Fluctuation theorems for quantum processes
We present fluctuation theorems and moment generating function equalities for generalized thermodynamic observables and quantum dynamics described by completely positive trace preserving (CPTP) maps, with and without feedback control. Our results include the quantum Jarzynski equality and Crooks fluctuation theorem, and clarify the special role played by the thermodynamic work and thermal equilibrium states in previous studies. We show that for a specific class of generalized measurements, which include projective measurements, unitality replaces microreversibility as the condition for the physicality of the reverse process in our fluctuation theorems. We present an experimental application of our theory to the problem of extracting the system-bath coupling magnitude, which we do for a system of pairs of coupled superconducting flux qubits undergoing quantum annealing.
pacs:05.30.-d 05.40.-a 05.70.Ln
Fluctuation theorems provide powerful analytical tools for nonequilibrium physics. Some years ago Jarzynski discovered an equality for classical processes that shows how to determine free energy changes by measuring only the work performed on the system, without the need to determine the accompanying entropy changes, and in particular without the requirement that the processes be quasistatic. Consider for example two thermal equilibrium states and of a system, each state with different macroscopic thermodynamic observables such as pressure and volume, but both at a fixed inverse temperature . The system is initially in state , and work is performed on the system according to some protocol to drive it to the macroscopic conditions of state . If the system is not allowed to equilibrate, the system may not reach the thermal equilibrium state . Nevertheless, for this forward process, the classical Jarzynski equality (CJE) Jarzynski (1997)
where , relates the statistical average of the work done on the driven system to the free energy difference of the final equilibrium state (whether this state is reached or not) and the initial thermal equilibrium (Gibbs) state . In particular, this result is independent of what protocol is used, which is one of its remarkable features. In the presence of feedback control (“Maxwell’s demon”) the efficacy parameter differs from unity, and characterizes the efficacy of feedback and the amount of information extracted Sagawa and Ueda (2010).
The CJE follows directly from the Tasaki-Crooks fluctuation theorem Crooks (1999); Tasaki (2000), which relates the probability density function (PDF) of work done in the forward process to the PDF of a reverse process :
The reverse process describes the evolution of the system starting from the thermal equilibrium state and applying an appropriately time-reversed work protocol on the system, although this may not correspond to the time-reversed evolution of the forward process. A key element of the fluctuation theorem is the requirement of a microreversibility condition, which relates the forward and reverse dynamics at any given instant in time. For example, for driven classical systems, microreversiblity relates the flow of phase space points under the forward driving protocol to the flow under the reversed driving protocol, via the heat absorbed by the system Stratonovich (1994); Campisi et al. (2011). For a specific pertinent statement of microreversiblity see, e.g., Eq. (5) in Ref. Crooks (1999). Many generalizations have been developed (for reviews see, e.g., Bustamante et al. (2005); Jarzynski (2011); Campisi et al. (2011)), with appropriate generalizations of the imposed microreversiblity condition. In particular, the classical results (see also Seifert (2005)) have been quantized, first for thermal states undergoing unitary evolution Kurchan (2000); Tasaki (2000), and subsequently for thermal states undergoing non-unitary, open system dynamics Yukawa (2000); Mukamel (2003); Monnai (2005); Talkner et al. (2007); Crooks (2008); Quan and Dong ; Campisi et al. (2009); Talkner et al. (2009); Deffner and Lutz (2011), including continuous monitoring Campisi et al. (2010) and quantum feedback Sagawa and Ueda (2008); Morikuni and Tasaki (2011); Sagawa and Ueda (2010).
