Energetic instability of passive states in thermodynamics

Within thermodynamics, heat engines are devices that operate in a thermal context so as to extract ordered energy in the form of work. The canonical scenario involves an engine that operates cyclically between two temperatures $T_{\text{hot}}$, $T_{\text{cold}}$ and performs a quantity of mechanical work. To do so the engine absorbs heat from the hot reservoir, converts some of this energy to mechanical work and releases heat into the cold reservoir in accordance with the Second Law of thermodynamics. The largest possible efficiency, $\eta=1-\frac{T_{\text{cold}}}{T_{\text{hot}}}$, occurs for the reversible Carnot engine carnot_reflections_1824 ; callen_thermodynamics_1985 and provides a fundamental thermodynamic bound on the amount of ordered energy that can be obtained. Carnot engines, and more in general heat engines, have been extensively studied in the microscopic regime scully_extracting_2003 ; scovil_three-level_1959 ; geusic_quantum_1967 ; alicki_quantum_1979 ; howard_molecular_1997 ; geva_classical_1992 ; hanggi_artificial_2009 ; feldmann_quantum_2006 ; rousselet_directional_1994 ; faucheux_optical_1995 ; linden_how_2010 ; horodecki_fundamental_2013 ; brunner_virtual_2012 ; tajima_optimal_2014 ; ito_optimal_2016 ; verley_unlikely_2014 ; gardas_thermodynamic_2015 ; uzdin_collective_2015 ; uzdin_quantum_2016 ; kosloff_quantum_2016 ; woods_maximum_2015 ; frenzel_quasi-autonomous_2016 ; lekscha_quantum_2016 ; niedenzu_operation_2016 (as well as, of course, in the macroscopic regime).

However, the issue of ordered energy extraction can also be considered in scenarios in which no notion of temperature exists, and can provide a broader notion of equilibrium states. For example, more general equilbrium states can occur in physical realisations when a system has been perturbed and has not had enough time to fully thermalise. They can also arise in the context of non-equilibrium steady states seifert2012stochastic . Given a quantum system in a state $\rho$ one can ask if it is possible to extract energy from it solely by performing a reversible unitary transformation on the system. The largest amount of ordered energy that can be extracted (the “ergotropy”, see Refs. allahverdyan_maximal_2004 ; alicki_entanglement_2013 ; sparaciari_resource_2016 ) depends non-trivially on the quantum state. If no energy can be extracted in this way then $\rho$ is called passive pusz_passive_1978 ; lenard_thermodynamical_1978 ; perarnau-llobet_extractable_2015 ; perarnau-llobet_most_2015 ; hovhannisyan_entanglement_2013 and constitutes a primitive form of equilibrium.

In this work, we consider a scenario that is intermediate between the above two contexts, and is motivated by the fact that a work extraction machine should be considered as a system which is involved in the process. Our core question is whether there exist passive states $\rho_{S}$ for which energy can be extracted if one performs a reversible unitary process over the system $S$ together with a second quantum system $M$, which starts and finishes in the same quantum state $\rho_{M}$. This second quantum system is the machine, such as the working body in a Carnot cycle, which undergoes a cyclic evolution. This class of processes is reminiscent of the ones taking place inside heat engines, and has been termed a catalytic thermal operation brandao_second_2013 .

Indeed, the system $S$ described by a passive state represents both the reservoirs, while the additional system $M$ is the machine which exchanges energy with the reservoirs in a cyclic manner. Due to the fact that microscopic heat engines can nowadays be realised in the laboratory scovil_three-level_1959 ; rousselet_directional_1994 ; faucheux_optical_1995 ; howard_molecular_1997 ; schulman_molecular_1999 ; scully_quantum_2002 , it seems reasonable to extend the analysis on passive states to the case in which this broader class of operations is allowed. Crucially, this analysis has fundamental implications for the notion of passivity. In fact, if energy can be extracted from a passive state with these reversible processes, and no entropy is generated, then it seems that associating passivity of the state is a restricted idealisation, unstable under this simple extension.

