Relations between dissipated work in non-equilibrium process and the family of Rényi divergences

In this paper, we establish a general relation which directly links the dissipated work done on a system driven arbitrarily far from equilibrium, a fundamental quantity in thermodynamics, and the family of Rényi divergences between two states along the forward and reversed dynamics, a fundamental concept in information theory. Specifically, we find that the generating function of the dissipated work under an arbitrary time-dependent driving is related to the family of Rényi divergences between a non-equilibrium state along the forward process and a non-equilibrium state along its time-reversed process. This relation is a consequence of the principle of conservation of information and time reversal symmetry and is universally applicable to both finite classical system and finite quantum system under arbitrary driving process. The significance of the relation between the generating function of dissipated work and the family of Rényi divergences are two fold. On the one hand, the relation establishes that the macroscopic entropy production and its fluctuations are determined by the family of Rényi divergences, a measure of distinguishability of two states, between a microscopic process and its time reversal. On the other hand, this relation tells us that we can extract the family of Renyi divergences from the work measurement in a microscopic process. For classical systems the work measurement is straightforward, from which the family of Rényi divergences can be obtained; for quantum systems under time-dependent driving the characteristic function of work distributions can be measured from Ramsey interferences of a single spin, then we can extract the family of Renyi divergences from Ramsey interferences of a single spin.


Introduction
The pioneering works by Clausius and Kelvin have established that the average mechanical work needed to move a system in contact with a heat bath at temperature T, from one equilibrium state A into another equilibrium state B, is at least equal to the free energy difference between these states: á ñ - where the equality holds only for a quasi-static process. In a remarkable development Jarzynski [1] discovered that for a classical system initialized in an equilibrium state the work done under a non-equilibrium change of control parameters is related to the equilibrium free energy difference between the initial and the final equilibrium states for the control parameters via Here b º T 1 is the inverse temperature T of the initial equilibrium state of the system and we take the Boltzmann constant º k 1 B , W is the work done on the system due to a driving protocol which varies the control parameter from l i to l f , b º --F Z ln 1 is the Helmholtz free energy of the system with Z being the equilibrium partition function and the angular bracket on the left of equation (1) denotes an ensemble average over realizations of the process. The Jarzynski equality connects equilibrium thermodynamic quantity, the free energy difference, to a non-equilibrium quantity, the work done in an arbitrary driving processes. It implies that we can determine the equilibrium free energy difference of a system by repeatedly performing work at any rate. Jarzynski equality and Crooks relation [2] from which it can be derived have been verified experimentally in various physical systems [3][4][5][6][7][8][9][10] and were also proven to hold for finite quantum mechanical systems [11][12][13][14][15]. The discovery of the Jarzynski equality has led to a very active field concerned with fluctuation relations in nonequlibrium thermodynamics [16][17][18].
The excess work -D W F that arises in irreversible processes is often referred to as the dissipated work, = -D W W F diss . Contrary to the reversible work, which only depends on the initial and final equilibrium states, the dissipated work depends on how the specific driven protocol is performed. Usually the driving protocol is realized by changing the control parameters in the Hamiltonian between initial and final values, which can in principle bring the system arbitrarily far out of equilibrium. Surprisingly, there exists a neat and exact microscopic fluctuation relation for the dissipated work. The central result of this paper is the following relation: where z is a finite real number, W diss is the dissipated work done on the system due to a driving protocol under which the control parameter changes from l i to l f , the angular bracket on the left hand side denotes an ensemble average over the realizations of the driving process and r t r is the order-z Rényi divergence between r t Q -Q -( ) t R 1 and r ( ) t F with Θ being the time reversal operation. For classical system, the order-z Rényi divergence between two distributions r ( ) X 1 and r ( ) X 2 is defined as [19,20] ò r r r r º - is the phase space density in the forward process at an arbitrary time t Î [ ] t 0, which was initialized in an equilibrium state at inverse temperature β and control parameter l i and r t -( ) t R is the phase space density in the reversed process at time tt which was initialized in an equilibrium state at inverse temperature β and control parameter l f . For quantum system, the order-z Rényi divergence betwen quantum states r 1 and r 2 is defined as [22,23] r r r r º - is the density matrix in the forward process at arbitrary time t Î [ ] t 0, which was initialized in the equilibrium state at inverse temperature β and force parameter l i and r t -( ) t R is the density matrix at time tt in the time reversed process which was initialized in the canonical equilibrium state at inverse temperature β and force parameter l f . It should be emphasized that because of the assumption that the system is isolated from the reservior before and in the course of driven, the right hand side of equation (2) can be written as the Rényi divergence between the state of the forward process and the state of backward process at same time t while in the case of an open system the probabilities of the entire process may appear. Equation (2) can be consider as a generalization of Jarzynski equality in equation (1) and includes Jarzynski equality as a special case. Jarzynski equality states that average of b e W on the work distribution measurement gives the free energy difference between the equilibrium states at the initial and final parameters. We are attempting to explore whether work distribution in a non-equilibrium driving can produce more knowledge about the physical process and physical system than that of Jarzynski equality tells us. Equation (2) is a useful result in this direction.