Here we aim to show that there exists a single unified framework from which all the quantum results can be derived as well as generalized, using only basic tools of the theory of open quantum systems Breuer and Petruccione (2002) and quantum information theory Nielsen and Chuang (2000). To this end we derive a general fluctuation theorem for quantum processes described by completely positive trace-preserving (CPTP) maps , with or without feedback. CPTP maps arguably represent the most general form of open quantum system dynamics, under the assumption of an initially uncorrelated system-bath state Breuer and Petruccione (2002). Our strategy leads to a general and simple recipe for writing down fluctuation theorems, not all of which must correspond to a measurable thermodynamic observable (note that work is not a quantum observable Talkner et al. (2007)), or involve a physical reverse process. We show that in order for a PDF for the reverse process to exist, the map must be unital, and we show that the map describing the reverse process is simply the dual map of the forward process, which leaves no ambiguity in defining the reverse process. In this sense unital channels emerge as playing a crucial role in fluctuation theory for any CPTP map, replacing the role typically played by the standard thermodynamic notion of microreversibility. Our work illuminates the special role played by the Gibbs state and work measurements, both of which feature prominently in the literature on fluctuation theorems.
We empirically determine the first moment of our integral fluctuation theorem in our theory via an experiment involving pairs of superconducting flux qubits on a programmable chip. The first moment turns out to be a measure of the information-geometric distance of the evolved state and the virtual final equilibrium state, where virtual here signifies that the equilibrium state is never actually reached by our evolution. We show that these experimental results can be well explained using a time-dependent Markovian master equation with a free adjustable parameter determining the system-bath coupling strength. As a novel application, our theory provides a meaningful optimization target that allows us to determine this parameter by fitting to the experimental data. We thus establish quantum fluctuation theorems as important tools for studying open quantum systems.
Ii General Fluctuation Theorem
ii.1 Review of Quantum Jarzynski Equality for Closed Systems with Projective Measurements
We briefly review the generalization of the CJE to the quantum case of thermal states undergoing unitary evolution Tasaki (2000), as it will help set the stage for our work (see also A). We consider a Hamiltonian that interpolates between two system states described by and , with an associated unitary time evolution operator . The system, described by a density matrix , is initially in the Gibbs state associated with :
where is the partition function associated with . A (projective) measurement of the eigenenergy (associated with ) is performed, which selects the energy state with probability . The system is then evolved according to , and another projective measurement of the eigenenergy (associated with ) is performed. The conditional probability of measuring the energy is given by:
Let us define the work performed during this evolution by (this definition applies since the system is closed). The PDF associated with is:
Using that and the Dirac delta properties , and , we note that upon multiplying both sides of the equation by , we have
where we have identified the right hand side with the PDF of the reverse process with work , for which the following temporally ordered sequence applies: (i) the system is initially in the Gibbs state associated with , (ii) a projective measurement is performed with outcome probability , (iii) the system evolves unitarily via , and (iv) a projective measurement is performed. Eq. (6b) is the closed system fluctuation theorem, the integration of which gives a closed system quantum Jarzynski equality exactly of the form of Eq. (1). Note that in this case the reverse process is simply the time-reversed process, a situation that will change in our more general analysis below.
ii.2 Quantum Jarzynski Equality for Generalized Measurements
We now generalize the previous section result beyond thermal states, unitary evolutions, and projective measurements. Consider a fiducial initial state , two sets of generalized measurements and (see also Morikuni and Tasaki (2011)), a CPTP map (see also Kafri and Deffner (2012); Vedral (2012)), and a fixed, yet arbitrary distribution , whose role we clarify later. The quintuple is the basic input data describing the problem. The measurement operators satisfy (the identity operator), and similarly for . Generalized measurements are the most general kind of measurements in quantum theory, and they include projective measurements, positive operator valued measures (POVM), and weak measurements as special cases Nielsen and Chuang (2000); Oreshkov and Brun (2005). The CPTP map has Kraus operators , i.e.,
We first consider the forward process depicted in Fig. 1.
