Decomposable coherence and quantum fluctuation relations

Erick Hinds Mingo1 and David Jennings1,2,3

1Controlled Quantum Dynamics Theory Group, Imperial College London, Prince Consort Road, London SW7 2BW, UK 2Department of Physics, University of Oxford, Oxford, OX1 3PU, UK 3School of Physics and Astronomy, University of Leeds, Leeds, LS2 9JT, UK. November 11, 2019

In Newtonian mechanics, any closed-system measurement outcomes are random, the state |ψi is a dynamics of a composite system in a microstate perfectly sharp state (a pure state, or state of maximal will leave all its individual subsystems in dis- knowledge) and so should be viewed exactly on a par tinct microstates, however this fails dramatically with a classical state of well-defined location |xii, say. in quantum mechanics due to the existence of More precisely, the structure of the state space in any quantum entanglement. Here we introduce the physical theory determines many of the distinct char- notion of a ‘coherent work process’, and show acteristics of the particular theory. State spaces are that it is the direct extension of a work process always convex sets determined entirely by the extremal in classical mechanics into quantum theory. This points of the set – the pure states of the physical the- leads to the notion of ‘decomposable’ and ‘non- ory. The admissible measurements that the theory al- decomposable’ quantum coherence and gives a lows are defined in relation to this state space (see [5] for new perspective on recent results in the theory more details) and so are secondary theoretical ingredi- of asymmetry as well as early analysis in the ents. Therefore a comparison of classical and quantum theory of classical random variables. Within the mechanics at a fundamental level should equate sharp context of recent fluctuation relations, originally microstates with pure quantum states |ψi, as opposed framed in terms of quantum channels, we show to say comparing measurement statistics. that coherent work processes play the same role In classical mechanics a system with initial position as their classical counterparts, and so provide and conjugate momentum (x0, p0) evolves in time along a simple physical primitive for quantum coher- a well-defined trajectory (x(t), p(t)) in phase space de- ence in such systems. We also introduce a pure termined by the Hamiltonian of the system. This evo- state effective potential as a tool with which to lution extends into quantum mechanics where a quan- analyze the coherent component of these fluc- tum state evolves under a unitary transformation. The tuation relations, and which leads to a notion unitary dynamics of such a system S can also be un- of temperature-dependent mean coherence, pro- derstood in terms of paths, however it is now de- vides connections with multi-partite entangle- scribed in terms of a path integral [6] where the tran- ment, and gives a hierarchy of quantum correc- sition amplitudes are given by integrating the func- tions to the classical Crooks relation in powers i R tional exp[ dt(p(t)x ˙(t)−HS[x(t), p(t)])] over all paths ~ of inverse temperature. (x(t), p(t)) in phase space consistent with the bound- ary conditions, and where HS[x(t), p(t)] is the classi- cal Hamiltonian for the system S. At the operator arXiv:1812.08159v3 [quant-ph] 8 Nov 2019 level this dynamics is described a unitary transforma- 1 Introduction tion on states |ψi → U(t)|ψi, for some unitary operator The superposition principle is at the core of what makes U(t). Thus, in both classical and quantum mechanics quantum mechanics so special [1–4]. Classically a par- the Hamiltonian plays a key role in the time evolution ticle might be located in a microstate at any one of a of a system. Moreover in quantum mechanics, if the number of spatial sites with sharp momentum, however system is energetically closed then we have [U, HS] = 0 quantum mechanics allows a fundamentally new kind for the unitary evolution of the system. In both classi- of state – the particle being a superposition of multiple cal and quantum mechanics an initial sharp (pure) state different locations x , x , . . . , x . For this, the particle is evolved to a final sharp state. 1 2 √ n P iθk is in a state |ψi = k e pk|xki where θk are phase The path integral perspective describes this unitary angles, and pk is the probability of a measurement of evolution as a sum over all consistent paths, and so re- position returning the classical outcome xk. While the ceives contributions from trajectories that respect nei-

Accepted in Quantum 2019-10-24, click title to verify 1 ther energy conservation nor the classical equations of and show that the classical notion of deterministic me- motion. However, in the “limit of ~ → 0” a stationary chanical work can be extended into quantum mechan- phase argument tells us that the dominant contributions ics in a way that exactly parallels the classical case, to the evolution come from those trajectories around and gives rise to connections with topics in contem- the classical phase space trajectory (xclass(t), pclass(t)), porary quantum physics. Firstly, it leads to a notion namely the one that obeys the Hamiltonian equations of decomposable and non-decomposable coherence and of motion [7] given by p˙ = −∂xHS, x˙ = ∂pHS. Thus we has an immediate description in terms of recent frame- recover classical mechanics in the limit. works for quantum coherence [8–11, 17, 18]. We then In the case of two quantum systems S and A, exactly extend these considerations in noisy quantum environ- the same formalism applies, and coherent dynamics of ments and show that this notion of a coherent work pro- the composite quantum system can be analysed without cess connects naturally with fluctuation relations [19– any further fuss – governed by a total Hamiltonian HSA 26]. The coherent fluctuation theorems we develop go for the joint system SA. However, we can now ask the beyond traditional relations, such as the Crooks rela- following question: tion, and allow the analysis of quantum coherence in a noisy thermal environment. Because quantum coher- What unitary transformations of a composite quantum ence is handled explicitly this also leads to connections system SA in a state |ψiSA are possible that (a) obey to many-body entanglement and our results provide a energy conservation over the system SA, but (b) give clean explanatory framework for recent experimental rise to a transformation of the system S from an proposals in trapped ion systems [25]. initial pure state and into a final pure state?

It is clear that in the classical limit the answer to this 1.1 Summary of the core results trivial with sharp microstates – all classical dynamics This paper’s aim is to extend classical notions of work to that conserve the energy H give rise to deterministic SA quantum systems and provide novel tools for the anal- transformations of S. However for coherent quantum ysis of quantum thermodynamics and the role that co- systems, which can become entangled with each other, herence plays. To a large extent, quantum thermody- the answer is less obvious. Indeed, as we shall see, the namics has taken inspiration from statistical mechanics above is directly linked to central results from the re- when studying notions of ‘work’ [27–30]. In the present source theory of asymmetry [8–12] and foundational re- analysis, we instead revisit the deterministic Newto- sults from the 1930s on classical random variables [13– nian concept of work and propose a natural coherent 16]. extension into quantum mechanics. We then show how The physical motivation for this is given by consider- this coherent notion of work automatically arises in the ing the elementary notion of a mechanical work process non-deterministic context of recent fully quantum fluc- on a classical system along some trajectory. For this, tuation theorems [23] and provides a fully quantum- a mechanical system S initially in some definite phase mechanical account of thermodynamic processes. space state (x , p ) is evolved deterministically to some 0 0 In Section 2 we begin by defining a coherent work final state (x1, p1) and the mechanical work w for the R process process is computed from w = F · dx, where F (t) ω path |ψ i −→ |ψ i , (1) is the force exerted on the system along the particular 0 S 1 S path [7]. However, the system in this case is not en- which is a deterministic primitive to describe coherent ergetically isolated, and so the quantity of energy w is energy exchanges. The evolution is constrained to con- only meaningful because it corresponds to an equal and serve energy microscopically and preserve the statistical opposite change in energy of some external classical de- independence of the systems. In particular, the exclu- gree of freedom A that is used to induce the process. sion of entanglement generating processes in this defi- For example, we could imagine some external weight of nition is a choice required in order to be the quantum- mass m that is initialised at some definite height h0 = 0 mechanical equivalent of Newtonian work involving a and finishes deterministically in some final height h1. deterministic transition of a ‘weight’ system. Coher- In order to associate the height value h1 for A to w the ent work processes are found to separate into coher- work done in the process on S (via w = mgh1) it is nec- ently trivial and coherently non-trivial types, with the essary that energy is exactly conserved between the two distinction appearing as a consequence of the notion systems S and A. However, if we now wish to consider of ‘decomposability of random variables’ in probabil- superpositions of work processes on S then this account ity theory [14, 16]. This connection enables us to show becomes non-trivial, and so provides the starting point that coherent states form a closed subset under coherent for our analysis. work processes. As a result, we find the classical limit of In this paper we shall address the above question this fragment of quantum theory is equivalent to New-

Accepted in Quantum 2019-10-24, click title to verify 2 tonian work processes under a conservative force. After shows that these two requirements are not compatible this, we explicitly provide a complete characterization for general coherent states. In particular, given any tu- of coherent work processes for quantum systems with ple (H(ti),H(tf ),U), it is impossible to always assign equal level spacings, and also prove that measurement POVM operators for the TPM scheme statistics on such systems always become less noisy un- (w) X der coherent work processes. MTPM = pj0|ipi|iihi| (3) In Section 3 we move to a thermodynamic regime in i,j0: which we now have a thermal mixed state described by w=Ej0 −Ei an inverse temperature β = (kT )−1. We derive a quan- tum fluctuation theorem using an inclusive picture, and that provide an average work from the expectation value P (w) frame the result on the athermal system. Under the as- of X = w wMTPM that is equal for the expectation † sumptions of time-reversal invariant dynamics and mi- value of the operator X = H(ti) − U H(tf )U. croscopic energy conservation, we derive a fluctuation Our results include the TPM scheme as a special case, theorem in terms of cumulant generating functions that but circumvent no-go results by distinguishing between has the classical Crooks result as its high-temperature a random variable obtained by a POVM measurement limit. However we find that the quantum fluctuation on a quantum state with coherence, and the quantum theorem admits an infinite series of corrections to the state itself as a complete description of a physical sys- Crooks form in powers of β. The higher order terms tem1. For us, ‘coherent work’ is a fully quantum me- account for the coherent properties of the initial pure chanical property and not a classical statistical feature. states of the system and gives an operational perspec- This stance is further supported by the fact that our tive on related works [23, 26]. There exists another nat- coherent work processes play precisely the same role in ural decomposition of the fluctuation theorem in terms the coherent fluctuation theorem, as do Newtonian work of ‘average change in energy’ and ‘average change in processes within the classical Crooks relation. coherence’ with the caveat that the average coherence must be carefully interpreted. We derive a closed ex- pression for the ‘average coherence’ using readily inter- 2 Coherent superpositions of classical pretable quantities. processes and decomposable quantum co- A key result of the work is in Section 4 where we show that the quantum fluctuation theorem becomes a func- herence. tion of the coherent work state in a precise sense, and so further justifies the choice of coherent work processes We now go into more detail on what is required in order as a coherent primitive. to have superpositions of processes that overall conserve energy and give rise to deterministic pure state trans- 1.2 Relation to no-go results formations on a subsystem. The criteria are the same as in the case of mechanical work discussed in the in- In the literature, one finds a range of different notions of troduction. We assume a process in which the following work. For example, given some closed system in a state hold: ρ, described by a time-dependent Hamiltonian H(t) and evolving under a unitary U, the average work is fre- 1. (Sharp initial state) The quantum system S begins † in a definite initial pure state |ψ i. quently expressed as W = tr(H(tf )UρU ) − tr(H(ti)ρ). 0 Two-point measurement schemes (TPM) are common ways of defining the work done in a process. In such a 2. (Deterministic dynamics) The system undergoes a scheme, one might define deterministic quantum evolution, given by some unitary U(t). X X W = w p 0 pi (2) j |i 3. (Conservation) Energy conservation holds micro- w i,j0: w=Ej0 −Ei scopically and is accounted for with any auxiliary ‘weight’ system A initialised in a default energy where pi is the probability of measuring |iihi| when the eigenstate |0i. system is in the state ρ, and pj0|i is the transition prob- ability of going from |ii → |j0i given |j0i is an eigenstate 4. (Sharp final state) The system S finishes in some of H(tf ). Within a TPM definition of work, one would final pure state |ψ1i. wish to recover the traditional results from classical sta- tistical mechanics when ρ is diagonal (fully incoherent) 1In the sense that quantum states in general do not admit state. However, a no-go result was obtained in [27] that satisfactory hidden variable theories.

Accepted in Quantum 2019-10-24, click title to verify 3 Since the dynamics is unitary, and S finishes in a pure state, this implies the system A must also finish in some pure state |ωi. In the classical case this change in state of the ‘weight’ exactly encodes the work done on the system S, and so by direct extension we refer to |ωi as the coherent work output of the process on S. For simplicity in what follows, we shall use the notation ω |ψ0iS −→ |ψ1iS, (4) to denote the coherent work process on a system S in which |ψ0iS ⊗ |0iA → U(t)[|ψ0iS ⊗ |0iA] = |ψ1iS ⊗ |ωiA under a energy conserving interaction. If separate co- ω ω0 herent work processes |ψ0iS −→ |ψ1iS and |ψ1iS −→ |ψ0iS are possible, the transformation is termed re- ω versible and denoted by |ψ0i ←→ |ψ1i. We are able to 0 Figure 1: Coherent superpositions of mechanical processes. label the two processes by ω instead of (ω, ω ) because What unitary transformations are possible on a joint system 0 the states |ωi and |ω i are related in a very simple way, that conserves total energy, but gives rise to a deterministic as we prove in Appendix A. transformation of a pure quantum state |ψ0i into some final The coherent work output is in general a quantum pure quantum state |ψ1i? Conservation laws place non-trivial state with coherences between energy eigenspaces, how- obstacles on quantum systems, and not all such processes can ever its form is essentially unique for a given initial state be superposed. This corresponds to the existence of quantum |ψ0i and given final state |ψ1i for the system S. This states that have non-decomposable coherence. States in the is discussed more in Appendix A. Finally, we note the set C of semi-classical states, defined in the main text, are assumption that the output system is the same as the infinitely divisible and in the ~ → 0 limit recover the classical input system S can be easily dropped, where we require regime. that SA → S0A0 under an isometry, with energy conser- vation defined over the composite input/output systems Beyond this simple example subtleties emerge and involved. one can obtain non-trivial interference effects in the We can now provide some concrete examples of co- work processes, as the following example demonstrates. herent work processes. Example 2.2. Consider again, a quantum system S as in the previous example. One can coherently and Example 2.1. Consider some finite quantum system deterministically do a work process where S with Hamiltonian H . We assume for simplicity that 1 ω 1 S (|0i + |1i + |2i + |3i) −→ √ (|5i + |6i), (5) the energy spectrum of HS is non-degenerate and given 2 2 by {0, 1, 2,... }. We write |ki for the energy eigenstate which can be viewed as a merging of classical work tra- of H with energy k. S jectories. The coherent work output for this process is Consider the following two incoherent transitions of |ωi = √1 (| − 3i + | − 5i). This process is irreversible. a system S between energy states. The first is |0i → |5i 2 and the second is |1i → |6i. In both cases the process The proof that this transition is possible determin- can be done by supplying 5 units of energy to the sys- istically and irreversibly follows directly from Theorem tem from an external source A, and thus |ωi = | − 5i. 2.6 below. Moreover there is a unitary V that can do both processes However quantum mechanics also has prohibitions coherently in superposition: that give rise to highly non-trivial constraints that rule V [(a|0i + b|1i) ⊗ |0i] = (a|5i + b|6i) ⊗ | − 5i, out certain processes, as the following illustrates. Example 2.3. Let S be the same quantum system as where a, b are arbitrary complex amplitudes for the in the above examples. It is impossible to superpose the quantum state. Note also, that for the case of |a| = √ two transitions |0i → |5i and |1i → |7i in a coherent |b| = 1/ 2 this coherent transition could equally arise work process, even if we allow A to finish in some super- as a superposition of |0i → |6i and |1i → |5i. For this posed state |ωi with amplitudes over different energies. realisation, a different energy cost occurs for each indi- vidual transition, however the net effect also gives rise This result again follows from Theorem 2.6 below, but to | − 5i on the external source system. This process is one way to see that this is impossible is that the swap- reversible. ping of the initial states |0i and |1i and the swapping of

Accepted in Quantum 2019-10-24, click title to verify 4 the two final states |5i and |7i do not correspond to the with its conjugate momentum p, which are quantised same change in energy, and so the “which way” infor- in the usual way with commutation relation [x, p] = mation for the process cannot be erased in a way that i~1. We define a coherent state for the system via does not leave an external energy signature√ (as occurs |αi = D(α)|0i √for any α ∈ C and where a|0i = 0, with in the first example for |a| = |b| = 1/ 2). a := (x + ip)/ 2 and with the displacement operator D(α) := exp[αa† −α∗a]. This defines the set of coherent 2.1 “Which-way” information for processes and states {|αi = D(α)|0i : α ∈ C} for the system. energy conservation constraints We now note that we always have the freedom to rigidly translate a coherent state in energy |ψi → k P We can make more precise what we mean by the en- ∆ |ψi, where ∆ := n≥0 |n + 1ihn| for some non- ergy conservation condition providing a restriction what negative integer k (∆ is occasionally referred to as the classical work processes can be superposed when the phase operator [31]). We will also allow arbitrary phase quantum state has some permutation symmetry. For shifts in energy |ni → eiθn |ni. We now define the states simplicity we can consider the same quantum system k S as in the above examples. Suppose it is in some |α, ki := ∆ |αi, (8) initial state |ψi that is invariant under swapping of S where |αi is a coherent state [32–36]. We denote by C energy eigenstates |ai and |bi. We can let F := a,b the set of quantum states obtained from any coherent (|aihb| + |biha| + P |kihk|) be the operator that k6=a,b state via rigid translations by a finite amount together swaps these levels, and thus F ⊗1 |ψi⊗|0i = |ψi⊗|0i. a,b A with arbitrary phase shifts in energy. Note that arbi- This must map to a corresponding operation on the out- trary energy eigenstates are included in the set C as a put state |φi ⊗ |ωi, which we denote F˜, and which must rigid-translation of the vacuum |α = 0i state. For os- obey VF ⊗ 1 = FV˜ , where V is the interaction uni- a,b A cillator systems the energy eigenstates are not usually tary that conserves energy and performs the transfor- considered as classical states, however their inclusion in mation. Therefore we have that F˜ = V (F ⊗ 1 )V †. a,b A the set C is required for consistency in the → 0 limit. However because of energy conservation we must also ~ Explicitly, the states take the form have that  −iLt C := |ψi = e |α, ki : k ∈ N0, α ∈ C, t ∈ R, [L, H] = 0 V |ai ⊗ |0i = |xi ⊗ |a − xi (6) V |bi ⊗ |0i = |yi ⊗ |b − yi, (7) where we allow the use of an arbitrary Hermitian ob- servable L that commutes with H to generate arbitrary for some integers x, y, and thus, relative phase shifts in energy. We now consider the harmonic oscillator, but it is F˜ = |x, a − xihy, b − y| + |y, b − yihx, a − x| expected that a similar statement for the classical ~ → 0 + other terms. limit can be made for more general coherent states. However in order for such a which-way symmetry to Theorem 2.4 (Semi-classical regime). Let S be a relate solely to the system S, we must have that F˜ fac- harmonic oscillator system, with Hamiltonian HS = † tors into a product operator of the form X ⊗ 1A so hνa a. Let C be the set of quantum states for S as that the invariance on the initial superposition is associ- defined above. Then: ated with a corresponding transformation on the output 1. The set of quantum states C is closed under all pos- state. Restricting just to the space spanned by |x, a−xi sible coherent work processes from S to S with fixed and |y, b − yi we see that this implies a − x = b − y, and Hamiltonian H for both the input and output. so y − x = b − a and the invariance is under the swap- S ping operation X = Fx,y. Thus, the invariance under 2. Given any quantum state |ψi ∈ C there is a unique, permutations in a superposition together with energy canonical state |α, ki ∈ C such that we have a re- conservation implies that a transformation such as the versible transformation one in example 2.3 is forbidden. |ψi ←→ωc |α, ki, (9) 2.2 Superposition of classical processes in a with |ωci = |0i and α = |α|. Moreover, the only semi-classical regime coherent work processes possible between canonical states are Having provided some initial examples for coherent |α, ki −→ω |α0, ki, (10) work processes, we next link with the concept of work in classical mechanics through the following result. For such that |α0| ≤ |α|. Modulo phases, the coherent this we consider a mechanical coordinate x together work output can be taken to be of the canonical form

