Fluctuations in Single-Shot -Deterministic Work Extraction
There has been an increasing interest in the quantification of nearly deterministic work extraction from a finite number of copies of microscopic particles in finite time. This paradigm, so called single-shot -deterministic work extraction, considers processes with small failure probabilities. However, the resulting fluctuations in the extracted work entailed by this failure probability have not been studied before. In the standard thermodynamics paradigm fluctuation theorems are powerful tools to study fluctuating quantities. Given that standard fluctuation theorems are inadequate for a single-shot scenario, here we formulate and prove a fluctuation relation specific to the single-shot -deterministic work extraction to bridge this gap. Our results are general in the sense that we allow the system to be in contact with the heat bath at all times. As a corollary of our theorem we derive the known bounds on the -deterministic work.
Thermodynamics has been historically developed as a discipline that deals with a large number of systems, e.g. an ensemble of particles in a box. In recent years, however, there has been significant interest in studying the relationship between thermodynamics and statistical physics, for finite number of copies of particles and finite application of operations on them. This limit is called single-shot thermodynamics Del Rio et al. (2011); Gemmer and Anders (2015); Horodecki and Oppenheim (2013); Åberg (2013). A particularly interesting question in single-shot thermodynamics is extraction of work from systems that are out of equilibrium Åberg (2013); Horodecki and Oppenheim (2013). In order to formulate work extraction, we first need to define work. The first law of thermodynamics splits energy into an ordered form and a disordered form. The ordered form of energy is called work. The challenge faced in the quantification of work in the single-shot regime is that, for small ensembles of microscopic systems, fluctuations dominate. Hence, determining the typical behaviour of systems, for which one would need large ensembles, is challenging. Thus, it is not clear how one can quantify ordered energy, i.e. work as perceived in standard thermodynamics. A way around this is to design a process which uses an amount of energy to lift the state of an external system, called ‘battery’ from a single energy level to another single energy level. This way one can define a notion of work that can be quantified in the single-shot regime, and that also corresponds to the notion of work as understood in standard thermodynamics. Nevertheless, since fully deterministic work extraction can be too stringent a constraint, one might also allow processes that fail with a small probability . Such a process is called ‘-deterministic’ which is to say ‘nearly deterministic’. Horodecki and Oppenheim in Horodecki and Oppenheim (2013) discussed a scenario involving a system out of thermal equilibrium and block diagonal in the energy eigenbasis, coupled to a bath and battery. The task is to lift the state of the battery from the ground state to its excited state -deterministically. If the task is done successfully, a certain amount of work is stored in the battery. In this scenario of a small failure probability they find an upper bound for the work extracted. The existence of a failure probability entails some fluctuations in the work extracted. Experiments which fail to extract exactly the desired amount of work are dismissed entirely. What has not been considered so far is that rather than dismissing these experiments one can quantify the alternative amount of work that has been extracted and how it fluctuates. Prior to the work presented here it was not known how to characterise these fluctuations. In this paper, we formulate and prove a theorem (Theorem 1) that characterises the fluctuations of the extractable work due to the failure probabilities in single shot thermodynamics.
To answer this, we use an idea developed in another domain of statistical physics. In statistical physics powerful tools called fluctuation theorems Campisi et al. (2011) have been developed to characterise fluctuations in quantities such as work or entropy. They put restrictions on the probability density function of these fluctuating quantities. In the case of work, for instance, these theorems compare the probability of work cost of a thermodynamic process with the probability of work gain of its reverse process. An example of fluctuation theorem is Crooks work relation Crooks (1999). Crooks theorem relates the probability, , of work cost, , of a process to the probability, , of work gain in the time-reverse of that process, where and are the process and its inverse respectively. The relation is formally expressed as , where is the inverse temperature of the environment divided by the Boltzmann constant, and is the equilibrium free energy difference of the initial and final state. A thermodynamic process, there, is defined by changing the Hamiltonian of the system at an arbitrary speed, according to a specific trajectory of the Hamiltonian. And the reverse process is one where the Hamiltonian is brought back to the initial setting, according to the reversed trajectory of the forward process. The ratio of these two probabilities equals the exponential of the work dissipated in the process, . Therefore, in a process where no work is dissipated, e.g. an isothermal expansion, these two probabilities are the same and no fluctuation in work occurs. As we see, this theorem characterises fluctuations in a thermodynamical quantity that meets the conditions of the theorem, by comparing the probability of the forward process to that of the backward process. In this paper we show that the main ideas of fluctuation relations can be used to characterise the fluctuations arising from the failure probabilities. However, since the currently existing fluctuation theorems have different assumptions to the ones in the single-shot work extraction, we formulate our theorem specifically for the single-shot scenario. This theorem will put the notion of fluctuations in -deterministic work extraction, and that in standard thermodynamics on an equal footing.
