# Comment on "Fluctuations in Extractable Work Bound the Charging Power of Quantum Batteries"

aa r X i v : . [ qu a n t - ph ] F e b Comment on “Fluctuations in Extractable WorkBound the Charging Power of Quantum Batter-ies”

In a recent Letter [1], Garc´ıa-Pintos et al. studied theconnection between the charging power of a quantumbattery and the ﬂuctuations of a “free energy operator”that characterizes the maximum extractable work of thebattery. The authors claimed that for a battery to havea nonzero charging power there must exist ﬂuctuationsof the free energy operator, and the ﬂuctuations upperbound the charging power of the battery. In this Com-ment, we point out that the analysis presented in Ref. [1]contains several misconceptions that call into questionthe validity of the results and conclusions.The central quantity in the development of Ref. [1]is the “free energy operator” F introduced by the au-thors for a quantum battery. Speciﬁcally, for a battery W with Hamiltonian H W and in the state ρ W it is de-ﬁned with respect to a reference heat bath at inversetemperature β as F := H W + β − log ρ W . The rationalebehind the deﬁnition is evident; the expectation valueof F in the battery state ρ W , i.e., hFi W := tr( ρ W F ),gives the free energy F ( ρ W ) = U W − β − S W of the bat-tery in the state ρ W at inverse temperature β . Here, U W and S W := − tr( ρ W log ρ W ) are the average en-ergy and von Neumann entropy of the battery, respec-tively. This rationale rests mainly on the study of statetransformation and work extraction in quantum ther-modynamics using the resource theory of athermality,also known as thermal operations [2, 3]. In particu-lar, in the IID regime, the maximum amount of workthat can be extracted from a quantum system (here,the battery) in the state ρ W and in thermal contactwith a heat bath at inverse temperature β is given by W max = β − S ( ρ W k τ β ) = F ( ρ W ) − F ( τ β ) [3]. Here, S ( ρ k τ ) := tr( ρ log ρ ) − tr( ρ log τ ) is the relative entropyof ρ to τ , τ β = e − βH W / tr(e − βH W ) is the thermal stateof the battery at inverse temperature β , and F ( τ β ) is theHelmholtz free energy of the battery.We note that the operator F is neither mathematicallywell deﬁned nor physically meaningful. Mathematically, F has logarithmic singularities and is not bounded inthe kernel of ρ W (the vector space spanned by the eigen-states of ρ W with eigenvalue 0) [4]. Physically, F is nota physical observable, a point correctly noted in Ref. [1].Hence its eigenvalues and corresponding eigenstates haveno physical meaning in terms of measurement outcomes.In the absence of measurements the quantum state of aparticular quantum system, be it a state vector | ψ i for apure state or a density matrix ρ for a mixed state, doesnot have any physical meaning in itself. This is exactlythe reason why we do not deﬁne the von Neumann en-tropy S of a quantum system as the expectation value ofthe “entropy operator” S := − log ρ or refer to ﬂuctua-tions of S . Both S and F suﬀer from the same math- ematical and physical problems. Therefore, ﬂuctuationsof F has no physical (and operational) meaning per se.The development of Ref. [1] went awry when the au-thors attempted to ﬁnd a “physical interpretation” of theoperator F and its ﬂuctuations that, as explained above,does not exist. The authors of Ref. [1] correctly identi-ﬁed the formal expression for the maximum extractablework W max from a battery in the state ρ W in terms ofthe expectation values of F as (see Eq. (5) of Ref. [1]) W max = hFi W − hFi β , (1)where hFi β := tr( τ β F ) = F ( τ β ) [5]. Unfortunately, theyfailed to recognize two essential physical conditions un-derlying this formal expression, leading to invalid analy-sis and incorrect conclusions.First, as clariﬁed above, W max is the maximum ex-tractable work of the battery when the latter is in thestate ρ W and in contact with a heat bath at inverse tem-perature β . However, the coupling of the reference heatbath to the battery is nowhere mentioned in Ref. [1].In the illustrative example, the reference heat bath at zero temperature is completely decoupled from the bat-tery and not even explicitly included. In doing so, F trivially reduces to H W and the formal expression (1)for W max can no longer hold. Second, and most impor-tantly, the ﬂuctuations of F have nothing to do with thecharging power of the battery. This is because, consis-tent with the second law of thermodynamics, the maxi-mum extractable work W max is achieved if and only if theextraction process is thermodynamically reversible (qua-sistatic and nondissipative) [3]. This implies the chargingpower P ( t ) = d W max / d t is zero. Thus to have a nonzerocharging power the charging eﬃciency cannot be max-imum [6]. Consequently, the bounds derived in Ref. [1](see Eqs. (12) and (18) thereof) reﬂect nothing more thanthe fact that the variance σ F of the operator F is non-negative, as it should be for a Hermitian operator.Shang-Yung Wang Department of PhysicsTamkang UniversityNew Taipei City 25137, Taiwan[1] L. P. Garc´ıa-Pintos, A. Hamma, and A. del Campo,Phys. Rev. Lett. , 040601 (2020).[2] F. G. S. L. Brand˜ao, M. Horodecki, J. Op-penheim, J. M. Renes, and R. W. Spekkens,Phys. Rev. Lett. , 250404 (2013).[3] M. Horodecki and J. Oppenheim,Nat. Commun. , 2059 (2013).[4] N. J. Higham, Functions of Matrices (SIAM, 2008).[5] Note that, while not explicit in the notation, the operator F in hFi β should read F = H W + β − log τ β .[6] C. Van den Broeck, Phys. Rev. Lett.95