Accepted in Quantum 2019-10-24, click title to verify 5 p 2 0 2 0 |ωiA = |λ, niA, where λ = |α| − |α | and n, k 2.3 Decomposable and non-decomposable quan- are any integers that obey n + k0 = k. tum coherence The topic of coherent work is directly related to whether a random variable in classical probability theory is ‘de- 3. In the classical limit of large displacements |α|  1, composable’ or not. In classical theory a random vari- from coherent processes on C we recover all classical able Xˆ is said to be decomposable if it can be written as work processes on the system S under a conserva- Xˆ = Yˆ + Zˆ for two independent, non-constant random tive force. variables Yˆ and Zˆ [13, 16]. This turns out to provide a different characterization of coherent work processes. For compactness, given a system X with Hamiltonian ˆ Note that we can weaken the assumption that the HX , denote by X the random variable obtained from output system is S and the Hamiltonian HS is the same the measurement of HX in some state |ψi. The proba- at the start and at the end. It is readily seen that bility distribution is given by p(Ek) = hψ|Πk|ψi, where any admissible output system S0 must have within its Πk is the projector onto the energy eigenspace of HX energy spectrum infinitely many discrete energies with with energy Ek. separations being multiples of hν, and the output state Within our quantum-mechanical setting we now say distribution is always Poissonian on a subset of these that a state |ψi has non-decomposable coherence if given ω discrete eigenstates. A similar degree of freedom ex- a coherent work process |ψi −→ |φi this implies that ists for the auxiliary system A, which also must have either |φi or |ωi are energy eigenstates of their respec- a Poissonian distribution on a subset of discrete energy tive Hamiltonians, and otherwise the state is said to levels. The output parameters α, α0, λ must obey the have decomposable coherence. We also refer to such same conditions as in Theorem 2.4. non-decomposable transformations as “trivial” coherent processes. With this notion in place, we obtain the fol- The proof of the above theorem is given in Appendix lowing result. A.3, and provides a simple correspondence between the coherent work processes and work processes in classical Theorem 2.5. Given a quantum system X with Hamil- Newtonian mechanics. The main aspect of this result tonian HX with a discrete spectrum, a state |ψi admits that is non-trivial is to show that C is closed under work a non-trivial coherent work process if and only if the processes, but follows from a non-trivial result in prob- associated classical random variable Xˆ is a decompos- ability theory. able random variable. Furthermore, for a coherent work process Care must be taken when defining the classical limit ω as we do in Theorem 2.4. Conservative forces are char- |ψiX −→ |φiY , (11) acterised by their ‘path-independence’, and so the en- the associated classical random variables are given by ergy change is dependent only upon the initial and final Xˆ = Yˆ +Wˆ , where Yˆ and Wˆ correspond to the measure- phase space co-ordinates (x0, p0) and (x1, p1) respec- ments of H and H in |φi and |ωi respectively. tively. But the distinction between quantum and clas- Y W Y W sical is that in classical physics we can assign a well- We see that the states in C have infinitely divisible defined pair of co-ordinates to our system, labelling the coherence, and thus admit an infinite sequence of non- position and momentum. In addition to this, the energy trivial coherent work processes. A random variable Yˆ is is also as sharp as instruments allow. We therefore use infinitely divisible if for any positive integer n, we can the fact that for |α| large, the standard deviation of the find n independent and identically distributed random position, momentum and energy of our system grow in- variables Xˆn that sum to Yˆ [14]. The proof of this is creasingly negligible compared to their expected values. provided in Appendix A.2. Furthermore, the energy change of the two phase space points entirely captures the work done in the process. The coherent state result shows that in a semi- classical limit we recover the familiar work behaviour. 2.4 General coherence decomposition and disor- However, the earlier examples show that once we de- der in measurement statistics viate from the semi-classical regime the structure of coherence in such processes becomes highly non-trivial We now specify exactly what coherent work processes and certain transformations are impossible. This is cap- are possible for a quantum system S with discrete, tured by the notion of decomposable coherence and non- equally-spaced energy levels. To avoid boundary effects, decomposable coherence, which we now describe. we assume for mathematical convenience that both S

Accepted in Quantum 2019-10-24, click title to verify 6 is possible with

X iθj √ |φiS = e qj|jiS (13) j √ X iϕk |ωiA = e rk|kiA (14) k

for arbitrary phases {θj} and {ϕk}, with output Hamil- P tonian HS as above and HA = n|niAhn|, if and n∈Z only the distribution (pn) over Z can be written as

X j pn = rj(∆ q)n. j∈Z X = rjqn−j (15) j∈Z

for distributions (qm) and (rj) over Z. Again, note that we may weaken the assumption that Figure 2: Decomposability of coherence. Shown is a the Hamiltonian of S is the same at the start and at the schematic of decomposable coherence. A state |ψ i for a quan- 0 end with essentially the same conclusion. For example, tum system S can be split into pure state coherence |ψ1i on S0 and a coherent work output |ωi on A under an energetically one may introduce unoccupied energy levels in the out- closed process. The probability distributions for the energy put system S. Moreover if we drop the assumption of measurements on both |ψ1i and |ωi always majorize the distri- the precise form of the Hamiltonian HA, it is readily bution for |ψ0i and so the energy measurement disorder on S checked that the output Hamiltonians of S and A could is always non-increasing. be shifted by equal and opposite constant amounts while respecting the conservation of energy condition. In each of these variants, however, the core structure of the out- put distributions is essentially the same. and A are doubly infinite ladders and make use of core Now the Birkhoff-von Neumann theorem states that results from asymmetry theory [10]. every bistochastic matrix is a convex combination of Note that for a finite d-dimensional system S with permutations [37, 38], and thus since equation (15) is constant energy gaps, essentially the same analysis ap- a convex combinations of translations (which are them- plies since one can always embed the finite system into d selves permutations) this means the two distributions levels of such a ladder, apply the below theorem and en- are related by a bistochastic mapping q → p = Aq, sure that the support of the output state does not stray where A is bistochastic. However this implies that q outside the d levels of the embedding. For systems with majorizes p, written p ≺ q, and which establishes a general energy spacings, again a similar analysis can very useful relation between the input distribution on be applied, since the required dynamics will not cou- S and output distributions on S and A. Specifically ple different Fourier modes in a quantum state and so it implies that in any such coherent work process we will only transform coherences between subsets of levels must have both p ≺ q and p ≺ r. Now if f is any con- with equal level spacings. cave real-valued function on probability distributions, then a standard theorem [38] tells us that f(p) ≥ f(q) For the core model of a ladder system with equally whenever p ≺ q. This leads us to the following result. spaced levels we have the following statement for coher- ent work processes, with the proof provided in Appendix Corollary 2.6.1 (Disorder in coherent processes A.4. never increases). Let fdis(ψ) be some real-valued con- cave function of the distribution (p ) over energy in √ k P iθk |ψi = k e pk|ki (such as the Shannon entropy Theorem 2.6. Let S be a quantum system, with energy function), then in any coherent work process |ψi −→ω |φi eigenbasis {|ni : n ∈ Z} with fixed Hamiltonian HS = P we have that n|nihn|. Given an initial quantum state |ψiS = Pn∈√Z i pi|iiS, a coherent work process max{fdis(φ), fdis(ω)} ≤ fdis(ψ). (16) The significance of this is that every measure of “dis- ω |ψiS −→ |φiS, (12) order” fdis for the energy statistics in a pure quantum

Accepted in Quantum 2019-10-24, click title to verify 7 state |φi will be a convex function of this form. The however, if we are concerned with transitioning from corollary thus shows that the energy statistics of both |ψiS to |φiS we can use the fact that if V is an energy the coherent work output and the final state of S are conserving unitary realising the process (17) then its always less disordered than that of the initial state of S inverse V † exists and is also energy conserving. This with respect to all measures. This also provides an in- implies that we can write the inverse transformation as tuitive perspective on the decomposability of coherence −ω in a such a process. |ψiS −→ |φiS, (18) Under repeated coherent work processes on the final state of S and the work output state, the disorder will which is realised through the unitary transformation become diluted but for a general scenario will stop when † one reaches non-decomposable components. However V [|ψiS ⊗ |ωiA] = |φiS ⊗ |0iA. (19) for the class of states in C there is no such obstacle This extension of notation allows us to define the nega- and the disorder can be separated indefinitely. Note tive coherent work output −ω, which physically means also that if in addition the function f is additive over dis the coherent quantum state |ωi required to realise the quantum systems, namely f (φ⊗ψ) = f (φ)+f (ψ), A dis dis dis transformation – in other words ω is the coherent work then the total disorder over all systems as measured by input to the process. If a coherent work process exists this function will remain constant throughout. in either direction between |ψiS and |φiS then we say the states |ψiS and |φiS are coherently connected. From 2.5 Multiple processes and coherently connected this definition we see that the whole set C is a coherently quantum states connected set in the sense that any two states in C are coherently connected. Note however that coherent con- The framework we have laid out involves an auxil- nectedness is not a transitive relation: if (|ψ1i, |ψ2i) and iary system initialised in an energy eigenstate. It is (|ψ2i, |ψ3i) are each coherently connected pairs then for this reason that the disorder dilutes among the it does not imply that (|ψ1i, |ψ3i) are coherently con- bipartite system, rather than being able to increase nected. in one subsystem. This has the advantage of isolat- ing the coherent manipulations on the primary system. 2.6 But shouldn’t ‘work’ be a number? Due to this, in a sequence of coherent work processes ω1 ωn |ψ1iS −→ |ψ2iS ··· −→ |ψn+1iS, there might only be a We have used the term ‘coherent work output’, however finite number of non-trivial processes possible before the the issue of what ‘work’ is in general, is subtle. Here system ends in a state with non-decomposable coher- we do not wish to use the term without justification, ence. However, a notable counter-example is the semi- and so now expand on why this is not an unreasonable classical result for which this never happens. use of language. In particular, we argue that the ap- The method we have chosen is the simplest concep- proach is fully consistent with operational physics and tually, but it is not the most general procedure possi- is a natural generalisation of what happens in both clas- ble. One could imagine an auxiliary system prepared in sical mechanics and statistical mechanics. For ease of an arbitrary state |φiA. In such a case, the approach analysis we list the main components of the argument: involving random variables must be slightly modified but the same tools can be applied. The most gen- • Work is done on a system S in a process when an eral coherent work process possible, excluding entan- auxiliary physical system A is required to trans- glement generation, is a transformation of the form form, under energy conservation from a default Nn Nm 0 state, so as to realise the process on S. i=1 |ψii −→ i=1 |ψii, with the only two constraints that the dynamics conserve energy globally, and the • Work on a system S in Newtonian mechanics is subsystems are left in a pure product state. In this mod- a number w ∈ R that operationally can be read ified framework, results such as Corollary 2.6.1 would no off from the transformed state of S or equivalently longer hold, since coherence can be concentrated from from the transformed auxiliary system A’s state. many subsystems to fewer subsystems. We leave this as an open question for later study. • Work on a system S in statistical mechanics is not a A special case that is relevant here is if we have two number, but is described mathematically by a ran- quantum states |ψiS and |φiS and the auxiliary system dom variable W :(dµ(x), x) on the real line. Op- A begins in a state |ωiA with coherence and terminates erationally, individual instances are read off from in some energy eigenstate |0iA. This can be denoted as measurement outcomes on S, and by energy con- servation are in a one-to-one correspondence with ω |φiS −→ |ψiS, (17) outcomes on A.

Accepted in Quantum 2019-10-24, click title to verify 8 to these observed outcomes, however this is a fundamen- tally different scenario – the state |ωi has no random- ness in itself, instead it has complementarity in observ- ables which is physically different. The measurement on |ωi, in contrast, introduces classical randomness and so steps out of the deterministic mechanical domain. Indeed, one can make the following comparison: the co- herent work output |ωi is to the measurement statistics p = (p(E1), p(E1), p(E2) ..., ) what a laser is to inco- Figure 3: When does “work” make sense? The two left- herent thermal light. We can obtain interference effects, hand cases concern deterministic processes, whereas the two entanglement and other non-classical phenomena from right-hand cases involve mixed states with unavoidable prob- abilistic elements. In Newtonian mechanics, work is a deter- a quantum state |ωi, but not from a probability distri- ministic number w and a largely unambiguous concept. Ex- bution p. tensions into statistical mechanics mean that work is no longer Note also, that essentially the same approach is taken described by a number, but instead is a classical random vari- for quantum entanglement. There is no measurement able W :(dµ(x), x). Classical mechanical work occurs when one can do corresponding to the “amount of entangle- this distribution is sharp, µ(x) = δ(x − w). However one may ment” in a state, despite it being a crucial physical extend the deterministic concept of work from Newtonian me- property of a state with dramatic effects. Instead, en- chanics in a natural way into quantum mechanics. This leads tanglement is quantified in terms of pure state ‘units’, to a coherent form of work, that coincides with a superposition for example the Bell state |ψ−i. In both the coherent of classical work processes and in the ~ → 0 limit recovers the work case and the entanglement case, one ultimately classical notion. It also it arises in fluctuation relations in the concerned with empirical statistics and not abstract same manner as its classical counterpart. quantum states. The reconciliation of this is that quan- tification in terms of the abstract quantum states will • Quantum mechanics is a non-commutative exten- determine the empirical statistics in a way that depends sion of statistical mechanics, in which classical dis- on the particular context involved. tributions dµ(x) are replaced with density opera- In the context of the next section, by adopting this tors ρ on some Hilbert space H. coherent description, we can evade recent no-go results [27] on fluctuation relations for thermodynamic systems • Complementarity in quantum theory implies that and provide insight into recent developments in this field general POVM measurements on a state ρ are in- [23, 25, 28, 39, 40] while also connecting naturally with compatible, and moreover “unperformed measure- well-established tools for the study of quantum coher- ments have no result” [2]. No underlying values are ence [8, 9, 41, 42]. assigned to a pure state |ωi in a superposition of different classical states. 3 Coherent Fluctuation Theorems The last two points are vital. One can of course consider random variables obtained from measurements on quan- In this section we show that the concept of coherent tum systems, but this is not the same thing and cannot work processes discussed in the previous section nat- describe quantum physics in any sensible way. Trying urally occurs within a recent fluctuation theorem [23– to use classical random variables to fully describe quan- 25] that explicitly handles all quantum coherences and tum physics is essentially a “hidden variable” approach, which is experimentally accessible in trapped ion sys- however we know from many results that it is impossi- tems [25]. We first introduce the core physical assump- ble to define a hidden variable theory for quantum me- tions involved and then explain how these naturally re- chanics unless one is willing to make highly problematic late to the concept of coherent work processes. But assumptions, such as violation of local causality [2]. first, we briefly re-cap on the classical Crooks fluctua- In our approach, the coherent work process does not tion relation, where phase space trajectories are center associate a single, unique number to the coherent work stage. output, instead we describe the output of the process in terms of a pair (|ωihω|,H) that generalizes the clas- 3.1 Classical, stochastic Tasaki-Crooks relation sical stochastic pair W :(dµ(x), x) into a fully coherent setting. In the most basic setting one has a bath subsystem B, e−βH0 Obviously one could measure the state |ωi in initially in some thermal equilibrium state γ0 = Z the energy eigenbasis to get measurement statistics with respect to an initial Hamiltonian H0, which is (p(E1), p(E1), p(E2) ..., ), and associate energy scales then subject to some varying potential that changes its

Accepted in Quantum 2019-10-24, click title to verify 9 Hamiltonian H0 → H(t) → H1 and finishes in some fi- nal Hamiltonian H1. Note that through an appropriate choice of units we can always arrange that a protocol starts at t = 0 and finishes at time t = 1. By doing sharp, projective measurements of the en- ergy at the start and end of the protocol one obtains stochastic changes in energy on B for the protocol that lead to a stochastic definition of work (see the review [22] and the recent paper [29] by Talkner and Hänggi for excellent discussions on the concept of work in statis- tical mechanics). In this setting one can readily derive a Tasaki-Crooks fluctuation relation that compares the work done during a forward protocol P : H0 → H(t) → H1 with the work done in the time-reversed protocol ∗ P : H1 → H(t)rev → H0, obtained by reversing the Figure 4: Classical fluctuation setting. In the classical set- time-dependent variation of the system’s Hamiltonian up, any particular phase space point (x, p) is taken along a under P and beginning in equilibrium with respect to unique trajectory (x(t), p(t)), and work is accumulated in the H1. The Tasaki-Crooks relation compares the probabil- process. Work originates as a change in Hamiltonian from a ity of doing work W = w in the forward protocol with time t0 to a time t1. This change is induced via an interaction the probability of doing work W = −w in the reverse with an implicit energy source, in which the total system can be protocol, and is given explicitly by described by a time-independent Hamiltonian. Note that if one follows a classical Crooks derivation but imposes a restriction to P [w|γ0, P] −β(∆F −w) coarse-grained resolutions of phase space (such as the shaded ∗ = e (20) P [−w|γ1, P ] regions shown) then it is expected that one will obtained similar results to the coherent fluctuation relations for quantum theory. where ∆F := F1 − F0 , and Fk is the free energy of the e−βHk equilibrium state γk = Z for k = 0, 1. Note that there is no assumption that B finishes in the equilibrium shown that these “negative probabilities” are in fact state γ1, and there is no assumption that the protocol a witness of contextuality – a fundamental feature of is in any way quasi-static. From this one can derive the quantum mechanics that includes quantum entangle- Jarynski relation and the Clausius relation hW i ≥ ∆F ment as a special case [48, 49] and is conjectured to from traditional thermodynamics. be the key ingredient providing speed-ups in quantum computers [50, 51]. However weak measurements are a very particular type of quantum measurement and so one can ask if an 3.2 From stochastic relations to genuinely quan- over-arching framework exists that includes the classical tum fluctuation relations stochastic setting, weak measurements and fully gen- While the formalism of the two-point measurement eral POVM measurements that lead to coherent fluc- scheme is applicable to both classical and quantum tuation relations. In [23] a general framework was de- systems alike, it must be emphasized that any gen- veloped by Åberg. The framework developed can be uine coherent features are destroyed and the operational viewed as a fully coherent version of the “inclusive” de- physics is entirely equivalent to classical stochastic dy- scription taken by Deffner and Jarzynski in [52] for the namics on a classical system (see [43] for a discussion thermodynamics of classical systems, and leads to an of why this stochasticity is essentially classical). Sev- extremely general fluctuation relation that handles co- eral works [24, 27, 28, 39, 44–46] have proposed vari- herence, and includes the classical fluctuation relations ants of the two-point scheme with the aim of providing as special cases. However the technical level of the anal- fluctuation relations that capture genuinely quantum- ysis is formidable and quite different from existing ap- mechanical features. Of note are the no-go results of proaches. The physical assumptions of the model are [27, 47] that prove that certain desirable features for a natural, however it was not clear how one should inter- measurement scheme are incompatible. One model that pret the broad, abstract results in more familiar terms. attempts to circumvent these obstacles was proposed in [28] where projective measurements were replaced by 3.3 The coherent fluctuation framework weak measurements, and in which quasi-probability dis- tributions arise. A related analysis in terms of quasi- Before proceeding, we make a brief comment on our probabilities was provided in [45], while in [47] it was notation. We shall use capital Roman letters (A, B, C)

Accepted in Quantum 2019-10-24, click title to verify 10 at the start of the alphabet to denote quantum systems 2. The microscopic description is energetically closed, that are initialized in quantum states with no coherence but not dynamically closed. between energy eigenspaces. For example, for an auxil- iary ‘weight’ systems initialized in |0i we use A. In this 3. Time-reversal symmetry holds for the microscopic analysis we are primarily concerned with coherent fea- dynamics of the composite system. tures and so we use the notation S for a quantum system that initially has coherence between energy eigenspaces. 4. The thermal system B is initially in some Gibbs state with respect to an initial Hamiltonian and S For fluctuation relation contexts it is typically the is in some arbitrary quantum state. case that we have a large thermal bath that we do not control, together with another quantum system that we The model therefore involves the initialisation of SB do control and which is initially in thermal equilibrium in some joint product state ρ⊗γ0 where B is in thermal with the bath. Following the above convention we label equilibrium, as above, while S is allowed to be in an these B1 (e.g.for the uncontrolled system) and B2 (e.g. arbitrary state ρ. The primary system is then subject for the controlled thermal system), and reserve S for to open system dynamics that transforms the state as additional coherent degrees of freedom that we can also ρ → E(ρ) where control and wish to study. † The approach taken in [23] can be viewed as a gen- E(ρ) = trBV (ρ ⊗ γ0)V , (21) eralisation of [52] to a fully quantum setting in which all energies and all coherences in energy are accounted in terms of a microscopic unitary V on SB that may be for explicitly. Moreover, it is exactly in the same spirit partially controlled by macroscopic parameters by an as the coherent work processes as introduced earlier – experimenter. Note that any protocol that an experi- quantum coherences in energy are always relational de- menter implements will in general be through macro- grees of freedom and so an inclusive approach is natu- scopic parameters, however any such transformation ral. Specifically, in addition to the primary system S will admit such a description via a Stinespring dilation one introduces an additional bath subsystem B and the [53]. We refer the reader to [23] for a discussion of this composite system SB is energetically closed in terms point. of energy flows and coherence flows, but not necessar- In the incoherent regime, the fluctuation theorem ily dynamically closed 2. Therefore we make use of an compares transition probabilities of ρ → σ with transi- inclusive “microscopic” description of the energetic de- tion probabilities of ΘσΘ† → ΘρΘ† where Θ is a time- grees of freedom with the aim of arriving at a fluctuation reversal operator. We shall find that when coherence is relation that handles arbitrary coherence. present, this must be generalised so that the core Crooks A key thing to highlight is that every fluctuation rela- construction can be implemented. tion (classical or quantum) involves such an additional system S, however this system is generally left implicit 3.4 Time-dependent Hamiltonians within a mi- as the external physical degrees of freedom that pro- vide energy and coherence so as to change the Hamilto- croscopic, inclusive constraint–description nian of B in some time-dependent manner. Such time- Since we must first account for microscopic degrees of dependent variations of a Hamiltonian interact non- freedom, any protocol for the change in Hamiltonian trivially with coherent structures and so this is a pri- H0 → H(t) → H1 should be handled with care. The mary reason why such an inclusive microscopic descrip- physical reason for this is that a general time-dependent tion is needed. Hamiltonian will interact non-trivially with the quan- A crucial point is the following: since the system SB tum coherence between energies and so to properly de- is closed in terms of energy and coherence flows this scribe the latter one must be careful with the former. means that all energy/coherence changes in S corre- At the microscopic level Hamiltonians are generators of spond in a one-to-one fashion to the energy/coherence time-translation, whereas any time-dependent Hamil- changes in B. tonian is always an effective description that arises The assumptions of the framework are as follows: through the interaction with some external system. Consider any time-dependent Hamiltonian H(t) sce- 1. A microscopic, inclusive description is taken for a nario that evolves from some initial H0 := H(0) to some thermal bath and quantum system. final H1 := H(1). Since the ‘t’ in H(t) corresponds to a physically discernible value this implies that it can 2This does not mean that the system is “autonomous” – an ex- always be made explicit within a Hilbert space descrip- perimenter can still implement general time-dependent protocols tion via a degree of freedom of the composite system by macroscopically varying internal parameters of the system. (otherwise the parameter is physically meaningless!).