To formulate a fluctuation theorem for single shot thermodynamics, we relax the original assumption of Crooks work relation that the system, both in the forward process and its inverse, starts in a state of thermal equilibrium with its environment. Our setup, in keeping with the setting in Horodecki and Oppenheim (2013), consists of a work system, a battery, and a thermal bath. The work system remains in contact with the bath throughout the process. The protocol extracts work by performing some operations on the work system and as a result lifts the battery to the excited state with failure probability. As a corollary of our result we obtain the bounds of the extractable work found by Åberg Åberg (2013).
It is worth noticing that as a byproduct of choosing the present setting, we avoid a potential complication in the fluctuation analysis of open systems. In particular, thermodynamic work, , is measured in a number of different ways in the context of fluctuation relations. Predominantly, the so-called scheme of Two-Measurement Protocol (TMP) is used were the eigenvalues of the initial and final Hamiltonians are measured, and work, , is defined as the difference between the two eigenvalues. Considering the first law of thermodynamics, we observe that for closed systems with no heat flow this quantity is the same as total work. However, in the case of open systems with strong coupling TMP does not provide a good measure of work. In our setup all of the work goes to lifting the state of the battery and despite the fact that the heat bath is attached to the work system at all times, a simple act of measuring the initial and final eigenvalues of the battery gives a correct measure of the work.
The tool with which we choose to prove the theorem is the so-called resource theory of states out of thermal equilibrium. Resource theories Gour and Spekkens (2008); Brandão et al. (2013) are mathematical formalisms, designed to answer questions such as ‘what state can be prepared in a given physical situation?’ or ‘is a transition from one state to another possible, and if yes what is the transition probability?’. By answering these questions, resource theories allow us to study what happens to the state of a system in a physical situation in full generality. Resource theories achieve this by restricting the allowed operations to a predefined set. A given set of operations one can prepare a set of states for free. Any state outside that set is considered a resource. For instance, it is a well-known fact that if we restrict ourselves to local operations and classical communications (LOCC), we can create separable states. For LOCC, therefore, an entangled state is considered a resource. The resource theory of athermal states is defined by restricting our allowed operations to the so-called thermal operations. Thermal operations, as explained concisely in Alhambra et al. (2016), are the set of maps characterised by the following rules a system with any Hamiltonian in the Gibbs state of that Hamiltonian can be added, any subsystem can be discarded through tracing out and any energy-conserving unitary, i.e. those unitaries that commute with the total Hamiltonian, can be applied to the global system. For these operations a state out of equilibrium is a resource and can be utilised to extract work. The commutation condition is imposed to ensure that total energy is conserved. As we see, the resource theory of athermal states provides a framework wherein one can study the most general thermodynamical state transformations that respect energy conservation. Specifically, in our theorem, as we are interested in thermodynamical processes that extract a certain amount of work, we use this framework to identify all possible processes that meet our conditions.
In our setup, initially the total system consists of the work system in state from which we intend to extract work, a bath in state and a battery in its pure ground state. The battery is a finite-dimensional many-level system, used for storing the ordered energy, i.e. work. A many-level battery is needed in our case in order to store the different possible amounts of the fluctuating work. A thermal operation is used to extract work from the work state. The thermal operation is generated by a global unitary operator on the initial state of the total system. This operator creates correlations between the work state, bath, and battery. As we can accept a small failure probability, the final state of the system can be slightly different every time we run the protocol. In particular the battery may be charged to the point , where here signifies different excited levels of the battery. The thermal operation is chosen such that it aims to transform the initial state of the work system to the Gibbs state . However, again, due to the failure probability, the work system might end in a state that is not exactly Gibbs, we call this state . A more detailed study of the final state in the setting we choose can be found in Gemmer and Anders (2015). To formulate our fluctuation theorem we also need to specify a backward process. The backward process is generated by the unitary operator, inverse of the global unitary used in the forward process. We choose the initial state of the backward process to be the correlated state of the work system and bath, such that the work state is left in a Gibbs state, after tracing out the state of the bath, as explained below. The initial state of the battery in the backward process is the same as its final state in the forward process. This guarantees that the work extracted in the forward process and the work cost in the backward process are the same. The forward and backward processes are illustrated in Fig. (1).
Our choice of the initial state of the backward process is due to a subtlety inherent in single-shot thermodynamics. In Brandão et al. (2013) it was shown that there is an intrinsic irreversibility in the single shot scenario in the sense that if a resource can be converted to a resource , in general one cannot assume that the resource can be converted back to the initial resource under the same conditions. In particular, in the first part of Horodecki and Oppenheim (2013) the maximum work extractable in the work extraction setting above was found, where work was extracted by transforming the resource state into the Gibbs state. In the second part, a reverse of that process was defined as a task of starting with a bath, a resource system in Gibbs state and a battery, all initially in tensor product with each other. It was shown that the work cost of returning the Gibbs state back to is greater than the work extracted in the first part in the single-shot scenario. This is due to the detailed correlations that are created between the system and the bath after the application of the global unitary that generates the thermal operations, which are significant for single-shot work extraction. Therefore, the backward process for a fluctuation relation described above has to begin in a correlated state of resource system and the bath, such that the correlations are produced by the operations of the kind used in the forward process.
In our theorem we use the -smooth Renyi relative entropy of order . The Renyi relative entropy of order is defined as , with being the support of . The -smooth version of this entropy is defined by maximising it over the distributions that are -close to the state of the system, formally defined as
Iii The Main Theorem
We now state the main result, a fluctuation theorem relating the ratio between the probability of extracting work in an -deterministic forward process and the probability of putting work back into the system in a reversed process.
Consider a total system consisting of a work system in some state , diagonal in energy eigenbasis, a thermal bath at temperature in state , and a many-level battery system in its pure ground state . Furthermore, assume -deterministic forward process , defined by a global unitary on the total system that commutes with its Hamiltonian, and a reverse process , defined by inversing the global unitary of the forward process. The initial state of the backward process is a correlated state of the bath and the resource system, such that the work state is in a Gibbs state, , after tracing out the bath. With these assumptions, the ratio of work extraction probability, , to work cost probability, , is given by
where is the work extracted. , where is the Boltzmann constant and is the temperature. is the smooth version of Renyi relative entropy of order . is the amount of deviation from the maximum extractable work, with .
We provide the proof of Theorem 1 in the Appendix. Theorem 1 characterises the fluctuations in the extracted work due to the failure probability in the single-shot -deterministic work extraction scenario. It states that if a quantum resource is used to extract work in an -deterministic process by thermal operations, the probability of the work extracted is related to the probability of the work put back into the system in its reversed process by an exponential factor. The term in the exponent of Eq. (3) is the analogue of dissipated work. This characterises a type of irreversibility in the finite-run behaviour of a microscopic system in the single-shot regime, akin to that of Crooks relation. As discussed before, the final state of the system in the forward process may be such that the state of the system is not exactly Gibbs. The term signifies this deviation. We show the details of this in the Appendix.
In the following corollary we shall observe how a known bound of work extraction follows from our result.
The work that can be extracted from a system out of equilibrium by thermal operations in an -deterministic process is bounded by
To see this, we first derive the lower bound and then the upper bound as follows. For the lower bound we multiply both sides of the Eq. (3) by . From the fact that we have
In the discussion section of Ref. Åberg (2013), Åberg leaves the question of the link between the fluctuations of the -deterministic work extraction paradigm to that of the fluctuation theorems paradigm open. Theorem 1 above has answered this question. One can see, as we show in the appendix, that in the absence of any fluctuations of the former type (the case where and are zero) the the dissipated work goes to zero and the forward and backward probabilities take the same value. This means there will be no fluctuations of the latter type. It is only through the introduction of the failure probabilities that we obtain a fluctuation theorem, and hence demonstrating the direct connection between the two. We have derived an equality in the same way Crooks relation does for work probability density function in the standard account of thermodynamics. The relation in Theorem 1 quantifies the ratio of the probability of work extraction in a process to that of work cost in its backward process, in terms of the smooth Renyi relative entropy of order . As a corollary we find the known bounds of work extraction protocols. The necessity of formulating and proving this variation of fluctuation theorem is that the single-shot regime has different assumptions to the standard thermodynamics, which renders the standard fluctuation relations inadequate for our purposes. In the standard fluctuation relations the work is determined by making several energy measurements at the beginning and the end of the processes and comparing the energy eigenvalues. This requires many measurements. In the single-shot scenario we require to be able to determine work by a single measurement. In our setting this is done by letting the process store all of the work in the battery. Then the work is measured by a single energy measurement on the battery state. Given the record in the experimental applications of the present fluctuation theorems Liphardt et al. (2002); Collin et al. (2005); Douarche et al. (2005), we believe our work will contribute to the experimental results on single-shot thermodynamics by providing a method to measure smooth Renyi entropies. As an outlook on further theoretical extensions of this work, one can think of generalisations or restrictions on the allowed operations, as well as consideration of quantumness. For instance Faist et. al. have shown that Gibbs-preserving maps outperform thermal operations in the quantum regime Faist et al. (2015). A further investigation into the hierarchy of operations and possible exploitation of quantum correlations and catalysts merits an extended discussion. Furthermore, it would be interesting to apply a version of Asymptotic Equipartition Theorem Tomamichel et al. (2009) to investigate the relationship between our theorem and the ones in the standard statistical mechanics. In this paper we have focused on the notion of -deterministic work. However, probabilistic work in the single-shot scenario is also an interesting concept and has been recently studied Alhambra et al. (2016).
Note added: Similar results were obtained independently by Dahlsten et al, using a different set-up and different starting assumptions, in Equality for worst-case work at any protocol speed.
Acknowledgements We would like to thank Johan Åberg, Jens Eisert, Oscar Dahlsten, Rodrigo Gallego, Henrik Wilming and Albert Werner for their comments and discussions. Part of this work was supported by the COST Action MP1209 ”Thermodynamics in the quantum regime”, Hong Kong Research Grant Council through Grant No. 17326616, National Science Foundation of China through Grant No. 11675136, HKU Postdoctoral Fellowship. K.W. acknowledges funding through EPSRC (EP/E501214/1).
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Appendix A Proof of the Theorem 1
Here, we prove Theorem 1 by deriving Eq. (3). However, to begin with we consider the simple case of a process which extracts the amount of work from the work system with probability one, i.e. with failure probability . For such a case, the following lemma holds. This does not give a fluctuation theorem directly, as without a failure probability there is no fluctuation. However, it is an important building block in proving the theorem. In this lemma we use Renyi relative entropy of order defined as , with being the support of the initial state and the thermal state of the system. In the following lemma we adopt the same sign convention for work as used in Horodecki and Oppenheim (2013); Åberg (2013). Here we have the same assumptions as in Horodecki and Oppenheim (2013) for the allowed operations to be thermal operations, which are obtained by performing some global unitaries on the total initial state. In the following lemma the reverse process is obtained from the inverse of the global unitary in the forward process. The initial state of the reverse process, here, is set to be the same as the final state of the forward process. Our assumptions on the bath are also the same as Horodecki and Oppenheim (2013) namely that (i) the spectrum of the heat bath is continuous, i.e. for an energy of the heat bath and two arbitrary energies of the system, and , there exists , such that , (ii) and around the energy the degeneracies can be written as (11) (12) where and . Notice that for a large enough bath the second and higher order terms in Eq. (11) and (12) can be neglected, which is the assumption we adopt throughout this paper.
Consider a total system consisting of a work system in some arbitrary state , diagonal in energy eigenbasis, subject to a time-independent Hamiltonian, a thermal bath at temperature in state , and a two-level battery system in its pure ground state . Here, the work is extracted by lifting the state of the battery from its ground state to the pure excited state . The gap between the ground and the excited state of the battery is set to . Furthermore, assume processes and consisting of Thermal Operations only. Then
with where is the Boltzmann constant.
In Horodecki and Oppenheim (2013) it was shown that for a fixed total energy if the bath is very large compared to the system, the state can be written as , where , is the degeneracy of the bath, and the identity acts on a -dimensional space. Similarly, the final state can be written as . Since the total system is block diagonal in energy eigenbasis, we can treat the eigenvalues of it as probabilities. Denote the energy levels of initial and final state of the work system by and , respectively. The eigenvalues of the initial and final state are and , respectively. Denote the transition current, i.e. the number of eigenstates that go from to by . Then occupation probabilities of the final state of the system are given by
Each summand on the RHS is the joint probability of the system initially occupying the th energy level and ending up in the th energy level in the final state. Since the protocol stores the energy difference between initial and final state, i.e. the work , in the battery state the probability of a value of work can be calculated by summing the RHS over . Notice that the total number of eigenstates in initial energy level is
and that the total number of eigenstates in final energy level is
We also need to consider the backward process. Recall that the global unitary for the forward process gives transition currents as the rate at which energy eigenstates are transformed from to . Therefore, as we are restricting ourselves to the diagonal case, this current can always be used to directly characterise the backward transition currents, , as the rate at which the energy eigenstates are transformed back from to . Notice, for the reversed process we have
which is the number of eigenstates in the final state of the forward process. In the setting of this lemma, this is also the number of eigenstates of the initial state of the backward process.
As in the standard Crooks relation, it is also important to fix the initial state of the backward process. For this lemma we use the final state of the forward process as the initial state of the backward process. Here, the final state of the bath and the work system is a correlated state such that the system is in thermal state after tracing out the bath.
Remark 1 – Formally speaking, a thermal operation is generated by a global unitary , acting jointly on the reservoir, system and battery, such that , where is the total state of the reservoir, system and battery, with a reduced state . The restriction on this global unitary is that it has to commute with the total Hamiltonian of the reservoir, system and battery. In the current case, the global unitary sends the initial state to the final state
such that the reduced state of the system, is the Gibbs state. These are the same initial and final states considered in the exact transformation case in Horodecki and Oppenheim (2013). Indeed, in Horodecki and Oppenheim (2013) it was shown that a reverse process starting with a fresh bath will cost more energy than the forward process. However, defining the backward process as the one produced by the inverse of the forward unitary, i.e. , and starting from the final state of the forward process, clearly completely reverses the process as . We shall make a remark on the backward process with non-zero epsilon later in this appendix, after we provide the proof of the theorem.
Below, we calculate the ratio of the work cost probability to that of work gain probability for such a process.
Using the fact that the system ends up in the Gibbs state, together with the assumption on the bath in Eq. (12) that , one can rewrite the term in the numerator as follows
where is the partition function of the Gibbs state.
Notice that each of the probabilities in the numerator and the denominator of the fraction in the LHS of Eq. (23) equal one, i.e.
since the sum of probabilities of the initial and final energy levels have to equal one. Therefore, the fully deterministic work extraction has only a trivial distribution of one value of extracted work and zero everywhere else. For a fluctuation theorem to say something about the work probability, it has to have a width. Below we extend the relation to the -deterministic case, thereby giving the proof of the Theorem. To achieve this, we use the technique of -ordering as introduced in Supplementary Note 4 of Horodecki and Oppenheim (2013) and as illustrated in Fig. (2). The graph plots the Gibbs rescaled probabilities against the truncated partition function for each energy level of the work system, , arranging states in decreasing order of their value . In this staircase function, the area of each rectangle is equal to the probability of the state, , and the total width is equal to , the partition function. Removing the maximum number of bins in Fig. (2), with the minimum total probability then corresponds to removing states from the right side of the -ordered spectrum. This procedure is called smoothing.
