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 Work Bound the Charging Power ofQuantum Batteries”
Stefano Cusumano ∗ and Lukasz Rudnicki
1, 2 International Centre for Theory of Quantum Technologies,University of Gda´nsk, Wita Stwosza 63, 80-308 Gda´nsk, Poland Center for Theoretical Physics, Polish Academy of Sciences, Aleja Lotnik´ow 32/46, 02-668 Warsaw, Poland
In the abstract of [Phys. Rev. Lett. , 040601 (2020)] one can read that: [...] to have a nonzerorate of change of the extractable work, the state ρ W of the battery cannot be an eigenstate of a ”freeenergy operator”, defined by F = H W + β − log ρ W , where H W is the Hamiltonian of the battery and β is the inverse temperature [...]. Contrarily to what is presented below Eq. (17) of the paper, weobserve that the above conclusion does not hold when the battery is subject to nonunitary dynamics. In [1], limits to the charging power of quantum bat-teries, defined as the rate of variation of the free energyof the battery, are scrutinized. In order to do so, a ”freeenergy operator” is defined, and bounds on the chargingpower P ( t ) = d hFi W /dt , where hFi W = Tr ( F ρ W ), arederived. The bounds are used to justify the main con-clusion presented in the abstract, namely, bound in Eq.(12) from [1] applies to unitary evolution while (16) and(18) from [1] also cover more general Lindblad dynamics.We believe the discussed paper presents an interestinganalysis of the problem. However, we have found a fewmistakes affecting the final conclusions of [1], which wenow aim to correct in order to facilitate follow up workwhich the paper deserves.In this comment we first rewrite Eq. (17) of [1], fromwhich the conclusion under discussion (further called a hypothesis ) is drawn, to amend for some imprecisions.We then show, using the rewritten equation, that the hy-pothesis concerning the null charging power of an eigen-state of the free energy operator when the battery istreated as an open system is not supported by (16).Then, we perform a direct calculation of the chargingpower in this special case, to show that the hypothesisindeed does not hold. As a by-product of this analysis,we violate the bound (18) from [1] showing it is false.In order to do so we work, as in the original paper,in the eigenbasis of the free energy operator, so that theoperators involved in the computation can be written as: δ F = F − hFi W = X i w i | i ih i | , ρ W = X i,k ρ ik | i ih k | , (1)and L j = P ik L ikj | i ih k | are the Lindblad operators. Weare interested in the quantity Θ j := h| [ δ F , L j ] | i , where | A | = AA † . Using Eq. (1) we can explicitly write:[ δ F , L j ] = X i,k w i L ikj | i ih k | − w i L kij | k ih i | , (2)[ δ F , L j ] † = X i,k w i (cid:0) L ikj (cid:1) ∗ | k ih i | − w i (cid:0) L kij (cid:1) ∗ | i ih k | . (3)Some algebra involving matrix multiplication and a care- ful permutation of indices gives | [ δ F , L j ] | == X i,k,ℓ L kij (cid:0) L ℓij (cid:1) ∗ ( w i − w i w ℓ − w k w i + w ℓ w k ) | k ih ℓ | , (4)from which it is straightforward to getΘ j = X i,k,ℓ ρ ℓk L kij (cid:0) L ℓij (cid:1) ∗ ( w i − w i w ℓ − w k w i + w ℓ w k ) . (5)Eq. (5) is a slightly corrected variant of Eq. (17) in [1].We are finally ready to verify the case in which thestate of the battery is an eigenstate of the free energyoperator, i.e., ρ W = | k ih k | or ρ ℓk = δ ℓk δ k k . Note thatthe hypothesis under discussion can be proved based onEq. (16) in [1] if and only if ∀ j Θ j = 0. However, we caneasily see that if ρ ℓk = δ ℓk δ k k then Eq. (5) givesΘ j = X i (cid:12)(cid:12)(cid:12) L k ij (cid:12)(cid:12)(cid:12) ( w i − w k ) . (6)The right hand side of the above equation is not iden-tically equal to zero, contrarily to what is stated belowEq. (17) in [1]. All Θ j can simultaneously vanish if andonly if H W = w k | k ih k | , or when all L j act trivially onthe range of H W .We have just shown that the hypothesis does not followfrom Eq. (16) in [1]. However, we can also see that, forthe master equation Eq. (14) of [1], the charging powerat t = t for ρ W ( t ) = | k ih k | can explicitly be given P ( t ) = X j γ j Tr ( D j [ ρ W ( t )] H W ) , (7)where D j [ ρ ] = L j ρL † j − n L † j L j , ρ o /
2. This is just be-cause F ( t ) = H W , the state | k i is by assumption aneigenstate of H W , and the time derivative of the von Neu-mann entropy of ρ W ( t ) at t = t is 0, provided that thedynamics is differentiable [2]. We therefore obtain P ( t ) = X j γ j X i (cid:12)(cid:12)(cid:12) L ik j (cid:12)(cid:12)(cid:12) ( w i − w k ) . (8)As a matter of fact, P ( t ) = 0 only when all Θ j do van-ish. This last conclusion also implies that the boundpresented as Eq. (18) in [1] is violated.We acknowledge support by the Foundation for Pol-ish Science (IRAP project, ICTQT, Contract No.2018/MAB/5, cofinanced by the EU within the SmartGrowth Operational Programme). We thank Luis PedroGarc´ıa-Pintos for several clarifications pertaining to thediscussed issue. ∗ Corresponding author: [email protected][1] L. P. Garc´ıa-Pintos, A. Hamma, and A. del Campo,
Fluc-tuations in Extractable Work Bound the Charging Powerof Quantum Batteries , Phys. Rev. Lett. , 040601(2020).[2] S. Das, S. Khatri, G. Siopsis, and M. M. Wilde,
Fun-damental limits on quantum dynamics based on entropychange , J. Math. Phys.59