Daemonic Ergotropy: Enhanced Work Extraction from Quantum Correlations

We investigate how the presence of quantum correlations can influence work extraction in closed quantum systems, establishing a new link between the field of quantum non-equilibrium thermodynamics and the one of quantum information theory. We consider a bipartite quantum system and we show that it is possible to optimise the process of work extraction, thanks to the correlations between the two parts of the system, by using an appropriate feedback protocol based on the concept of ergotropy. We prove that the maximum gain in the extracted work is related to the existence of quantum correlations between the two parts, quantified by either quantum discord or, for pure states, entanglement. We then illustrate our general findings on a simple physical situation consisting of a qubit system.

The thermodynamic implications of quantum dynamics are currently helping us build new architectures for the superefficient nano-and micro-engines, and design protocols for the manipulation and management of work and heat above and beyond the possibilities offered by merely classical processes [1]. Exciting experimental progress towards the achievement of such paramount goals is currently ongoing [2]. Quantum coherences are believed to be responsible for the extraction of work from a single heat bath [3], while weakly driven quantum heat engines are known to exhibit enhanced power outputs with respect to their classical (stochastic) versions [4].
Despite such evidences, the identification of the specific features of quantum systems that might influence their thermodynamic performance is currently a debated point. In particular, the role that quantum correlations and coherences in schemes for the extraction of work from quantum systems appears to be quite controversial [5,6]. Yet, the clarification of the relevance of genuinely quantum features would be key for the grounding of quantum thermodynamics as a viable route towards the construction of a framework for quantum technologies [1]. Indeed, the very tight link between thermodynamics and quantum entanglement [7] cries loud for the clarification of the role of quantum correlations as a resource for coherent thermodynamic processes and transformations [8].
In this paper, we make steps towards the clarification of the role of quantum correlations in work extraction processes by investigating a simple ancilla-assisted primitive. We address the concept of ergotropy, i.e. the maximum work that can be gained from a quantum state, with respect to some reference Hamiltonian, under cyclic unitaries [9]. We consider the joint state of a system and an isodimensional ancilla, which can be measured in an arbitrary basis, and show that quantum correlations are related to a possible increase of the extracted work. More precisely, we demonstrate that if system and ancilla share no quantum quantum discord [10], then the information gathered through the measurements performed on the state of the ancilla cannot help in catalyzing the extraction of work from the system. We extend this result to the case of quantum entanglement, thus establishing a tight link between enhanced work-extraction performances and a clearcut resource in quantum information processing. We illustrate our findings for the relevant case where system and ancilla are both embodied by qubits, showing the existence of a families of states that provide attainable (upper and lower) bounds to the gain in extractable work at a set degree of quantum correlations between system and ancilla. Not only do our results shed light on the core role that quantum correlations have in thermodynamically relevant processes they also open up the pathway towards the study of the implications of the structure of generally quantum correlated resources for ancilla-assisted work extraction schemes and the grounding of the technological potential of the thermodynamics of quantum systems.
