Quantum nonequilibrium equalities with absolute irreversibility

We derive quantum nonequilibrium equalities in absolutely irreversible processes. Here by absolute irreversibility we mean that in the backward process the density matrix does not return to the subspace spanned by those eigenvectors that have nonzero weight in the initial density matrix. Since the initial state of a memory and the postmeasurement state of the system are usually restricted to a subspace, absolute irreversibility occurs during the measurement and feedback processes. An additional entropy produced in absolute irreversible processes needs to be taken into account to derive nonequilibrium equalities. We discuss a model of a feedback control on a qubit system to illustrate the obtained equalities. By introducing $N$ heat baths each composed of a qubit and letting them interact with the system, we show how the entropy reduction via feedback control can be converted into work. An explicit form of extractable work in the presence of absolute irreversibility is given.

1 Introduction forward backward absolutely conƟnuous singular Figure 1: Schematic illustration of an absolutely irreversible process. We start with an initial state described by the density matrix ρ ini = x p(x)|ψ(x) ψ(x)|, where its support is restricted in the subspace H X . Here the label X is a set of variables x satisfying p(x) = 0; thus the subspace H X is spanned by the set of orthonormal states {ψ(x)} x∈X . We consider the case in which the forward process is given by the time evolution via a unitary operator U . We next prepare the initial state of the backward process described by the density matrix ρ r = y p r (y)|φ(y) φ(y)|. The backward process is given described by the time reversal of the forward process via applying the unitary operator U † . Then, with nonzero probability, the density matrix of the backward protocol evolves in time into the space outside of the subspace H X . In terms of path probabilities, the forward and backward probabilities are given by p(x, y) and p r (x, y), respectively. By using Lebesgue's decomposition theorem [44,45], we can uniquely decompose the backward probability into two parts: p r (x ∈ X, y) and p r (x ∈ X, y) which are absolutely continuous and singular with respect to p(x, y), respectively. We call such a process absolutely irreversible in the sense that there is no one-to-one correspondence between the forward and backward probabilities, and the formal definition of the entropy production diverges for the singular part, i.e., σ(x ∈ X, y) = ln p(x ∈ X, y) − ln p r (x ∈ X, y) = −∞.
It is known that the Jarzynski equalities are inapplicable to such cases as free expansion [30][31][32][33][34] and feedback control involving projective measurements [23] because in these cases there exist those forward paths with vanishing probability that have nonvanishing corresponding backward probabilities. We shall call such processes absolutely irreversible. Recently, nonequilibrium equalities were obtained that can be applied to absolutely irreversible processes, including the processes mentioned above [35,36]. We extend this idea to quantum systems and derive quantum fluctuation theorems and Jarzynski equalities with absolute irreversibility. For the quantum case, absolute irreversibility occurs when the initial state ρ ini = x p(x)|ψ(x) ψ(x)| (with p(x) = 0 for x ∈ X) is restricted to the subspace H X (spanned by {|ψ(x) } x∈X ) of the total Hilbert space, and the density matrix of the backward process is not confined in that subspace, i.e., λ = x ∈X ψ(x)|ρ |ψ(x) = 0, whereρ is the final density matrix of the backward process. (See Fig. 1). Then, the initially localized state expands into a space larger that the initial subspace, as it happens in free expansion, causing a dissipation, and that is quantified by the probability λ. Absolute irreversibility is likely to occur in the measurement and feedback processes since the initial state of the memory and the postmeasurement state are localized in general, and the projective measurement (decoherence in the measurement basis) on the memory and the effect of (inefficient) feedback control let those states expand to a space larger than the initial state, resulting in additional entropy production. By subtracting the absolute irreversible part in a mathematically well-defined manner, we derive those nonequilibrium equalities for measurement and feedback processes which give stronger restrictions on entropy productions or work compared with previously known results [24,37,38]. In Ref. [23], a quantum Jarzynski equality under feedback control with projective measurement was obtained, where the issue of absolute irreversibility was circumvented by introducing classical errors on measurement outcomes. This paper is organized as follows. In Sec. 2, we derive nonequilibrium equalities without feedback control for quantum systems. We introduce the idea of absolute irreversibility and discuss how the nonequilibrium equalities are modified by this effect. In Sec. 3, we derive nonequilibrium equalities with feedback control in the presence of absolute irreversibility during feedback control and the measurement process. In Sec. 4, we give an example of the feedback control on a qubit system to illustrate our work. In Sec. 5, we summarize the main results of this paper.