The paper is structured as follows. In Sec. II we provide the definition of passive and completely passive states, together with their description in terms of virtual temperatures. Any three levels of a passive state can be thought of as containing a subsystem which acts as a virtual hot reservoir, and one which acts as a virtual cold one. Sec. III provides the description of a protocol involving finite-sized engines which enable us to extract work in a cyclic way from a single passive state. In Sec. IV we prove that in fact any passive state which is not thermal can be activated by such an engine. Finally in Sec. V we show that such an activation can be done optimally via a quasi-static, reversible process with zero generation of entropy. We provide a simple expression for the amount of work which can be extracted from an arbitrary passive state, Eq. (38), and show that our machine can operate at the efficiency of a Carnot engine operating between two virtual heat reservours. We conclude that the only states in the single-shot regime that are stable under general reversible processes are thermal states.

Consider a finite-dimensional quantum system associated with the Hilbert space $\mathcal{H}\equiv\mathbb{C}^{d}$ (a qudit), with Hamiltonian $H=\sum_{i=0}^{d-1}E_{i}\ket{i}\bra{i}$, and described by the state $\rho$. We say that the state $\rho$ is passive iff its average energy cannot be lowered by acting on it with unitary operations, that is,

This implies that no work can be extracted from the state via a unitary process, since by conservation of energy, lowering the energy of a system would mean that this energy has been transfered to a work storage device.

We can also introduce a more restrictive notion of passivity. Let us consider $n\in\mathbb{N}$ independent and identically distributed (i.i.d.) copies of our system, with a total Hamiltonian $H^{(n)}=\sum_{i=1}^{d}H_{i}$, where each $H_{i}$ is a single-system Hamiltonian acting on a different copy of the system. The state of this global system is described by $\rho^{\otimes n}$. Then, we say that the state $\rho$ is completely passive if and only if the state $\rho^{\otimes n}$ is passive for all $n\in\mathbb{N}$. It can be shown pusz_passive_1978 that the completely passive states of a system with Hamiltonian $H$ are the ones satisfying the KMS condition kubo_statistical-mechanical_1957 ; martin_theory_1959 ; haag_equilibrium_1967 . Specifically, these states are the ground state and the thermal states with temperature $\beta\geq 0$, that is, $\tau_{\beta}=e^{-\beta H}/Z$ with $Z=\mathrm{Tr}\left[{e^{-\beta H}}\right]$. Any state which is not of this form, is called athermal.

A characterization of all passive states can be easily obtained. A system in a passive state is such that the ground state has the highest probability of being occupied, and the probability of occupation decreases as the energy associated with the eigenstate of $H$ increases, Fig. 1, left plot. Specifically, a state $\rho$ is passive iff $\rho=f(H)$, where $f$ is a monotone non-increasing function. Simply put, this means that the state can expressed as

where $\left\{\ket{i}\right\}_{i=0}^{d-1}$ are the eigenvectors of $H$, ordered so that $E_{i}\leq E_{i+1}$ for all $i$ (for the case of equal energies $E_{i}=E_{i+1}$ we must make an additional stability assumption to ensure that $p_{i}=p_{i+1}$).

We can describe the probability distribution of the passive state $\rho$ by using virtual temperatures brunner_virtual_2012 ; skrzypczyk_passivity_2015 . In fact, for any given passive state, we can associate a (non-negative) virtual temperature with each pair of its eigenstates. For example, if we consider the pair $\left(\ket{i},\ket{j}\right)$, we define the virtual temperature associated with them as the $\beta_{ij}^{-1}\geq 0$ such that

where $\mathrm{p}_{i}$ is the probability of occupation of the state $\ket{i}$, and $E_{i}$ is the energy associated with the state (similarly for $j$). Thus, each pair of states can be regarded as an effective thermal state at a specific temperature. When all pairs of states has the same virtual temperature, we have that the passive state is completely passive, that is, it is the thermal state of $H$ at that temperature.

We now introduce an engine that extracts work by acting individually on a passive state. The engine is composed by two main elements, namely, a qudit “machine” system with trivial Hamiltonian ($H=0$), and a particular passive state. It suffices to consider qutrit systems, as a similar construction works more generally. The qutrit is assumed to have a Hamiltonian

and $E_{i}\leq E_{i+1}$. The qutrit system is described by the state

where $\mathrm{p}_{i}\geq\mathrm{p}_{i+1}$ (Fig. 1, left plot).