Ergotropy.-We start by introducing the ergotropy, which is the maximum amount of work that can be extracted from a quantum system in a given state by means of a cyclical unitary transformation [9]. Consider a system S with Hamiltonian H S and density matrixρ S given by, with k the energy of the k th eigenstate | k ofĤ S and r k the population of the eigenstate |r k ofρ S . If ρ S is a passive state (i.e., if [ρ s ,Ĥ s ] = 0 and r n ≥ r m whenever n < m ), no work can be extracted by means of a cyclical variation of the Hamiltonian parameters (Ĥ S (0) =Ĥ S (τ ) =Ĥ S ) over a fixed time interval [0, τ ] [11][12][13]. If the initial stateρ S is not passive with respect toĤ S , then work may be extracted cyclically, and its maximal amount, the ergotropy W, has been shown by Allahverdyan to be given by Daemonic work and quantum correlations.-In order to connect with the theory of quantum correlations, we extend the framework for maximal work extraction by introducing a noninteracting ancilla A and assume that system and ancilla are initially prepared in the joint stateρ SA . The intuition behind the protocol, that will be discussed below, is that shouldρ SA bring about correlations between S and A, a measurement performed on the ancilla would give us information about the state of S, which could then be used to enhance the amount of work that can be extracted from its state. Within such a generalized framework, the amount of extractable work crucially depends on the measurements performed on A, that we describe through a set of orthogonal projectors {Π A a }. Upon the measurement of A with outcome a, the state of the system collapses onto the conditional density matrixρ S|a = Tr A [Π A aρSAΠ A a ]/p a with probability p a = Tr[Π A aρSA ]. The time evolution of state ρ S|a then follows a cyclic unitary processÛ a conditioned on the outcome of the measurement. By averaging over all of the possible outcomes of the measurement, the work extracted from the state of S reads . This quantity explicitly depends on the specific control strategy determined by the outcomes of the measurements {Π A a }. We can thus proceed to maximise the extracted work by performing the optimal ergotropic transformation for each of theρ S|a such that with {Π A a } a set of orthogonal projective measurements, and r a k the eigenvalues ofρ S|a . We call this quantity the Daemonic Ergotropy.
On the other hand if we do not use the information obtained upon measuring the ancilla, and thus control the system in the same way, independently of the measurement outcomes (i.e.Û a =Û for any a), the maximum extractable work would be given by the ergotropy W associated with stateρ S = Tr A {ρ SA } = k r k |r k r k |. In the Appendix we have shown that the information acquired through the measurements allows to extract more work than in the absence of them, that is W {Π A a } ≥ W and that it provides a tighter upperbound on the ergotropy than the one derived in [9]. If we call W th the work extracted when the final state is the Gibbs state e −βĤ S (λ0) /Tr[e −βĤ S (λ0) ] with the same entropy asρ S , then W th ≥ W {Π A a } ≥ W. The characterization of the efficiency of work extraction scheme, though, should take into account the energetic cost of the measurements ∆E meas , whose quantification depends on several factors. However, it cannot be smaller than the average variation in the energy of A, so that a lower value can be established as In what follows the main object of our attention will be the difference W {Π A a } − W, which is expected to be related to the (nature and degree of) correlations between S and A. For instance, should S and A be initially statistically independent, i.e.ρ SA =ρ S ⊗ρ A , the measurements on the ancilla would not bring about any information on the state of S, as we would haveρ S|a =ρ S for any set {Π A a } and outcome a. Consequently, there would be no gain in work extraction and W {Π A a } = W. However, besides such a rather extreme case, other instances of no gain in work extraction (from correlated ρ SA states) might be possible, and our goal here is to characterize such occurrences.
In order to achieve this goal, we introduce the quantity which we dub, from now on daemonic gain in light of its ancilla-assisted nature. Clearly, δW ≥ 0 because of the considerations above and the optimization entailed in Eq. (5).
Our aim is to connect δW to quantum correlations. To this end, we notice that δW is invariant under local unitary transformations: any unitary transformation on S can be incorporated in the transformations used for the extraction of work, while any unitary on A is equivalent to a change of measurement basis. Then, we consider quantum discord [10] as the figure of merit to quantify the degree of quantum correlations shared by system and ancilla. For measurements performed on the system S, discord is defined as where I SA is the mutual information between S and A, and − → J SA is the one-way classical information associated with an orthogonal measurement set {Φ A a } performed on the system [10]. Explicit definitions are given in the Appendix. We believe the choice of Eq. (6) is well motivated in light of the explicit asymmetry of both δW and − → D SA with respect to the subject of the projective measurements. We are now in a position to state one of the main results of our work, which we present in the form of the following Theorem: Theorem 1. For any system S and ancilla A prepared in a stateρ SA , we have with δW and − → D SA as defined in Eq. (5) and (6), respectively.
