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 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.
Nonequilibrium equalities [1, 7, 2, 3, 4, 5, 6, 14, 8, 9, 10, 11, 12, 21, 13, 15, 16, 17, 18, 19, 20, 22] such as fluctuation theorems and Jarzynski equalities have attracted a great deal of interest in the field of nonequilibrium statistical mechanics. They give general insights into thermodynamic quantities in nonequilibrium processes irrespective of details of individual systems. For example, the Jarzynski equality relates work done on the system in a nonequilbrium process to the equilibrium free-energy difference. Nonequilibrium equalities have been obtained in both classical [7, 1, 2, 3, 4, 5, 6] and quantum [8, 9, 10, 11, 12, 13, 14, 15, 16, 21, 17, 18, 19, 20, 22] systems, and generalized to situations involving feedback control [23, 24, 25, 26, 27]. Due to recent advancement in experimental techniques, nonequilibrium equalities have been experimentally verified for classical systems such as a single-molecule RNA [29, 28], and they have been vindicated in a quantum regime using a trapped ion system . Also, feedback control on a Brownian particle was carried out to experimentally demonstrate Maxwell’s demon , and the generalized Jarzynski equality for a feedback-controlled system was verified .
It is known that the Jarzynski equalities are inapplicable to such cases as free expansion [34, 33, 35, 36, 37] and feedback control involving projective measurements  because in these cases there exist those forward paths with vanishing probability that have the corresponsing backward paths with nonzero probabilities. We shall call such processes absolutely irreversible. We give examples of absolutely irreversible processes in Sec. 4. Recently, nonequilibrium equalities were obtained that can be applied to absolutely irreversible processes, including the processes mentioned above [41, 40]. 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 (with for ) is restricted to the subspace (spanned by ) of the total Hilbert space, and the density matrix of the backward process is not confined to that subspace, i.e., , where is the final density matrix of the backward process (see Fig. 1). Then, the initially localized state expands into a larger space, as happens in free expansion, with the probability . Absolute irreversibility is likely to occur in measurement and feedback processes since the initial state of the memory and the postmeasurement state are localized in general, and the projective measurement on the memory and the effect of (inefficient) feedback control let these states expand to a space larger than the subspace of the initial state, resulting in additional entropy production. By subtracting the absolutely 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 [42, 43, 27]. We note that a quantum Jarzynski equality under feedback control with projective measurement was obtained in Ref. , 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 concept 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 quantum piston to analyze the free expansion of the gas with absolute irreversibility. We also 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
Let the initial state of the system be and let the system evolve in time according to a unitary evolution:
where 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 is the von Neumann entropy and is a reference state which can be chosen arbitrarily . Because of Eq. (2), 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 . Different choices of the reference states lead to different entropy productions . Here we give two examples.
Examples of the choice of reference states and the corresponding entropy productions
1. Dissipated work We defined the dissipated work in terms of the work done by the system (or work that can be extracted from the system)
and assume the initial state to be the canonical distribution
where . If we choose the reference state to be the canonical distribution with respect to the final Hamiltonian
then Eq. (3) reduces to the dissipated work
where we define work during the nonequilibrium process by the energy change of the system:
The above argument applies to an isolated system. In the presence of a heat bath, the total Hamiltonian in Eq. (1) is given by
where is the system Hamiltonian depending on time via external control parameters, is the Hamiltonian of the heat bath, and is the interaction between the system and the heat bath. In Eq. (10), we assume that the interaction is either turned off at the initial and final states, i.e.,
or the interaction is independent of time and very weak, i.e.,
Later, we discuss the validity of these assumption we made for this system-heat bath interaction. We also use the abbreviations and . We first consider the case (11). Then
holds and the choice of reference state in Eq. (7) leads to
where the work appearing in Eq. (15) is given by
Here, we interpret the heat as the energy that is transfered from the heat bath to the system during the process:
Now let us consider the validity of the assumption (11) we made for the interaction . We have in mind a system that is attached to the heat bath at the initial time and detached at the final time. One example of a system satisfying this condition is a cavity field interacting with a sequence of atoms passing through the cavity, where atoms can act as a heat bath to the cavity field . There are some subtlety for the definition of work in this case because there might be a contribution to work (16) from the action of switching on and off the interaction. We can avoid this subtlety by assuming that the interaction is turned on and off adiabatically. We also note that we can adopt the framework of continuous measurement (by considering many heat baths interacting with a system one by one and) by taking the limit in which the interaction time with each environment is sufficiently small, but the coupling strength is assumed to scale as the inverse of the square root of the interaction time. In this limit, the stochastic master equation was derived in Ref., and fluctuation theorems were obtained.