A mixed state ensemble is prepared by measuring , so that the normalized state
has probability 111For a given generalized measurement , the set of possible values of the probabilities is in general constrained, even if the initial state is arbitrary (see B for an example of this).. Next evolves under , and finally the measurement is performed. The conditional probability of observing outcome given outcome is then
The marginal probability distribution of outcomes is , where
and where is the joint probability distribution. In the last equality we used Bayes’ rule for the joint probability . We therefore have:
Note that the transition matrix of the forward process is column stochastic:
where we used the normalization condition of the generalized measurement and the fact that is a trace preserving map.
Let the random variable be a real-valued function parametrized by the measurement outcomes . will play the role of a generalized thermodynamic observable, where we use the term ‘observable’ in a loose sense since it is typically an abstract quantity and only is a thermodynamic observable in special cases. The PDF associated with is
Let us now choose
(note that such a form will give us the expression found in Eq. (6b)) and also define the generalized reverse thermodynamic observable . This choice of requires that . Then, using the Dirac delta properties , and , we find
where and where we used the dual map with Kraus operators , i.e., , for which for any pair of operators and .
Comparing Eqs. (13b) and (15b), it is tempting to identify the latter with a PDF associated with the dual map , however there are important differences. First, while the map acts on a normalized state , the “state” acted upon by dual map is not necessarily normalized. Second, while , so that the set forms a POVM, is not necessarily equal to the identity operator, so the set cannot always be identified with a POVM. For this reason we have for the time being used the notation in Eq. (15b). We revisit this issue in subsection II.3, where we show under which conditions can be interpreted as the PDF of a reverse process.
We define the efficacy222We use the term efficacy loosely here. In the case of the classical Jarzynski equality with feedback Sagawa and Ueda (2010), the right hand side of the equality is indeed a measure of the efficacy of the feedback protocol. Here we make no such claims. Sagawa and Ueda (2010); Kafri and Deffner (2012) as
which upon integration gives the QJE with .
Using Jensen’s inequality we have and thus find a generalized 2nd law of thermodynamics (we clarify this claim below):
We can substantially generalize the QJE Eq. (17) in terms of the moment generating functions for the map and its dual,
Multiplying Eq. (15a) by and integrating, we find:
Moreover, using we find (details can be found in C)
where is the relative entropy (Kullback-Leibler divergence), and is the Shannon entropy.
ii.3 Fluctuation Theorem for “Microreversible” Generalized Measurements and Unital Maps
Recall that in the discussion immediately following Eq. (15) we stressed that cannot always be interpreted as the PDF associated with the dual map . Let us now restrict ourselves to a class of generalized measurements and that satisfy additional constraints so that such an interpretation becomes possible:
We call generalized measurements and that satisfy Eq. (24) “microreversible” for reasons that will shortly become apparent. Rank- projective measurements trivially satisfy these constraints, but are not necessary333An example of generalized measurements, which are not rank-1 projective measurements, satisfying Eq. (24) are and and ..
Let denote the Hilbert space dimension; we prove in D that if the constraints (24) are satisfied then , i.e., that none of the probabilities or can vanish. With these additional constraints, can be identified as a normalized state, and we can define a new measurement such that:
where is an arbitrary unitary operator, is a virtual final state, and the probability now takes the value such that the mixed state ensemble is generated by measuring 444Since the input data specifies rather than , it is more natural to think of each measurement operator as specifying the corresponding , i.e., . The virtual final state should then be full rank in order for its inverse to exist.. Similarly, we can write the state in terms of a new generalized measurement :
where is an arbitrary unitary operator. Thus
associated with the dual map acting on the state , followed by the generalized measurement . We have therefore arrived at our fluctuation theorem for CPTP maps:
A bound on the value of is presented in E.
One more condition must be imposed in order for to become a PDF, i.e., for [Eq. (16)] to equal , namely, should itself be a CP map. This is the case if is unital . If it is unital, then Eq. (29) is a fluctuation theorem relating a physical forward and reverse process, where we can interpret as the probability density associated with the following reverse process (depicted in Fig. 2): i) prepare the state by measuring , ii) evolve via , iii) measure . We emphasize that here, the forward and reverse process are described by different measurements, namely and , respectively, related via Eqs. (25) and (26).
Therefore, Eq. (29) represents a physical fluctuation theorem for unital CPTP maps, where unitality replaces the role typically occupied by microreversibility. Consequently, upon integration of the fluctuation theorem, we obtain (the observation that unital channels yield was first stated in Ref. Kafri and Deffner ).
Why are unital channels singled out? Recall that the transition matrix of the forward process is in general column stochastic [Eq. (12)]. Under the additional assumptions of microreversible generalized measurements and unitality, it becomes bistochastic:
Thus, whereas classical microreversibility imposes a specific relation between forward and time-reversed phase space paths Crooks (1999), unitality gives rise to a form of microreversibility relating the forward and reverse probabilities:
where Eq. (31) shows that is a proper conditional probability.555 Another unique and suggestive feature of unital channels is that they can always be written as an affine combination of reversible channels Mendl and Wolf (2009), i.e., , where is unitary, and . In the special case when is a probability distribution, the affine combination becomes a convex one, and the unital channel can be interpreted as representing the unitary evolution occurring with probability , while its dual becomes the time-reversed unitary occurring with the same probability.
We remark that once we fix the operators in the forward process, then the operators are uniquely defined only for the case of rank-1 projective measurements; otherwise they are defined up to a unitary operator and the full-rank virtual final state [see Eqs. (25) and (26)]. Therefore, beyond rank-1 projective measurements, there is no unique reverse process, but all allowed choices give a fluctuation theorem relating the forward and reverse processes.
To summarize, the derivation essentially involved only the Kraus representation of CPTP maps, the standard form of generalized measurements, Bayes’ rule, and a judicious choice of forward and reverse generalized thermodynamics variables. The Kraus representation formalism allows for an unambiguous definition of the reverse process in terms of the dual map of the forward process. However, the choice (14) is by no means unique. For example, if we choose (a type of mutual information, as in Vedral (2012)) then Eq. (13a) yields
ii.4 The Case of Projective measurements
When the measurements and are projective, our results for the QJE and fluctuation theorem simplify. We prepare and let , with and rank-1 projectors (pure states), and we have and , i.e. the forward and reverse processes are described by the same measurements, and Eq. (32) provides a standard microreversibility condition. Furthermore, in this case, and can be made arbitrary. We refer to as the “virtual final state” since the final state reached at the conclusion of the forward protocol (see Fig. 1) is in general different from the state .
where the quantum relative entropy , and is the von Neumann entropy. This generalizes the result for the mean irreversible entropy production of Refs. Deffner and Lutz (2010); Dorner et al. (2012). If (e.g., when is unitary) then . If is -invariant  and is unitary (as in a quantum quench), then . If then . If the evolution is adiabatic with initial Hamiltonian and final Hamiltonian such that , then and . Therefore, for the mixed state ensembles and , we have
where are the probability distributions associated with and respectively.
Eq. (35) has an interesting interpretation, illustrated in Fig. 3. Referring first to Eq. (35a), side of the triangle represents the von Neumann entropy change occurring in the physical process enacted by , while side is an information-geometric measure of the distance between the evolved state and the “virtual” one .
On the other hand, referring to Eq. (35b), side can also represent , which is again related to the information-theoretic distance666Note that the relative entropy is not strictly a distance (since it is not symmetric and does not satisfy the triangle inequality), but a divergence. between the evolved and virtual state. Side (3) represents the von Neumann entropy change, i.e., .
Known quantum fluctuation theorems follow from our formalism. For example, we show in A how Eq. (35a) reduces to the standard statement of the 2nd law for isothermal processes, , where is the mean irreversible entropy production Deffner and Lutz (2010, 2011), after we choose and as Gibbs distributions and as a unitary evolution. We also note that a calculation of would yield a generalized fluctuation-dissipation theorem.