The remainder of this paper is organized as follows: in section 2, we briefly review the formalism for nonequilibrium classical thermodynamics, including the concept of work in classical system, the forward driving process and its time reversed process and then derive the relation between the generating function of dissipated work and the family of Rényi divergences between two phase space distributions along the forward and reversed process. In section 3, we review the formalism for quantum thermodynamics and then derive the relations between the generating function of dissipated work in driven quantum system and the family of Rényi divergences between two quantum states along the forward and its time reversed process. In section 4, we establish the formalism for extracting the family of Rényi divergences for two quantum states from the Ramsey interference of a single spin. In section 5 we use the one-dimensional (1D) transverse field Ising model in a sudden quench process to demonstrate our theoretical finding, equation (2), between the generating function of dissipated work and the family of Rényi divergences between two quantum states and then we use a single spin undergoing a sudden quench process to demonstrate the method of extracting the family of Rényi divergences from the Ramsey interference of a single spin. In section 6, we summarize our findings and discuss the possible related future problems.

Non-equilibrium classical thermodynamics
We consider a finite classical system with Hamiltonian q p q p q p , ; , ; ; , denotes collectively the coordinates and momenta of all the N particles in the system, λ is a parameter controlled by an external agent. For a classical system with time-dependent Hamiltonian, the microscopic reversibility [24] is illustrated in figure 1.
We first introduce the forward process for a classical system under time-dependent driving. We assume that the classical system is initialized in the equilibrium state at inverse temperature b = T 1 with the control parameter l i , which is described by the Boltzmann distribution in phase space r 0 i being the initial partition function. Then the classical system is isolated and driven by an external agent, which varies the control parameter λ from an initial value l i to a final value l f in a time duration τ according to a specified protocol l t 0, . According to Liouville theorem, which states that the phase space distribution is invariant along any trajectory of the system [25], one has, for where X(t) is the resulting phase space point at time t under the dynamics of forward Hamiltonian if it was initially at X 0 at t=0 (see the upper red line in figure 1). According to first law of thermodynamics, the work done associated with the trajectory in the forward process is ; . 4 Now we consider the reversed process. In the reversed process, the classical system is initialized in a canonical equilibrium state at inverse temperature β at the value l f of the control parameter, d e H X f ; f . Then the classical system is completely isolated and is driven by the reversed Hamiltonian for a time duration τ. According to Liouville theorem [25], we have t is the resulting phase space point at time t under the dynamics of Hamiltonian in the reversed if it was at QX 1 at t=0 (see the lower blue line in figure 1). Combing equations (3), (4) and (5), we obtain [26] r t r l n , 1 being the Helmholtz free energy, t Î [ ] t 0, is arbitrary time points. Note that on the right hand side of equation (6), the phase space densities are observed at the same phase space point X(t). Equation (6) is a consequence of Liouville theorem in classical mechanics.