We want to express the LHS of Eq. (3) in terms of the work extracted from the system. We first start by giving the ratio of the work cost probability to the probability of its reverse process for the case maximum possible work was extracted. Then we use that to prove the fluctuation theorem in the general case.
Starting with extracting the maximum work in the -deterministic forward process we need to find a minimal set of eigenstates such that their total probability is at least . Removing an weight of the initial state amounts to such a smoothing.
To achieve this we -order the eigenstates of the initial state as explained in the main text. Then we remove as many eigenstates from the low weight end of the spectrum as possible while staying within the limit. This is done by choosing an index , such that
Now we map these eigenvalues of weight to . For such a mapping, the ratio of the work gain distribution to its work cost counterpart is
In Eq. (26) we used Eqs. (24) and (25) to count the number of states with their specific weights that are mapped to the work extracted. As opposed to the setting in Lemma 1, here th initial state of the reverse process and the final state of the forward process are not the same. In Gemmer and Anders (2015) it was discussed that the final state of the system in an -deterministic process may not be exactly a thermal state. To formulate the fluctuation theorem for these processes, we choose the initial state of the backward process to be the same as that in the Lemma 1, where the backward process starts by the system correlated with the bath, such that the reduced state, describing the system is in the Gibb state. Using this, in Eq. (27), the probability of the system being in energy state is re-written in terms of the probabilities in the backward process. In the backward process the work is returned from the battery to the system also via a -deterministic process. It was shown in Åberg (2013) that for -deterministic processes, the thermal distribution has a maximum work content of . This means our reverse process costs this amount less than a deterministic reversal. Therefore we subtract this amount from the work stored in the battery when calculating the total number of states in the backward process in order to conserve the total energy. Hence, defining we have
which gives the denominator in Eq. (28). As we see, Eq. (28) gives ratio the probability of maximum work extraction to that of its reversed process in terms of the number of the eigenstates that are mapped to the work content. However, in an -deterministic scenario we do not always extract the maximum work possible, which is the motivation for having a fluctuation theorem. We would like to have the ratio above for any possible work that can be extracted in an -deterministic process. Let us consider an unknown parameter to characterise a possible work extracted. A work is obtained by mapping a number of eigenstates in Eq. (28) to work. Since the battery in that case is charged by amount, the ratio above generalises to
Eq. (30) follows from a similar argument where we derived Eq. (23) from Eq. (21), albeit with smoothing. For the definition of smooth Renyi relative entropy see Eq. (1). Smoothing maximises Renyi relative entropy over all states that are -close to the initial state. As we discussed, in our situation, this is done by removing the eigenstates that have the lowest probability, with the total weight of . The expression in the big brackets in the numerator of Eq. (28) corresponds to times that entropy and is the same expression used in Horodecki and Oppenheim (2013). ∎
Remark 2 – For clarity of the proof we have used the notation for the work extracted. This corresponds to in the main text and in particular in the statement of Theorem 1.
Remark 3 – To formalise the fact that the backward process is the reversal of the forward process in the presence of the epsilon-error, notice that we require that the battery has to be charged the amount required to perform the backward process. Although here the global unitary that generates the thermal operation is not unique, one can still decompose any such unitary , as a completely positive trace-preserving map on the reservoir and the system, , and the corresponding translation unitary map on the battery as . This means that defining the backward process by the inverse of the forward unitary will send the battery from its charged level to the ground state. Finally, in order for this process to be a valid backward process, we require that the map send the final state of the forward process of Lemma 1 to a state that is close to our initial state. Suppose the unitary that has been implemented is one that takes the states that are on the surface of the ball of -close states to into , i.e. the final state achieved in Lemma 1. If the process fails to extract the maximum work, the final state will not exactly be . Nevertheless, the backward process, defined by the inverse of the unitary will map the state given by the final state of Lemma 1, to a state that is on the surface of the ball of -close states to . This is of course the farthest that the final state of the backward process can be from . Suppose the unitary is implemented, such that mapping does not use all the eigenstates in the range depicted as in Fig. (2). In that case the inverse process will map the final state of Lemma 1 into a state within the ball of -close states to .