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 , 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. contains several misconceptions that call into questionthe validity of the results and conclusions.The central quantity in the development of Ref. 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 ) . 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) . Physically, F is nota physical observable, a point correctly noted in Ref. .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.  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.  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. ) W max = hFi W − hFi β , (1)where hFi β := tr( τ β F ) = F ( τ β ) . 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. .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) . 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 . Consequently, the bounds derived in Ref. (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 L. P. Garc´ıa-Pintos, A. Hamma, and A. del Campo,Phys. Rev. Lett. , 040601 (2020). F. G. S. L. Brand˜ao, M. Horodecki, J. Op-penheim, J. M. Renes, and R. W. Spekkens,Phys. Rev. Lett. , 250404 (2013). M. Horodecki and J. Oppenheim,Nat. Commun. , 2059 (2013). N. J. Higham, Functions of Matrices (SIAM, 2008). Note that, while not explicit in the notation, the operator F in hFi β should read F = H W + β − log τ β . C. Van den Broeck, Phys. Rev. Lett.95