The asymmetry of the daemonic gain is well reflected into the impossibility of linking δW to the discord associated with measurements performed on the ancilla. That is The proof of both Theorem 1 and the corollary statement in Eq. (8) are presented fully in the Appendix, while a scheme of principle is presented in Fig. 1. It is important to observe that, in general, the inverse of Theorem 1 does not hold, i.e. ← − D SA = 0 or − → D SA = 0 δW = 0 as there can well be classically correlated states associated with a non-null daemonic gain. However, a remarkable result is found whenρ SA is pure, for which the only possible quantum correlations are embodied by entanglement.
Theorem 2. For any system S and ancilla A prepared in a pure stateρ SA = |ψ ψ| SA we have and δW = k r k k − 1 , where r k are the Schmidt coefficients of |ψ SA and k are the eigenvalues ofĤ S , ordered such that r k ≥ r k+1 and k ≤ k+1 .
Theorem 2 is a thermodynamically motivated separability criterion for pure bipartite states in arbitrary dimensions and an explicit quantitative link between the theory of entanglement and the thermodynamics of information. Illustrations in two-qubit systems.-The statements in Theorems 1 and 2 are completely general, and independent of the nature of either S or A, which could in principle live in Hilbert spaces of different dimensions. However, in order to illustrate their implications and gather further insight into the relation between the introduced daemonic gain and both discord and entanglement, here we focus on the smallest non-trivial situation, which is embodied by a two-qubit system.
We start with the implications of Theorem 1 and compare δW with discord − → D SA . Since both these quantities are invariant under local unitary transformations onρ SA , without loss of generality we can consider the system Hamiltonian H S = −σ z . In Fig. 2 we show the distribution of randomly generated two-qubit states over the δW-versus-− → D SA plane.
Such an extensive numerical analysis reveals that, for any statê where . The monotonicity of h(x) implies that growing values of quantum correlations are associated with a monotonically increasing daemonic gain: for the states lying on such lower bound, quantum correlations form a genuine resource for the catalysis of thermodynamic work extraction. Moreover, as lim x→1 h(x) = 0, a two-qubit system with − → D SA = 0 (i.e. a classically correlated state) can achieve, in principle, any value of daemonic gain up to the maximum that, for this case, is δW = 1. On the other hand, the daemonic ergotropy is maximized by taking pure two-qubit states with growing degree of entanglement.
We can now address Theorem 2 and its consequences for two-qubit states. Similarly to what was done above, we have studied the distribution of random two-qubit states in the daemonic ergotropy-versus-entanglement plane, choosing quantum concurrence C as a measure for the latter [14]. The results are illustrated in Fig. 3. As before, a lower bound to the amount of daemonic ergotropy at set value of concurrence can be identified. We have that, for any stateρ SA with concurrence C ing of quantum correlations between a system and an ancilla that is subjected to suitably chosen projective measurements. Our approach allowed us the introduce of a new form of information-enhanced ergotropy, which we have dubbed daemonic, that acts aptly as a witness for quantum correlations in general, and serves as a necessary and sufficient criterion for separability of bipartite pure states. We have characterised fully the distribution of quantum correlated twoqubit states with respect to the figure of merit set by the daemonic ergotropy, finding that quantum correlations embody a proper resource for the work-extraction performances of the states that minimize δW. Our work opens up interesting avenues for the thermodynamic interpretation of quantum correlations, clarifies their resource-role in ancilla-assisted information thermodynamics and opens up possibilites to understand the role of correlations in the charging power of quantum batteries [17].
Acknowledgements.-J. G. would like to sincerely thank F. Binder, K. Modi and S. Vinjanampathy for discussions related to this work. G. Francica thanks the Centre for Theoretical Atomic, Molecular, and Optical Physics, School of Mathematics and Physics, Queen's University Belfast, for hospitality during the completion of this work. We acknowledge support from the EU FP7 Collaborative Projects QuProCS and TherMiQ, the John Templeton Foundation (grant number 43467), the Julian Schwinger Foundation (grant number JSF-14-7-0000), and the UK EPSRC (grant number EP/M003019/1). We acknowledge partial support from COST Action MP1209.