In the thermal case and (where is the initial/final Hamiltonian and is the initial/final partition function), we find . Here represents the heat exchange in the (virtual) undriven relaxation between the evolved and virtual states, (for a proof see C). In the thermal case amounts to a thermodynamical entropy change . Thus in this case, using Eq. (19) and assuming unitality (), we have the 2nd law in the Clausius form . This clarifies why Eq. (19) can be interpreted as a generalized 2nd law of thermodynamics.
ii.5 Including feedback
Suppose we repeat the forward protocol of Fig. 1 in the projective measurement case, but denote the CPTP map by and the final measurement by . Depending on the measurement outcome , we apply an additional CP map to the resulting state . This constitutes a feedback step, and generalizes earlier work which considered only unitary feedback maps Sagawa and Ueda (2008); Morikuni and Tasaki (2011); Sagawa and Ueda (2010). Next we apply another projective measurement , labeled by the outcomes . Thus from Fig. 1 is replaced by , and by . Our generalized feedback control protocol is illustrated in Fig. 4.
Given an initial state , the probability of observing outcomes and is , and the joint distribution is . As in the feedback-free case, we can construct the PDF associated with the generalized thermodynamic observable as:
where the notation () is shorthand for (). We now choose , where is an arbitrary, fixed distribution, associated with the (virtual) state . Then
Integrating, we find a generalized integral fluctuation theorem in the presence of feedback:
We show in H how these results allow us to recover known quantum fluctuation theorems with feedback. Although we have identified with , it is important to note that does not coincide with the reverse process because (unlike in the feedback-free case) there is now no unique association between the initial and final measurement outcome indices: the same pair is connected via different values of the intermediate measurement outcomes.
Iii Experimental application of the open-system QJE
Compared to the classical case (e.g., Wang et al. (2002); Collin et al. (2005); Junier et al. (2009); Garnier and Ciliberto (2005); Douarche et al. (2005); Toyabe et al. (2010); Utsumi et al. (2010); Küng et al. (2012); Saira et al. (2012)), there has been relatively little work on experimental tests or applications of the quantum version of the Jarzysnki-Crooks relations. Existing data on x-ray spectra of simple metals Heyl and Kehrein (2012), and experiments on a driven single qubit (defect center in diamond) Schuler et al. (2005) have been used to verify the previously derived quantum fluctuation relations. There have also been recent proposals that showed that viability of single-qubit interferometry to verify quantum fluctuation theorems Dorner et al. (2013); Mazzola et al. (2013). Although we are not able to test the generalized QJE or the generalized fluctuation theorem, we present an application of our generalized QJE (specifically, the first moment given by Eq. (35)) to the problem of extracting the system-bath coupling magnitude, using both numerical simulations and an experiment using a commercially-available quantum annealing processor comprising superconducting flux qubits (see I for further details about the experimental system). Since the system Hamiltonian is time dependent, the formalism of the QJE provides meaningful observables in this setting.
The processor performs a quantum annealing protocol to find the ground state of a classical Ising Hamiltonian. The protocol is described by the transverse-field Ising Hamiltonian
and are standard Pauli operators acting on the th qubit. The magnetic fields and couplings (superconducting inductances) are programmable. The annealing functions and satisfy , where is the total annealing time. The annealing protocol amounts to starting with the transverse field turned on and the Ising Hamiltonian turned off, and then slowly reversing their role until only the Ising Hamiltonian remains. The processor is performed at mK, with the qubits in contact with a thermal environment. It is ideally suited to measuring in Eq. (35) since it performs the process described in Fig. 1 (but without the measurement ) with the initial state being the Gibbs state.