Next, let us consider the case where we assume Eq. (12) for the interaction . We note that when the interaction is always present, the initial state has a correlation between the system and the heat bath. However, in the weak coupling limit, Eq. (13) is correct up to the second order of the coupling strength:
When we use the definition of work and heat as given in Eqs. (16) and (17), we must assume that the energy change due to the interaction energy is small. We assume that the total energy change is divided into the energy change of the system and that of the heat bath for weak coupling as discussed in Ref. :
The main results of the rest of this paper is based on the assumption (11) for the interaction, but the same result can be derived if we assume (12) instead of (11) (in particular, a quantum Jarzynski equality without absolute irreversibility was derived in Ref.  for the weak coupling interaction.)
2. Total entropy production To relate the entropy production to the total entropy production, we consider a system composed of a system and a heat bath, and use the same Hamiltonian as 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:
where is a change in the von Neumann entropy of the system and is the heat defined in Eq. (17). If we interpret heat as the entropy produced in the heat bath, Eq. (22) expresses the entropy that is produced for the total system during the protocol; 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 dissipation in a given process . The process is thermodynamically reversible if and only if the equality in (4) holds (for example, if the dissipated work or the total entropy production vanishes).
2.2 Quantum fluctuation theorem
Next, we derive quantum fluctuation theorems by expressing the initial state in the diagonal basis.m and the initial state as , where is an orthonormal basis set, and the reference state as . The entropy production can then be calculated as
is the transition probability from the state to via the unitary operator . Such a transition is characterized by a set of labels . In deriving the third line in Eq. (23), we used the relation
From Eq. (23), we define the following unaveraged entropy production:
Next, we introduce the reference probability distribution
is the transition probability from to via . Equation (28) gives the probability of the backward process that starts from the reference state and evolves in time via . It follows from Eq. (29) 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 (27). 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 in Eq. (30) diverges. To deal with such singular situations, we divide the reference probability into two parts:
where . Since we can take the ratio between the forward and reference probabilities for the first term of the right-hand side of Eq. (32), we have
gives the total probability of those backward processes that do not return to the subspace spanned by . 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 paths, i.e., the entropy production is finite
By rewriting Eq. (33), we obtain a quantum fluctuation theorem in the presence of absolute irreversibility:
By using the Jensen’s inequality, i.e., , we obtain the following 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 , giving a stronger constraint compared with the second law-like inequality . Note that only when there is no absolute irreversibility, i.e., , the conventional fluctuation theorem is reproduced: .
In the classical case, a decomposition similar to Eq. (32) can be carried out in a general framework using the probability measure . To see this, let us denote the forward and reference probability measures in phase space as and , respectively. According to Lebesgue’s decomposition theorem [50, 51], is uniquely decomposed into two parts: , where and are absolutely continuous and singular with respect to , respectively. 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 with nonvanishing forward probabilities () and the variables with vanishing forward probabilities (). Then, corresponds to and corresponds to , and this decomposition is unique as ensured by Lebesgue’s decomposition theorem.
Note that the absence of absolute irreversibility and the requirement for the “ergodic consistency” discussed in Ref.  are different concepts. The ergodic consistency requires that for all initial phase space with nonzero probability , the corresponding initial probability distribution of the time-reversed process is nonzero , i.e., , where is the probability distribution of the system in the phase space point at time . Therefore, the ergodic consistency requires that for all nonzero forward path probabilities, the corresponding reference (backward) path probabilities are nonzero. In contrast, the absence of absolute irreversibility requires that for all nonzero reference (backward) path probabilities, the corresponding forward path probabilities are nonzero. The two conditions are different in the sense that in the former case, the backward protocol (especially, the initial state of the backward process) is fixed and thus the condition on the forward path probability is imposed, whereas and in the latter case, the forward protocol is fixed and thus the condition on the backward path probability is imposed.
2.3 Quantum Jarzynski equality
We now derive the quantum Jarzynski equality by assuming that the initial state is given by the canonical distribution (13) and by taking the reference state as given in Eq. (14). For convenience, we use the notation and , where the subscript refers to the system and to the heat bath. By assumption, we have
where and are energy eigenstates of the initial and final Hamiltonians of the system, respectively, and is the energy eigenstate of the heat bath. Now the unaveraged entropy production (27) is related to work by
is the unaveraged work done by the system and is the unaveraged heat flowing into the system.
Since the canonical distribution is full rank, i.e.,
there is no absolute irreversibility, i.e., . 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 restricts the extractable work. For simplicity, let us divide the Hamiltonian of the system into two parts and prepare the initial state as the canonical distribution that is restricted to the subspace corresponding to :
where is the energy eigenstate of the Hamiltonian . Then, is the energy eigenstate of the Hamiltonian . Now is given by the total probability of the backward process that the system returns to the subspace spanned by :
where is given by the right-hand side of Eq. (14). 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 .
3 Nonequilibrium equalities with feedback control
3.1 Formulation of the problem
The total system consists of the system (), the memory (), the bath (), and the interactions between them ( and ). 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 after the measurement step (b) at time :
We denote the initial Hamiltonian of the system by .
(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 transformation between the system and the memory followed by a projection on the memory, where T is the time-ordering operator. Here and is an orthonormal basis set of . The postmeasurement state for the measurement outcome is given by
where is the probability of obtaining outcome . The reduced density matrix of the system is given by
is the measurement operator satisfying completeness relation
Here and in Eq. (51) are given by the spectral decomposition of the initial state of the memory: . Note that for the special case of and , 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 depending on the measurement outcome . 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 and discribed by the unitary operator , which may, in general, depend on .
Now we introduce reference states for each subsystem and define entropy production-like quantities which measure the amount of entropy of () that is reduced (or produced) due to the feedback control (or measurement). The reference states of each subsystem is given by
where is the canonical distribution, and and is the eigenenergy and energy eigenstate of the heat bath, respectively.
We define the following quantity that measures the amount of entropy reduction of due to feedback control:
is the energy change of the heat bath which we identify as heat transfered from to . Note that if we choose the reference state as the final density matrix of , i.e., , Eq. (61) is nothing but the total entropy change of due to feedback control:
is a change in the von Neumann entropy of the system during the entire protocol.
We also define the following quantity which measures the amount of entropy produced in due to measurement:
where is the final density matrix of and . If we choose the reference state as the canonical distribution, Eqs. (61) and (65) are related to work and the free-energy difference, respectively, as shown in the next section.
The entropy production-like quantities (61) and (65) 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 [46, 47, 42]:
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 , i.e., , where the equality holds if and only if the measurement is given by a projective measurement using the diagonal basis of . Moreover, the information gain is nonnegative for any premeasurement state if the measurement is a pure measurement (53) as discussed in Ref. .
Extracting the information gain from entropy production-like quantities (61) and (65), we obtain the measures of irreversibility during measurement and feedback processes. For the feedback process, we have
where 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 , which is the equality condition of the last inequality (67), that is .
Similarly, for the measurement process, we have
is the average postmeasurement state over measurement outcomes, and
is a change in the von Neumann entropy due to projection and the inequality results from the fact that von Neumann entropy does not decrease under projection measurements. The nonnegativity in Eq. (68) shows the irreversibility of the measurement process.
Next, let us consider the following spectral decompositions of the initial states of the system, the heat bath, and the memory
Let us also decompose the postmeasurement state of the system as follows:
Using the above decompositions, we calculate Eqs. (67) and (68) and define an unaveraged form of Eqs. (61), (65) and (66), along a line similar to what we did in deriving the unaverage form of the entropy production in Eq. (23). From Eq. (67), we obtain
where we introduce the forward probability distribution corresponding to the forward process of the total system: