Mandelstam and Tamm (MT)-and Margolus and Levitin (ML)-type bounds on speed limit time (Mandelstam and Tamm, 1945; Margolus and Levitin, 1998) are estimated by using the geometric approach, using the Bures angle, to quantify the closeness between the initial and final states. This approach is used to provide a bound for the initial pure state ρ ₀ = |ψ ₀⟩⟨ψ ₀| on the quantum speed limit time (τ _QSL ) (Deffner and Lutz, 2013) as τ QSL = max 1 Λ t op , 1 Λ t tr , 1 Λ t hs sin 2 B , (1) where B ( ρ 0 , ρ t ) = arccos Tr ρ 0 ρ t ρ 0 2 , and Λ t op,tr,hs = 1 t ∫ 0 t d s | | L ρ s | | op,tr,hs . (2) The three norms ||⋅||_op, ||⋅||_tr, and ||⋅||_hs are the operator, trace, and Hilbert–Schmidt norms, respectively. L is the Liouvillian superoperator acting on ρ. From the norm inequalities, it can be shown that the operator norm of the generator provides a tighter bound on the quantum speed limit time.

Here, we make use of the Wigner–Yanase information metric given by B ρ 0 , ρ t = arccos Tr ρ 0 ρ t , (3) in the expression of the QSL time given in Eq. 1. The QSL time obtained using this metric is τ ̃ QSL . In (Pires et al., 2016), it was shown that by using this metric, one can obtain an upper bound on the QSL time in the case of mixed states.

A bound on the QSL time for the open quantum systems can be given based on the relative purity measure given in (del Campo et al., 2013). The bound to the required time of evolution, in this case, is given as t ≥ τ QSL ′ = 4 θ 2 ⁡ Tr ρ 0 2 π 2 Tr L † ρ 0 2 ̄ , (4) where θ = cos⁻¹[f(t)]. f ( t ) = Tr ( ρ t ρ 0 ) / Tr ( ρ 0 2 ) is the relative purity measure, ρ ₀ is the initial state, and ρ _t is the state evolved to time t. In the denominator, we have ( ⋅ ) ̄ = t − 1 ∫ 0 t ( ⋅ ) d s .

There are several widely known (basis dependent) quantum coherence measures, such as the relative entropy of coherence, the l ₁ norm of coherence, the geometric coherence, and the robustness of coherence. The relative entropy of coherence is used because of its operational meaning as distillable coherence. In addition, it is also easier to work and compute compared to some other measures of coherence. For a given state ρ, the relative entropy of coherence C(ρ) defined as C ρ = S ρ D − S ρ , (5) where ρ ^D = ∑ _i ⟨i|ρ|i⟩|i⟩⟨i| is the density operator that is diagonal in the reference basis, obtained by dephasing off-diagonal elements of ρ, and S(ρ) = − Tr[ρ log  ρ] is the von Neumann entropy of the state ρ.

For an arbitrary quantum dynamics of a finite-dimensional quantum system describable as the time evolution of its state, the minimum time needed for the state ρ _t to attain coherence C(ρ _t ), starting with the initial coherence C(ρ ₀), is lower bounded by the quantum speed limit time for coherence (τ _CSL ) given by (Mohan et al., 2022) t ≥ τ CSL = | C ρ t − C ρ 0 | Λ t rms , D | | ln ⁡ ρ s D | | H S 2 ̄ + Λ t rms | | ln ρ s | | H S 2 ̄ , (6) where Λ t rms , D = 1 t ∫ 0 t | | L s ( ρ s D ) | | H S 2 d s , Λ t rms = 1 t ∫ 0 t | | L s ( ρ s ) | | H S 2 d s , | | ln ⁡ ρ s D | | H S 2 ̄ = 1 t ∫ 0 t | | ln ⁡ ρ s D | | H S 2 d s , and | | ln ρ s | | H S 2 ̄ = 1 t ∫ 0 t | | ln ρ s | | H S 2 d s . ||⋅|| _HS is the Hilbert–Schmidt norm, and L s is the Liouvillian superoperator acting on ρ.