Accepted in Quantum 2019-10-24, click title to verify 11 Therefore, there always exists a Hilbert space descrip- determine the time t. In the event of the first outcome, tion of a time-dependent Hamiltonian in terms of a com- the system B is updated via the projector Π0 that en- 0 posite SB system with a Hilbert space that takes the sures it is entirely constrained to HB and so C0 simply form amounts to the statement: the system B has Hamilto- 0 1 other HSB = HS ⊗ (HB ⊕ HB ⊕ HB ), (22) nian H0. 0 0 Thermalisation with respect to the constraint C0 rep- where HB is the span of the eigenstates {|Eki} of the 1 resented by a projector Π0 is now defined [23] by the initial Hamiltonian H0, and similarly for HB and H1. other transformation The subspace HB corresponds to any other physical degrees of freedom that may be accessed at intermediate e−βHB /2Π e−βHB /2 3 0 times 0 < t < 1 of the protocol . Therefore any protocol Π0 → Γ(Π0) = −βH (24) tr(e B Π0) H0 → H(t) → H1 on a quantum system can be under- −βH0 stood at a microscopic level as the evolution of the bath e = = γ0, (25) subsystem B from being constrained solely to the sub- Z space H0 , at the start of the protocol (t=0), to it being B and thus Γ(Π ) represents the statement that B has constrained to the subspace H1 at the end of the pro- 0 B Hamiltonian H and is thermalised with respect to it, tocol (t=1). The underlying Hamiltonian for the com- 0 at inverse temperature β. The mapping Γ should be posite system is therefore H = H ⊗ 1 + 1 ⊗ H , SB S B S B viewed as transforming a Hamiltonian constraint into with a thermodynamic constraint. In exactly the same way, HB = H0 ⊕ H1 ⊕ Hother, (23) we also have that Γ(Π1) = γ1. where we simply combine all possible intermediate The particular form of the coherent thermalisation Hamiltonians H(t) into the term Hother simply for com- transformation, which we shall simply call Gibbs rescal- pactness. This underlying description in terms of con- ing, is required for various reasons, however certain im- straints, while appearing strange from the traditional portant cases should first be highlighted. If the con- H(t) formulation, is entirely consistent with the usual straint C is “The system B is in the energy eigenstate story. More importantly, it turns out to provide a pow- |Eki of H0” then this is a much stronger constraint. 0 0 erful perspective in the context of fluctuation relations Again, this is represented via a projector |EkihEk|, how- with coherence. ever now we find that

e−βHB /2|E0ihE0|e−βHB /2 3.5 Coherent thermalisation of a system with re- Γ(|E0ihE0|) = k k = |E0ihE0|. k k −βHB 0 0 k k tr(e |EkihEk|) spect to constraints (26) In our analysis, the constraint–description for the time- Thus if B is constrained to be exactly in the sharp eigen- 0 dependent protocol turns out to connect with the tra- state |Eki then there are no remaining degrees of free- ditional notion of thermodynamic constraints in phe- dom to thermalise, and the state remains the same. nomenological settings (see for example Callen [54]). A more interesting case is if the only thing we know Specifically, we can talk of a “coherent thermalisation” is that B is in pure state, with a uniform superposi- of B with respect to a constraint C at inverse tempera- tion over all the eigenstates of H0. This condition is 1 1 ture β, as we now describe through examples. described by the projector | 0ih 0| where Let C be the constraint “B is constrained to H0 ”. 0 B 1 Mathematically this constraint is described by the pro- 1 X 0 | 0i := √ |Eki, (27) P 0 0 0 d0 jector Π0 = k |EkihEk| onto the subspace HB. Exper- k imentally this corresponds to a POVM measurement on SB given by {1S ⊗ Π0, 1S ⊗ Π1, 1S ⊗ Πother} that asks: where d0 is the dimension of H0. It is readily checked does B have a Hamiltonian H0, the Hamiltonian H1, or that for this case some intermediate Hamiltonian H(t)? In actual experi- ments it simply amounts to the experimenter looking at Γ(|10ih10|) = |γ0ihγ0|, (28) what the classical dials of the apparatus are set to, or if 1 P −βE0/2 0 the time-varying protocol is fixed, looking at a clock to where |γ0i = √ e k |E i is the coherent Z0 k k Gibbs state with respect to the initial Hamiltonian 3The description of a continuous infinity of ‘t’ values in this H0. Or, in terms of entanglement we can consider the vein has technical subtleties, namely the mathematical awkward- d−1 maximally entangled state |φ+i = √1 P |E i ⊗ ness of non-separable Hilbert spaces. Experimentally however, AB d k=0 k A physics always involves a finite resolution scale and so this sub- |EkiB on two quantum systems of dimension d. In this tlety has no physical content here. case, coherent thermalization of the maximally entan-

Accepted in Quantum 2019-10-24, click title to verify 12 √ gled state with respect to A leads to factors of e−βEk/2/ Z that Gibbs re-scale all probabil- ity distributions. The above mapping should be viewed Γ(|φ+ihφ+|) = |φ˜ihφ˜| (29) as the extension of this to a fully coherent setting. The 1 X √ ˜ −βEk transformation Γ(X) also arises in the context of the |φi = √ e |EkiA ⊗ |EkiB, (30) Z k Petz recovery map [58] for quantum channels, and of course in the paper [23] by Åberg, where it is dis- which is the thermofield double state [55] in high energy cussed in more detail. Its form also arises in quantum- physics and condensed matter. mechanical settings for the transition between micro- These examples justify X 7→ Γ(X) as describing a scopic and macroscopic descriptions of thermodynamic formal coherent thermalisation: if no coherences are systems [59], which is perhaps the most appropriate per- present Γ provides the thermal Gibbs state; pure states spective in light of the above constraint discussion. It are always sent to pure states; and coherent superpo- would be of interest to obtain an independent opera- sitions over energy are re-weighted with Gibbs factors tional analysis of this transformation (perhaps as a co- at inverse temperature β. However, it is important to herent form of a maximum entropy principle). We do emphasise that Γ is not a physical transformation but not expand any more on this here, but instead refer the an invertible mapping on states that pairs (ρ, Γ(ρ)), as reader to [26] and [23] for more discussion. being distinguished – in the same way as time-reversal pairs (ρ, ΘρΘ†). The time-reversal operator pairs a forward temporal- 3.6 A key relation between forwards and reverse direction with a backward temporal-direction, while Γ protocols for quantum systems pairs a Hamiltonian constraint with a thermodynamic Given the preceding discussion we can derive the co- constraint. It might seem strange that we pair a me- herent fluctuation relation, however this follows from a chanical constraint with one that has an explicit ther- core structure that is describable on a single quantum modynamic feature (namely a temperature), but the system. This was discussed in [23], but here we provide need for this arises from not necessarily having sharp a slightly modified approach that will prove useful for microstate properties in quantum superpositions and it what follows. Let S be any quantum system, with is this indeterminacy that requires the Gibbs-rescaling. tot Hamiltonian H . We consider the following sequence Such a scenario would also arise in a classical exam- tot that abstracts the classical notion of a ‘trajectory’ to a ple where instead of doing a Crooks relation on mi- coherent form for S : crostates (x, p) one follows the same construction on tot distinguished distributions {qk(x, p)}, e.g. if we took 1. A measurement is done on Stot with outcome given coarse-grained resolutions of finite sized support on by a (not necessarily rank-1) projector Π0. phase space. The qk(x, p) would also have to be Gibbs- rescaled in order to obtain a coarse-grained Crooks re- 2. Coherent thermalisation of Π0 with respect to Htot lation (this is the classical analog of the above example and at inverse temperature β updates the system of projector Π that is not of rank-1). to the state Γ(Π0). For the case where the constraint is the above Π0, 3. The system Stot evolves unitarily under a unitary the pairing (Π0, Γ(Π0)) could potentially be viewed as V that commutes with Htot. a statement of the equivalence of the microcanonical and canonical ensembles in a similar vein to recent clas- 4. A final measurement is done on Stot with projective sical results [56]. This perspective can be formulated outcome Π1 (again not necessarily rank-1). in terms of dualities between constrained and uncon- We denote the probability of this sequence as P [Π |Π ], strained optimization problems, and we conjecture that 1 0 and is given explicitly as the present Gibbs-rescaling could be naturally described in such terms. We leave this to future analysis. ˜ † P [Π1|Π0] = tr[Π1V (Γ(Π0))V ]. (31) Beyond its role with time-reversal in identifying the correct trajectories to pair, we note that the transforma- The expression can be viewed as the probability of a tion Γ is not a linear physical map but instead amounts particular ‘trajectory’ under the above protocol. to an change in our description of the system. In par- The fluctuation setting also requires a notion of a ticular, one that is covariant under energy conserving time-reversal of a trajectory. In quantum mechan- unitaries V , with V Γ(ρ)V † = Γ(V ρV †). This structure ics, time-reversal is an anti-unitary transformation at arises naturally in fluctuation settings and scenarios in the level of the Hilbert space. A state transforms as which one reverses a general quantum operation. In |ψi → Θ|ψi where Θ is both anti-linear (e.g. Θ(α|ψi) = [57] Crooks considered the reversal of a Markov pro- α∗Θ|ψi for any α ∈ C) and Θ†Θ = ΘΘ† = 1. How- cess, and found the resulting transformation generates ever any anti-unitary Θ can be written as KU where

Accepted in Quantum 2019-10-24, click title to verify 13 U is some unitary and K is complex conjugation in a preferred basis. At the level of Hermitian operators (e.g. projectors, observables, quantum states...) for which we have that X = X†, this complex conjuga- tion X → K†XK = X∗ is equivalent to simply tak- ing a transpose of the operator X → XT . For the case of time-reversal, we can identify the time-reversal of a state ρ as ρ 7→ ρ∗ = ρT . An inspection of the off-diagonal components of ρ in the energy eigenbasis (which is the relevant basis for the harmonic oscillator system below) makes this clearer: a typical component Figure 5: Global system schematic. A system S, initially −iωij t uncorrelated with a thermal system B, evolve under a micro- ρij will oscillate in time as e and thus by taking the complex conjugation one obtains an oscillation of scopically energy conserving unitary V , to produce open-system dynamics on S. e+iωij t for the matrix component – in other words one swaps the positive and negative modes in the quantum state. 3.7 Coherent Tasaki-Crooks relation. Since tr[X] = tr[XT ] for any Hilbert space operator X, and (XY )T = Y T XT , we see that Equation (31) As already mentioned, we adopt an inclusive descrip- can be re-written as tion in which the total system Stot is actually composed of two subsystems S and B, and the total system SB is ˜ † T P [Π1|Π0)] = tr[(Π1V Γ(Π0)V ) ] (32) energetically closed, but not dynamically closed. As dis- † T T T T cussed earlier, one should view the initially incoherent B = tr[(V ) Γ(Π0) V Π1 ]. (33) system as sub-divided into two components B = B1B2 where we have introduced the short-hand notation X˜ := where B1 is a large, uncontrolled thermal bath while B2 Γ(X) for any operator X. We now make the final as- corresponds to a quantum system for which we can con- sumption that V , in addition to [V,Htot] = 0, is also trol its degrees of freedom – for example we can vary its invariant under the above time reversal transformation Hamiltonian in some manner. The total Hamiltonian – namely V = V T . Informally, this condition can be for SB is as before and has eigenspaces given by the interpreted as assuming that the unitary V does not decomposition in Equation (22). inject in any microscopic time-asymmetry into the sys- We now consider an initial constraint C0 for t = 0 and tem, and thus any differences in probabilities between an final constraint C1 for t = 1, given by forward and reverse trajectories are purely due to the thermodynamic structure. C0 : B has Hamiltonian H0 and S is in a state |ψ0i. T ∗ T Using that V = V , and writing X = X for any C1 : B has Hamiltonian H1 and S is in a state |ψ1i. Hermitian X we see that Mathematically, these are represented by ΠSB,0 = ˜ † T ∗ P [Π1|Π0] = tr[(V )Γ(Π0) V Π1] |ψ0ihψ0| ⊗ Π0 and ΠSB,1 = |ψ1ihψ1| ⊗ Π1 respectively. 1 k −βHtot/2 ∗ −βHtot/2 ∗ † where Πk is now the projector onto HB for the system = −βH tr[e Π0e V Π1V ] tr(e tot Π0) B. tr(e−βHtot Π∗) The coherent thermalisation for the initial set-up 1 ∗ ∗ † ˜ ˜ ˜ ˜ = −βH tr[Π0V Γ(Π1)V ] gives Γ(ΠSB,0) = |ψ0ihψ0| ⊗ γ0 where |ψ0ihψ0| := tr(e tot Π0) ΓS(|ψ0ihψ0|) where ΓS is Gibbs-rescaling purely with −βHtot tr(e Π1) ∗ ˜ ∗ respect to the Hamiltonian HS. If we now substitute = −βH P [Π0|Π1], (34) tr(e tot Π0) these components into Equation (35) we obtain

T where we used the fact that Htot = Htot to write −βHtot ∗ −βHtot tr(e Π1) = tr(e Π1). We can therefore ex- press the ratio of an abstract ‘forward trajectory’ to its ‘reversed trajectory’ as P [Π |Π˜ ] tr(e−βHSB |ψ ihψ | ⊗ Π ) SB,1 SB,0 = 1 1 1 −βH ∗ ˜ ∗ −βHSB P [Π |Π˜ ] tr(e tot Π ) P [ΠSB,0|ΠSB,1] tr(e |ψ0ihψ0| ⊗ Π0) 1 0 = 1 . (35) ∗ ∗ −βH −βH1 −βHS P [Π |Π˜ ] tr(e tot Π0) tr(e ) tr(e |ψ ihψ |) 0 1 = 1 1 tr(e−βH0 ) tr(e−βHS |ψ ihψ |) This is the key relation that we will use to obtain the 0 0 −β∆F −∆Λ fluctuation relations. = e , (36)

Accepted in Quantum 2019-10-24, click title to verify 14 −βH where βFk = − log tre k is the free energy of B at distribution pk over energy. Under the assumption that time t = k, ∆F = F1 − F0, and ∆Λ := Λ(β, ψ1) − all cumulants are finite, one immediately sees that in the Λ(β, ψ0), where we define Λ(β, ρ) the effective potential high temperature regime one recovers a classical form of a state ρ at inverse temperature β as for the coherent Crooks relation in which the first order β term dominates. Λ(β, ρ) := − log tr(e−βHS ρ). (37) Corollary 3.1.1. In the high temperature limit, With the understanding that the Hamiltonian of B ˜ P [ψ1|ψ0] 2 changes from H0 to H1 we can simply write = eβ(WB −∆F )+O(β ) (41) P [ψ∗|ψ˜∗] P [ψ |ψ˜ ] 0 1 1 0 =−β∆F −∆Λ, (38) ∗ ˜∗ where WB := hHBit=1 − hHBit=0 = −[hHSiψ − P [ψ0 |ψ1 ] 1 hHSiψ0 ]. as the final coherent Crooks relation for the ratio of the probabilities of a given forward coherent trajectory to Note that since the composite system SB is energet- the probability of its reverse trajectory. ically closed a change in energy in B, as measured by It is important to note that if one ranges over all pos- the first moment hHBi, is identified with a correspond- sible POVMs on S then the above relation is equivalent ing change in energy in S. A more interesting question is how to interpret the to the abstract channel relation that was first derived n n (−1) β ∆κn by Åberg in [23]. The difference here is that we focus higher order corrections n! for n > 1. These on projective measurements that admit a simple con- vanish if the state of S is an energy eigenstate, and so straint interpretation and introduce the effective poten- arise from the non-trivial coherent structure of the input tial. Both of these turn out to be key in our physical and output pure states on S. One might suspect that analysis of the relation. We now state the core quantum they are measures of quantum coherence, but this turns fluctuation theorem in terms of the effective potential out to not be the case, for example κ3 can both increase of the quantum states. and decrease under incoherent quantum operations and therefore cannot be a genuine measure of coherence [41, Theorem 3.1 (Effective potential form). Let H = 61]. H ⊗ 1 + 1 ⊗ H be a microscopic, time-reversal S B S B Despite the fact that a general cumulant κn is not invariant Hamiltonian and assume B begins in a ther- a measure of coherence it turns out that κ2 is distin- mal state at inverse temperature β = 1/(kT ). Addition- guished and admits such an interpretation. The second ally, assume the dynamics admit a microscopic, time- cumulant κ2 is the variance of energy in a quantum state reversal invariant unitary V such that [V,H] = 0. Then |ψi, and it has been shown that if one restricts to pure ˜ ∗ ˜∗ with P [ψ1|ψ0] and P [ψ0 |ψ1 ] defined as above, the tran- quantum states then in the asymptotic regime of many sition probabilities satisfy: copies of a state |ψi there is an essentially unique way ˜ to quantify coherence [9] between different eigenspaces, P [ψ1|ψ0] = e−β∆F −(Λ(β,ψ1)−Λ(β,ψ0)), (39) and is given by χ(ψ) := 4πκ (ψ). With this in mind, P [ψ∗|ψ˜∗] 2 0 1 we next provide a decomposition of the effective poten- where ∆F = (F1 − F0) is the difference in free energies tial Λ(β, ψ) of a pure quantum state into energy and with respect to the final and initial Hamiltonians of B, coherence contributions. and Λ(β, ρ) is the effective potential for any state ρ on S at inverse temperature β. 3.8 Separation of the effective potential into en- The effective potential Λ(β, ψ) can be viewed as a log- ergetic and coherent contributions arithm of the Laplace transform for the quantum state We can show that the general fluctuation relation takes |ψi of S with respect to HS, and as such it corresponds a particularly simple form by exploiting how Λ(β, ψ) to the cumulant generating function [60] for the mea- varies as a function of the inverse temperature β. We as- surement statistics of energy in the quantum state |ψi. sume that Λ(β, ρ) is differentiable to second-order with Thus by expansion in terms of the cumulants, the fluc- respect to β. Firstly, it is clear that Λ(0, ρ) = 0 for any tuation relation can be re-expressed as quantum state ρ, and moreover that

˜ P n βn P [ψ1|ψ0] −β∆F − (−1) ∆κn = e n≥1 n! , (40) ∂βΛ(β, ρ)|β=0 = hHiρ. (42) P [ψ∗|ψ˜∗] 0 1 2 Looking at the second derivative ∂βΛ(β, ρ) we find that th where ∆κn = κn(p1) − κn(p0), and κn(pk) is the n 2 −βH  −βH 2 cumulant for the random variable obtained if one mea- 2 tr[ρH e ] tr[ρHe ] − ∂βΛ(β, ρ) = − . (43) sured HS in the state |ψki, and which has probability tr(e−βH ρ) tr(e−βH ρ)

