state by σ σ where σ S ( )is the von Neumann entropy of σ. With this, we now show that the definitions above obey the first and second laws of thermodynamics, and that optimal work extraction is possible.

First and second laws
Consider a quantum machine whose Hilbert space and Hamiltonian are as described above, and take as assumptions the following identities: . Note that these are all satisfied by the constructions thus far. The first law of thermodynamics is trivially satisfied by equation (15a), which implies that 〈 〉 H e is a conserved quantity 4 , and so To prove the second law, we construct a Kelvin-Planck statement [19,20], showing that it is impossible to extract positive work in a cyclic process. From equation (15b), we know H int must satisfy ub w c int int 2 As detailed in the appendix, this means the reduced state on ⨂ u bis a mixture of unitaries V x p ( , )applied on the initial state, This means the entropy of ⨂ u bnever decreases where Δ denotes a difference between times 0 and t, and we have used the subadditivity of the entropy and the assumption that the initial state is a product state.
Since the initial bath state has minimum free energy for a given temperature, one finds This means one cannot extract more energy than the reduction in free energy of the subsystem. Clearly, if the thermodynamic properties of the subsystem are the same in its initial and final state, Δ ρ = F ( ) 0 u and hence positive work cannot be extracted from the bath. That is, 'one cannot turn heat purely into work'.
Note that this result is a direct consequence of the assumptions in equation (15), and not of any implementation details. So the first and second laws hold given an arbitrary protocol which follows these assumptions, even one which is far from optimal, or which acts on a different initial state from the one it was designed for.

Optimal transformations
We now show that this framework allows thermodynamically optimal state transformation protocols. In particular, a protocol exists such that the work extracted is arbitrarily close to the reduction in free energy of the subsystem, and the final state of the subsystem is arbitrarily close to the desired final state. Here, a protocol represents a unitary operation on ⨂ ⨂ u b wor, equivalently, an H int on ⨂ ⨂ ⨂ u b w c. This section is based on the protocols in [2,4], but our framework is slightly different and involves a simpler proof strategy. Given the above results, we need only find a unitary V s on = ⨂ s u bwith the desired effect. Then, the existence of an interaction Hamiltonian which implements this unitary relies only on the weight state being good enough.
Let us write the initial state of the subsystem as u n n n n and the desired target state of the subsystem by , so the interacting initial state becomes the freely evolving target state. For simplicity we assume that σ u is full rank. If the final state has lower rank, we can instead transform the subsystem to a state σ′ s with full rank and close to σ u in trace distance.
The desired subsystem-bath unitary, = V V ub s , is composed of three stages. The first stage acts only on u, rotating ρ u to be diagonal in its energy basis. That is ub n n u n b (1) where | 〉 E n u are the eigenstates of H u . After this step the state of the subsystem will be ∑ p E E .
(24) n n n u n u Note that entropy is conserved in this stage, so the total energy change is equal to the change in free energy of ρ u . The second stage, V ub (2) , takes this diagonal state into another diagonal state, and has been studied before [1][2][3][4]. We present here a proof of its energy efficiency which is simpler than previous ones. The unitary is divided into many small steps, acting on ⨂ u band gradually changing the probabilities of the subsystemʼs energy levels from p n to q n , by performing the swap operation on the subsystem state and a similar state from the bath. The first step uses a thermal state in the bath of dimension d u , whose probabilities ′ p n lie between p n and q n and are very close to the former, i.e., δ | − ′| < p p p n n for all n.
Since this thermal state has minimum free energy, ΔF b must be of order δp 2 , so Furthermore, due to the swap operation, Δ Δ = − S S b u , and so the energy variation of this step is n n for all n, we can choose a unitary V ub (2) such that only δ − p 1 steps are necessary, and so the total energy variation must be  (1) . Note that the protocol also involves free evolution, see equation (7), so the subsystemʼs state becomes the freely evolving σ u . That is, . If one desires to have exactly σ u at a specific time t′, one simply needs to add a fourth step to the unitary, Looking at the energy balance, we find that the energy change of the subsystem and bath is as close as desired to the free energy change of the subsystem, making it a thermodynamically optimal transformation. That is

Discussion
Here, we have shown it is possible to exactly perform any energy-conserving unitary on a closed quantum system by attaching it to a quantum clock via a static interaction Hamiltonian. We have extended this so that any unitary can be approximated by attaching this system to a weight. Furthermore, this framework was proven to always satisfy the laws of thermodynamics, and to be a viable implementation of optimal quantum thermal machines. It was also shown that neither the clock nor the weight are degraded by the procedure, so they can be repeatedly used to transform a succession of systems. This addresses the question of whether quantum thermal machines, composed of nothing but timeindependent Hamiltonian evolution, can be as efficient as externally controlled ones. Remarkably, even if the clock has a broad initial state, this poses no thermodynamic cost in principle. There is also no intrinsic cost in using the weight, albeit there is a stronger restriction on the initial state. Namely, one needs it to be narrow in momentum space in order to achieve arbitrary precision.
For simplicity, we have considered the domain in the clockʼs position space to be , but the same results can be achieved with a periodic clock as long as the timescales involved are smaller than the period. It is an open question whether one can derive similar results with a more physical Hamiltonian, such as one whose energies are bounded from below or one which is finite dimensional. One way of doing so could be to look for physical Hamiltonians H c that are similar to vP c whenever the state of the clock lies in a certain region of the state space. In this way, it seems like it should be possible to approximate our results sufficiently well, by choosing the interaction Hamiltonian and initial state of the clock such that the system stays inside this region with high probability.
So far, we used the clock as a quantum mechanical way of controlling the engine, not as a quantum-timemeasuring device. If one desires to use it as such, the unitary is still implemented exactly, but there are two relevant scenarios to consider with regards to knowing the state of the subsystem. For instance, consider the initial state of the clock to have support inside −K , so our precision is dependent on how narrow the clock state is. In addition, having to perform a measurement could impose additional thermodynamic costs.
Throughout this work, we have considered unitaries and interaction Hamiltonians which commute with the free Hamiltonian of the engine. Previous work [2] also allowed unitaries which preserve the average energy of the specified initial state, without commuting with the Hamiltonian. This allowed for optimal protocols which were independent of the state of the weight. However, in order to implement such unitaries with the clock it seems that one would need to place strong conditions on its initial state. This is one of the reasons why we chose interactions which commute with the engine Hamiltonian.