Ergotropy refers to the maximum amount of work that can be extracted from a quantum system through a cyclic unitary transformation of the initial state. The problem of maximal extraction of work was discussed in (Allahverdyan et al., 2004). For a state governed with a time-dependent Hamiltonian H _S + V(t), the time-dependent potential V(t) corresponds to the transfer of work to external sources. The process is cyclic when the external source is connected at time t = 0 and disconnected at time t = t ₀, i.e., the time-dependent potential has the form V(0) = V(t ₀) = 0. One then looks out for the maximum work that can be extracted from the system for an arbitrary V(t). To this effect, among all final states ρ t 0 reached from the initial state ρ ₀, we look for the state with the lowest final energy E f = Tr [ ρ t 0 H S ] . The thermal equilibrium state ρ t 0 e q = exp ( − β H S ) / Tr [ exp ( − β H S ) ] (where β = T ⁻¹, with T being the Temperature) under invariant von Neumann entropy is the standard answer to this problem. The largest amount of work that can be extracted in this case is W t h = E ( ρ 0 ) − T S ( ρ 0 ) + T ⁡ log ( Z ) (Allahverdyan et al., 2004), where S(⋅) is the von Neumann entropy, and Z = Tr[exp(−βH _S )] is the partition function.

Above, two arguments are made: one where we assume that the initial state can be brought to a thermal equilibrium state, and the other is that the entropy of the system is invariant during this evolution. This is allowed for the macroscopic systems where dissipative processes within the system may occur while the system evolves under the influence of V(t). However, in the case of the finite systems, in general, the action of only the potential V(t) is not sufficient for a state to reach the thermal state in time t ₀. Here, for the unitary evolution of the finite systems, not only the von Neumann entropy but also the eigenvalues of ρ are conserved. Unlike thermodynamic systems, finite systems have the memory of their initial state, and relaxation mechanisms are not involved. Therefore, in these systems, the maximum amount of work extracted W , called ergotropy, is generally smaller than W t h . Below, we outline the method to obtain ergotropy of the system. We assume that we are given a quantum state ρ ₀ with its internal Hamiltonian H _S such that they have the following spectral decomposition: ρ 0 = ∑ i r i | r i 〉 〈 r i | , (17) and H S = ∑ i ε i | ε i 〉 〈 ε i | , (18) where ordering of the eigenvalues for ρ ₀ and H _S is in the decreasing, r ₁ ≥ r ₂ ≥…, and increasing, ɛ ₁ ≤ ɛ ₂ ≤ …, order, respectively. Since unitary dynamics is considered, any decrease in the internal energy of the system at hand, with respect to its self-Hamiltonian H _S , will be extracted as work. Thus, in order to find the ergotropy, one aims to minimize the internal energy of the final state: W ρ 0 = Tr ρ 0 H S − min Tr U ρ U † H S , (19) where minimization is performed over all possible unitaries.

It can be shown that the final state, ρ t 0 = ρ f = U ρ U † , that achieves this minimum should commute with H _S and have the same eigenvalues as ρ ₀, that is, ρ _f = ∑ _j r _j |ɛ _j ⟩⟨ɛ _j |. This is known as the passive state because no work can be extracted from this state. The intuition for the order of eigenvalues comes from the interpretation that the highest occupation fraction r ₁ of ρ ₀ should occupy the lowest level. A unitary operator that performs such a transformation is U = ∑ _j |ɛ _j ⟩⟨r _j |. Now we can rewrite Eq. 19 as W ρ 0 = ∑ j , i r j ε i | r j ε i 〉 | 2 − δ i j . (20) The ergotropy W depends only on the initial state and Hamiltonian of the system. The upper bound on ergotropy is given by W t h ≥ W ≥ 0 , where W t h is equal to W when r _j = −βɛ _j − log(Z). Furthermore, ergotropy has already been studied for open quantum systems in (Kamin et al., 2020; Çakmak, 2020; Touil et al., 2021). In the previous section, we discussed a simple open quantum system modeling a non-Markovian amplitude damping channel. To calculate the ergotropy of the system, we make use of the solution ρ(t) obtained after solving Eq. 11 for any time t, and we supply it as the initial state in the Eq. 19 to find the maximum work that can be extracted from this state after a cyclic unitary transformation. Physically, the work is extracted from the system after it is detached from the bath and is now subjected to the transformation stated above.