Making use of equation (6), we have ò r Figure 1. Schematic illustration of the time reversal symmetry for driving classical system. The upper red line with arrow towards right is a trajectory in the forward process, it starts at X 0 and is driven by , 0 , and arrives in X(t) at time t and finally ends in X 1 at time τ. The lower blue line with arrow towards left is its time reversed trajectory, which starts at QX 1 (time reversed state of X 1 ) and is driven by t after time t and finally returns to QX 0 at time τ.
where z is a finite real number and ò r r r r º - is the order-z Rényi divergence of two probability distributions r ( ) X 1 and r ( ) X 2 [19,20]. From equations (7) to (8) we have applied the Liouville theorem = X X d d 0 and equation (6). Identifying the dissipated work = -D W W F diss in equations (7)-(10), we consequently obtain equation (2) for a classical system. Now we give several comments on equation (2) for classical system: (1) For z=1, equation (2) for driven classical system returns to the Jarzynski equality [1] for classical system.
(2) The fluctuation of the dissipated work is independent of time t because the densities on equation (10) can be evaluated at any intermediate time.
While the fact that the dissipated work is independent of t can also be seen from [26,29]. This time independence is a consequence of the Liouville equation in Hamilton dynamics.
(3) For < < z 0 2, the Rényi divergence for any two probability distributions r 1 and r 2 and any classical channel ε satisfies [21] e r e r r r Therefore the Rényi divergence for < < z 0 2 is a valid measure of distinguishability. Thus the family of Rényi divergence appears in equation (2) for classical system is a quantification of the breaking of time reversal symmetry between the forward process and its time reversed one.
(4) It relates the fluctuation of dissipated work or entropy production to the family of Rényi divergences between the phase space density in the forward process and the phase density in its reversed process. Entropy production is a macroscopic quantity while the Renyi divergences between two states is essentially a microscopic quantity, which quantifies the distance between two states or to what extent the time reversal symmetry is breaking. The various moments of the dissipated work are given by, is the relative entropy [31] between forward phase space density distributions and the reversed phase space density distributions.
(5) It is an exact relation between the generating function of the dissipated work in a driving process and the family of Rényi divergences between two non-equilibrium states. Work distributions has been measured in various classical systems [3-5, 16, 17], from which one can extract the family of Rényi divergences between two classical phase space densities along the forward and reversed process.

Non-equilibrium quantum thermodynamics
Let us consider a finite quantum system governed by a Hamiltonian l ( ) H and λ is a parameter controlled by an external agent. We illustrate the time reversal symmetry for quantum system under time-dependent driving in figure 2.
Let us first define the forward process in quantum system under time-dependent driving. We initialize the quantum system in canonical equilibrium state at inverse temperature b = T 1 at a fixed value of control parameter l i , which is described by the density matrix r r bl being the partition function. Then we isolate the system and drive it by the Hamiltonian l ( ( )) H t for a time duration τ, where the force protocol l t 0, brings the parameter from l i at t=0 to l f at a later time τ. Then the state at t in the forward process is given by with  being the time ordering operator. In general r t ( ) , . Work in quantum system is defined by two projective measurements [14,18]. We assume, for any λ, and the symbol n labels eigenenergy. At t=0, the first projective measurement of . Simultaneously the initial equilibrium state projects into the state , the system is isolated and driven by a unitary evolution operator ( ) U t, 0 F and the state at So the probability of obtaining l ( ) E n i for the first measurement and followed by obtaining . Thus the work distribution in the forward process is given by [14,18] The quantum work distribution P F (W) encodes the fluctuations in the work that arise from thermal statistics and from quantum measurement statistics over many identical realizations of the protocol. The characteristic function of quantum work distribution is given by Tr e e e . 17 Now we define the reversed process in quantum system. In the reversed process, we initialize the quantum system in the time reversed state of the canonical equilibrium state at inverse temperature b = T 1 at value l f of the control parameter, being the canonical partition function. Then we drive the system by the Hamiltonian in the reversed process for a time duration τ which brings the force parameter from l f at t=0 to l i at a later time τ. The time evolution operator for the forward process and its the reversed process are related by [32] . Then the state at t in the reversed process is given by are far from equilibrium states, they satisfy the following lemma due to time reversal symmetry: Lemma. The density matrices in the forward driving process at arbitrary time t Î [ ] t 0, and its time reversed process at time tt satisfy, for any finite real numbers Tr 0 . 20 (18) and (19), we have Schematic illustration of the time reversal symmetry for driving quantum system. The upper red line with arrow towards right denotes the forward process: it starts at an arbitrary initial state ñ |i and evolves in time under the unitary evolution generated by 0 , for time duration τ. Then the state at time t is and finally becomes ñ | f at time t . The lower blue line with arrow towards left is the time reversed process: it starts at Q ñ | f (time reversed state of ñ | f ) and evolves under Hamiltonian are related to each other by a unitary transformation ( ) U t, 0 F . They must be equal under the trace. Thus we have proved equation (20).