Accepted in Quantum 2019-10-24, click title to verify 15 The right-hand side of this only depends on the dis- the amount of coherence in the physically prepared ini- ˜ tribution p = (pk) over energy eigenstates for ρ. We tial state Γ(ψ) = ψ after Gibbs re-scaling. However therefore define pk = tr[Πkρ] where Πk is the projector more generally, χm is the amount of coherence in the ˜ onto the energy eigenspace with energy Ek, and so pure state ∝ exp[−(βm −β)HS]|ψi with βm determined !2 by the energy statistics of the quantum state. We go P 2 −βEk P −βEk [pkE e ] [pkEke ] into more detail on χ in the next section. −∂2Λ(β, ρ) = k k − k . m β P −βEj P −βEj j(e pj) j(e pj) In terms of the mean coherence the core fluctuation (44) relation can be re-stated as follows. Now define the distribution Theorem 3.2 (Mean coherence & mean energy e−βEk decomposition). Let the assumptions Theorem 3.1 p˜k = pk for all k. (45) P −βEj hold. Then with χm(˜ρ) defined as before at an inverse j e pj temperature β, for any two pure states |ψ0i and |ψ1i, This is a Gibbs re-scaling of the distribution pk at the fluctuation theorem takes the form inverse temperature β. In particular, we see that ˜ 2 2 P [ψ1|ψ0] −β∆F −βW + β ∆χ ∂βΛ(β, ρ) = −κ2(p˜), the variance of energy under the = e S 8π , (50) ∗ ˜∗ re-scaled distribution p˜k. P [ψ0 |ψ1 ] We can now apply the second–order Mean Value The- ˜ where WS = hHSiψ1 − hHSiψ0 and ∆χ = χm(ψ1) − orem to Λ(β, ρ) to deduce that for some βm ∈ [0, β] we ˜ have χm0 (ψ0). This fully separates the change in the first moment Λ(β, ρ) = Λ(0, ρ) + β∂βΛ(β, ρ)|β=0 (46) hHSi from the higher order corrections. Note that 1 ˜ ˜ + β2 ∂2Λ(β, ρ)|  χm(ψ1) and χm0 (ψ0) are in general evaluated at two 2 β β=βm different temperatures βm and βm0 respectively. Such a 1 = βhHi + β2 ∂2Λ(β, ρ)|  , (47) difference depends on β and also the original statistics ρ 2 β β=βm of each state |ψ0i and |ψ1i. and thus we have that in general 1 3.9 An explicit form for the mean coherence at Λ(β, ρ) = βhHi − β2χ (˜ρ), (48) ρ 8π m inverse temperature β where we have used that χ(ρ) = 4πκ2(p), with the The above form of χm, and its dependence on tempera- short-hand notation ρ˜ = Γ(ρ). Here χm(˜ρ) is non-trivial ture is slightly opaque, however one can obtain a simple and should be interpreted with care. It involves com- expression for computing it that makes clear the differ- puting χ on a Gibbs re-scaling of the quantum state, ent contributions. To this end we can make use of the and then evaluated at an effective inverse temperature following result. βm ≤ β that is determined by the statistics of the orig- inal state. Lemma 3.3. Given a quantum system S with Hamil- The utility of this is that for a pure state |ψi, the tonian HS and {Πk} the projectors onto the energy ˜ eigenspaces of HS, let us denote the de-phased form of re-scaled state |ψi is also pure, and thus χ, the vari- P ance in energy, is always a genuine measure of coher- state in the energy basis by D(ρ) = k ΠkρΠk. Then, ence when restricted to pure states [10]. Therefore, the for any quantum state ρ, the effective potential Λ(β, ρ) higher order cumulants for n > 1 arise from the co- is given by herent structure of |ψi and while individual cumulants Λ(β, ρ) = min{βhHSiσ + S(σ||D(ρ))}, (51) are not coherence measures, one can still deduce that σ the sum of all higher order terms can be reduced to a where the minimization is taken over all quantum states single coherence measure of a rescaled pure quantum σ of S, and S(σ||ρ) = tr[σ log σ − σ log ρ] is the rel- ˜ state. We refer to χm(ψ) as the mean coherence at in- ative entropy function. Moreover, the minimization is verse temperature β of the pure quantum state |ψi, and attained for the state σ = Γ(D(ρ)). it allows the effective potential to split in a remarkably simple way as The proof of this is provided in the Appendix B.3. If the probability distribution over energy of ρ is p = (mean energy) (mean coherence) Λ(β, ψ) = − , (49) (pk) then this is unaffected by the dephasing ρ 7→ D(ρ). kT 8π(kT )2 Denoting by p˜ the Gibbs rescaling of p, we therefore ˜ have that however the way in which χm(ψ) corresponds to “mean coherence” is subtle. When βm = β then χm is precisely Λ(β, ρ) = βhHSip˜ + S(p˜||p), (52)

Accepted in Quantum 2019-10-24, click title to verify 16 where the relative entropy term is now the classical rel- [8, 42, 63]. In addition, given any elements M0,M1 P ative entropy for distributions and hHSip˜ := k Ekp˜k that occur in some POVM M = {Mk}, one can always is the expectation value of energy for the classical dis- include them in a covariant measurement on the sys- tribution p˜. We thus have the following. tem, by simply considering the orbit of each under the adjoint action X → U(t)XU(t)† for all t and if these Theorem 3.4. Given a quantum system S with Hamil- are not in the measurement set, expanding the set to tonian HS, the mean coherence at inverse temperature include them. Thus, a POVM M = {Mk} exists that β of a quantum state |ψi is given by contains both M0 and M1 and the total non-selective 2 evolution is time-translation covariant. Therefore there β ˜ χm(ψ) = β(hHSip − hHSip˜) − S(p˜||p). (53) is no coherence being exploited to perform the measure- 8π ment [64] and so over the whole procedure on S we have where p = (pk) is the distribution over energy of ψ and that the coherence in S can never increase. p˜ is the distribution over energy for |ψ˜ihψ˜| = Γ(|ψihψ|).

Everything on the right-hand side of Equation (53) 4 Coherent work processes, quantum is easily computable, and its form sheds light on what fluctuation relations and the semi- the mean coherence at inverse temperature β actually depends on. The first term quantifies the degree to classical limit. which the coherent thermalisation raises or lowers the expected energy of |ψi in units of β, while the second We can now show that the earlier notion of a coherent term quantifies the degree to which the coherent ther- work process fits neatly into the inclusive quantum fluc- malisation makes the state |ψ˜i distinguishable from the tuation setting. Recall that the classical Crooks relation state |ψi (in the sense of hypothesis testing optimized takes the form over all possible quantum measurements [62]). Since the P [w|γ0, P] −β(∆F −w) left-hand side is always non-negative, we also see that ∗ = e (56) P [−w|γ1, P ] energy term is never smaller than the relative entropy term. that relates the probability of an amount of work w for a forward trajectory with −w for the reversed trajec- Example 3.5. Consider the system S in the uni- tory. As already discussed this quantity w is always as- form superposition over d energy eigenstates |ψi = sociated with some auxiliary system within an inclusive d−1 √1 P |E i. Then, as discussed earlier, this trans- d n=0 n description. It is readily seen that the previous quan- forms under Γ to the coherent Gibbs state |ψ˜i = |γi. tum fluctuation relations can be re-cast in the following Then by direct calculation with equation (53), we find form that provides a natural extension of the classical that result. β2  1  χ (|γihγ|) = β[F 1 − F (γ)], (54) Theorem 4.1 (Fluctuations and coherent work 8π m d d processes). Let S be a quantum system with Hamilto- where F (ρ) := hHiρ − kBTS(ρ) is the free-energy of nian HS and let |ψ0iS and |ψ1iS be two pure states of the state ρ and γ is the Gibbs state. Thus the mean S that have energy statistics with finite cumulants of all coherence is proportional to the difference between the orders. Then the following are equivalent: free energy of S with a trivial Hamiltonian HS = 0 and the free energy of S with non-trivial Hamiltonian 1. The pure states |ψ0iS and |ψ1iS are coherently con- nected with coherent work output/input ω on an HS 6= 0 at inverse temperature β. auxiliary system A. Finally, note that while χ is a genuine measure of 2. Within the fluctuation relation context with a ther- coherence for any pure quantum state, this is different mal system B and quantum system S we have that from χm being a monotone under the general dynamics of the setting. While this is not immediately obvious ˜ P [ψ1|ψ0] −β(∆F ±kT Λ(β,ω)) from the discussion, we highlight that the reduced dy- = e , (57) P [ψ∗|ψ˜∗] namics for any state ρ of S is of the form 0 1 for all inverse temperatures β ≥ 0, for some state E(ρ) = tr V (ρ ⊗ γ )V †, (55) B 0 |ωiA with finite cumulants on an auxiliary system A and some choice of sign before Λ(β, ω). and it is readily seen that this quantum operation is time-translation covariant, and so the total coherence We note that for |ψ0iS and |ψ1iS being energy eigen- between energy eigenspaces of S can never increase states we have that |ωiA is an energy eigenstate |wiA,

Accepted in Quantum 2019-10-24, click title to verify 17 verge, and therefore we have that the coherent frame- work is a faithful and consistent extension of the classi- cal work relations into quantum mechanics.

4.2 Mixed state coherent work output/input The connection with coherent work processes can be made more explicit by observing that the Gibbs state γ0 is a probabilistic mixture of energy eigenstates and therefore the unitary part of the dynamics can be writ- ten as

0 −βEk Figure 6: Coherent work processes (CWP) and thermal in- † X e 0 0 † V (ψ0 ⊗ γ0)V = V (|ψ0ihψ0| ⊗ |EkihEk|)V . teractions (TI). We introduced coherent work processes which Z0 are deterministic processes that transition a pure state to an- k other pure state. Thermal interactions are stochastic, hence (58) the output is a probabilistic mixture of pure states. In the In general |ϕ i = V [|ψ i⊗|E0i] is not a product state context of the fluctuation relation, we post-select a particular k SB 0 k pure state. The coherent work output/input |ωi associated to over S and B, and so does not describe a coherent work the initial and final quantum states contributes Λ(β, ω) to the process in itself, however the projective measurement 1 fluctuation relation and generalises the classical term βw from on S will collapse |ϕki to a product state |ψ1i ⊗ |φki the standard Crooks relation. and so the overall effect is the probabilistic transition 0 1 |ψ0i ⊗ |Eki → |ψ1i ⊗ |φki. Thus the particular energy 0 eigenstate |Eki in the Gibbs ensemble acts as a refer- for which Λ(β, ω) = βw for some sharp value w ∈ , and 0 1 R ence energy level and |Eki → |φki can be viewed as a so the above expression reduces to a classical Crooks 0 coherent work output conditioned on |Eki. relation form. The proof of the theorem is provided in We can now link with more abstract formalism by Appendix B.4. considering the evolution E of S and looking at what is called the complementary channel [65] E¯ from S into B 4.1 Physical interpretation of the result given by E¯(ρ) := tr V (ρ ⊗ γ )V †. (59) Interpreting this theorem requires some care. State- S 0 ment 1 is simply a deterministic transformation between This provides a stochastic mixture of coherent work out- pure states with coherence on a quantum system S. puts, from the system S into the bath system B. In contrast, statement 2 involves a stochastic thermal In fact this perspective can be taken as a more gen- process in which initial coherence in S interacts with a eral formulation of coherent work processes in which a thermal environment B and projective measurements general state ρ of a system S transforms through some at the start and at the end is assumed. Moreover this energy-conserving interaction with a second system A coherence is not outputted to some ordered degree of in a default energy eigenstate |0ih0| as ρ → E(ρ) = † freedom but is dissipated into the large thermal envi- trAV (ρ ⊗ |0ih0|)V . Then we define a general coherent ronment B. The appearance of an auxiliary system A work process as ω in statement 2 is independent of the actual thermody- ρ −→ E(ρ). (60) namic context and takes no part in the process. where ω = E¯(ρ) is now the mixed state coherent output This shows that coherent work processes and coher- for the process. ently connectedness relate to the coherent fluctuation The set of such quantum transformations are simply in a simple way, via the coherent work output/input the time-translation covariant quantum operations [8, ωA. Moreover, the numerical contribution of ωA to the 42, 63], and so in the same way as before we can say that fluctuation relation exponent is given simply by the ef- two quantum states ρ and σ are coherently connected if fective potential of ωA. From a structural perspective there exists a time-translation covariant channel E from we may therefore summarise this conclusion as follows: one state to the other. The coherent work output/input ¯ Coherent work processes are to coherent fluctuation is given via the complementary channel E acting on the relations what deterministic Newtonian work processes input system. We leave the study of such features to are to classical fluctuation relations. future work, where coherent work processes could prove useful tools in studying asymmetry theory. Moreover in the semi-classical regime, where coherence In the next section, we return to more concrete sys- becomes negligible, the two fluctuation relations con- tems, and briefly outline how coherent work processes

Accepted in Quantum 2019-10-24, click title to verify 18 are experimentally accessible in existing trapped ion The interpretation of this is quite natural, with only proposals. the term W¯ needing care. We first note that W¯ is the average of two changes in pure state energy: the energy 4.3 Coherent fluctuation relations in the experi- change in |α0i → |α1i and its Gibbs rescaled version |α˜0i → |α˜1i. Since Gibbs rescaling leaves energy eigen- mental semi-classical regime states unchanged this implies that W¯ reduces to the We have shown that our analysis of coherent work pro- classical sharp energy transition in the case of zero co- cesses, in which classical deterministic transitions are herences, otherwise the coherences provide a non-trivial superposed together while respecting energy conserva- distortion for a semi-classical ‘work’ term4. By compar- tion, appears naturally within the fluctuation relation. ing with the previous analysis one sees that W¯ is not We can now re-visit the case that the quantum system simply a change in first moments of H, and thus incor- S is a harmonic oscillator system, and restrict to states porates part of the mean coherence contribution from in C from Section 2.2, which are closed under coherent χm. work processes. This is particularly of interest because a The reason that this form of the fluctuation relation fluctuation relation for this system can be demonstrated is natural is that coherent states are the “most classi- within existing trapped ion systems [25], and therefore cal” states for the quantum system (they saturate the shows that the theoretical analysis presented here is of Heisenberg bound), and so gives rise to a fluctuation relevance to existing experimental work. relation that is as close to the classical one as possible We can without loss of generality restrict to the while still having coherent structure. This is reinforced canonical states |α, ki in the set C, and to simplify by the fact that all the terms on the right-hand side, things further we consider the subset formed of pure except for the ‘work’ term W˜ are simple equilibrium coherent states (with k = 0), although the fully gen- properties of a quantum system – as in the standard eral case can also be easily computed. This gives rise Crooks relation. The principle difference is that the to the same fluctuation relation as in [25], which fol- equilibrium temperature term kT for the work is re- lowed from the analysis of [23]. For a coherent state placed by hνth which is a quantum mechanical equilib- |αi of a quantum harmonic oscillator with Hamiltonian rium property, related to the de Broglie thermal wave- † 1 length. This shows that in the high temperature regime HS = hν(a a+ 2 ), the effective potential is readily com- puted and takes the form where kT  hν, the equilibrium thermal fluctuations dominate the de Broglie thermal wavelength and we re- 1 1  1  cover the classical regime hν → kT . In an intermedi- Λ(β, α) = βhν + hH i − hν (1 − e−βhν ), th 2 hν S α 2 ate temperature regime we get a smooth temperature- (61) dependent distortion of the standard Crooks relation, 2 where hHSiα := hα|HS|αi = hν(|α| + 1/2). while for very low temperatures we have hνth → 1/2hν A straightforward calculation leads to the follow- in a high coherence regime – namely it is now the vac- ing quantum fluctuation relation restricted to coherent uum fluctuations that dominate in the fluctuation rela- states of an oscillator system S. tion. One can also readily compute the mean coherence χm Theorem 4.2 (Semi-classical relation [25]). Let the for the oscillator system, and find that for any coherent assumptions of Theorem 3.1 hold, and let S is a har- state |αi it takes the form † 1 monic oscillator with Hamiltonian HS = hν(a a + ). 2 2 2 −βhν Then for two coherent states of the system |α0i and χm = 8π|α| (kT ) [βhν + e − 1]. (64) |α1i, the following holds Expansion of the exponential gives that χm =   4π|α|2[(hν)2 − 1 β(hν)3 + ··· ] and so we see the tem- P [α1|α˜0] ∆F W¯ B 3 ∗ ∗ = exp − + , (62) perature dependence only begins at cubic order in hν. P [α0|α˜1] kT hνth We also see that limh→0 χm = 0, for all temperatures T , and so the mean coherence χm is a purely quantum- where hνth = hHSiγ is the average energy of a Gibbs state γ of a quantum harmonic oscillator, related to the mechanical feature that disappears in the classical limit. Perhaps more interestingly, we see that χm vanishes in thermal de Broglie wavelength λdB(T ) via the T → 0 limit also. Therefore, since χm only con- h2 1 tributes when we simultaneously have both a non-zero hν = + hν, (63) th mλ (T )2 2 dB 4We do not refer to this as “the work” since it is purely re- stricted to the semi-classical case of coherent states, and there is ¯ 1 ˜ and WB := − 2 (WS + WS), WS = hHSiα1 − hHSiα0 , no reason to believe that a physically sensible and unique “work” ˜ quantity should exist for general quantum systems. WS = hHSiα˜1 − hHSiα˜0 .

Accepted in Quantum 2019-10-24, click title to verify 19 Planck’s constant h and non-zero temperature T , the In the large n limit we know from the central limit mean coherence is a genuinely quantum-thermodynamic theorem that the distributions over energy will tend to property of the system. a Gaussian in the neighborhood around the mean en- ergy. As we show in the Supplementary Material, the 4.4 The macroscopic regime and multipartite fluctuation theorem takes the following form: for any entanglement.  > 0 there is an M such that for all n > M we have ˜ P [ψn|φn] −n(β∆f+β∆µ− 1 β2∆σ2) One method of characterising the classical regime is by − e 2 ≤ , (68) ∗ ˜∗ letting thermal fluctuations dominate quantum fluctu- P [φn|ψn] ations, as we showed. But one expects a notion ‘clas- however with the proviso that the scaling β = √β0 is sicality’ to also arise in the large system limit, where nλ adopted, where β0 is a dimensionless unit and λ a char- the state of the system is given in terms of a small set acteristic energy scale – which is technically required in of intrinsic variables. In this section we show that if order to apply the central limit theorem rigorously, and the system S is composed of many independent, iden- amounts to a choice of units for temperature for a fix tically distributed (IID) systems S1, ..., Sn, all higher scale n. This captures the informal statement that for order corrections bar the variance vanish in the limit the IID case the states “become more like Gaussians” n → ∞, under an appropriate scaling of energy units. and Given a macroscopic regime in which a quantum sys- ˜ P [ψn|φn] n→∞ −n(β∆f+β∆µ− 1 β2∆σ2) ∼ e 2 (69) tem with no correlations is well-described by a bounded P [φ∗ |ψ˜∗ ] list of intrinsic parameters, the total quantum state can n n ⊗n Thus in the macroscopic limit, the central limit theo- be written as ρtot ≈ ρ , for some effective subsys- tem state ρ that encodes the intrinsic parameters. We rem can be used to truncate the exponent in the fluctu- ⊗n ation relation to the second-order cumulants, with the can now consider the macroscopic states |ψni := |ψi ⊗n usual provisos in this statement. and |φni := |φi on the reference system, for two pure non-energy eigenstates |ψi and |φi and see how the fluc- As in the asymptotic IID regime one reduces to only tuation relation behaves. the first two cumulants, we can use the second cumu- In particular, we consider a state |ψi = lant to place a bound on many-body entangled systems. √ Suppose we start with a state |φi⊗n and perform any Pd−1 p |mi of a d-dimensional system, where m=0 m unitary U that is block diagonal in the total Hamilto- {|mi} are the eigenstates of the Hamiltonian nian H = P H then in general the state U|φi⊗n will H =  Pd−1 m|mihm|. Looking at n copies of i i 1 m=0 be entangled between the n subsystems. But it is read- the system, † ily seen that Λ(β, UφnU ) = Λ(β, φn) and thus (a) in n(d−1) the large n limit will have a Gaussian profile in energy ⊗n X √ and (b) will generically be entangled between subsys- |ψi = cj|ejin, (65) j=0 tems. When the total system S is a large multi-partite sys- where we have decomposed |ψi⊗n into a superposition tem composed of n spin–1/2 particles one can exploit of energy eigenstates {|ejin} of n copies of the system the fact that the variance V ar(ψ, σi) of a Pauli operator [9], where |ejin has total energy j and it is assumed Htot σi in a pure quantum state |ψi is upper bounded when is the sum of the non-interacting individual Hamiltoni- limited to k–producible states. A state |ψi is called ans. The coefficients {cj} are then described by multi- k–producible if it can be written in the form: nomial coefficients |ψi = |φ i ⊗ |φ i ⊗ ... ⊗ |φ i, (70)   1 2 m n k c = pk0 pk1 ...p d−1 (66) j k ...k 0 1 d−1 where m ≥ n/k and each pure state |φji involves at 0 d−1 most k particles [66] (which captures the notion of mul- P P where j = i i ki and n = i ki. Furthermore, if the tipartite entanglement). It is clear from the defini- spectrum {pj} is gapless, with pj 6= 0 for 0 ≤ j ≤ d − 1, tion that 1–producible states are product states with then in the asymptotic limit the distribution cj over m = n, containing no entanglement. A pure state energies becomes [9] |ψi has genuine k–partite entanglement if |ψi is k– producible but not (k − 1)–producible. Now for such 2 1 − (j−µn(ψ)) 2σ2 (ψ) −1
a state, the variance of a many-body Pauli spin opera- cj = e n + O n , (67) p 2 (n) (1) (n) 2πσn tor Σi = σi + ··· + σi , where i = x, y, z and the superscript denotes the qubit label, obeys where µn(ψ) = hψn|Htot|ψni = nhψ|H1|ψi = nµ and 2 2 (n) 2 2 σn(ψ) = nσ is the variance of Htot evaluated on |ψni. 4Var(ψ, Σi ) ≤ sk + (n − sk) (71)