For the state ρ(t) given in Eq. 14, one can analytically calculate the ergotropy of the system with the inherent system Hamiltonian H _S defined by Eq. 8. To this end, the eigenvalues of the state ρ(t) are 1 2 ( 1 ± | a ( t ) | ) , where | a ( t ) | = x ( t ) 2 + z ( t ) 2 . Note that we have dropped y(t) from the calculations because it is zero for the case considered here, as given in Eq. 16 The spectral decomposition of the H S = ω 0 1 1 + 0 0 0 can now be used to find the passive state ρ f ( t ) = r 1 0 0 + r 2 1 1 , where 2 r 1 = 1 + | a ( t ) | ≥ 1 − | a ( t ) | = 2 r 2 . The ergotropy of the system now boils down to W ρ t = Tr ρ t H S − Tr ρ f t H S , = ω 0 1 + z t 2 − ω 0 1 − | a t | 2 = ω 0 2 | a t | + z t , = ω 0 2 x t 2 + z t 2 + z t . (21) In the above equation, z(t) denotes the population terms, and x(t) denotes the coherence terms of the state at any time t Eq. 14 The value of the ergotropy is non-zero whenever the coherence term x(t) is present in the state. It becomes zero only when x(t) is zero and z(t) is less than or equal to zero. Using the Eq. 16 and the initial state ρ(0) = |ψ ₀⟩⟨ψ ₀|, where | ψ 0 〉 = 3 2 | 0 〉 + 1 2 | 1 〉 , we can write the ergotropy in terms of the function G(t) as W ρ t = ω 0 4 − 2 + G t 2 + 4 − G t 2 + G t 4 . (22)

It was recently recognized (Francica et al., 2020; Sone and Deffner, 2021) that quantum ergotropy can be separated into two different (coherent and incoherent) contributions: W = W i + W c . (23) The incoherent ergotropy W i denotes the maximal work that can be extracted from ρ without changing its coherence, which is defined as W i ρ = Tr ρ − σ H S , (24) where σ is the coherence invariant state of ρ, which has property: Tr σ H S = min U ∈ U i Tr U ρ U † H S , (25) where U ( i ) is the set of unitary operations without changing the coherence of ρ. Alternatively, the incoherent ergotropy can be calculated using the state ρ after erasing all its coherence terms by applying a dephasing map and then using the same method used in calculating the full ergotropy for this dephased state. That is, consider the state ρ ^D = ∑ _i ⟨i|ρ|i⟩|i⟩⟨i|; the incoherent ergotropy W i ( ρ ) for state ρ is equivalent to the ergotropy W of the state ρ ^D . The passive state ρ f D , in this case, can be found in a similar way to that done previously for the calculation of ergotropy. Therefore, W i ρ = W ρ D = Tr ρ D − ρ f D H S . (26) The dephased state ρ ^D (t) for the state ρ(t) in Eq. 14 is given by 1 2 ( 1 + z ( t ) ) | 1 〉 〈 1 | + ( 1 − z ( t ) ) | 0 〉 〈 0 | , and the corresponding passive state ρ f D ( t ) depends on the sign of z(t). Considering the case z(t) < 0, we find that 1 + z(t) is lesser than 1 − z(t). To this end, the passive state ρ f D ( t ) is found to be the same as ρ ^D (t); therefore, the incoherent ergotropy W i ( ρ ( t ) ) becomes zero. However, if z(t) ≥ 0, then 1 + z(t) is greater than 1 − z(t) and ρ f D ( t ) = 1 2 ( 1 − z ( t ) ) | 1 〉 〈 1 | + ( 1 + z ( t ) ) | 0 〉 〈 0 | . In this case, for the system Hamiltonian H _S = ω ₀|1⟩⟨1|, the incoherent ergotropy is given by W i ρ t = ω 0 z t . for  z t ≥ 0 0 . for  z t < 0 (27) Upon comparing the above equation with Eq. 21, we observe that for the positive z(t) and zero coherence term x(t)/2, the incoherent ergotropy W i becomes equal to the ergotropy W . Furthermore, the incoherent ergotropy is always positive here and obtains contribution only from the population terms of ρ(t). The incoherent ergotropy in terms of the function G(t) and the initial state ρ(0) defined above Eq. 22 is given by W i ρ t = ω 0 G t 2 − 2 2 . for  G t 2 ≥ 2 0 . for  G t 2 < 2 (28)