2
Nonequilibrium equalities without feedback control

Setup
Let the initial state of the system be ρ ini and let the system evolve in time according to the unitary evolution: where H(t) is the time-dependent Hamiltonian. The final state of the system is given by We define the entropy production, which measures the irreversibility of the process, as where ρ r is a reference state which can be chosen arbitrarily [39]. Note that the entropy production defined here is nothing but the quantum relative entropy between the final state and the reference state: where the inequality results from the nonnegativity of the quantum relative entropy [43]. Different choices of the reference states lead to different entropy productions [6]. Here we give two examples.
2.1.1 Examples of the choice of reference states and the corresponding entropy productions 1. Dissipated work Let us relate the entropy production with dissipated work W d which is defined in terms of the work W done on the system and the equilibrium free energy ∆F = F fin − F ini as We assume that the initial state is given by the canonical distribution where H ini = H(0), and choose the reference state as the canonical distribution with respect to the final Hamiltonian H fin = H(t fin ): Then, Eq. (3) becomes equal to the dissipated work where we define work during the nonequilibrium process by the energy change of the system: The above argument is based on an isolated system. We can also introduce a heat bath and do the same argument. In this case, the total Hamiltonian in Eq. (1) is given by where the interaction V SB (t) is turned off at the initial and final states, i.e., V SB (0) = V SB (t fin ) = 0. We also use the abbreviations H S ini = H S (0) and H S fin = H S (t fin ). Then and the choice of reference state in Eq. (7) leads to Combining Eqs. (3), (11) and (12), we reproduce Eq. (8): where the work appearing in Eq. (13) is given by Here, we interpreted the heat Q as the energy that is transfered from the heat bath during the process: 2. Total entropy production To relate the entropy production to the total entropy production, we consider a composite system composed of a system and a heat bath, and use the same Hamiltonian in Eq. (10). We assume that the initial state of the heat bath is given by the canonical distribution and choose the reference state as follows: Combining Eqs. (3), (16) and (17), we obtain where ∆S = S(ρ S fin ) − S(ρ S ini ) is a change in the von Neumann entropy of the system and Q is the heat defined in Eq. (15). If we interpret heat as the entropy produced in the heat bath, Eq. (18) expresses the entropy that is produced for the total system during the protocol; σ tot is therefore called the total entropy production.
Equation (4) leads to second-law like inequalities for entropy productions (e.g., for dissipated work and total entropy production), and the nonnegativity of the entropy production shows that there is a dissipation in a given process [39]. The process is thermodynamically reversible if and only if the equality in (4) holds (for example, if the dissipated work W d or the total entropy production σ tot is zero).

Derivation of quantum fluctuation theorems
Next, we derive quantum fluctuation theorems by expressing the initial state in the diagonal basis. We perform the spectral decomposition of the initial state as ρ ini = x p ini (x)|ψ(x) ψ(x)|, where {|ψ(x) } is an orthonormal basis set. We then calculate the entropy production by using the spectral decomposition of the reference state; ρ r = y p r (y)|φ(y) φ(y)|. The entropy production can be calculated as where p(x, y) = p ini (x)p(y|x) (20) and is the transition probability from the state |ψ(x) to |φ(y) via the unitary operator U . Such a transition is characterized by a set of labels (x, y). When deriving the third line in Eq. (19), we used the relation φ(y)|ρ fin |φ(y) = φ(y)|U ρ ini U † |φ(y) From Eq. (19), we define the following unaveraged entropy production dependent on the specific transition labeled by x and y: Next, we introduce the reference probability distribution is the transition probability from |φ(y) to |ψ(x) via U † . Equation (24) gives the probability of the backward process that starts from the reference state and evolves in time via U † . It follows from Eq. (25) that the entropy production is expressed in terms of the forward and reference probabilities as follows: Now we derive the quantum fluctuation theorem by using the above definition of entropy production (23).