Accepted in Quantum 2019-10-24, click title to verify 20 on the set of k–producible states [67, 68]. Here s is the 6 Acknowledgements integer part of n/k and the bound ranges from n up to n2. We would like to thank Johan Åberg, Zoë Holmes, By using the above bound we see that the largest Hyukjoon Kwon, Myungshik Kim, Jack Clarke and possible changes in variance will be ±[sk2 + (n − sk)2]. Doug Plato for useful discussions. EHM is funded by Therefore for a system S composed of n particles, with the EPSRC Centre for Doctoral Training in Controlled each subsystem having a Hamiltonian Hi = σz giving Quantum Dynamics. DJ is supported by the Royal So- (n) ciety and also a University Academic Fellowship. a total Hamiltonian HS =  Σz ,

˜ 2 √ P [ψ|φn] β(W −∆F ) (β) (sk2+(n−sk)2)+O(1/ n) ≤ e B e 8 . References ∗ ˜∗ P [φn|ψ ] (72) [1] P.A.M. Dirac. The Principles of Quantum Me- chanics. Comparative Pathobiology - Studies in where WB = −[hHSiψ − hHSiφn ] is the change in en- ergy of the bath subsystem. This holds for any large n the Postmodern Theory of Education. Clarendon IID product state |φni and any k–producible state |ψi Press, 1981. ISBN 9780198520115. URL https:// with genuine k–partite entanglement. While this bound books.google.co.uk/books?id=XehUpGiM6FIC. (which turns out to be tight [67]) grows weaker and [2] A. Peres. Quantum Theory: Concepts and weaker as n increases, it is nonetheless striking that one Methods. Fundamental Theories of Physics. can apply the fluctuation relation to provide non-trivial Springer Netherlands, 2006. ISBN 9780306471209. statements on transitions between quantum states with URL https://books.google.co.uk/books?id= different multipartite entanglement and it would be of pQXSBwAAQBAJ. interest to further develop this connection further. [3] Anton Zeilinger. Experiment and the founda- tions of quantum physics. Rev. Mod. Phys., 71: S288–S297, Mar 1999. DOI: 10.1103/RevMod- Phys.71.S288. URL https://link.aps.org/doi/ 5 Outlook 10.1103/RevModPhys.71.S288. [4] C. Monroe, D. M. Meekhof, B. E. King, and D. J. Wineland. A “Schrödinger cat” superposi- We started by extending the core defining character- tion state of an atom. Science, 272(5265):1131– istics of deterministic work in classical mechanics into 1136, 1996. ISSN 0036-8075. DOI: 10.1126/sci- quantum mechanics, which lead to the notion of coher- ence.272.5265.1131. URL http://science. ent work processes. It turns out that the set of coher- sciencemag.org/content/272/5265/1131. ent processes is non-trivial for quantum systems, and it [5] Giulio Chiribella and Robert W. Spekkens, edi- would be of interest to extend this analysis to determine, tors. Quantum Theory: Informational Foundations for example, what quantum systems admit infinitely de- and Foils, volume 181 of Fundamental Theories of composable quantum coherence. Because of Theorem Physics. Springer Netherlands, 1st edition, 2016. 2.5 this can be determined from the large literature on DOI: 10.1007/978-94-017-7303-4. the decomposability of classical random variables. [6] R.P. Feynman, A.R. Hibbs, and D.F. Styer. Quan- Another open question is whether a mixed state ver- tum Mechanics and Path Integrals. Dover Books sion of coherent work processes leads to interesting on Physics. Dover Publications, 2010. ISBN physics, or could be related to on-going research in ther- 9780486477220. URL https://books.google. modynamics. The asymmetry theory perspective on co.uk/books?id=JkMuDAAAQBAJ. coherent work processes means that this is expected [7] H. Goldstein, C.P. Poole, and J.L. Safko. Clas- to have non-trivial structure, and so applying exist- sical Mechanics: Pearson New International Edi- ing asymmetry results within a thermodynamic context tion. Pearson Education Limited, 2014. ISBN would be of interest. 9781292038933. URL https://books.google. The fluctuation relations and the effective potentials co.uk/books?id=Xr-pBwAAQBAJ. discussed here suggest that a wide range of tools (e.g. [8] Iman Marvian and Robert W. Spekkens. Modes from many-body physics [69]) could be brought to bear of asymmetry: The application of harmonic on coherent fluctuation relations. It is essential to de- analysis to symmetric quantum dynamics and velop greater connection with existing fluctuation rela- quantum reference frames. Phys. Rev. A, tions, but it is conjectured that essentially any of the 90:062110, Dec 2014. DOI: 10.1103/Phys- existing extensions of the classical stochastic Crooks re- RevA.90.062110. URL https://link.aps.org/ lation could be faithfully represented in this setting. doi/10.1103/PhysRevA.90.062110.

Accepted in Quantum 2019-10-24, click title to verify 21 [9] Michael Skotiniotis and Gilad Gour. Alignment [20] Denis J. Evans and Debra J. Searles. Equi- of reference frames and an operational interpreta- librium microstates which generate second law tion for theG-asymmetry. New Journal of Physics, violating steady states. Phys. Rev. E, 50: 14(7):073022, jul 2012. DOI: 10.1088/1367- 1645–1648, Aug 1994. DOI: 10.1103/Phys- 2630/14/7/073022. URL https://doi.org/10. RevE.50.1645. URL https://link.aps.org/doi/ 1088%2F1367-2630%2F14%2F7%2F073022. 10.1103/PhysRevE.50.1645. [10] Gilad Gour and Robert W Spekkens. The resource [21] Christopher Jarzynski. Equalities and inequali- theory of quantum reference frames: manipula- ties: Irreversibility and the second law of ther- tions and monotones. New Journal of Physics, modynamics at the nanoscale. Annual Re- 10(3):033023, mar 2008. DOI: 10.1088/1367- view of Condensed Matter Physics, 2(1):329– 2630/10/3/033023. URL https://doi.org/10. 351, 2011. DOI: 10.1146/annurev-conmatphys- 1088%2F1367-2630%2F10%2F3%2F033023. 062910-140506. URL https://doi.org/10.1146/ [11] J. A. Vaccaro, F. Anselmi, H. M. Wiseman, annurev-conmatphys-062910-140506. and K. Jacobs. Tradeoff between extractable [22] Michele Campisi, Peter Hänggi, and Peter Talkner. mechanical work, accessible entanglement, and Colloquium: Quantum fluctuation relations: Foun- ability to act as a reference system, under ar- dations and applications. Rev. Mod. Phys., bitrary superselection rules. Phys. Rev. A, 83:771–791, Jul 2011. DOI: 10.1103/RevMod- 77:032114, Mar 2008. DOI: 10.1103/Phys- Phys.83.771. URL https://link.aps.org/doi/ RevA.77.032114. URL https://link.aps.org/ 10.1103/RevModPhys.83.771. doi/10.1103/PhysRevA.77.032114. [23] Johan Åberg. Fully quantum fluctuation [12] Cristina Cirstoiu and David Jennings. Irreversibil- theorems. Phys. Rev. X, 8:011019, Feb ity and quantum information flow under global and 2018. DOI: 10.1103/PhysRevX.8.011019. local gauge symmetries. arXiv:1707.09826, 2017. URL https://link.aps.org/doi/10.1103/ [13] E. Lukacs. Characteristic Functions. Griffin PhysRevX.8.011019. books of cognate interest. Hafner Publishing Com- [24] Álvaro M. Alhambra, Lluis Masanes, Jonathan pany, 1970. URL https://books.google.co.uk/ Oppenheim, and Christopher Perry. Fluctuat- books?id=uGEPAQAAMAAJ. ing work: From quantum thermodynamical iden- [14] Eugene Lukacs. A survey of the theory of char- tities to a second law equality. Phys. Rev. acteristic functions. Advances in Applied Prob- X, 6:041017, Oct 2016. DOI: 10.1103/Phys- ability, 4(1):1–38, 1972. ISSN 00018678. URL RevX.6.041017. URL https://link.aps.org/ http://www.jstor.org/stable/1425805. doi/10.1103/PhysRevX.6.041017. [15] D. Raikov. On the decomposition of Gauss and [25] Zoë Holmes, Sebastian Weidt, David Jennings, Poisson laws. Izv. Akad. Nauk SSSR Ser. Mat., 2 Janet Anders, and Florian Mintert. Coher- (1):91–124, 1938. ent fluctuation relations: from the abstract [16] I. Z. RUZSA. Arithmetic of probability distribu- to the concrete. Quantum, 3:124, February tions. Séminaire de Théorie des Nombres de Bor- 2019. ISSN 2521-327X. DOI: 10.22331/q-2019- deaux, pages 1–12, 1982. ISSN 09895558. URL 02-25-124. URL https://doi.org/10.22331/ http://www.jstor.org/stable/44166410. q-2019-02-25-124. [17] Andreas Winter and Dong Yang. Operational [26] Hyukjoon Kwon and M. S. Kim. Fluctuation resource theory of coherence. Phys. Rev. Lett., theorems for a quantum channel. Phys. Rev. 116:120404, Mar 2016. DOI: 10.1103/Phys- X, 9:031029, Aug 2019. DOI: 10.1103/Phys- RevLett.116.120404. URL https://link.aps. RevX.9.031029. URL https://link.aps.org/ org/doi/10.1103/PhysRevLett.116.120404. doi/10.1103/PhysRevX.9.031029. [18] Benjamin Morris and Gerardo Adesso. Quan- [27] Martí Perarnau-Llobet, Elisa Bäumer, Karen V. tum coherence fluctuation relations. Journal Hovhannisyan, Marcus Huber, and Antonio of Physics A: Mathematical and Theoretical, 51 Acin. No-go theorem for the characteriza- (41):414007, sep 2018. DOI: 10.1088/1751- tion of work fluctuations in coherent quantum 8121/aac115. URL https://doi.org/10.1088% systems. Phys. Rev. Lett., 118:070601, Feb 2F1751-8121%2Faac115. 2017. DOI: 10.1103/PhysRevLett.118.070601. [19] G. E. Crooks. Nonequilibrium measurements URL https://link.aps.org/doi/10.1103/ of free energy differences for microscopically re- PhysRevLett.118.070601. versible markovian systems. J. Stat. Phys., 90, [28] A. E. Allahverdyan. Nonequilibrium quan- 1998. DOI: 10.1023/A:1023208217925. URL tum fluctuations of work. Phys. Rev. E, https://doi.org/10.1023/A:1023208217925. 90:032137, Sep 2014. DOI: 10.1103/Phys-

Accepted in Quantum 2019-10-24, click title to verify 22 RevE.90.032137. URL https://link.aps.org/ ume 54 of Graduate Studies in Mathematics. Amer- doi/10.1103/PhysRevE.90.032137. ican Mathematical Society, 2002. [29] Peter Talkner and Peter Hänggi. Aspects [39] P. Solinas, H. J. D. Miller, and J. Anders. of quantum work. Phys. Rev. E, 93:022131, Measurement-dependent corrections to work dis- Feb 2016. DOI: 10.1103/PhysRevE.93.022131. tributions arising from quantum coherences. Phys. URL https://link.aps.org/doi/10.1103/ Rev. A, 96:052115, Nov 2017. DOI: 10.1103/Phys- PhysRevE.93.022131. RevA.96.052115. URL https://link.aps.org/ [30] Peter Talkner, Eric Lutz, and Peter Hänggi. Fluc- doi/10.1103/PhysRevA.96.052115. tuation theorems: Work is not an observable. Phys. [40] Juliette Monsel, Cyril Elouard, and Alexia Auf- Rev. E, 75:050102, May 2007. DOI: 10.1103/Phys- fèves. An autonomous quantum machine to mea- RevE.75.050102. URL https://link.aps.org/ sure the thermodynamic arrow of time. npj doi/10.1103/PhysRevE.75.050102. Quantum Information, 4(1):59, 2018. DOI: [31] Leonard Susskind and Jonathan Glogower. 10.1038/s41534-018-0109-8. URL https://doi. Quantum mechanical phase and time oper- org/10.1038/s41534-018-0109-8. ator. Physics Physique Fizika, 1:49–61, Jul [41] Iman Marvian Mashhad. Symmetry, Asymmetry 1964. DOI: 10.1103/PhysicsPhysiqueFizika.1.49. and Quantum Information. PhD thesis, 2012. URL URL https://link.aps.org/doi/10.1103/ http://hdl.handle.net/10012/7088. PhysicsPhysiqueFizika.1.49. [42] Matteo Lostaglio, Kamil Korzekwa, David [32] S Sivakumar. Studies on nonlinear coherent Jennings, and Terry Rudolph. Quantum states. Journal of Optics B: Quantum and Semi- coherence, time-translation symmetry, and classical Optics, 2(6):R61–R75, nov 2000. DOI: thermodynamics. Phys. Rev. X, 5:021001, Apr 2015. DOI: 10.1103/PhysRevX.5.021001. 10.1088/1464-4266/2/6/02. URL https://doi. URL org/10.1088%2F1464-4266%2F2%2F6%2F02. https://link.aps.org/doi/10.1103/ PhysRevX.5.021001. [33] G. S. Agarwal and K. Tara. Nonclassical properties [43] David Jennings and Matthew Leifer. No of states generated by the excitations on a coherent return to classical reality. Contempo- state. Phys. Rev. A, 43:492–497, Jan 1991. DOI: rary Physics, 57(1):60–82, 2016. DOI: 10.1103/PhysRevA.43.492. URL https://link. 10.1080/00107514.2015.1063233. URL https: aps.org/doi/10.1103/PhysRevA.43.492. //doi.org/10.1080/00107514.2015.1063233. [34] A. Mahdifar, E. Amooghorban, and M. Ja- [44] Harry J D Miller and Janet Anders. Time-reversal fari. Photon-added and photon-subtracted co- symmetric work distributions for closed quantum herent states on a sphere. Journal of Math- dynamics in the histories framework. New Jour- ematical Physics, 59(7):072109, 2018. DOI: nal of Physics, 19(6):062001, jun 2017. DOI: 10.1063/1.5031036. URL https://doi.org/10. 10.1088/1367-2630/aa703f. URL https://doi. . 1063/1.5031036 org/10.1088%2F1367-2630%2Faa703f. [35] Luke C. G. Govia, Emily J. Pritchett, Seth T. [45] Nicole Yunger Halpern. Jarzynski-like equality Merkel, Deanna Pineau, and Frank K. Wilhelm. for the out-of-time-ordered correlator. Phys. Rev. Theory of Josephson photomultipliers: Optimal A, 95:012120, Jan 2017. DOI: 10.1103/Phys- working conditions and back action. Phys. Rev. RevA.95.012120. URL https://link.aps.org/ A, 86:032311, Sep 2012. DOI: 10.1103/Phys- doi/10.1103/PhysRevA.95.012120. RevA.86.032311. URL https://link.aps.org/ [46] Tameem Albash, Daniel A. Lidar, Milad Mar- doi/10.1103/PhysRevA.86.032311. vian, and Paolo Zanardi. Fluctuation the- [36] Luke C G Govia, Emily J Pritchett, and orems for quantum processes. Phys. Rev. Frank K Wilhelm. Generating nonclassical states E, 88:032146, Sep 2013. DOI: 10.1103/Phys- from classical radiation by subtraction measure- RevE.88.032146. URL https://link.aps.org/ ments. New Journal of Physics, 16(4):045011, doi/10.1103/PhysRevE.88.032146. apr 2014. DOI: 10.1088/1367-2630/16/4/045011. [47] Matteo Lostaglio. Quantum fluctuation the- URL https://doi.org/10.1088%2F1367-2630% orems, contextuality, and work quasiproba- 2F16%2F4%2F045011. bilities. Phys. Rev. Lett., 120:040602, Jan [37] David G. Kendall. On infinite doubly-stochastic 2018. DOI: 10.1103/PhysRevLett.120.040602. matrices and birkhoffs problem 111. Journal of URL https://link.aps.org/doi/10.1103/ the London Mathematical Society, s1-35(1):81–84, PhysRevLett.120.040602. 1960. DOI: 10.1112/jlms/s1-35.1.81. [48] Simon Kochen and E. P. Specker. The problem [38] Alexander Barvinok. A course in convexity, vol- of hidden variables in quantum mechanics. Jour-

Accepted in Quantum 2019-10-24, click title to verify 23 nal of Mathematics and Mechanics, 17(1):59–87, 131, Mar 1986. DOI: 10.1007/BF01212345. URL 1967. ISSN 00959057, 19435274. URL http: https://doi.org/10.1007/BF01212345. //www.jstor.org/stable/24902153. [59] Philippe Faist. Quantum coarse-graining: An [49] Ernesto F. Galvao. Foundations of quantum information-theoretic approach to thermodynam- theory and quantum information applications. ics. arXiv:1607.03104, 2016. arXiv:quant-ph/0212124, 2002. [60] W. Feller. An introduction to probability the- [50] Robert Raussendorf. Contextuality in ory and its applications. Number v. 1 in Wi- measurement-based quantum computation. Phys. ley series in probability and mathematical statis- Rev. A, 88:022322, Aug 2013. DOI: 10.1103/Phys- tics. Probability and mathematical statistics. Wi- RevA.88.022322. URL https://link.aps.org/ ley, 1968. ISBN 9780471257080. URL https:// doi/10.1103/PhysRevA.88.022322. books.google.co.uk/books?id=mfRQAAAAMAAJ. [51] Markus Frembs, Sam Roberts, and Stephen D [61] Iman Marvian and Robert W. Spekkens. Asym- Bartlett. Contextuality as a resource for metry properties of pure quantum states. Phys. measurement-based quantum computation be- Rev. A, 90:014102, Jul 2014. DOI: 10.1103/Phys- yond qubits. New Journal of Physics, 20(10): RevA.90.014102. URL https://link.aps.org/ 103011, oct 2018. DOI: 10.1088/1367-2630/aae3ad. doi/10.1103/PhysRevA.90.014102. URL https://doi.org/10.1088%2F1367-2630% [62] Tomohiro Ogawa and Hiroshi Nagaoka. 2Faae3ad. Strong Converse and Stein’s Lemma in [52] Sebastian Deffner and Christopher Jarzyn- Quantum Hypothesis Testing, pages 28–42. ski. Information processing and the second DOI: 10.1142/9789812563071_0003. URL law of thermodynamics: An inclusive, hamil- https://www.worldscientific.com/doi/abs/ tonian approach. Phys. Rev. X, 3:041003, 10.1142/9789812563071_0003. Oct 2013. DOI: 10.1103/PhysRevX.3.041003. [63] Matteo Lostaglio, David Jennings, and Terry URL https://link.aps.org/doi/10.1103/ Rudolph. Description of quantum coherence in PhysRevX.3.041003. thermodynamic processes requires constraints be- [53] W. Forrest Stinespring. Positive functions on c*- yond free energy. Nature Communications, 6(1): algebras. Proceedings of the American Mathemat- 6383, 2015. DOI: 10.1038/ncomms7383. URL ical Society, 6(2):211–216, 1955. ISSN 00029939, https://doi.org/10.1038/ncomms7383. 10886826. URL http://www.jstor.org/stable/ 2032342. [64] Mehdi Ahmadi, David Jennings, and Terry [54] H.B. Callen. Thermodynamics and an Introduc- Rudolph. The wigner–araki–yanase theorem tion to Thermostatistics. Wiley, 1985. ISBN and the quantum resource theory of asymme- try. New Journal of Physics, 15(1):013057, 9780471610564. URL https://books.google. jan 2013. DOI: 10.1088/1367-2630/15/1/013057. co.uk/books?id=MFutGQAACAAJ. URL https://doi.org/10.1088%2F1367-2630% [55] Run-Qiu Yang. Complexity for quantum field 2F15%2F1%2F013057. theory states and applications to thermofield double states. Phys. Rev. D, 97:066004, Mar [65] John Watrous. The Theory of Quantum Informa- 2018. DOI: 10.1103/PhysRevD.97.066004. tion. Cambridge University Press, 2018. DOI: URL https://link.aps.org/doi/10.1103/ 10.1017/9781316848142. PhysRevD.97.066004. [66] Otfried Gühne, Géza Tóth, and Hans J Briegel. [56] M. Costeniuc, R. S. Ellis, H. Touchette, and Multipartite entanglement in spin chains. New B. Turkington. Generalized canonical ensem- Journal of Physics, 7:229–229, nov 2005. DOI: bles and ensemble equivalence. Phys. Rev. 10.1088/1367-2630/7/1/229. URL https://doi. E, 73:026105, Feb 2006. DOI: 10.1103/Phys- org/10.1088%2F1367-2630%2F7%2F1%2F229. RevE.73.026105. URL https://link.aps.org/ [67] Géza Tóth. Multipartite entanglement and high- doi/10.1103/PhysRevE.73.026105. precision metrology. Phys. Rev. A, 85:022322, [57] Gavin E. Crooks. Quantum operation time Feb 2012. DOI: 10.1103/PhysRevA.85.022322. reversal. Phys. Rev. A, 77:034101, Mar URL https://link.aps.org/doi/10.1103/ 2008. DOI: 10.1103/PhysRevA.77.034101. PhysRevA.85.022322. URL https://link.aps.org/doi/10.1103/ [68] Zeqian Chen. Wigner-Yanase skew information PhysRevA.77.034101. as tests for quantum entanglement. Phys. Rev. [58] Dénes Petz. Sufficient subalgebras and the relative A, 71:052302, May 2005. DOI: 10.1103/Phys- entropy of states of a von neumann algebra. Com- RevA.71.052302. URL https://link.aps.org/ munications in Mathematical Physics, 105(1):123– doi/10.1103/PhysRevA.71.052302.