Moreover, it is straightforward now to obtain the expression for the coherent ergotropy W c = W − W i for the state ρ(t) given in Eq. 14 and system Hamiltonian H _S in Eq. 8. This is given by W c ρ t = ω 0 2 x t 2 + z t 2 − z t . for  z t ≥ 0 ω 0 2 x t 2 + z t 2 + z t . for  z t < 0 (29) The coherent ergotropy W c is the work that is exclusively stored in the coherence. This can be verified from the above equation. We note here that whenever the coherence term x(t)/2 of the state ρ(t) is zero, the coherent ergotropy of the system also becomes zero. Moreover, the coherent ergotropy becomes the full ergotropy for the negative values of z(t). The expression of coherent ergotropy W c in terms of the function G(t) and for the initial state ρ(0) can be obtained in a similar manner to that for ergotropy and incoherent ergotropy using Eq. 16 Furthermore, we outline the relationship between the coherent ergotropy and the relative entropy of coherence C(ρ) defined in Eq. 5. The expression was derived in (Francica et al., 2020) and can be given via quantum relative entropy as β W c ρ = C ρ + D ρ f D | | ρ e q − D ρ f | | ρ e q , (30) where D ( σ ‖ ρ ) = Tr σ log ⁡ σ − log ⁡ ρ is the quantum relative entropy, ρ f D is the passive state of the dephased state ρ ^D , and ρ _f is the passive state of ρ. The state ρ ^eq is the Gibbs state ρ e q = exp − β H S Z with Z = Tr exp − β H S , (31) and β = 1/k _B T, with T being the temperature. Despite the fact that in Eq. 30, temperature T is present explicitly, upon substituting the values of all the terms in the right-hand side using Eq. 14 and dividing with β, we get the same expression as in Eq. 29, i.e., free of T.

We next establish a relation between the coherent ergotropy and another quantifier of the coherence of a quantum state given by the l ₁ norm of coherence (Streltsov et al., 2017), defined as C l 1 ρ = ∑ i , j , i ≠ j | ρ i , j | , (32) For the state of the form of Eq. 14 with y(t) being zero, the value of C l 1 ( ρ ( t ) ) is given by C l 1 ρ t = | x t | . (33) Using the above equation and Eq. 29, for the case studied here, we can write a coherent ergotropy W c in terms of C l 1 ( ρ ( t ) ) as W c ρ t = ω 0 2 C l 1 ρ t 2 + z t 2 − z t . for  z t ≥ 0 ω 0 2 C l 1 ρ t 2 + z t 2 + z t . for  z t < 0 (34) We note that in case the value of z(t) becomes zero, the coherent ergotropy W c becomes directly proportional to the value of C l 1 ( ρ ( t ) ) .

The instantaneous charging power is defined by available work in the battery as P t = lim Δ t → 0 W t + Δ t − W t Δ t = d W d t . (35) Using Eq. 21, we can write the instantaneous power P ( t ) as P t = ω 0 2 x t x ̇ t + z t z ̇ t x t 2 + z t 2 + z ̇ t . (36) It is also possible to define the average power-to-energy transfer given by P a v = W t − W t 0 t − t 0 , (37) where t − t ₀ refers to the charging time of the battery.

We will now see the connection between various forms of the quantum speed limit, particularly the QSL time for coherence and using the relative purity measure, and the ergotropy, with the dynamics being generated by the NMAD model.