APPENDIX
Here we define the notion of discord used in the paper, present details of the core results discussed in the main body of the paper and the formal proofs of both Theorem 1 and 2. Discord.-We recall the definition of quantum discord ← − D SA associated with orthogonal measurementsΦ A a performed over the ancilla [10] where I SA is the mutual information is the so-called one-way classical information and S(ρ) = −Tr[ρ log 2ρ ] is the von Neumann entropy of the general stateρ. The maximization inherent in Eq. (12) is over all the possible orthogonal measurements on the state of A. Similarly we define the discord − → D SA associated with measurements performed over the state of the system S as Eq. (12) with the role of S and A being swapped.
Theorem 1.-In order to provide a full-fledged assessment of Theorem 1, we should first discuss the following Lemma.
Lemma 1. For any set of orthogonal projective measurements {Π A a } performed over an ancilla A prepared with a system S in a stateρ SA , we have W {Π A a } ≥ W Proof. In order to show this statement, we observe that Eq. (13) implies that r k = a p a j r a j | r k | r a j | 2 . As W {Π A a } − W = k k (r k − a p a r a k ), we have that due to the fact that k,j r a j k | r k | r a j | 2 − δ k j ≥ 0, as this is the ergotropy ofρ S|a relative to the Hamiltonian k k |r k r k |.
We are now in a position to provide the full proof of Theorem 1, which we restate here for easiness of consultation: Theorem 1. For any system S and ancilla A prepared in a stateρ SA , we have with δW and − → D SA as defined in Eq. (i) There is a measurement outcomeā such thatρ S|ā = k rā k |r k r k | with rā k ≥ rā k+1 . Then (16) given that k,j rā j k | r k | rā j | 2 − δ kj is the ergotropy ofρ S|ā relative to the Hamiltonian k k |r k r k |, and is zero if and only if ρ S|ā = k rā k |r k r k |.
(ii) For every a ρ S|a = k r a k |r k r k | with r a k ≥ r a k+1 . In this case W {Π A a } − W = 0. However, asρ SA is such that − → D SA = 0, it is always possible to identify another set {Π A a } such that W {Π A a } − W > 0. In order to show how this is possible, we note thatρ SA can be written aŝ with the condition p a C aa kk = r a k δ kk . As − → D SA = 0, there are two measurement outcomesā andā such that Cāā kk = Cāā k δ kk . Should this be not true, we would have − → D SA = 0, and thus a contradiction. Therefore, as − → D SA = 0, the matrix A ā|ρ SA |ā A cannot be diagonal in the basis {|r k S } (here |ā A is the eigenstate ofΠ A a with eigenvaluesā). Ifā =ā , case (ii ) cannot occur.
However, ifā =ā , we can define the new set of projectors {Π A a } with elementsΠ Ā a = (|ā + |ā )( ā| + ā |)/2,Π Ā a = (|ā − |ā )( ā| − ā |)/2 andΠ A a = Π A a for a =ā,ā . Then, the density matrixρ S|ā = Tr A {Π Ā aρSA }/p ā readŝ which shows thatρ S|ā is not diagonal in the basis {|r k S }. Therefore ρ S|ā = k r ā k |r k r k | S with r ā k ≥ r ā k+1 . So, proceeding in a similar way as for case (i ), we conclude that Having proven Theorem 1, we can provide a justification of two important Corollaries Proof. It is enough to consider the statê where {|φ a A } is a non orthogonal set of states. Under such conditions, we have ← − D SA = 0. If we choose q ak such that q ak ≥ q ak+1 , we have W {Π A a } − W = 0 for any set {Π A a }, asρ S|a = k r a k |r k r k | S with r a k = a q a k | φ a | a A | 2 /p a ≥ r a k+1 ).

Corollary 2.