Since the sum of reference probability is unity, we have The entropy production is given by the ratio between the forward and reference probabilities. However, if the forward probability vanishes and the corresponding reference probability does not, the logarithm of the ratio as in Eq. (26) diverges. To treat such situations, we divide the reference probability into two parts: where X = {x|p i (x) = 0}. Since we can take the ratio between the forward and reference probabilities for the first term of the right-hand side of Eq. (28), we have where λ = x ∈X,y gives the total probability of those backward processes that do not return to the subspace spanned by {|ψ(x) } x∈X . In an ordinary irreversible process, the process is stochastically reversible in the sense that the backward path returns to the initial state with nonzero probability since there is a one-to-one correspondence between the forward and backward probabilities, i.e., the entropy production is non-divergent for all (x, y) in Eq. (26). However, the path labeled by the set of variables (x ∈ X, y) is not even stochastically reversible since the formal definition of the entropy production diverges, i.e., σ(x ∈ X, y) = ln 0 pr(x,y) = −∞, and we call this type of irreversibility as absolute irreversibility [36]. A schematic illustration of absolutely irreversible process is shown in Fig. 1.
By rewriting Eq. (29), we obtain a quantum fluctuation theorem with absolute irreversibility: By using the Jensen inequality, i.e., e x ≥ e x , we obtain the inequality for the entropy production: This result shows that in the presence of absolute irreversibility the entropy production must be positive and not less than − ln(1 − λ) ≥ 0, giving a stronger constraint compared with the second law-like inequality σ ≥ 0. Note that only when there is no absolute irreversibility, i.e., λ = 0, the conventional form of the fluctuation theorem is reproduced: e −σ = 1.
In the classical case, the decomposition similar to Eq. (28) can be carried out in a general framework using the probability measure [36]. To see this, let us denote the forward and reference probability measures in phase space as M and M r , respectively. According to Lebesgue's decomposition theorem [44,45], M r is uniquely decomposed into two parts: M r = M r AC + M r S , where M r AC and M r S are absolutely continuous and singular with respect to M. Provided that the probability distribution of a quantum process in this setup is labeled by discrete variables, the decomposition of the reference probability is carried out by dividing variables into two parts: the variables corresponding to the nonvanishing forward probabilities (x ∈ X) and the variables corresponding to the vanishing forward probabilities (x ∈ X). Then, M r AC corresponds to p r (x ∈ X, y) and M r S corresponds to p r (x ∈ X, y), and this decomposition is unique as ensured by Lebesgue's decomposition theorem.

Derivation of the quantum Jarzynski equality
We now derive the quantum Jarzynski equality by assuming that the initial state is given by the canonical distribution (11) and by taking the reference state as given in Eq. (12). For convenience, we use the notation x = (x 1 , x 2 ) and y = (y 1 , y 2 ), where the subscript 1 refers to the system and 2 to the heat bath. By assumption, we have where |E S ini (x 1 ) and |E S fin (y 1 ) are energy eigenstates of the initial and final Hamiltonians of the system, respectively, and |E B (x 2 ) is the energy eigenstate of the heat bath. Now the (unaveraged) entropy production (23) is related to work by where is the work done by the system. Substituting Eq. (34) into Eq. (31), we obtain the quantum Jarzynski equality with absolute irreversibility: and substituting Eq. (34) into Eq. (32), we obtain the second-law like inequality Since the canonical distribution is full rank, i.e., there is no absolute irreversibility, i.e., λ = 0. However, if we prepare the initial state in a local equilibrium state, there is a possibility that the process is absolutely irreversible and the effect of nonzero λ restricts the extracted work. For simplicity, let us divide the Hamiltonian of the system into two parts and prepare the initial state as the canonical distribution restricted to the subspace corresponding to H Sa ini : where ini . Now λ is given by the total probability of the backward process that the system returns to the subspace spanned by where ρ S can,fin ⊗ ρ B can is given by right-hand side of Eq. (12). When the initially localized state expands into the total Hilbert space, the process would be absolutely irreversible and a positive entropy is produced during this process. The effect of absolute irreversibility is to lower the extractable work by k B T | ln(1 − λ)|.

3
Nonequilibrium equalities with feedback control

Formulation of the problem
We consider the following protocol to realize a general measurement and a feedback protocol, which is basically the same as the one considered in Ref. [24] and schematically illustrated as Fig. 2.