In the following we assume that the passive state $\rho_{P}$ is described by the virtual temperature $T_{\text{hot}}=\beta_{\text{hot}}^{-1}>0$ associated with the pair of eigenstates $\left(\ket{0}_{P},\ket{1}_{P}\right)$, and the virtual temperature $T_{\text{cold}}=\beta_{\text{cold}}^{-1}>0$ associated with the pair $\left(\ket{1}_{P},\ket{2}_{P}\right)$. We assume for simplicity that $T_{\text{hot}}>T_{\text{cold}}$, but a similar analysis applies for $T_{\text{hot}}<T_{\text{cold}}$. In the Supplemental Material, Sec. I, the cycle is presented in full detail. The relation between the probability distribution of $\rho_{P}$ and the temperatures $T_{\text{hot}}$ and $T_{\text{cold}}$ is given by

where $\Delta E_{10}=E_{1}-E_{0}\geq 0$, and $\Delta E_{21}=E_{2}-E_{1}\geq 0$. Thus, the pair of states $\left(\ket{0}_{P},\ket{1}_{P}\right)$ can be viewed as representing a “hot virtual reservoir”, while the pair of states $\left(\ket{1}_{P},\ket{2}_{P}\right)$ represent a “cold virtual reservoir”. It is worth noting that the other pair of states, $\left(\ket{0}_{P},\ket{2}_{P}\right)$, is associated with a virtual temperature that is intermediate between $T_{\text{cold}}$ and $T_{\text{hot}}$, as we can easily verify from Eqs. (6).

The engine extracts work by means of the following cycle. A single system, described by the state $\rho_{P}$, is put in contact with the machine, described by the state $\rho_{M}=\sum_{j=0}^{d-1}\mathrm{q}_{j}\ket{j}\bra{j}_{M}$. Then, we perform $m$ swaps between the hot virtual reservoir of the passive state and $m$ different pairs of states of $\rho_{M}$, followed by $n$ swaps between the cold virtual reservoir and other $n$ different pairs of states of $\rho_{M}$. In order to perform the swaps on different pairs of states, we need the machine to have at least $m+n$ levels, and therefore we fix $d=m+n$. Specifically, we apply the following unitary operation to the global system

where the operator $S_{(a,b)}^{(c,d)}$ is a swap between system and machine, performed through the permutation $\ket{a}_{P}\ket{d}_{M}\leftrightarrow\ket{b}_{P}\ket{c}_{M}$. A graphical representation of this global operation is shown in Fig. 2, where each swap is depicted by an arrow acting over the states of the machine. Although in the figure we represent the eigenstates of $\rho_{M}$ in a ladder, they are all associated with the same energy, and therefore the order in which we present them is only functional for visualising the cycle $S_{m,n}$.

In order for the engine to be re-usable, we need the local state of the machine to end up in its initial state. Therefore, we impose the following constraint on the state of the machine,

Through Eq. (8) we can express the probability distribution of the machine in terms of the passive state $\rho_{P}$ (see the Supplemental Material for further details). In our model we do not explicitly include an additional system (a battery) for storing the energy we extract from the passive state. Instead, we implicitly assume the existence of this work-storage system, and we define the work extracted, $\Delta W$, as the difference in average energy between the initial and final state of the main system (as the machine $M$ has a trivial Hamiltonian, and no interaction terms are present between system and machine). Thus, we have that

where the final state of the system is

It is worth noting that the final state of system and machine will in general develop correlations. These correlations are classical, and without them work would not be extracted during the cycle. However, they do not compromise the re-usability of the machine if applied to another uncorrelated quantum system.

For a given system Hamiltonian $H_{P}$ and a given cycle $S_{m,n}$ we can investigate the amount of work we extract from the state $\rho_{P}$. In the Supplemental Material we provide all the necessary steps to evaluate $\Delta W$ in terms of the probability distribution of $\rho_{P}$. We can express this quantity as

where $\alpha$ is a positive coefficient depending non-trivially on the probability distribution of $\rho_{P}$. For the class of passive states we are considering (namely, the one in which $\beta_{\text{cold}}>\beta_{\text{hot}}$), we find that work can be extracted ($\Delta W>0$) iff

1. 1.