Under the premises of Theorem 1, we have that Proof. We consider the stateρ SA = k r kΠ S k ⊗Π A k , wherê Π A k andΠ S k are orthogonal projectors of rank one. Although such state has zero discord, the quantity Therefore, δW > 0.
Theorem 2.-We can now provide a proof of Theorem 2, which we state again for easiness of consultation. Theorem 2. For any system S and ancilla A prepared in a pure stateρ SA = |ψ ψ| SA we have and δW = k r k k − 1 , where r k are the Schmidt coefficients of |ψ SA and k are the eigenvalues ofĤ S , ordered such that r k ≥ r k+1 and k ≤ k+1 .
Proof. We make use of the instrumental result embodied by Corollary 2 and consider the pure stateρ SA = |ψ SA ψ SA | whose Schmidt decomposition reads |ψ SA = k √ r k |r k S ⊗ |φ k A with r k ≥ r k+1 . Corollary 2 has shown that δW = k r k k − 1 . Therefore, δW = 0 it must be 1 = k r k k , which implies r k = δ 1k . This implies that the state has a single Schmidt coefficient, and is thus separable. The proof of the reverse statement is trivial.
Analysis of the two-qubit case.-We provide additional details on the analysis performed on the two-qubit case illustrated in the main body of the paper.
In what follows, with no loss of generality, we choose the system HamiltonianĤ S = −σ z . As stated in the main body of the paper, we choose concurrence as the entanglement measure to be used in our analysis. For a bipartite qubit state, concurrence is defined as [14] where λ k are the square roots of the eigenvalues ofρρ witĥ ρ = (σ y ⊗σ y )ρ * (σ y ⊗σ y ), ordered so that λ k ≥ λ k+1 . In the main body of the paper we have proven that the ergotropic gain of any stateρ SA with concurrence C is larger than, or equal to The states locally equivalent tô with x = (1 ± √ 1 − C 2 )/2, which have concurrence C, are such that δW = δW min (C). These states belong to the class parametrized as p φ η On the other hand, the states locally equivalent tô which also have concurrence C, are such that δW = 1, and thus embody the upper bound to the daemonic ergotropy at set value of concurrence.
In order to show this, we parameterize the projectorsΠ A 1 andΠ A 2 that are needed to calculate the daemonic ergotropy in terms of the angles θ ∈ [0, π] and φ ∈ [0, 2π) such that Π A 1 = cos 2 (θ/2) e −iφ sin(θ/2) e iφ sin(θ/2) sin 2 (θ/2) and Π A 2 = 1 1 − Π A 1 . An extensive numerical analysis of the distribution itself has shown that the states lying on the lower boundary belong to the class of so-called x-states of the form where a, b, c, w, z are positive numbers such that bc ≥ w 2 , ad ≥ z 2 . This class plays a key role in the characterisation of the states that maximize quantum correlations at set values of the purity of a given bipartite qubit state [15,16]. The ergotropy W for such class of states is On the other hand, we have W {Π A a } = 1 − 2(a + b) + (X + + X − )/2 with X ± = [2(a + b) − 1 ± (1 − 2b − 2c) cos θ] 2 +4 we −iφ + ze iφ 2 sin 2 θ 1 2 .

(28)
The associated concurrence is C = 2max{0, z − √ bc, w − √ ad}. We make the ansatz that a state as in Eq. (24) with x real and positive, minimizes δW at a fixed value of C. Then, from the positivity of the density matrix, x must satisfy the condition C ≤ 2 x(1 − x) with x ∈ [0, 1].
In order to show that the class in Eq. (25) is such that δW = 1, it is enough to observe that, for such state, W = 0. In fact, we trivially have ρ S = 1/2 and, by choosing for instancê Π A 1 = |0 0| A , we get pure post-measurement states, and thus W {Π A a } = 1. Therefore δW = 1 regardless of the value taken by C.
As mentioned above, the validity of the ansatz used here is justified by an extensive numerical investigation based on 10 6 random bipartite states generated uniformly according to the Haar measure.