From the definition of quantum work distribution, we have (29) to (30), we have used the lemma proved above. In the last step, we have made use of definition of the order-z quantum Rényi divergence of two density matrices r 1 and r 2 [22,23], r r r r º - , which is information theoretic generalization of standard relative entropy [19]. If we identify -D W F as the dissipated work W diss in equations (26) and (31), we therefore obtain equation (2) for quantum system. Now we make several comments on equation (2) for quantum system: (1) When z=1, equation (2) for quantum system recovers the Jarzynski equality for quantum system.
(2) The fluctuation of the dissipated work in quantum system is independent of time t because the density matrices on the right hand side of equation (31) can be evaluated at any intermediate time.
While the fact that it is independent of t can also be seen from [29,34]. This time independence is a consequence of the time reversal symmetry.
(3) For < < z 0 1, the Rényi divergence for any two density matrices r 1 and r 2 and any quantum channel ε satisfies [20,22] e r e r r r Therefore the quantum Rényi divergence for < < z 0 1 is a valid measure of distinguishability. Thus the family of Rényi divergence appears in equation (2) for quantum system is a quantification of the breaking of time reversal symmetry between the forward process and its time reversed one.
(4) It relates fluctuations of the dissipated work or entropy production to the family of Rényi divergences between two non-equilibrium quantum states along the forward process and its time reversed process. Entropy production is a macroscopic quantity while the Renyi divergences between two states is essentially a microscopic quantity, which quantifies the distance between two states or to what extent the time reversal symmetry is breaking. Various moments of the dissipated work for quantum system under time-dependent driving are given by, where T is the temperature, = n 1, 2, 3, and  n is an ordering operator which sorts that in each term of the binomial expansion of 2 is the von Neumann relative entropy [31,33] between two density matrices. Recently the relation in equation (34) was experimentally demonstrated by using a nuclear magnetic resonance study of an isolated spin-1/2 system following fast quenches of an external magnetic field [36].
(5) It is an exact relation between the generating function of the dissipated work and the family of Rényi divergences between two non-equilibrium states. Thus we can extract the family of Rényi divergences from the work measurement in a driven quantum system. Since the characteristic function of quantum work distribution can be measured from the Ramsey interference of a single spin [37][38][39], one can also measure the family of Rényi divergences between two quantum states. We shall establish the formalism for extracting the family of Rényi divergences from Ramsey interference of a single spin in next section. Note that two recent related works connect fluctuation relations to information-theoretic concepts with more information-theoretic setting [40,41].