   The Hamiltonian $H_{P}$ is such that $m\,\Delta E_{10}>n\,\Delta E_{21}$.

2. 2.

   The temperatures of the two virtual reservoirs are such that $\beta_{\text{cold}}>\frac{m\,\Delta E_{10}}{n\,\Delta E_{21}}\beta_{\text{hot}}$.

Thus, for a fixed cycle (defined by the parameters $m$ and $n$), and for a fixed Hamiltonian $H_{P}$, we find that work can only be extracted if the virtual temperature $T_{\text{cold}}$ is lower than $T_{\text{hot}}$ by a multiplicative factor which depends on the energy gaps of the Hamiltonian, see Fig. 3, left plot, for an example. In Sec. IV we show that, for a given Hamiltonian $H_{P}$, work can be extracted by any passive (but not completely passive) state, and we characterise the cycle which allows for this extraction.

If we analyse in a more detailed way the cycle, we find that the same amount of energy is gained during each swap between the machine $M$ and the hot virtual reservoir, that is

where $\alpha$ is the same positive coefficient of Eq. (11). Moreover, the same amount of energy is spent during each swap between the machine $M$ and the cold virtual reservoir,

Knowing the amount of energy exchanged during each swap allows us to evaluate the heat exchanged with the virtual reservoirs. In fact, if we identify the pair of levels $\left(\ket{0}_{P},\ket{1}_{P}\right)$ with the hot virtual reservoir, then the energy exchanged during a swap with these levels can be considered as heat coming from the hot virtual reservoir. In this way, the total heat absorbed by the machine is

while the total heat provided to the cold virtual reservoir is

From Eqs. (14) and (15) we obtain that the work extracted can be expressed as $\Delta W=Q_{\text{hot}}-Q_{\text{cold}}$, as in a standard heat engine exchanging energy between two reservoirs. Once $Q_{\text{hot}}$ and $Q_{\text{cold}}$ are defined, we can evaluate an efficiency of this cycle, that is

The efficiency of the engine (when the machine is finite-dimensional) is sub-Carnot in the virtual temperatures, see Fig. 3, right plot. In fact, work can only be extracted when conditions 1 and 2 are satisfied, and these conditions implied $0<\eta<1-\frac{T_{\text{cold}}}{T_{\text{hot}}}$. When we consider the case of an infinite-dimensional machine, we find that by a judicious choice of parameters we may obtain Carnot efficiency.

Once the cycle $S_{m,n}$ is ended, the local state of the main system is moved to a less energetic state. By solving Eq. (8), we find that the final state of the main system $\tilde{\rho}_{P}$ has the following probability distribution

where the unit of probability $\Delta\mathrm{P}$ depends on the initial state $\rho_{P}$, and it is given by

with $\alpha$ the same positive coefficient of Eq. (11). Due to condition 2, the unit $\Delta\mathrm{P}>0$, so that the probability of occupation of $\ket{1}_{P}$ is reduced in favour of the probabilities $\mathrm{p}_{0}$ and $\mathrm{p}_{1}$. Thus, energy is extracted from the passive state when $m\,\Delta\mathrm{P}$ is moved from $\mathrm{p}_{1}$ to $\mathrm{p}_{0}$ (during the hot swaps), and part of this energy is used to move the probability $n\,\Delta\mathrm{P}$ from $\mathrm{p}_{1}$ to $\mathrm{p}_{2}$ (during the cold swaps).

We now show that, for a given Hamiltonian $H_{P}$, work can be extracted from any passive but not completely passive state. In particular, we first show this for qutrit passive states, and we then generalise to the qudit case. Work extraction is achieved with the cycle presented in Sec. III, for specific values of the parameters $m$ and $n$. In what follows, we represent the passive state with the probabilities of occupation $\left\{\mathrm{p}_{0},\mathrm{p}_{1},\mathrm{p}_{2}\right\}$, as opposed to the previous case in which the virtual temperatures were used. In this way, we can consider all possible scenarios, and we are not limited to the case in which a specific pair of eigenstates has a colder (hotter) virtual temperature than the other pair.

The Hamiltonian of the system $H_{P}$ is defined in Eq. (4), where the energy gap between ground and first excited state is $\Delta E_{10}$, and the gap between first and second excited states is $\Delta E_{21}$. We assume that

that is, we ask the ratio between the two energy gaps to be rational. Notice that, even if the ratio is irrational, we can find a suitable $N$ and $M$ such that the condition is approximatively satisfied. Once the relation between energy gaps is defined, we can divide the set of passive states into three different subsets, namely

The union of these three subsets gives the set of all passive states. In particular, one can verify that the subset $R_{3}$ contains all the completely passive states, that is, the thermal states of $H_{P}$ at any temperature $\beta^{-1}\geq 0$. Moreover, $R_{1}$ corresponds to the set of passive states with $\beta_{\text{hot}}$ associated with the pair of eigenstates $\ket{0}_{P}$ and $\ket{1}_{P}$, and $\beta_{\text{cold}}$ associated with the pair $\ket{1}_{P}$ and $\ket{2}_{P}$. The set $R_{2}$, instead, contains the passive states with opposite hot and cold virtual temperatures. Since we are considering qutrit systems, we can represent the set of passive states in a two-dimensional diagram, using their probability distribution. Each point in this diagram represents a passive state. In Fig. 4, left plot, we show the three subsets of Eqs. (20).

In the previous section, we have seen that a cycle defined by the parameters $m$ and $n$ can activate a passive state $\rho_{P}$ with Hamiltonian $H_{P}$ if conditions 1 and 2 are satisfied. These conditions apply to the case in which the passive state is described by Eqs. 6, with $\beta_{\text{hot}}<\beta_{\text{cold}}$. In the present, more general scenario we find that work is extracted by the cycle if and only if the passive state belongs to the following subset

where these conditions can be obtained by analysing the general expression of the extracted work, see Supplemental Material. We now show that, by tailoring the value of the parameters $m$ and $n$, we can make $R^{+}_{m,n}$ to (asymptotically) cover either the region $R_{1}$ or $R_{2}$. Here, we focus on $R_{1}$ solely, since $R_{2}$ follows from similar arguments. As a first step, we ask $m\,\Delta E_{10}-n\,\Delta E_{21}>0$, which implies $m>\frac{M}{N}n$, due to Eq. (19). Then, in order to satisfy this condition, we set $m=\frac{M}{N}n+1$, where we ask $n$ to be large enough for $m$ to be an integer. The set of passive states activated by the cycle is such that

We notice that, since $\rho_{P}$ is passive, $\mathrm{p}_{0}\geq\mathrm{p}_{1}$, which implies that

Thus, Eq. (22) and (23) together assure that $R^{+}_{\frac{M}{N}n+1,n}\subset R_{1}$. Moreover, if $n\rightarrow\infty$, we have that $M+\frac{N}{n}\rightarrow M$, which implies that $R^{+}_{\frac{M}{N}n+1,n}\rightarrow R_{1}$. Thus, we have that, for a given Hamiltonian $H_{P}$, and a given passive state $\rho_{P}\in R_{1}$, there exist a cycle $S_{m,n}$ such that $\rho_{P}\in R^{+}_{m,n}$. However, the closer (in trace norm) the state $\rho_{P}$ is to the set of completely passive states ($R_{3}$), the larger the parameters $m$ and $n$ have to be, that is, the larger the machine has to be (see Fig. 4, right plot).

We can now establish our central claim: that any athermal passive state is energetically unstable under a reversible process that does not generate entropy. We analyse the evolution of a passive state which sequentially interacts with an infinite-dimensional machine $M$, and find that the system moves through a continuous trajectory of passive states towards the set of minimum-energy states, that is, the set of the states jaynes_information_1957 .