Accepted in Quantum 2019-10-24, click title to verify 24 [69] John Goold, Francesco Plastina, Andrea Gam- bassi, and Alessandro Silva. The Role of Quan- tum Work Statistics in Many-Body Physics, pages 317–336. Springer International Publishing, Cham, 2018. ISBN 978-3-319-99046-0. DOI: 10.1007/978- 3-319-99046-0_13. URL https://doi.org/10. 1007/978-3-319-99046-0_13. [70] A. S. Said. Some properties of the poisson distribution. AIChE Journal, 4(3):290–292, 1958. DOI: 10.1002/aic.690040311. URL https://aiche.onlinelibrary.wiley.com/ doi/abs/10.1002/aic.690040311. [71] Sudhakar Prasad, Marlan O. Scully, and Werner Martienssen. A quantum description of the beam splitter. Optics Communications, 62 (3):139 – 145, 1987. ISSN 0030-4018. DOI: https://doi.org/10.1016/0030-4018(87)90015-0. URL http://www.sciencedirect.com/science/ article/pii/0030401887900150. [72] Dénes Petz. A survey of certain trace inequalities. Banach Center Publications, 30(1):287–298, 1994. DOI: 10.4064/-30-1-287-298. URL http://eudml. org/doc/262566. [73] Pascal Massart. Concentration Inequalities and Model Selection: Ecole d’Eté de Probabilités de Saint-Flour XXXIII - 2003. Lecture Notes in Mathematics. Springer, 2007. [74] P. Billingsley. Probability and Measure. Wiley Series in Probability and Statistics. Wiley, 1995. ISBN 9780471007104.

Accepted in Quantum 2019-10-24, click title to verify 25 A Coherent work processes Proposition A.2. A coherent work process is trivial if and only if it is reversible. Here we gather the proofs of the statements for coherent work processes. First we show that for auxiliary ‘weight’ Proof. Let V |ψ0iS ⊗ |E0iA = |ψ1iS0 ⊗ |ωiA be a co- systems with non-degenerate spectra, the coherent work herent work process. In terms of random variables output is effectively unique. with respect to Hamiltonians HS and HA, an equiv- √ alent condition for the existence of this process is Sˆ = P iθk Proposition A.1. Let |ωiA = k e pk|kiA be Sˆ0 + Aˆ, where S,ˆ Sˆ0, Aˆ are the random variables for the coherent work output from a coherent work process ω |ψ0iS, |ψ1iS0 , |ωiA respectively. If it is a trivial coher- |ψiS −→ |φiS where {|kiA} is the eigenbasis of the non- ent work process, then at least one of Sˆ0 or Aˆ is a con- degenerate Hamiltonian HA. Then the distribution {pk} stant random variable. Without loss of generality, we is uniquely determined, while the phases {θk} are arbi- assume Aˆ = a is constant and so we must have that trary. 0 0 a Sˆ = Sˆ − a and Sˆ = Sˆ + a. This gives |ψ0iS −→ |ψ1iS0 −a and |ψ i 0 −→ |ψ i . Proof. Assume |ωiA is the coherent work output of the 1 S 0 S ω Conversely, assume we have a reversible coherent process |ψSi −→ |φSi, enacted by the unitary U. Define iHX t work process the characteristic function ϕX,η(t) := tr(e |ηihη|X ). Then under a coherent work process, 0 V |ψ0iS ⊗ |E0iA1 = |ψ1iS ⊗ |ω1iA1 , (76) i(HS +HA)t 0 ϕS,ψ(t)ϕA,0(t) = tr(e |ψihψ|S ⊗ |0ih0|A) 0 U|ψ1iS ⊗ |E0iA2 = |ψ0iS ⊗ |ω2iA2 , (77) (73)

i(HS +HA)t † = tr(e U U|ψihψ|S ⊗ |0ih0|A) where we introduce a different auxiliary system for each ˆ ˆ0 ˆ ˆ0 ˆ ˆ (74) process. Then S = S + A1 and S = S + A2. As the random variables are all independent, we have both = ϕS,φ(t)ϕA,ω(t), (75) 0 0 Var(Sˆ) = Var(Sˆ ) + Var(Aˆ1) and Var(Sˆ ) = Var(Sˆ) + Var(Aˆ2). Therefore where U is a unitary and [U, HS + HA] = 0 by the construction of coherent work processes. Due to the ˆ ˆ uniqueness property of characteristic functions [13], Var(A1) + Var(A1) = 0. (78) ϕA,ω(t) = ϕA,ω0 (t) for all t ∈ R if and only if the distri- bution for ω and ω0 are equal. Therefore with ψ and φ Due to the non-negativity of the variance for real- ˆ pre-determined, the distribution of ω is uniquely deter- valued random variables this implies that Var(A1) = ˆ ˆ ˆ mined. Var(A2) = 0. This implies that both A1 A2 are con- Since the only constraint on U is that it is en- stant random variables and so both processes are trivial ergy conserving, then for any Hermitian operator LA as required. such that [LA,HA] = 0, we can define a unitary −iLAθ V = (1S ⊗ e )U that also conserves energy, with [V,H +H ] = 0. Furthermore, with appropriate choice The consequence of this result is that any reversible S A coherent work process can simply be labeled by the con- of LA, any set of phases {θk} can be selected. ω stant value of Aˆ1. For example, if we have |ψ0iS −→ ω0 It is necessary to constrain the spectrum of A to be |ψ1iS0 and |ψ1iS0 −→ |ψ0iS then this implies |ωi = |Eki 0 non-degenerate, as given a degenerate Hamiltonian HA0 and |ω i = | − Eki for some energy Ek (if E0 = 0), 0 Ek for a system A , one could enact a unitary T that lo- then we label the reversible process simply as |ψ0iS ←→ cally conserves energy ([HA0 ,T ] = 0) and can arbitrarily |ψ1iS0 . change the state |ωiA in the degenerate subspace. Though we describe the coherent work output as ef- fectively unique, one is always free to enact evolution A.1 Decomposition of Poisson distributions locally on the auxiliary system that will induce a set of relative phases. In this way, any phases gained from The main theorem used in this section is Theorem A.3 the joint evolution between S and A can be arbitrarily due to Raikov and later generalized by others, but we changed with local evolution. Since relative phases are first re-cap on some background details [13]. a signature of time-evolution, the result distinguishes The cumulative distribution function (CDF) of a ran- between global and local effects of evolution on states. dom variable Xˆ on the real line is defined as F (x) := We can also prove that all reversible transformations P rob[Xˆ ≤ x]. If Xˆ has CDF F1 and Yˆ has CDF F2, and involve trivial coherent work processes. Xˆ and Yˆ are independent, then the CDF of the random

Accepted in Quantum 2019-10-24, click title to verify 26 variable Zˆ := Xˆ + Yˆ has CDF given by the convolution Poisson distributed random variable Zˆ shifted by b ∈ R, such that Zˆ0 = (Zˆ−b) ∼ Pois(ν). If we assume Yˆ and Zˆ Z ∞ are independent, then the sum of the random variables FZˆ(x) = F1(x − y)dF2(y) −∞ is Z ∞ dF2 = F1(x − y) (y)dy Xˆ = Yˆ + Zˆ (85) −∞ dy Z ∞ = (Yˆ − a) + (Zˆ − b) + (a + b). (86) = F1(x − y)f2(y)dy, (79) −∞ Since a and b are constants, we are able to re-arrange this to find where f2(y) is the probability density for Yˆ . For com- pleteness of notation, we also define the step function ˆ ˆ ˆ (x) = 1 if x ≥ 0 and zero otherwise. X − (a + b) = (Y − a) + (Z − b). (87) With this notation in place, we can now state the However, the right hand side of this equation corre- theorem by Raikov. sponds to the sum of two unshifted Poisson distributed Theorem A.3 (Raikov, 1938). The Poisson law has random variables and thus Xˆ − (a + b) = Yˆ 0 + Zˆ0 ∼ cumulative distribution function Pois(µ + ν), where we deduce this from Raikov’s theo- rem. From this we see that Xˆ is a Poisson distributed   x − α random variable shifted by an amount (a + b). Now let F , λ , σ > 0, α ∈ R, λ ≥ 0, (80) σ us choose a = α+β and b = α−β. Thus a+b = 2α and we expect the sum of the shifted random variables to where have a shifting of 2α, rather than the α that is claimed. X λk F (x; λ) = e−λ . (81) Another consistency check is to consider the limit of a k! k≤x Poisson cumulative distribution function. We have that

(for x < 0 the sum is empty and F (x; λ) = 0). F (x; λ) k x − α  X λ corresponds to the Poisson cumulative distribution for lim F ; λ = lim e−λ (88) λ→0 σ λ→0 k! x−α a random variable that takes only integer values at non- k≤ σ k negative values k with probability λ e−λ. When λ = 0 ( k! 1, if x ≥ α we obtain the singular distribution law (x − α). = (89) The integral equation 0, otherwise

Z ∞ x − α  where we make use of the limit of a Poisson probability F1(x − y)dF2(y) = F , λ , (82) ˆ 0 σ mass function [70]. Now suppose Y ∼ Pois(µ) and −∞ 0 0 Yˆ = Yˆ + a. Then limµ→0 Yˆ = limµ→0 Yˆ + a = a. 0 0 where F1(x) and F2(x) are cumulative distribution func- If we similarly define Zˆ ∼ Pois(ν), with Zˆ = Zˆ + b, tions only has solutions then limν→0 Zˆ = b. Thus when we sum the two shifted random variables, limµ,ν→0 Yˆ + Zˆ = a + b, as expected. x − α  F (x) = F 1 , µ (83) 1 σ A.2 Proof for decomposability theorem x − α  F (x) = F 2 , ν , (84) 2 σ Theorem A.4. Given a quantum system X with Hamiltonian HX with a discrete spectrum, a state |ψiX where µ ≥ 0, ν ≥ 0, µ + ν = λ and α1 + α2 = α, with admits a non-trivial coherent work process if and only if α1, α2 ∈ R. the associated classical random variable Xˆ is a decom- We refer the reader to [15] for the proof. posable random variable. Furthermore, for a coherent work process We note that the original statement of Raikov’s the- ω orem found in [15] has a typographical error to do with |ψiX −→ |φiZ , (90) the shifting parameters α and β. Namely, F1(x) = the associated classical random variables are given by x−α−β x−α+β F ( σ , µ) and F2(x) = F ( σ , ν), with β ∈ R. Xˆ = Zˆ+Wˆ , where Zˆ and Wˆ correspond to the measure- This error can be seen in the following manner. Sup- ments of HZ and HW in |φiZ and |ωiW respectively. pose we have a Poisson distributed random variable Yˆ shifted by a ∈ R, such that Yˆ 0 = (Yˆ − a) ∼ Pois(µ), Proof. Given quantum systems X, W, Y, Z with corre- where Pois(µ) denotes a Poisson distribution on the sponding Hamiltonians, suppose first the state |ψiX has non-negative integers. Similarly, let us define another decomposable coherence. This implies that there exists

Accepted in Quantum 2019-10-24, click title to verify 27 an auxiliary system Y with zero energy eigenstate |0iY First observe that and an isometry V from H ⊗ H into H ⊗ H with X Y Z W |Ω| X √ † |φiZ ⊗ |ωiW = rzsw|EziZ ⊗ |EwiW . (95) V (HX ⊗ 1Y + 1X ⊗ HY ) = (HZ ⊗ 1W + 1Z ⊗ HW )V , (91) z,w=1 such that However Xˆ = Zˆ + Wˆ , with Zˆ and Wˆ independent ran- dom variables. This implies that the distribution (px) V [|ψiX ⊗ |0iY ] = |φiZ ⊗ |ωiW , (92) of Xˆ over Ω is given by the convolution and where neither |φiZ nor |ωiW are energy eigenstates X px = qzrw. (96) of their respective Hamiltonians. The associated classi- z,w: cal random variables for the input systems are Xˆ and Ex=Ez +Ew Yˆ , while the associated classical random variables on Therefore we instead write the above expression for the the output systems are Zˆ and Wˆ . output states as Now since Yˆ = 0 the distribution for Xˆ coincides ˆ ˆ |Ω| with the distribution X + Y , which is simply the total X √ energy of the input systems. However equation (91) |φiZ ⊗ |ωiW = rzsw|EziZ ⊗ |EwiW implies that V is block diagonal in the total energy and z,w=1 X X √ thus preserves the measurement statistics for the total = rzsw|EziZ ⊗ |EwiW . ˆ ˆ ˆ energy. Therefore X = Z + W . Since |φiZ and |ωiW Ex z,w: are not energy eigenstates this means that both Zˆ and Ex=Ez +Ew (97) Wˆ have non-trivial distributions and are independent since the composite state on the output is a product We let Πx denote the projector onto the Ex eigenspace state, and thus Xˆ is a decomposable random variable. of HX , and so for those x for which px 6= 0 we have Conversely, given an initial state |ψi with associ- X √ ated classical random variable Xˆ, now suppose that Xˆ Πx|ψiX = px|ψxiX , (98) is decomposable. This means there exist classical ran- dom variables Zˆ and Wˆ , both with independent, non- which defines a normalised pure state |ψxiX within the trivial distributions, such that Xˆ = Zˆ + Wˆ . Since Ex eigenspace of HX . Thus

HX is assumed to have a discrete spectrum the clas- |Ω| ˆ X √ sical random variable X only has support on a discrete |ψi = p |ψ i . (99) set of values, and thus Zˆ and Wˆ only take on discrete X x x X x=1 values too. Let E1,E2,... denote the values that the random variables X,ˆ Z,ˆ Wˆ take on, and define the set We next define a linear mapping V from Hsup := Ω = {E1,...,Ed,... }. Denote by (rk) and (sk) respec- span{|ψxiX ⊗ |0iY }x into HZ ⊗ HW for which tively the distributions of Zˆ and Wˆ over Ω. 1 X √ Given these distributions, we first introduce two V [|ψ i ⊗ |0i ] = √ r s |E i ⊗|E i , x X Y p z w z Z w W quantum systems Z,W , each of countable dimension x z,w: Ex=Ez +Ew |Ω| with Hamiltonians HZ ,HW and each with an en- (100) ergy spectrum lying in Ω. Define the states for all x = 1,... |Ω| for which px 6= 0. If px 6= 0 for some x then equation (96) implies that there is at least one |Ω| X √ pair (z, w) such that qz and rw are non-zero and so for |φi := r |E i (93) Z k k Z such an x the image under V is a non-zero vector. k=1 V is an isometry: It is readily checked, using the |Ω| X √ orthogonality of the eigenstates of the output systems |ωiW := sk|EkiW , (94) and equation (96) that k=1 hψ | ⊗ h0| V †V |ψ i ⊗ |0i = 1 (101) on quantum systems Z and W where we label the eigen- x X Y x X Y states of each Hamiltonian with their corresponding for all x for which px 6= 0. Note that since different en- eigenvalues. We also introduce a third system auxil- ergy eigenspaces are orthogonal, we have both that the iary Y of arbitrary dimension, but with a zero energy input states {|ψxiX ⊗ |0iY }x are an orthonormal basis eigenstate |0i for some Hamiltonian H . We now con- Y Y for Hsup, and also the output states are orthonormal struct an isometry Vtot from HX ⊗HY to HZ ⊗HW that † sends |ψiX ⊗ |0iY to |φiZ ⊗ |ωiW . hψx|X ⊗ h0|Y V V |ψyiX ⊗ |0iY = 0, (102)

Accepted in Quantum 2019-10-24, click title to verify 28 0 for any x 6= y. The above provide the matrix compo- can construct (non-unique) tuples (S , |φiS0 ,HS0 ) and ˆ0 nents for V and imply that V is an isometry from Hsup (A, |ωiA,HA) associated to the random variables S and into HZ ⊗ HW . Aˆ respectively. On these quantum systems we then V is energy conserving: Let HXY := HX ⊗1Y +1X ⊗ show that a coherent work process can be constructed. HY be the initial Hamiltonian. Now since HX ⊗ HY = The precise statement and proof are as follows. ⊥ Hsup ⊕ Hsup and since |ψxiX ⊗ |0iY are eigenstates of HXY we can write HXY = HXY,s ⊕ HXY,p, where HXY,s is simply the restriction of the total Hamil- tonian to the subspace Hsup and HXY,p is the com- ponent on H⊥ . Now each pair (z, w) in equation sup √ √ (100) contributes a vector rz|EziZ ⊗ sw|EwiW that is an Ex energy eigenvector of the total Hamiltonian HZW := HZ ⊗ 1W + 1Z ⊗ HW , and so V |ψxiX ⊗ |0iY is an Ex energy eigenstate of HZW for any x. Thus V sends Ex eigenstates of HXY,s to Ex eigenstates of H as required by energy conservation. ZW Figure 7: Raikov’s theorem and coherent work processes. Finally, by rearranging equation (100) and summing Given any quantum system S with Hamiltonian HS in a state over x we see that V [|ψi ⊗|0i ] = |φi ⊗|ωi . To get X Y Z W |ψiX there is a unique classical random variable XˆS corre- an isometry on the full space, we again write HX ⊗HY = sponding to the energy measurement on S. Conversely, given ⊥ ˆ Hsup ⊕Hsup, and extend the systems Z and W and their a random variable XS we may construct a (non-unique) tuple Hamiltonians so that now (S, |ψiS ,HS ) associated to this random variable.

HZ ⊗ HW = span{|EziZ ⊗ |EwiW }z,w ⊕ Hother (103) Theorem A.5. Let S be a harmonic oscillator system, ⊥ † with dim(Hother) ≥ dim(Hsup). The final isometry on with Hamiltonian HS = hνa a. Let C be the set of the input system is then Vtot = V ⊕Vother, where Vother is quantum states for S as defined above. Then: an arbitrary energy conserving isometry from H⊥ into sup 1. The set of quantum states C is closed under all pos- Hother . This constructs a globally energy conserving sible coherent work processes from S to S with fixed Vtot such that Hamiltonian HS for both the input and output.

Vtot|ψiX ⊗ |0iY = |φiZ ⊗ |ωiW , (104) 2. Given any quantum state |ψiS ∈ C there is a and since both of Zˆ and Wˆ are non-trivial random unique, canonical state |α, kiS ∈ C such that we variables we have that neither of the output states have a reversible transformation

|ωiW , |φiZ is an energy eigenstate of its respective ωc |ψiS ←→ |α, kiS, (105) Hamiltonian. Thus the state |ψiX has decomposable coherence as required. with |ωciA = |0iA and α = |α|. Moreover, the only coherent work processes possible between canonical states are ω 0 A.3 Proof of semi-classical coherent work pro- |α, kiS −→ |α , kiS, (106) cesses theorem such that |α0| ≤ |α|. Modulo phases, the coherent The proof of the semi-classical theorem relies on work output is of the form |ωiA = |λ, niA, where p 2 0 2 0 Raikov’s theorem, which is a result for classical proba- λ = |α| − |α | and n, k are any integers that 0 bility distributions. The core idea behind the proof is obey n + k = k. outlined in Fig. 7. For the non-trivial direction, one 3. In the classical limit of large displacements |α|  1, starts with a tuple (S, |ψiS,HS) with S a quantum sys- from coherent processes on C we recover all classical tem with Hamiltonian HS, and |ψiS a state of the sys- work processes on the system S under a conserva- tem that admits a non-trivial coherent work process. tive force. Any state |ψiS uniquely determines a classical random iLS θ variable Sˆ, although the converse is not true since we Proof. (Proof of 1.) Let |ψiS = e |λ, miS be a state † have freedom in, for example, choosing phases or the in C defined with respect to a Hamiltonian HS = hνa a, ˆ † ˆ dimension of S. If S is decomposable, then Raikov’s with [LS,HS] = 0, LS = LS and θ ∈ R. Denote by S theorem tells us the relationship between the compo- the classical random variable obtained from an energy 0 nents of the decomposition Sˆ = Sˆ + Aˆ. From this we measurement of S, which takes values in {sn = nhν :

Accepted in Quantum 2019-10-24, click title to verify 29 2 n ∈ Z}, and with distribution p = (pn = |hψ|ni| ). We By suitable constant shifts of the Hamiltonians HS0 can expand the state |ψiS in the energy basis as and HA, that shift the zero energy level, we can en- 0 0 ∞ r sure that cS0 = m hν for some integer m , and thus X λn 0 |ψi = e−λ/2 eiθn |n + mi, (107) cA = (m − m )hν. This implies that both (qn) and (ra) n! n=0 are both Poissonian distributions on a discrete set of energy eigenvalues with constant separation hν. for some arbitrary phases {θn} and some m ∈ N. De- S0 A Let Πn and Πa denote the projectors for the discrete note by σ = hν the energy spacing of HS. Then the en- energy eigenvalues in the energy measurements of S0 ergy measurement distribution is Poissonian and given and A with probabilities qn and ra on |φiS0 and |ωiA by 0 respectively. We now define the states |n + m iS0 and n 0   −λ λ |a + m − m iA via pn = P Sˆ = (n + m)hν = e , (108) n! √ 0 S0 qn|n + m iS0 := Π |φiS0 (115) for n ≥ 0 and p = 0 for n < 0. The associated cumu- n n √ 0 A lative distribution function is given by ra|a + m − m iA := Πa |ωiA, (116) x − mhν  with n, a ≥ 0 and where we have ensured trivial phases F (x) = F , λ . (109) Sˆ σ via the definition of the states. Therefore we can write the two output states as The set C is clearly closed under trivial coherent work processes. Now suppose that |ψi admits some non- ∞ r n S −µ/2 X µ 0 |φi 0 = e |n + m i 0 trivial coherent work process given by S n! S n=0 ω |ψi −→ |φi 0 . (110) ∞ r S S X κa |ωi = e−κ/2 |a + m − m0i . (117) ˆ ˆ0 ˆ A a! A By Theorem 2.5 this implies that S = S + A, where a=0 Sˆ0 and Aˆ are non-trivial, independent classical ran- dom variables. From these we can construct tuples Recall that the translated coherent states are given by 0 0 0 (S , |φiS .HS ) and (A, |ωiA,HA) that realise the asso- 2 ∞ n − |α| X α ciated coherent work process, as detailed in the proof of D(α)|0i = |αi = e 2 |ni (118) 0 n! Theorem 2.5. For any |ψiS ∈ C we have that Sˆ = Sˆ +Aˆ n=0 ∞ is a decomposable Poisson random variable, and thus by 2 n − |α| X α |α, mi = e 2 |n + mi. (119) Raikov’s theorem the cumulative distribution functions n! for Sˆ0 and Aˆ are also Poissonian, with n=0 √   and so we can define real constants β = µ and x − α1 1 √ 0 FSˆ0 (x) = F , µ , (111) β2 = κ that allow us to write |φiS0 = |β1, m iS0 and σ 0   |ωiA = |β2, m − m iA, in terms of the notation for C x − α2 states. Therefore |φi 0 , |ωi ∈ C and so C is closed FAˆ(x) = F , κ , (112) S A σ under coherent work processes. where ν, κ ≥ 0 and constrained as µ + κ = λ, α1 + α2 = (Proof of 2.) Any state in C can be expressed in the form mhν and α1, α2 ∈ R. As σ appears identically in all cumulative distribution ∞ r λn −λ/2 X iθn functions this implies that both H 0 and H must have |ψi = e e |n + mi, (120) S A S n! within their spectral decompositions, countably infinite n=0 discrete energy eigenstates with equal spacings hν, cor- where m is some non-negative integer, λ ≥ 0 and {θ } responding to the outcomes of the energy measurements n are phases. We now define a unitary of the form V = for the constructed states |φi 0 and |ωi . We write the S A V ⊗ 1 where corresponding distributions for Sˆ0 and Aˆ as S A n ∞ m−1 0 −µ µ X −iθn X q := P [Sˆ = nhν + c 0 ] = e VS = e |n + mihn + m| + |kihk|, (121) n S n! a n=0 k=0 −κ κ ra := P [Aˆ = ahν + cA] = e , (113) a! which√ realises the transformation V [|ψiS ⊗ |0iA] = | λ, miS ⊗ |0iA with λ ≥ 0. It is clear that both V and for integers n, a ≥ 0, and where cS0 and cA are any † constants that obey its inverse V are energy conserving and so we have a reversible coherent work process with |ωciA = |0iA as cS0 + cA = mhν. (114) claimed.

Accepted in Quantum 2019-10-24, click title to verify 30 0 As constructed in the proof of part 1, we saw that |α| ≥ |α | and therefore hHSiα ≥ hHSiα0 – meaning the modulo arbitrary phases between the energy eigen- process on S is work extracting only for the coherence states, any coherent work process on a state |α, miS component of |α, ki. However variations in the k label takes the form are always sharp and can involve either positive or neg- ω 0 ative work and it is clear that for large displacements |α, mi −→ |β , m i 0 (122) S 1 S the fractional energy separation of levels tends to zero 0 and with |ωiA = |β2, m − m iA. With a freedom to and one has an effective continuum of energy values that shift the energies of S0 and A by a constant factor recovers the classical case. ±c respectively. Moreover, since the Poisson parame- 2 2 ters are given, as above, by λ = |α| , µ = |β1| and 2 κ = |β2| , Raikov’s theorem tells us that we must have A.4 Proof of Theorem II.6 (coherent work pro- 2 2 2 p 2 2 |α| = |β1| + |β2| , and therefore |β2| = |α| − |β1| cesses on a ladder system) as claimed. (Proof of 3.) For a classical work process under a Theorem A.6. Let S be a quantum system, with en- conservative force, we need two conditions to hold: (i) ergy eigenbasis {|ni : n ∈ Z} with fixed Hamiltonian P the position, momentum and energy are sharp and (ii) HS = n|nihn|. Given an initial quantum state Pn∈√Z the process depends only on the initial and final en- |ψiS = i pi|iiS, a coherent work process ergies, not the path. For the first, consider the posi- q ω ~ † |ψiS −→ |φiS, (126) tion and momentum operators q = 2mω (a + a) and q ~mω † p = i 2 (a − a) with the canonical commutation is possible with relation [q, p] = i~, and ~ω = hν. It is straightfor- X iθj √ 2 ~ |φiS = e qj|jiS (127) ward to verify that the variances satisfy σq = 2mω and σ2 = ~mω for any coherent state. Then coherent states j p 2 √ X iϕk are minimal uncertainty states with Heisenberg uncer- |ωiA = e rk|kiA (128) 2 2 2 ~ k tainty relation σq σp = 4 . For a coherent state |αi, the expectation value for the position and momentum q q for arbitrary phases {θj} and {ϕk}, with output Hamil- ~ ∗ ~mω ∗ are hqiα = 2mω (α + α) and hpiα = i 2 (α − α). P tonian HS as above and HA = n∈ n|niAhn|, if and Therefore Z only the distribution (pn) over Z can be written as σq 1 = ∗ , (123) X j hqiα (α + α) pn = rj(∆ q)n. (129) σp i j∈Z = ∗ . (124) hpiα (α − α ) for distributions (qm) and (rj) over Z. In the limit of large displacement |α|  1, the values for ω hqiα and hpiα become arbitrarily sharp with the ratio of Proof. Assume |ψiS −→ |φiS. Then there exists a uni- standard deviation to value tending to zero. Similarly, tary V such that V |ψiS ⊗ |0iA = |φiS ⊗ |ωiA. Further- † for a Harmonic oscillator Hamiltonian HS = ~ωa a, the more, the unitary satisfies the global energy constraint 2 2 2 2 [V,H + H ] = 0, where H and H are the Hamilto- variance of energy is σH = ~ ω |α| leading to the ratio S A S A of the standard deviation to the mean nians of S and A respectively. We expand |φiS and |ωiA in their energy eigenbases σH 1 = , (125) as hHSiα |α|

X iθj √ which again becomes sharp in the limit in the same |φiS = e qj|jiS manner. Thus, all energy transfers, positions and mo- j menta become sharp in the limit of large displacements X √ |ωi = eiϕk r |ki . and the classical mechanics of a harmonic oscillator is A k A k recovered. For (ii), the energy transfer depends only on the ini- If V provides a valid coherent work process then so too 0 iBS θ iCAϕ tial and final states of the system |αiS and |α iS. As does the unitary UV where U = e ⊗ e for any all coherent states are included in C, then coherent work Hermitian BS and CA such that [BS,HS] = [CA,HA] = processes on C states with large displacement are classi- 0. By choosing the operators BS and CA appropriately cal work processes under a conservative force. Further- one can freely vary the phases θj and ϕk. We have ω 0 more, for the transition |αiS −→ |α iS, we have that that A has the same Hamiltonian ladder structure as

Accepted in Quantum 2019-10-24, click title to verify 31 S. Energy conservation implies that for any n ∈ Z, the where U † is a phase unitary that generates the phases action of V on the state |niS ⊗ |0iA gives θk, ϕj and which is diagonal in the energy eigenba- sis. Again, we let Πn denote the projector onto the X iαn,j √ V |niS ⊗ |0iA = e mj|n|n − jiS ⊗ |jiA. (130) n–eigenspace of the system SA and so j∈Z √ Π |ψi ⊗ |0i = p |ii ⊗ |0i for some distribution (m ) over j, and phases (α ). n S A n S A j|n j n,j X √ This implies that ΠnU|φiS ⊗ |ωiA = qn−mrm|n − miS|miA. (141) m∈Z X iαn,j √ |φiS ⊗ |ωiA = e mj|npn|n − jiS ⊗ |jiA, n,j∈Z Thus we have that (131) † X √ hφ|S ⊗ hω|AU ΠnU|φiS ⊗ |ωiA = qn−mrm = pn, X i(θk+ϕl) = e qkrl|kiS ⊗ |liA. (132) m k,l∈Z (142) and thus the vectors Π |ψi ⊗|0i and Π U|φi ⊗|ωi Let Π be the projector onto the energy eigenspace of n S A n S A n both lie in the n–eigenspace of the total hamiltonian and total energy n ∈ for the joint system SA. We thus Z have equal norms. Therefore, we may define a unitary have that Vn within each energy eigenspace such that X hφ|S ⊗ hω|AΠn|φiS ⊗ |ωiA = mj|npn, (133) VnΠn|ψiS ⊗ |0iA = ΠnU|φiS ⊗ |ωiA. (143) j∈Z X = qn−jrj, (134) For those n–eigenspaces outside the support of |ψiS ⊗ j∈Z |0iA we simply let Vn be the identity unitary on that subspace. We now write V := ⊕ V to obtain an energy However P m = 1 and so we have that the distri- n n j j|n conserving unitary on the full space, and using the fact bution (p ) decomposes as n that U commutes with the energy projectors we have X X j that pn = rjqn−j = rj(∆ q)n, (135) U †V |ψi ⊗ |0i = |φi ⊗ |ωi , (144) j∈Z j∈Z S A S A † as required. Conversely, suppose we have a distribution where U V is an energy conserving unitary on the joint system. Therefore the required coherent work process (pn) over Z which can be decomposed as exists. X X j pn = rjqn−j = rj(∆ q)n, (136) j∈Z j∈Z for two distributions (rj) and (qk). From these we define A.5 Physical example of decomposable coher- for ladder systems S and A the states ence X √ In the main text, an example of infinitely divisible co- |ψiS = pi|iiS, (137) i herence was provided in the form of a coherent state. Here we provide a physical system in which one might and encounter such examples of coherent state manipula- √ X iθj tions. Consider a beam splitter set up, with a coherent |φiS = e qj|jiS (138) j state of light injected as the input on one mode and the √ vacuum on the other mode, such that the initial state X iϕk |ωiA = e rk|kiA (139) is |αiS ⊗ |0iA. We use the beam splitter unitary k θ(ab†−a†b) where θj and ϕk are arbitrary phases. The joint system B(θ) = e (145) SA therefore has the two states X √ where a and b are the annihilation operators for S and A |ψiS ⊗ |0iA = pi|iiS ⊗ |0iA respectively [71]. Under the action of a beam splitting, i which conserves photon number, we want to achieve √ X i(θk+ϕl) |φiS ⊗ |ωiA = e qkrl|kiS ⊗ |liA. B(θ)|αi ⊗ |0i = |βi ⊗ |ωi, (146) k,l∈Z † X √ = U qkrl|kiS ⊗ |liA, (140) for some tunable parameter θ. It is well known that the k,l∈Z two mode annihilation operators satisfy the algebraic

Accepted in Quantum 2019-10-24, click title to verify 32 property: classical distribution pk over the energy eigenstates. For a projective measurement of energy in this eigenbasis we † B (θ)aB(θ) = a cos θ + b sin θ (147) can denote by Xˆ the classical random variable for the B†(θ)bB(θ) = b cos θ − a sin θ. (148) energy obtained in the measurement, then it is readily seen that Λ(β, ψ) is essentially the negative cumulant Since coherent states are displaced vacuum states, we generating function for Xˆ in the inverse temperature β, have that X (−1)nβn Λ(β, ψ) = − κ (Xˆ). (154) n! n n≥1 − 1 |α|2 αa† −α∗a B(θ)|αi ⊗ |0i = B(θ)e 2 e e |0i ⊗ |0i (149) − 1 |α|2 αa† † We shall make use of the following theorem for our = e 2 B(θ)e B (θ)|0i ⊗ |0i (150) analysis. − 1 |α|2 αB(θ)a†B†(θ) = e 2 e |0i ⊗ |0i (151) Theorem B.1. If A is Hermitian and Y is strictly pos- − 1 |α|2 α(a† cos θ−b† sin θ) = e 2 e |0i ⊗ |0i itive, then (152) ln tr(eA+ln Y ) = max{tr(XA) − S(X||Y ): X is positive = |α cos θi ⊗ | − α sin θi (153) & trX = 1}. where we used the fact that B(θ)|0i ⊗ |0i = |0i ⊗ |0i Alternatively, if X is positive with Tr(X) = 1 and B is and B(θ) is unitary. Here, r = cos θ and t = sin θ are Hermitian, then the transmission and reflection coefficients. The con- sequence is that we have achieved the desired coherent S(X||eB) = max{tr(XA) − log tr(eA+B): A = A†}. state splitting with a photon number conserving uni- tary. The second mode that began in the vacuum there- The proof of this can be found in [72]. fore carries information of the coherent energy transfer. We now establish basic properties of the effective po- As expected, the condition |α|2 = |α cos θ|2 + |α sin θ|2 tential Λ(β, ρ). As a function of the quantum state ρ, is satisfied. the effective potential has the following properties. If one were to perform measurements on the ancilla sys- tem, they would extract information on the energetics Lemma B.2. Given a quantum system S with Hamil- and coherences. However, by performing the measure- tonian H , and inverse temperature β > 0. We have ment, the state would no longer be available for ma- that nipulations that involve interference effects. For this 1. (Invariance) Let ρ(t) = e−iHtρe+iHt for all t ∈ reason, the coherent work output is the state |α sin θi R. For any state ρ and any t ∈ R we have rather than a post-processed state. Λ(β, ρ(t)) = Λ(β, ρ(0)). Moreover, we have that P Λ(β, ρ) = Λ(β, D(ρ)) where D(ρ) = k ΠkρΠk is the dephasing of ρ in the energy basis, with Πk be- B Fluctuation relations and the effective ing the projector onto the k’th energy eigenspace of potential H.

Here we gather together the proofs for the fluctuation 2. For any state ρ, we have that relation analysis. β2 βhHi − (Emax−Emin)2 ≤ Λ(β, ρ) ≤ βhHi . (155) ρ 8 ρ B.1 Basic properties of the effective potential. 3. Let Emin and Emax be the smallest and largest It is worth describing some properties of the function eigenvalues of H respectively in the support of ρ. Λ(β, ρ) to shed light on the fluctuation relations. Then We first note that the effective potential has an in- βEmin ≤ Λ(β, ρ) ≤ βEmax. (156) variance under the group generated by the observable H. More precisely, let U(t) be the unitary repre- 4. (Variational form) For states ρ of full rank and sentation of time-translations generated by H. Then [ρ, H] = 0, we have that Λ(β, U(t)ρU(t)†) = Λ(β, ρ) for any t ∈ R, and so we have that Λ(β, ρ(t1)) = Λ(β, ρ(t2)) for any two times Λ(β, ρ) = min [βhHiσ + S(σ||ρ)] , (157) σ t1, and t2. Thus, Λ is a constant of motion under the free Hamiltonian evolution. For a pure quantum state where the minimization is taken over all quantum √ P iθk |ψi = k e pk|Eki, Λ is purely a function of the states σ of the system S.

Accepted in Quantum 2019-10-24, click title to verify 33 min 5. (Additivity) For a bipartite system S1S2 with to- By assumption, E is the smallest eigenvalue of H min −β(En−E ) tal Hamiltonian H12 = H1 ⊗ 12 + 11 ⊗ H2 and in the support of ρ and therefore e ≤ 1 P a bipartite product state ρ12 = ρ1 ⊗ ρ2 we have for all n. From the property i pi = 1, the argument Λ(β, ρ1 ⊗ ρ2) = Λ1(β, ρ1) + Λ2(β, ρ2), where the ef- of the logarithm is smaller than one and therefore fective potentials for subsystems are defined solely the latter term is positive. It therefore follows that with respect to the corresponding Hamiltonian on βEmin ≤ Λ(β, ρ) ≤ βEmax. that system. 4. With an appropriate choice of operators A = Proof. 1. For any t ∈ we have Λ(β, ρ(t)) = R −βH and Y = ρ, Theorem B.1 implies Λ(β, ρ) = − ln tr(e−βH e−iHtρe+iHt) = Λ(β, ρ), by cyclicity of the − max {−βhHi − S(σ||ρ)} = min {βhHi + S(σ||ρ)} trace. Moreover, we can evaluate the trace in an energy σ σ σ σ for states σ. However as ln ρ will diverge for non-full eigenbasis of H, and so denoting by Π the projector k rank states, and so the assumption that ρ must be full onto the k’th eigenspace of H we have that rank is required. −βH X −βH 5. Since H12 = H1 ⊗ 12 + 11 ⊗ H2 we obtain − ln tr(e ρ) = − ln tr( ΠkΠk(e ρ)) k −β(H1⊗1+1⊗H2) Λ(β, ρ1 ⊗ ρ2) = − ln tr(e ρ1 ⊗ ρ2) X −βH = − ln tr( Πk(e ρ)Πk) (163) k = − ln Tr(e−βH1 ρ )tr(e−βH2 ρ ) (164) X 1 2 = − ln tr( e−βH Π ρΠ ) k k = Λ1(β, ρ1) + Λ2(β, ρ2). (165) k = − ln tr(e−βH D(ρ)) = Λ(β, D(ρ)). (158) The effective potential has the following properties in 2. The proof of the upper bound is a direct result of terms of the parameter β. −βH −βhHiρ Jensen’s inequality, he iρ ≥ e , which implies Lemma B.3. Given a quantum system S with Hamil- tonian H. For any state ρ of S, the effective potential −βhHiρ Λ(β, ρ) ≤ − ln e = βhHiρ, (159) Λ(β, ρ) obeys achieving the desired result. 1. Λ(β, ρ) is concave in β for all β ≥ 0. The lower bound is obtained as follows. Given a state ρ, let us define the zero mean observable H = 2. Λ(β, ρ) is monotone increasing in the variable β, 0 for all states ρ, if and only if H ≥ 0. H − hHiρ1. It is readily seen that the effective poten- −1 tial satisfies Λ(β, ρ) = Λ0(β, ρ) + βhHiρ, where Λ0(β, ρ) 3. (High temperature regime) lims→0 s Λ(sβ, ρ) = is evaluated for the Hamiltonian H0. Let us denote the βhHiρ. random variable Xˆ0 with sample space eigs(H0), dis- min 4. (Low temperature regime) lim s−1Λ(sβ, ρ) = tributed by ρ. Then Xˆ0 has zero mean and E ≤ s→∞ max βEmin, where Emin is the smallest eigenvalue of Xˆ0 ≤ E almost surely. Thus we can apply Hoeffd- ing’s lemma [73], which states H in the support of ρ.