The experiment can be described by an adiabatic Markovian master equation derived in Ref. Albash et al. (2012) (see J for essential details). The CPTP map generated by the master equation is not unital. We consider projective measurements that prepare , , where are the instantaneous eigenenergies of , so that our generalized thermodynamic observable [Eq. (14)] is given by , where the free energy . Note that for an open quantum system, does not correspond to the work done on the system. Equations (16) and (17) yield
where denotes the Gibbs state associated with , with and , is a random variable taking values in the set , and . Equation (35a) gives:
where we have denoted .
For concreteness we consider a two-qubit system with . We checked the equality expressed in Eq. (43) by independently numerically simulating its two sides for this model using our master equation, and find essentially perfect agreement. The same holds for Eq. (35a) (see K for details). We tested the same two-qubit system experimentally using the quantum annealing processor. For each experimental run, the system is initialized in the Gibbs state of , and after performing the annealing protocol , a projection onto the computational basis (eigenstates of the operators, i.e., the final energy eigenstate basis of ) is performed. Therefore, for each run, a single energy eigenstate of is measured. We then repeat the quantum annealing process thousands of times to build up the statistics necessary to determine the relative occupancy of each final energy eigenstate. The empirical relative occupancy corresponds directly to the probability of measuring energy . Therefore, this allows us to experimentally determine . We also know from the initial Gibbs state and the value of . We thus determine
Our master equation has two free parameters: the high-frequency cut-off , which we set to Albash et al. (2012), and the system-bath coupling magnitude , where is the system-bath coupling constant and characterizes the Ohmic bath (see J for details). is a quantity that combines both the statistics and the energy spectrum of the system, making it more system-specific. Remarkably, by simultaneously minimizing the deviation between the numerical solution of our master equation and the experimental data for as a function of and allowed us to extract the system-bath coupling magnitude from Eq. (45a) (see I for details). Therefore, we find that provides a valuable optimization target in addition to its information theoretic content, which the statistics of the experiment alone may not provide.
Why does display a minimum as a function of (main panel of Fig. 5)? In our experiment , so that the Gibbs state is almost pure, i.e., both and . Therefore we are effectively measuring the information-theoretic distance . Increasing at fixed temperature is like decreasing while fixing . Thus the system requires more time to equilibrate as grows, but we keep fixed. On the other hand, as becomes very small the ground and first excited states become degenerate, so the excitation probability increases, and the system is again farther from equilibrium. Also, as we increase the annealing time, becomes closer to the Gibbs state, causing to decrease as observed in the inset.
To conclude, we presented fluctuation theorems and moment generating functions for CPTP maps, thus generalizing previous work on the Jarzynski-Crooks relations and the 2nd law for open quantum systems, including processes with feedback. We performed an experiment using superconducting flux qubits that matches the fluctuation theorem protocols, and used this experiment to extract the system-bath coupling for an adiabatic Markovian master equation that nicely matches the experimental results. Our work ties together key ideas from statistical mechanics, quantum information theory, and the theory of open quantum systems, and paves the way to experimental tests and applications of fluctuation theorems in the most general setting of open quantum system dynamics.
Acknowledgements.We thank S. Deffner, P. Talkner, and I. Marvian for useful comments. This research was supported by the ARO MURI grant W911NF-11-1-0268, the Lockheed Martin Corporation, and by NSF grant numbers PHY- 969969 and PHY-803304 (to P.Z. and D.A.L.).
Appendix A Recovering Known Fluctuation Theorems
Since this map is unital, its dual is also a CPTP map (the actual time reversed process) given by:
where , and the Hamiltonian has instantaneous eigenenergies . Using Eq. (16) this yields .
Recall that in our formalism we need to also specify the fiducial initial state , the measurements and , and the distribution . We pick these so that they generate the Gibbs distributions
at inverse temperature , where is the partition function corresponding to , and is the corresponding Gibbs state. For example, we can assume that , , and .
If we let this then corresponds to the choice:
where is the free energy given by . This corresponds to identifying with the (dimensionless) work.
We thus find from Eq. (17) the QJE for a closed quantum system: