The tight Second Law inequality for coherent quantum systems and finite-size heat baths

In classical thermodynamics, the optimal work is given by the free energy difference, what according to the result of Skrzypczyk et al. can be generalized for individual quantum systems. The saturation of this bound, however, requires an infinite bath and ideal energy storage that is able to extract work from coherences. Here we present the tight Second Law inequality, defined in terms of the ergotropy (rather than free energy), that incorporates both of those important microscopic effects – the locked energy in coherences and the locked energy due to the finite-size bath. The former is solely quantified by the so-called control-marginal state, whereas the latter is given by the free energy difference between the global passive state and the equilibrium state. Furthermore, we discuss the thermodynamic limit where the finite-size bath correction vanishes, and the locked energy in coherences takes the form of the entropy difference. We supplement our results by numerical simulations for the heat bath given by the collection of qubits and the Gaussian model of the work reservoir.


Supplementary Note 1 -Energy and time states of the weight
We assume that the energy storage is given by the weight. In particular, this implies that its energy spectrum is continuous, namelyĤ We further define time states |t W as the conjugate states with respect to the energy vectors |ε W , i.e. transition from one to another basis is given by the Fourier transform: Next, we consider a shift operator which is defined by its action on energy eigenstates:Γ δ |ε W = |ε + δ W . From this relation it follows that: i.e. time states are eigenstates of the shift operator. According to this definition one can further define an operator: which is the generator of shifts, i.e.Γ δ = e −i∆ W δ . This obeys the canonical commutation relation with the HamiltonianĤ W in the form: [Ĥ W ,∆ W ] = i.

Supplementary Note 2 -Energy-conserving and translationally-invariant unitary
We consider a quantum system coupled to a heat bath, prepared in a Gibbs state, and the energy storage given by the weight. Then, we investigate a unitary protocolρ SW ⊗τ B →Ûρ SW ⊗τ BÛ † , such that the evolution operator is the energy-conserving and translationally-invariant unitary, i.e. it satisfies the following commutation relations: 2, 3 whereĤ k is a free Hamiltonian of k = S, B,W subsystem and∆ W is the generator of energy shifts (5). It was proven that unitaryÛ obeying conditions (6) can be always written in the form: 3, 4 whereV SB is some unitary acting on the system and bath Hilbert space, 1 W is the identity operator acting on the weight, andŜ is a kind of control-shift operator defined as follows: where |ε i S is an eigenstate of the system HamiltonianĤ S and ε j B is an eigenstate of theĤ B .

1/3 Supplementary Note 3 -Control-marginal state
We define the so-called control-marginal state: 4 Now, we would like to derive an alternative form of the control-marginal operatorσ SB for the product states. Firstly, for a product stateρ SBW =ρ S ⊗τ B ⊗ρ W , we havê Then, let us represent the density matrix of the weight in the time states basis, i.e.ρ W = dt ds |t t| Wρ W |s s| W . Putting it into the above formula we obtain: where

Supplementary Note 4 -Optimal work extraction
From the Eq. (7) further follows that work is equal to: where the first line follows from the energy-conservation [Û,Ĥ SB +Ĥ W ] = 0 and we also used a fact that [Ŝ,Ĥ SB ] = 0. According to this relation, one can further straightforwardly formulate the following proposition defining the optimal work. Proposition 1. For arbitrary transitionρ S →ρ S = Tr BW [Ûρ SW ⊗τ BÛ † ] the work extracted by the weight is equal to: Moreover, there exist a unitaryV SB such that R(V SBσS ⊗τ BV † SB ) = 0, and then we have: Proof. According to Eq. (12) we have The second part follows from the fact that the arbitrary stateσ SB can be unitarly transformed to the passive state (i.e. with zero ergotropy).

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Supplementary Note 5 -Ergotropy and free energy Proposition 2. Letρ S andξ S are arbitrary quantum states andτ B is the Gibbs state. Further,ρ p andξ p are passive states (i.e. with minimal energy) obtained through the unitary channel fromρ S ⊗τ B andξ S ⊗τ B , respectively. Then, if F(ξ p ) ≤ F(ρ p ) it implies that Proof. From a definition of the free energy and assumption F(ξ p ) ≤ F(ρ p ) we obtain: Further, we have S(ρ p ) = S(ρ S ⊗τ B ) and S(ξ p ) = S(ξ S ⊗τ B ) such that the above inequality can be rewritten in the form: Proof. Let us take the stateξ S as a Gibbs state in the same temperature asτ B , i.e.ξ S =τ S , thenξ p =τ S ⊗τ B =ξ S ⊗τ B . Moreover, for arbitrary stateρ S it is satisfied an inequality F(ρ p ) ≥ F(ξ p ), sinceξ p is a Gibbs state for which free energy has minimum. Finally, since R(τ S ⊗τ B ) = 0 and from inequality (16) follows what was to be shown.
Finally, we obtain