2 Proof. 1. Let a, b > 0 satisfy a+b = 1. For two different λXˆ λ (Emax−Emin)2 [e 0 ] ≤ e 8 (160) E inverse temperatures β1 and β2 and Hamiltonian H, we have: for any λ ∈ R. Taking the logarithm and re-ordering 2 β max min 2 −aβ1H −bβ2H the inequality, Λ0(β, ρ) ≥ − 8 (E − E ) , which Λ(aβ1 + bβ2, ρ) = − ln tr(ρe e ). (166) gives the lower bound. −aβ H max Let us define the operators X = e 1 and Y = 3. From 2, we know that Λ(β, ρ) ≤ βhHiρ ≤ βE −bβ2H as the expectation value cannot exceed the greatest e , where X,Y ≥ 0. Then Hölder’s inequal- ity tells us expectation values satisfy E[|XY |] ≤ eigenvalue. To prove the lower bound, consider a state p 1/p q 1/q 1 1 min ( [|X| ]) ( [|Y | ]) , where + = 1. Choosing ρ with diag(ρ) = p and assume Ek = E =⇒ pk 6= 0 E E p q 1 1 without loss of generality. Then p = a and q = b , we get:

min min −βE X −β(Ej −E )
Λ(aβ1 + bβ2, ρ) ≥ aΛ(β1, ρ) + bΛ(β2, ρ). (167) Λ(β, ρ) = − ln e [pk + pje ] j6=k Therefore Λ is concave in the inverse temperature. (161) 2. First suppose that H ≥ 0, then min min X −β(Ej −E )
= βE − ln pk + pje (162) ∂Λ(β, ρ) tr(He−βH ρ) = (168) j6=k ∂β tr(e−βH ρ)

Accepted in Quantum 2019-10-24, click title to verify 34 For β ∈ R, the denominator is always strictly positive. sumption, and therefore The nominator can be expanded using the spectral de- min P −1 X −sβ(Ei−E ) composition H = EkΠk and so lim (−s ) ln(ρk + e ρi) k s→∞ i6=k

−βH X −βEk −1 tr(He ρ) = Eke tr(ΠkρΠk). (169) = lim (−s ) ln(ρk) = 0. (177) s→∞ k which implies that lim s−1Λ(sβ, ρ) = βEmin as re- Since H ≥ 0 all terms in this expression are non- s→∞ quired. negative for all β and thus ∂β(Λ(β, ρ) ≥ 0 which implies the function is monotonely increasing in β. Conversely, assume ∂βΛ(β, ρ) ≥ 0 for all states ρ. This implies that tr(He−βH ρ) ≥ 0 for all states ρ. Again, using the spectral decomposition of H and the fact that we have that

X −βEk Eke pk ≥ 0 (170) k for all distributions pk over the eigenvalues Ek of H. For p = (pk) sharp on an eigenvalue Em the above equation implies that Em ≥ 0, and since this holds for all m this in turn implies that H ≥ 0 as required. 3. To prove this, we Taylor expand in orders of s. We find:

−1 −1 −sβH lim s Λ(sβ, ρ) = lim(−s ) lnhe iρ (171) s→0 s→0 −1 2 = lim(−s ) ln[1 − sβhHiρ + O(s )] s→0 (172) −1 2 = lim(−s )[−sβhHiρ + O(s )] s→0 (173)

= βhHiρ (174) as claimed. min 4. Let E = Ek be the smallest energy eigenvalue in the support of ρ, and let ρi := tr(Πiρ) for any i. Then: ! −1 −1 X −sβEi lim s Λ(sβ, ρ) = lim (−s ) ln e ρi s→∞ s→∞ i (175) min min −1 −sβE X −sβ(Ei−E )
= lim (−s ) ln e [ρk + e ρi] s→∞ i6=k min −1 min X −sβ(Ei−E ) = lim (−s )[−sβE + ln(ρk + e ρi)] s→∞ i6=k min min −1 X −sβ(Ei−E ) = βE + lim (−s ) ln(ρk + e ρi). s→∞ i6=k (176)

−sβ(Ei−Emin) Now since Ei − Ek > 0 we have that e < 1, for all i 6= k and for all s > 0, β > 0 and this term decreases to zero as s → ∞. Moreover ρk 6= 0 by as-

Accepted in Quantum 2019-10-24, click title to verify 35 B.2 Mean coherence representation the respective probability distributions over energy S(p˜||p). Finally, using equation (178) we have that As in the main text, we have that χm, the mean coher- ence at inverse temperature β, is given by the equation 1 β2χ (ψ˜) = β(hH i − hH i ) − S(p˜||p), (184) 8π m S p S p˜ 1 2 Λ(β, ρ) = βhHiρ − β χm(˜ρ), (178) 8π as required. where χm is evaluated at the inverse temperature βm ≤ β determined from the Mean Value Theorem. From this the fluctuation relation in Theorem 3.1 can be stated in the following way. Theorem B.4. Given a quantum system S with Hamil- P Theorem B.5. Given the assumptions to Theorem 3.1, tonian HS with spectral decomposition HS = k EkΠk, the mean coherence at inverse temperature β of a quan- let pk and p˜k denote the probability distribution for an ˜ tum state |ψi is given by energy measurement of the state |ψki and |ψki respec- tively. Then 2 β ˜ χm(ψ) = β(hHSip − hHSip˜) − S(p˜||p). (179) ˜ P [ψ1|ψ0] ˜ 8π = e−β∆F −βWS eS(p˜0||p0)−S(p˜1||p1), (185) ∗ ˜∗ P [ψ0 |ψ1 ] where p is the distribution pk := tr[Πkψ] over energy of ˜ ψ and p˜ is given by p˜k := tr[Πkψ], the distribution over where W˜ S := hHSi ˜ − hHSi ˜ and S(p||q) is the energy for |ψ˜ihψ˜| = Γ(|ψihψ|). ψ1 ψ0 Kullback-Leibler divergence. Proof. From Lemma B.2 we have that Λ(β, ρ) = Λ(β, D(ρ)) where D(ρ) is the dephased state across the B.3 Variational expression for the effective po- energy eigenspaces, and so because of this it will suffice tential to restrict to this dephased state, as we now show. Let σ = D(ρ) for compactness. Noting that [σ, H] = The following is the same as Lemma 3.3 in the main 0 we now expand S(˜σ) as text.

S(˜σ) = −tr[˜σ lnσ ˜] Lemma B.6. Given a quantum system S with Hamil-   1  tonian HS and {Πk} the projectors onto the energy = −tr σ˜ ln e−βH/2σe−βH/2 tr(e−βH σ) eigenspaces of HS. Let us denote the de-phased form of P state in the energy basis by D(ρ) = ΠkρΠk. Then, h n −βH/2 −βH/2 oi k = −tr σ˜ ln(e σe ) + Λ(β, σ) for any full rank quantum state ρ, the effective potential h i Λ(β, ρ) is given by = −tr σ˜ ln(e−βH/2σe−βH/2) − Λ(β, σ)

Λ(β, ρ) = min{βhHSiτ + S(τ||D(ρ))}, (186) = −tr [˜σ(−βH + ln σ)] − Λ(β, σ) τ = βhHi − tr[˜σ ln σ] − Λ(β, σ) σ˜ where the minimization is over all quantum states τ of = βhHiσ˜ + S(˜σ) + S(˜σ||σ) − Λ(β, σ), (180) S, and S(τ||ρ) = tr[τ log τ − τ log ρ] is the relative en- tropy function. Moreover, the minimization is attained which implies that for the state τ = Γ(D(ρ)). Λ(β, σ) = βhHi + S(˜σ||σ). (181) σ˜ Proof. From Lemma B.2, we see that for any full rank

Now since σ = D(ρ) we have that tr[Πkσ] = tr[Πkρ] ρ we have and likewise it can be checked that tr[Πkσ˜] = tr[Πkρ˜], where we use that D(Γ(ρ)) = Γ(D(ρ)) for any ρ. Thus Λ(β, ρ) = Λ(β, D(ρ)) (187) the preceding equation can be written = min[βhHSiτ + S(τ||D(ρ))]. τ

Λ(β, D(ρ)) = Λ(β, ρ) = βhHiρ˜ + S(˜σ||σ). (182) However equation (181) says

Restricting to ρ = ψ, a pure state, and using pk = ˜ Λ(β, ρ) = Λ(β, σ) = βhHSiσ˜ + S(˜σ||σ), (188) tr[Πkψ] and p˜k = tr[Πkψ] we have that where σ = D(ρ). Comparing the two equations we see Λ(β, ψ) = βhHip˜ + S(p˜||p), (183) that the minimization is realised for τ =σ ˜ = Γ(σ) = where we use that σ˜ commutes with σ and so S(˜σ||σ) Γ(D(ρ)) as claimed. can be replaced with the classical divergence between

Accepted in Quantum 2019-10-24, click title to verify 36 B.4 Coherent work and the quantum fluctuation 2. Within the fluctuation relation context with a ther- relation. mal system B and quantum system S we have that

We now prove the coherent work form of the coherent P [ψ |ψ˜ ] 1 0 = e−β(∆F ±kT Λ(β,ω)), (190) fluctuation theorem and provide an account of how it P [ψ∗|ψ˜∗] naturally connects with the resource theory of asym- 0 1 metry, inducing a natural measure for coherent work. for all inverse temperatures β ≥ 0, for some state However, first we require the conditions in which the |ωiA with finite cumulants on an auxiliary system moments of a probability distribution uniquely deter- A and some choice of sign before Λ(β, ω). mine the distribution. Proof. First assume that |ψ0iS and |ψ1iS are coherently Theorem B.7 ([74]). Let µ be a probability measure on R ∞ k connected states. Suppose there is a coherent work pro- the line having finite moments mk = x µ(dx) of ω −∞ cess transforming |ψ0iS −→ |ψ1iS. Without loss of P∞ rk all orders. If the power series k=1 k! mk has positive generality we can choose the initial state |0iA of the radius of convergence, then µ is the unique probability auxiliary system A to have zero energy. measure with the moments m1, m2,... . By unitary invariance of the effective potential we have that When the full set of moments uniquely define the probability distribution, then so too do the cumulants. Λ(β, |ψ i ⊗ |0i ) = Λ(β, V (|ψ i ⊗ |0i )) This follows since in such cases the moments can be ex- 0 S A 0 A A pressed in terms of the cumulants. Furthermore, finite = Λ(β, |ψ1iA ⊗ |ωiA), (191) moments correspond to finite cumulants since once can where V is a unitary that realises the pro- obtain the cumulants from a recursive formula that is cess. This conserves energy and so commutes with polynomial in the moments. Thus the full set of cu- 1 1 e−β(HS ⊗ A+ S ⊗HA). Since the effective potential is ad- mulants uniquely determines a distribution when the ditive for independent Hamiltonians and product states, moment generating function of the distribution has a we have that positive radius of convergence. When the moment gen- erating function is finite and non-zero, then the cumu- Λ(β, ψ ) = Λ(β, ψ ) + Λ(β, ω). (192) lant generating function is finite and non-zero as it is 0 1 obtained by taking the logarithm. Therefore the effective potential change in the fluctua- In the result that follows, we assume the moment tion theorem can be expressed in terms of the coherent generating function and by extension the effective po- work output |ωi as tential, have infinite radius of convergence and therefore always exist and uniquely specify the distribution. This eΛ(β,ψ0)−Λ(β,ψ1) = eΛ(β,ω). (193) choice is for simplicity in the result, though one could generalise it to hold for any positive radius of conver- and so the fluctuation theorem can be expressed as gence. Requiring an infinite radius of convergence is a restriction on the states permitted. For example, the P [ψ |ψ˜ ] e−β∆F 1 0 = eΛ(β,ω)−β∆F = . (194) negative binomial distribution NB(r, p) has finite radius ∗ ˜∗ −βHA P [ψ0 |ψ1 ] hω|e |ωi of convergence and therefore any state with such a dis- tribution would not be included in the statement. This ω If instead |ψ1iS −→ |ψ0iS, then the same argument can be seen from the moment generating function M(t) applies with the only difference being the sign in front ˆ for a random variable X ∼ NB(r, p), with [13] of Λ(β, ω).  p r Conversely, suppose the above statement 2 holds for M(t) = , (189) all inverse temperatures β ≥ 0, for some pure state |ωi 1 − (1 − p)et A on a system A, and choice of sign in the exponent. The which is valid for t < − ln(1 − p). exponent in Theorem 3.1 implies that we have

Theorem B.8. Let S be a quantum system with Hamil- Λ(β, ψ0) = Λ(β, ψ1) ∓ Λ(β, ω), (195) tonian HS and let |ψ0iS and |ψ1iS be two pure states of S that have energy statistics with finite cumulants of for all inverse temperatures β ∈ [0, ∞). In the event of all orders. Then the following are equivalent: the minus sign, we write this as 1. The pure states |ψ i and |ψ i are coherently con- 0 S 1 S Λ(β, ψ ) + Λ(β, ω) = Λ(β, |ψ i ⊗ |ωi ) nected with coherent work output/input ω on an 0 0 A A auxiliary system A. = Λ(β, |ψ1iS ⊗ |0iA), (196)

Accepted in Quantum 2019-10-24, click title to verify 37 where we can assume that A has a zero energy eigen- |α, ki ∈ C and assume k = 0, as the extension is simple state |0iA. However, since all the above cumulant gen- but provides little benefit. The effective potential for a erating functions have infinite radius of convergence and coherent state |αi: are finite, the energy distributions of measuring |ψ0iS, βhν 2 −βhν |ψ1iS and |ωiA are uniquely determined. Moreover the Λ(β, α) = + |α| (1 − e ). (203) distributions of a joint energy measurement of compos- 2 ite system SA in the state |ψ0iS ⊗|ωiA and |ψ1iS ⊗|0iA Now consider the average energy of a thermally dis- are identical. This implies that tributed quantum harmonic oscillator

P 1 −βhν(n+ 1 ) Sˆ + Aˆ = Sˆ , (197) hν(n + )e 2 0 1 hH i = n≥0 2 (204) S γ P −βhν(m+ 1 ) m≥0 e 2 where Sˆ0, Sˆ1, Aˆ are the associated random variables for measuring |ψ i , |ψ i , |ωi respectively in their energy hν 1 + e−βhν  0 S 1 S A = . (205) eigenbases. By Theorem 2.5 this implies there exists a 2 1 − e−βhν non-trivial coherent work process Let us define a thermal frequency hνth := hHSiγ . Then ω the effective potential change can be expressed as |ψ1i −→ |ψ0i, (198) 2 2 −βhν and so the states are coherently connected. In the event Λ(β, α1) − Λ(β, α0) = (|α1| − |α0| )(1 − e ) of the plus sign in the exponent we instead write (206) ν 2 2 −βhν Λ(β, ψ0) = Λ(β, ψ1) + Λ(β, ω), (199) = (|α1| − |α0| )(1 + e ). 2νth (207) and then express the argument of the left-hand side us- ing |ψ0iS ⊗ |0iA as before, using the additivity of the Since the Gibbs rescaled coherent state |α˜i = effective potential for the left-hand side. The same argu- |αe−βhν/2i, the change in effective potential can be ex- ment as before then implies the existence of a coherent pressed in terms of the unscaled and rescaled coherent work process ω state energy transfers, |ψ0i −→ |ψ1i, (200) W = hH i − hH i (208) and again the states are coherently connected, which S S α1 S α0 ˜ completes the proof. WS = hHSiα˜1 − hHSiα˜0 , (209) as   1 WS + W˜ S B.5 Proof of semi-classical fluctuation theorem Λ(β, α1) − Λ(β, α0) = . (210) hνth 2 Theorem B.9 (Semi-classical relation [25]). Let the assumptions of Theorem 3.1 hold, and let S be a har- which gives the claimed result upon insertion into Theo- † 1 rem 3.1. For the breakdown of hν in terms of λ (T ), monic oscillator with Hamiltonian HS = hν(a a + ). th dB 2 we refer to [25]. Then for two coherent states of the system |α0i and |α1i, the following holds For completeness, we can also show the form of the   fluctuation theorem using the coherent work represen- P [α1|α˜0] ∆F W¯ B ∗ ∗ = exp − + , (201) tation. Consider an auxillary weight system A with P [α0|α˜1] kT hνth † Hamiltonian HA = hνa a. Once again, this system ex- ω ists as a hypothetical process in which |ψ0i −→ |ψ1i where hνth = hHSiγ is the average energy of a Gibbs state γ of a quantum harmonic oscillator, related to the under a coherent work process. We note that thermal de Broglie wavelength λdB(T ) via −βHS Λ(β,ω) hα1|e |α1i e = −βH . (211) h2 1 hα0|e S |α0i hνth = 2 + hν, (202) mλdB(T ) 2 From the semi-classical result Theorem 2.4, we know ω that |α0iS −→ |α1iS under a coherent work process if ¯ 1 ˜ and WB := − 2 (WS + WS), WS = hHSiα1 − hHSiα0 , and only if |α1| ≤ |α0|. Assuming this condition to be ˜ WS = hHSiα˜1 − hHSiα˜0 . true, then we can simply write down:

Proof. Consider a system with Hamiltonian HS = P [α1|α˜0] |α|2(1−e−βhν )−β∆F † 1 ∗ ∗ = e (212) hν(a a + 2 ). For simplicity, we take an arbitrary state P [α0|α˜1]

Accepted in Quantum 2019-10-24, click title to verify 38 2 2 2 √ where |α| = |α0| −|α1| and ||α|i is the coherent work is dimensionless and we count in energy units of nλ output of the hypothetical process. Owing to the sim- when we have n systems. It might look like the inverse ple form of the effective potential for coherent states, temperature is decreasing as we increase the number of this is fairly trivial to write down. However generally, systems, but this is not what is intended by this – in the effective potential fails to obtain such a simplified the description of the sequence of systems one scales form, and simplifying the change in effective potential the units of energy as we increase n when we specify Λ(β, ψ1) − Λ(β, ψ0) may be harder than finding the an- the temperature. Note also that the first and second alytic form for Λ(β, ω). cumulants scale differently with energy, and it is this that makes the scaling trick needed to obtain a state- B.6 Macrosopic limit of the quantum fluctuation ment containing both in the large n limit. relation Expressed in these units we have β √ For simplicity, we choose to make the random variable Λ(√ 0 , ψ ) = β µ n + Λ(˜ β µ , ψ ), (224) nλ n 0 0 0 0 n dimensionless. We have that hψi|Hi|ψii = µ, which is 1 µ independent of i. We define the operator Xi := Hi−1, µ where µ0 := λ is dimensionless, and β0 does not vary which has mean zero in the state |ψii. Note that the in n. variance of Xi is also dimensionless and given by This holds for all n, and in the n → ∞ limit we have that 2 2 2 hXi i − hXii = hXi i (213)   β0 √ 1 2 2 2 1 2 2 lim Λ(√ , ψn) − β0µ0(ψ) n = − σ˜ψβ0 µ0(ψ) , 1 n→∞ = h 2 Hi − Hi + i (214) nλ 2 µ µ (225) 1 = hH2i − 2 + 1 (215) and also µ2 i  β √  1 1 √ 0 2 2 2 = (hH2i − µ2) (216) lim Λ( , φn) − β0µ0(φ) n = − σ˜φβ0 µ0(φ) , µ2 i n→∞ nλ 2 2 (226) σ 2 Var(Xi) = ≡ σ˜ . (217) where we have included the state variables to distin- µ2 guish the energies of the two states. This implies that for any  > 0 there is an integer The cumulant generating function of the total state M such that for all n > M we have the following two is conditions hold P −Λ(β,ψ ) −β Hi ⊗n e n = tr[e i ψ ] (218) β0 √ 1 2 2 P Λ(√ , ψn) − [β0µ0(ψ) n − σψβ0 ] ≤  (227) −βµ (Xi+1) ⊗n nλ 2 = tr[e i ψ ] (219)

P β0 √ 1 −βnµ −βµ Xi ⊗n 2 2 = e tr[e i ψ ] (220) Λ(√ , φn) − [β0µ0(φ) n − σφβ0 ] ≤ , (228) nλ 2 √ P 1 −βnµ −β nµ √ Xi ⊗n = e tr[e i n ψ ] (221) where we also used the definition of σ˜2. We therefore ˜ √ = e−βnµe−Λ(β nµ,ψn). (222) have that for any  > 0 there is an M > 0 such that for all n > M we have ˜ Here Λ is the CGF corresponding to the dimensionless β β √ 1 1 0 0 2 2 mean zero observable √ (X +···+X ). Thus we have |Λ(√ , ψn)−Λ(√ , φn)−[β0 n∆µ0− β ∆σ ]| ≤ 2. n 1 n nλ nλ 2 0 0 that (229) ˜ √ Λ(β, ψn) = βnµ + Λ(β nµ, ψn), (223) Thus, for any  > 0 there is an M such that for all for all β, n. n > M we have √ A subtlety comes in here – if one wants to take a limit P [ψn|φn] 1 2 2 − n(β0∆f0+β0∆µ0)+ 2 β0 ∆σ0 and obtain a Central Limit Theorem (CLT) statement ∗ ∗ − e ≤  (230) P [φn|ψn] we need to scale β also in order to get a non-trivial ∆F statement. This is a mathematical re-formulation so as where ∆f0 := nλ is a dimensionless number. We could to maintain the correct scaling to obtain a non-trivial put back in β, but only with the proviso that it is as- statement the mean value of the sequence (in n) of ran- sumed to scale with n in the way stated. This gives dom variables. ˜ Since β has units of reciprocal energy we can set P [ψn|φn] −n(β∆f+β∆µ− 1 β2∆σ2) − e 2 ≤  (231) √β0 P [φ∗ |ψ˜∗ ] β = nλ , for some arbitrary energy scale λ, where β0 n n

Accepted in Quantum 2019-10-24, click title to verify 39 √β0 with the proviso that the scaling β = nλ is adopted. This captures the informal statement that for the IID case the states “become more like Gaussians” and thus the fluctuation theorem behaves as ˜ P [ψn|φn] n→∞ −n(β∆f+β∆µ− 1 β2∆σ2) ∼ e 2 . (232) ∗ ˜∗ P [φn|ψn]

Accepted in Quantum 2019-10-24, click title to verify 40