We consider a cycle composed of infinitely many hot swaps, $m\rightarrow\infty$, and infinite many cold swaps, $n\rightarrow\infty$, with the assumption that $n=\alpha\,m$, where $\alpha$ is a parameter taking values in a specific range we will describe shortly. Let us now consider the situation in which the main system is a qutrit with Hamiltonian $H_{P}$ given in Eq. (4), described by the passive state $\rho_{P}$ whose probability distribution satisfies the equalities of Eqs. (6). Then, $\rho_{P}$ belongs to the subset $R_{1}$ defined in Eq. (20a), and the cycle $S_{m,n}$ has to satisfy conditions 1 and 2 in order to extract work from it. These conditions are reflected in the allowed range of the parameter $\alpha$, that is

If we set $\alpha$ equal to a value inside the range specified by the previous equation, and we send $m\rightarrow\infty$, we find that (see Supplemental Material for details) the state of the machine as obtained from Eq. (8) is given by a mixture of two “thermal” states, one with effective temperature $\beta_{\text{hot}}^{-1}$, the other with effective temperature $\beta_{\text{cold}}^{-1}$ (note we still have $H_{M}=0$ for the machine). These distributions have support in two different subspaces, and their weight depends non-trivially on the energy gaps of $H_{P}$ and on the virtual temperatures of $\rho_{P}$. In fact, we can loosely interpret the state of the machine in terms of a thermal mixture

where

and

Notice that in order to define these “thermal states” we have introduced two fictitious Hamiltonians, namely, $H_{\text{hot}}$ and $H_{\text{cold}}$. These operators are necessary if we want to consider the distribution of the machine as the mixture of two thermal distributions, but they do not enter in any way in the derivation of the extractable work. Indeed, as we have specified at the beginning of Sec. III, the machine $M$ can have any Hamiltonian (it does not modify the amount of work we extract during the cycle), and we choose to use a trivial one $H_{M}=0$, so that the machine acts as a memory. The weight $\lambda$ in the mixture is given by

Thus, during the cycle, the passive state $\rho_{P}$ first interacts with the “hot reservoir”, by performing a sequence of swaps between the pair of states $\ket{0}_{P}$ and $\ket{1}_{P}$ and the levels of $\tau_{\beta_{\text{hot}}}$. Then, the state interacts with the “cold reservoir”, performing a sequence of swaps between the pair $\ket{1}_{P}$ and $\ket{2}_{P}$ and the levels of $\tau_{\beta_{\text{cold}}}$.

In this scenario, we find that the probability distribution of the passive state $\rho_{P}$ is infinitesimally modified, and consequently the work extracted is infinitesimally small. In particular, we find that the unit of probability, defined in Eq. (18), tends to $0$ with an exponential scaling, $\Delta\mathrm{P}\propto e^{-\beta_{\text{hot}}m\Delta E_{10}}$ for $m\rightarrow\infty$. Let us consider the probability distribution of the final state of the system $\tilde{\rho}_{P}$. Since the distribution only changes infinitesimally during the cycle, we can recast Eqs. (17) as a set of differential equations (see the Supplemental Material for further details). Thus, we can imagine the situation in which infinite many machines are present, so that we can keep infinitesimally changing the state of the main system. In this case, the evolution of the state $\rho_{P}$ is governed by the following equation

where the parameter $t$ provides a continuum label for the sequence of cycles we perform on the passive state. We can then solve this equation for extremal cases for the function $\alpha(p_{0},p_{1})$. When the parameter function $\alpha(p_{0},(t),p_{1}(t))$ is equal to one of its limiting values, Eq. (35) assumes a clear meaning. In fact,

• •

  when $\alpha(p_{0}(t),p_{1}(t))=\frac{\Delta E_{10}}{\Delta E_{21}}$, then the differential equation can be recast as a condition over the average energy of the system, that is,

  $$\mathrm{Tr}\left[{H_{P}\,\rho_{P}}\right]=\mathrm{Tr}\left[{H_{P}\,\tilde{\rho}_{P}}\right].$$

  (36)

  Then, for $\alpha$ taking this value, the passive state evolves along a trajectory that conserves the energy of the system.