The total system consists of the system (S), the memory (M ), the bath (B), and the interactions between them (SM and SB). The corresponding Hamiltonian is given by where the interaction between the system and the heat bath is turned off until the thermalization process (e) starts. The Hamiltonian of the system is controlled by the protocol that depends on the measurement outcome k after the measurement step (b) at time t = t meas : We denote the initial Hamiltonian of the system by H S ini = H S k (t = 0). (a) Let the initial state of the system and the memory be (b) A general quantum measurement on the system is implemented by performing a unitary transfor- dt between the system and the memory followed by a projection is an orthonormal basis set of M . The postmeasurement state for the measurement outcome k is given by where is the probability of obtaining outcome k. The reduced density matrix of the system ρ S (k) := Tr M [ρ SM (k)] is given by where is the measurement operator satisfying completeness relation Here |ψ M (a) and p M ini (a) in Eq. (46) are given by the spectral decomposition of the initial state of the memory: the measurement is a pure measurement (which maps a pure state into a pure state) and the postmeasurement state is given by (c) We perform a unitary transformation U S k depending on the measurement outcome k. Here the unitary operator is given by We note that the above unitary operation associated with the measurement outcome is nothing but the feedback control. The density matrix of the system after the feedback control is given by (d) Finally, we let the system and heat bath interact with each other so that the reduced entropy of the system via feedback control is converted to heat. Here, we assume that the initial state of the heat bath is given by the canonical distribution, i.e., The final state is given by where the interaction between S and B discribed by the unitary operator U SB Now we introduce reference states for each subsystems and define entropy production-like quantities which measure the amount of entropy of SB (M ) that is reduced (or produced) due to the feedback control (or measurement). The reference states of each subsystem is given by is the canonical distribution, and E B (j) and |ψ B (j) is the eigenenergy and energy eigenstate of the heat bath, respectively.
We define the following quantity that measures the amount of entropy reduction of SB due to feedback control: where is the energy change of the heat bath which we identify as heat transfered from B to S. Note that if we choose the reference state as the final density matrix of S, i.e., ρ S r (k) = ρ S fin (k), Eq. (56) is nothing but the total entropy change of SB due to feedback control: where is a change in the von Neumann entropy of the system during the whole protocol. We also define the following quantity which measures the amount of entropy produced in M due to measurement: where ρ M fin = k p k ρ M (k) is the final density matrix of M and ρ M r := k p k ρ M r (k). If we choose the reference state as the canonical distribution, Eqs. (56) and (60) are related to work and the free-energy difference, respectively, as shown in the next section.
The entropy production-like quantities (56) and (60) contain not only the effect of dissipated entropy due to irreversibility of the process but also the effect of entropy change due to information processing (measurement and feedback control), and they can take either positive or negative values depending on the process. The effect of the information exchange between the system and the memory can be expressed by the information gain (quantum-classical mutual information) of the system S [37,40,41]: which is the amount of entropy that is reduced from the system due to the measurement. The information gain is bounded from above by the Shannon entropy H = − k p k ln p k , i.e., I ≤ H, where the equality holds if and only if the measurement is given by a projective measurement using the diagonal basis of ρ S ini . Moreover, the information gain is nonnegative for any premeasurement state if the measurement is a pure measurement (48) as discussed in Ref. [41].
Extracting the information gain from entropy production-like quantities (56) and (60), we obtain the measures of irreversibility during measurement and feedback processes. For the feedback process, we have k is the final density matrix of the backward process by reversing the thermalization and feedback control protocols. Note that the feedback protocol of the system (and the heat bath) is reversible if and only if ρ S (k) ⊗ ρ B can =ρ SB (k) [47], which is the equality condition of the last inequality (62), that is σ SB + I = 0.
Similarly, for the measurement process, we have where is the average postmeasurement state over measurement outcomes, and is a change in the von Neumann entropy due to projection P M k and the inequality results from the fact that von Neumann entropy does not decrease under projection measurements. The nonnegativity in Eq. (63) shows the irreversibility of the measurement process.
Next, let us consider the following spectral decompositions of the initial states of the system, heat bath, and the memory Let us also decompose the postmeasurement state of the system as follows: where we introduce the forward probability distribution corresponding to the feedback (and thermalization) processes: and is the transition probability between the state labeled by k, y, h to the state labeled by z, j during the feedback and the thermalization protocol. We also follow the same procedure for the memory using Eq. (60) and obtain where the forward probability of the measurement process is defined as where is the transition probability between the state labeled by x, a to the state labeled by k, y, b during the measurement process. From Eqs. (61) (70) and (73), we define unaveraged entropy production-like quantities and the corresponding information content as follows: where Q(h, j) = E B (h) − E B (j) is the heat transfered from the heat bath to the system. Since the entropy production relates the forward and reference probabilities as in Eq. (26), we have similar relations for the combinations σ SB + I and σ M − I: σ SB (x, h, k, j, z) + I(x, k, y) = ln p fb (h, k, y, j, z) p fb r (h, k, y, j, z) , where p fb r (h, k, y, j, z) = p k p S r (z|k)p B can (j)p(z, j|k, y, h) and p meas r (x, a, k, y, b) = p k p S (y|k)p M r (b|k)p(k, y, b|x, a) are the reference probabilites of the feedback and measurement processes, respectively. Note that Eq. (82) gives the probability of the system and the heat bath returning to the postmeasurement state of the system and the initial state of the heat bath |ϕ S k (y) ⊗ |ψ B (h) when we start from the initial state of the backward process ρ S r (z|k) ⊗ ρ B can and do the reverse of the thermalization and feedback control U †S k U †SB k , as shown in the gray dashed upward arrow in Fig. 2. Also, Eq. (83) gives the probability of the system and the memory returning to the initial state |ψ S (x) ⊗ |ψ M (a) when we start from the initial state of the backward process k p k ρ S (k) ⊗ ρ M r (k) and let the system and the memory undo the correlation by applying a unitary operation U †SM , as shown by the gray dashed upward arrow in Fig. 2. We use the definitions of entropy production-like quantities (77) and (78) and the information content (79) to derive quantum fluctuation theorems for both the feedback-controlled system and the measurement device.

Derivation of quantum fluctuation theorems
We derive quantum fluctuation theorems for both the feedback-controlled system and the measurement device based on the fact that the sum of the reference probabilities is unity for both the feedback control process (82) and the measurement process (83): 1 = x,a,y,k,b p meas r (x, a, k, y, b).
As in Eq. (28), we decompose Eqs. (84) and (85) into two parts; one is the part where we can take the ratio between the forward and reference probabilities, and the other is the part where the corresponding forward probability vanishes. (See dashed upward arrows in Fig. 2

. )
We introduce a set of labels corresponding to the non-vanishing probability distributions as follows: we introduce Y as a set of labels y satisfying p(y|k) = 0, and A as a set of labels (x, a) satisfying both p S ini (x) = 0 and p M 0 (a) = 0. Then, the support of the postmeasurement state of the system belongs to the subspace H S k,Y , which is spanned by {|ϕ S k (y) } y∈Y , and the support of the initial state of the system and the memory belongs to the subspace H SM A , which is spanned by {|ψ S (x) ⊗ |ψ M (a) } (x,a)∈A . Using the above notations, we decompose the reference states and derive quantum fluctuation theorems as follows: for the system and the heat bath, we have is the sum of the reference probabilities such that the density matrix of the backward process ends up outside of the subspace H SM A and the overlap with the initial state of the forward process vanishes.
Rewriting Eqs. (86) and (88), we obtain the quantum fluctuation theorems for the system and the memory: Using the Jensen inequality, we can reproduce second law-like inequalities by using Eqs. (90) and (91): and where the presence of absolute irreversibility (nonzero λ) imposes stronger lower bounds on σ SB and σ M compared with the previous results (62) and (63) given in Ref. [39]. Since λ fb gives the total probability of the density matrix of the backward process ending up outside of the subspace H S k,Y , it measures the degree of absolute irreversibility of the feedback protocol.
If the measurement on the system is given by projective measurements |k k| S , the situation becomes simple. In this case, λ fb takes the following form: which is the sum of the probability of the backward protocol for each measurement outcome k that does not end in the state |k S . If the unitary operator takes the postmeasurement state |k S into the reference state ρ S (k) for all k, the feedback (and thermalization) process is reversible and λ fb vanishes; otherwise the irreversibility of the process reduces the efficiency of the feedback gain. Note that Eq. (90) holds even for projective measurements on the system, where the previous results in Ref. [24] are inapplicable, since we take into account the effect of absolute irreversibility. Although the obtained information is given by the Shannon entropy and is maximal for projective measurements, feedback protocol is likely to be absolutely irreversible since the postmeasurement state is sharply localized in the Hilbert space; it is given by a pure state |k . Similarly, λ meas measures the absolute irreversibility of the measurement process since it is nonzero when the density matrix of the backward protocol ends up outside of the subspace H SM A . Now let us compare the obtained equalities (90) and (91) with the quantum fluctuation theorems of a total system by using the total entropy production σ tot (x, h, k, j, a, b) = σ SB (x, h, k, j, a) + σ M (a, k, b). The total entropy production can be written in the form where is the probability distribution of the forward process and is the transition probability between states labeled by x, a, h and k, b, j, z, and is the probability distribution of the backward process. Since the sum of the reference probability (98) is unity, we can derive the quantum fluctuation theorem for the total system: Since the obtained fluctuation theorem is applicable to the total system, the effect of information exchange between S and M is canceled out, and the information content does not appear in Eq. (99). Moreover, λ tot measures the aboslute irreversibility of the combined process of the measurement and feedback control, whereas from Eqs. (90) and (91) we can separately obatin the information about the absolute irreversibility in measurement and feedback.

Derivation of the quantum Jarzynski equalities
In this subsection, we derive the quantum Jarzynski equality for the feedback-controlled system by assuming that the initial and reference states are given by canonical distributions and respectively, where H S ini and H S fin (k) are the initial and final Hamiltonians of the system. Then, the orthogonal bases {|ψ S (x) } and {|φ S k (z) } are given by the set of energy eigenfunctions: . Now σ SB is related to the work done by the system as follows: where is the work done by the system, and ∆f S (k) = F S k − F S ini is the free-energy difference. We now derive the following quantum Jarzynski equality for a feedback-controlled system by using Eq. (90): Using the Jensen inequality, Eq. (104) reproduces the generalized second law under feedback control: where is the averaged work done by the system. Imperfect feedback control leads to nonzero λ fb , which lowers the extractable work from the system as shown in Eq. (105). Next, we derive the quantum Jarzynski equality for the memory that acquires the measurement results by assuming that the initial and reference states are given by canonical distributions and We assume that the initial state of the memory (107) is given by the local equilibrium state defined as the canonical distribution using a local Hamiltonian H M 0 , where the Hamiltonian of the memory is decomposed [38]. Here, the spectral decomposition of each local Hamiltonian is given by For convenience, let us relabel a as a = (a 1 , a 2 ) so that |ψ M (a) = |φ M a1 (a 2 ) . Then p M ini (a) = 0 if a = (0, a 2 ) and zero otherwise for the initial state defined in Eq. (107). Now σ M is related to the work done on the memory as follows: where is the work done on the memory, ∆f M (k) = F M k − F M 0 is the free-energy difference, and H(k) = − ln p k is the (unaveraged) Shannon entropy. We now derive the following quantum Jarzynski equality for the memory by using Eq. (91): Using the Jensen inequality, Eq. (111) reproduces the generalized second law for the memory: where is the average work done on the memory and H := − k p k ln p k is the Shannon entropy. A nonzero λ meas increases the work cost of the measurement due to absolute irreversibility as shown in Eq. (112). Using the setup of our Hamiltonian of the memory in this section, λ meas can be expressed as is the canonical distribution corresponding to the initial state of the backward process, and A = {(x, a)|a = (0, a 2 )} since p S can (x) = 0 for all x in this setup. From Eq. (113), we note that λ meas is the total probability that the backward process ends in the subspace |φ M a1 (a 2 )} a1 =0,a2 , which was not occupied by the initial local equilibrium state (107). The origin of the absolute irreversibility for this measurement process is due to the projection on the memory, which turns the initially localized low-entropy states into decohered high-entropy states.

Quantum fluctuation theorems for feedback-controlled systems and unavailable information
In this section, we consider the effect of absolute irreversibility during the feedback process in more detail.
Without absolute irreversibility, the extra work beyond the conventional second law of thermodynamics that can be extracted from the system is bounded from above by k B T times the obtained information I . However, if the feedback process is absolutely irreversible, we cannot fully utilize the information to extract work. We introduce the amount of information that is unavailable for use in extracting work for a given feedback control protocol as which was originally introduced in Ref. [35] for classical systems. Here, we introduce the total probability of the reference probability that does not go back to the postmeasurement state conditioned on the measurement outcome k: λ fb (k) = 1 p k h,k,y ∈Y,j,z p fb r (h, k, y, j, z).
Then we start from the following relation: Let us multiply both sides of Eq. (117) by e −Iu(k) and sum over k. By using Eq. (80), we obtain e −σ SB −(I−Iu) = 1.
Using Jensen's inequality, we obtain the inequality for σ SB in the presence of unavailable information: If we use the same assumptions (100) and (101) when we derive the quantum Jarzynski equality, we obtain and The obtained inequalities (119) and (121) give bounds on the entropy reduction of SB and extractable work from the system, where they take into account the inefficiency of the feedback control by subtracting the unavailable information I u from the obtained information I . From the convexity, the unavailable information is bounded from above by so that inequality (119) gives a tighter bound compared with inequality (92).

Examples: Feedback control on qubit systems
In this section, we apply the quantum Jarzynski equality (104) to qubit systems. Let us prepare an initial state given by where V is the energy difference between the two states |0 S and |1 S . Let us perform a projective measurement with respect to the basis set {|0 S , |1 S }. The probability p k of obtaining the measurement outcome k is given by The acquired knowledge of the system is given by the Shannon entropy: Depending on the measurement outcome, we perform the following feedback control which has the effect of flipping the state if the post-measurement state is |1 S : After the feedback control, we obtain which is independent of the measurement outcome k. The averaged density matrix is given by ρ S fb = k p k ρ S fb (k) = |0 0| S . The energy change of the system during the feedback is given by We model the thermalization process by introducing N +1 different heat baths, each of which is composed of a qubit, as schematically illustrated in Fig. 3. A similar model is discussed in Ref. [46]. The Hamiltonian of each heat bath is given by where E 0 = N ∆V + V is the energy difference between two states of the zeroth heat bath. The initial state of the entire heat bath is given by the tensor product of the canonical distributions: We consider the following N + 1 steps of the protocol to thermalize the system. (a) We quench the Hamiltonian of the system so that the energy difference of the system is changed from E to E 0 . Note that this process preserves the energy of the system since the excited state |1 S is not populated during this process. Next, we perform the following unitary transformation between the system and B 0 : which swaps the populations between S and B 0 : During this process, the energy flow occurs from B 0 to S. The energy change ∆E = E fin − E ini of B 0 and S can be explicitly calculated as where the total energy change is zero: ∆E B0 + ∆E S 0 = 0. (b) We quench the Hamiltonian of the system so that the energy difference is changed from E 0 to E 0 −∆V . During this process, the energy change of the system is given by Next, we let S interact with B 1 via the unitary transformation which has the same form of Eq. (132). After the swap, the density matrix is given by and the energy changes of B 1 and S are given by (c) For the n-th step (2 ≤ n ≤ N ), we quench the Hamiltonian of the system so that the energy difference is changed from E 0 − (n − 1)∆V to E 0 − n∆V . During this process, the energy change of the system is given by Next, we interact S and B n using the unitary transformation which has the same form of Eq. (132). After the swap, the density matrix is given by and the energy change is given by After the N -th step, the system returns to the canonical distribution, which is the final state of this protocol: We use the short-hand notation where h n takes the value 0 or 1, and |h n Bn describes the energy eigenstate of the n-th heat bath. We also use the notation U SB = U SBN U SBN−1 · · · U SB0 , which is the total unitary operation performed on the total system during the thermalization process. Now we explicitly calculate the left-hand side of Eq. (104): 1 where and Noting that the work in Eq. (144) is given by Due to the reverse protocol given above, the density matrix of the system returns to the state Using Eq. (150), we have an explicit form of Eq. (144): where λ fb is the total probability of the backward process not returning to the postmeasurement state |k S as given in Eq. (94): Using Jensen's inequality and Eq. (151), we can derive the upper bound on extractable work from the system via feedback control, that is, The right-hand side of (153) can be explicitly calculated as We can also calculate the work defined in Eq. (106): where ∆E S and ∆E B are the total energy change of S and B, respectively. As the system returns to the initial state at the end of the protocol, ∆E S = 0. The total energy change of the heat bath is given by We can also interpret work as the energy extraction during the quench process during (a)-(c) combined with the energy extraction during the flipping process of the feedback control, that is, which gives the same amount of work compared with the extracted work defined in Eq. (155). As we fix E 0 = N ∆V + V and take the limit ∆V → 0 (and N → ∞), the right-hand side of Eq. (156) reaches Since ∆E B ≤ ∆E B | ∆V →0 , inequality (153) is valid and the equality condition is achieved when in the limit of ∆V → 0 and N → ∞: If we consider a finite ∆V , the density matrix of S jumps from ρ S n to ρ S n+1 during the process, causing dissipation. This dissipation is due to the ordinary irreversibility of the process and no absolute irreversibility, since only the relative weight of two states |0 S and |1 S are changed during the protocols between (a) and (c).
The effect of absolute irreversibility depends on the parameter E 0 for this model, since the protocol (a) brings the pure state |0 S of the postmeasurement state into a thermal state. If we take the limit E 0 → ∞, we have no absolute irreversibility (λ fb = 0) and the backward process corresponding to the forward protocol (a) makes the density matrix return to |0 S . In this limit, one can extract work up to the amount commensurate with information obtained via measurement: and the equality is achieved again in the ∆V → 0 limit: where the acquired information is fully utilized to extract work. The protocol we consider (in the limit of ∆V → 0 and E 0 → ∞, so that the system interacts with infinitely many heat baths) gives a quasi-static process of the isothermal expansion of the system in the sense that is achieved, where ∆S S = S(ρ S can ) − S(|0 0| S ) gives a change in the von Neumann entropy of the system during the thermalization process (a) -(c) and is the heat taken from the heat baths. We relate the energy change of the heat baths to heat because the total change in the von Neumann entropy of the heat bath satisfies the thermodynamic relation As a result, Eq. (161) is satisfied and the reduced entropy of the system via feedback is fully converted into work by this quasi-static process. This result is to campared with the classical Szilard engine that achieves Eq. (161) via quasistatic isothermal expansion of the box [48]. Next, let us consider the opposite limit of E 0 = V . In this case, we do not quench the energy level of the system. We only attach a single heat bath, letting the postmeasurement state of the system transform into a thermal state by a single jump (the protocol (a)). The effect of absolute irreversibility is maximal in this limit: and the work gain takes the smallest value W E0=V = V e −βV 1 + e −βV . (166)

Conclusion
We have derived the quantum fluctuation theorems (31) and Jarzynski equalities (36) in the presence of absolutely irreversible processes, where the density matrix of the backward process does not return to the subspace spanned by the eigenvectors that have nonzero weight of the initial density matrix. We have also derived equalities for feedback and measurement processes (90) and (91). The effect of absolute irreversibility limits the work gain via (inefficient) feedback control and also gives additional entropy production due to the projection on the memory (92) and (93). We have also discussed a model of the feedback control on a qubit system to illustrate the obtained equalities.  Figure 3: Schematic illustration of the thermalization protocol. We consider a system composed of a qubit, and N + 1 different heat baths B 0 , · · · , B N , each composed of a qubit. Here p(E) = (1 + exp(βE)) −1 denotes the the occupation probability of the state |1 , that is, the density matrix is given by the canonical distribution ρ can (E) := (1 − p(E)) |0 0| + p(E) |1 1|. The feedback control brings the state of the system to a pure state ρ S fb = |0 0| S . We consider the following protocols (a)-(c) that transforms a pure state into the canonical distribution ρ S can (V ). Protocol (a): We quench the energy level of the state |1 S to E 0 . Next, we prepare a heat bath in the canonical distribution with the energy level E 0 . We swap the density matrices ρ S fb and ρ B0 can (E 0 ) by applying U SB0 , where energy is transfered from B 0 to S during this process. After the swap, the density matrix of the system is given by ρ S can (E 0 ). Protocol (b): We quench the system and lower the energy level by ∆V , and energy is extracted from the system. We prepare a heat bath B 1 in the canonical distribution ρ B can (E 0 − ∆V ) and swap the density matrices between S and B 1 , where energy is transfered from B 1 to S. After the swap, the density matrix of the system is given by ρ S can (E 0 − ∆V ). Protocol (c): We repeat the protocol which is similar to the protocol (b) by lowering the energy level by ∆V (quench) and swapping the density matrices between S and B n (2 ≤ n ≤ N ) and the density matrix of the system is given by ρ S can (E 0 − n∆V ). After the N -th protocol, the density matrix of the system is transformed into the canonical distribution ρ S can (V ) and the energy level of the system is returned to V , which completes the thermalization process .