• •

  when $\alpha(p_{0}(t),p_{1}(t))=\frac{\beta_{\text{hot}}(t)\Delta E_{10}}{\beta_{\text{cold}}(t)\Delta E_{21}}$, instead, the differential equation can be recast as a condition over the entropy of the system, that is,

  $$S\left(\rho_{P}\right)=S\left(\tilde{\rho}_{P}\right),$$

  (37)

  where $S(\rho)=-\mathrm{Tr}\left[{\rho\log\rho}\right]$ is the Von Neumann entropy. Then, for this $\alpha$, the passive state evolves along a trajectory that conserves the entropy of the system.

For $\alpha$ taking values inside the allowed range, we have that any trajectory between the two presented above is possible, and the set of achievable states is shown in Fig. 5, left plot. It is possible to show that the evolution of the system moves the passive state toward the set of thermal states, which are the stationary states of this dynamic. In Fig. 5, right plot, we show the same set of achievable states, represented this time in the energy-entropy diagram sparaciari_resource_2016 . It is clear that, through this evolution, we can obtain any passive state with a smaller average energy and a bigger entropy than $\rho_{P}$. In Supplemental Material, we show that these states are also the only ones that we can reach with our engines (and with a broader class of maps, called activation maps).

It is interesting to consider the limiting values of work extraction that can be achieved following the scheme suggested in this section. In particular, when the system evolves along the energy-preserving trajectory, the final state we obtain is the thermal state of $H_{P}$ at temperature $\beta_{\text{min}}^{-1}$, that is, $\tau_{\beta_{\text{min}}}$, where the temperature is such that $\mathrm{Tr}\left[{H_{P}\,\rho_{P}}\right]=\mathrm{Tr}\left[{H_{P}\,\tau_{\beta_{\text{min}}}}\right]$. In this case, it is easy to see that the engine is not extracting any work, and its only effect consists in raising the entropy of the system. If we consider the efficiency of this cycle, Eq. (16), we see that $\eta=0$, as expected. The opposite limit is obtained when the system evolves along the entropy-preserving trajectory. In this case, the final state is $\tau_{\beta_{\text{max}}}$, that is, the thermal state of $H_{P}$ at temperature $\beta_{\text{max}}^{-1}$, such that $S\left(\rho_{P}\right)=S\left(\tau_{\beta_{\text{max}}}\right)$. In this case, the work extracted by the cycle is

that is the maximum amount one can extract alicki_entanglement_2013 ; sparaciari_resource_2016 . Significantly, in this case the efficiency is equal to the Carnot one, $\eta_{\text{Carnot}}=1-\frac{\beta_{\text{hot}}}{\beta_{\text{cold}}}$.

In the paper we have presented an engine that is able to extract work from any single copy of an athermal passive state. The engine utilises an ancillary system for the work extraction, and the local state of this system is recovered at the end of the cycle. In this way, the cycle can be run multiple times, and each time it acts on a new copy of the passive state. We show that, for any given Hamiltonian, work can be extracted from any passive, but not completely passive state. Moreover, we show that, in order to extract work from passive states close to the set of completely passive states, we need an ancillary system of large dimension. With an infinite dimension machine we can also evolve a passive state smoothly toward the set of thermal states. In particular, optimal work extraction can be obtained in this case, and it is achieved by mapping the initial state into the thermal state with the same entropy.

The present work provides some evidence that a resource theory for thermodynamics with an imperfect thermal reservoir presents non-trivial challenges. Such a resource theory could be realised by providing passive states for free. However, an obvious restriction we should make in this resource theory consists in the fact that we could not provide more than $k-1$ copies of a $k$-activatable passive state, otherwise work might be extracted with unitary operations from this free state. Moreover, our results show that, even in the case in which a single passive state is provided, an ancillary system exists such that work can be extracted from the individual passive state. Then, in order to build a sensible resource theory, passive states should be always provided at a work cost, equal to the optimal amount of energy extractable from them when a machine is present.

It might also be interesting to analyse which passive states allow for the extraction of the highest amount of work in our cycle. In particular, this problem has been studied in the case of passive states and unitary evolution perarnau-llobet_most_2015 , and a comparison between that class of states and the class of states which allow for maximal work extraction during our cycle might be interesting.