Extracting the family of Rényi divergences from Ramsey interference of a single spin
In the recent formulation of laws in quantum thermodynamic, the family of Rényi divergences between a state and the thermal equilibrium state play a central role [42][43][44][45]. To verify the laws in quantum thermodynamics, it is also critical to measure the family of Rényi divergences. However it is quite difficult to measure the Rényi divergences from the tomographic measurement of the quantum state due to scalability problem. In section 3, we have derived the relations between dissipated work and the family of Rényi divergences. Thus we can extract the family of Rényi divergences from quantum work measurement. Work measurement in a driven quantum system is quite difficult because it requires two projective measurements of Hamiltonian [14]. However the characteristic function of quantum work distribution can be measured from Ramsey interference of a single spin [37][38][39]. Here we shall establish the formalism for extracting the family of Rényi divergences from Ramsey interference of a single spin. In terms of the characteristic function of the quantum work distribution, our finding, equation (2), can be written as Since the evaluation of the Rényi divergences is independent of time t, for convenience we shall take t = t . Thus the family of Rényi divergences can be expressed from the characteristic function of work distribution with complex arguments, The characteristic function of work distribution with real arguments can be measured from the Ramsey interference of a single spin [37][38][39]. To extract the family of Rényi divergences, we need to obtain the characteristic function of work distribution with complex arguments. We are now to demonstrate that the characteristic function of work distribution with complex arguments can be obtained from the characteristic function of work distribution with real arguments. This can be achieved by studying the analytic properties of the characteristic function of work distribution G(u) in the complex plane of its argument u. Our results are based on the following theorems: The imaginary part of the characteristic function is It is straightforward to prove that G R (u) and G I (u) satisfy the Cauchy-Riemann condition (appendix A), namely Thus we proved that G(u) is an analytic function in the complex plane of u. The theorem 1 is proved.  (45) is a complex number on the upper half complex plane while the argument of ( ) G u R on the right hand side of equation (45) is a real number. Equation (45) tells us that G(u) with I > u 0 are determined by ( ) G u R . Thus we proved theorem 2. Because ( ) G u R with real arguments can be measured by Ramsey interference of a single spin [37][38][39], we can The other two segments parallel to the real axis, with imaginary parts being zeros and approaching to +¥ respectively, and the real parts extended from -¥ to +¥. The read arrows in each line label the direction of the integration contour. experimentally measure G(u) with complex arguments from the Ramsey interference measurement according to theorem 2.
Jarzynski equality for quantum system establishes that one can measure the free energy differences from non-equilibrium work measurement.

In terms of characteristic function the left hand side of the above equation is
According to theorem 2, b ( ) G i can be obtained from G(u) with the real arguments. This leads to Corollary 1. The free energy difference for the equilibrium states of the initial Hamiltonian H i and final Hamiltonian H f can be obtained from the characteristic function of work distribution in the non-equilibrium process which begins with the equilibrium state of the the initial Hamiltonian and ends at non-equilibrium state of final Hamiltonian (48) is the exact formula for extracting the free energy difference between the equilibrium states for the initial and final control parameters from the Ramsey interference of a single spin.
According to equation (36) and theorem 2, we have In equations (45), (48) and (49), we only need to measure ( ) For the quantum system which satisfies that R and the period T is determined by the energy difference of the system, then equation (45) can be simplified into Now we show how to measure the characteristic function of work distribution with real arguments from Ramsey interference of a single spin [37][38][39]. We use a probe spin s º ñá -ñá | | | | 1 1 0 0 z to couple the system with the following Hamiltonian s l l = + D + ñá Ä - The procedure for measuring the Ramsey inference of the probe spin is: (1) Initialize the system in the equilibrium state Hamiltonian H i and the probe spin in ñ |0 state; (2) Apply a p 2-pulse along the y direction of the probe spin, which then transform the probe spin in a superposition state ñ + ñ (| | ) 0 1 2; (3) The probe spin and the system evolve together by the Hamiltonian H T for a time interval t = + t u ; R (4) Apply a p 2-pulse to the probe spin along the y direction; (5) Measure s á ñ z and s á ñ y , which gives us that Tr e . 56 Tr e . 57 Thus we can measure the characteristic function of work distribution from Ramsey interference of a single spin. To extract the free energy difference or the family of Rényi divergences we need to know ( ) However in any realistic experiment we can only measure ( ) G u R for a finite time duration, say Î [ ] t T 0, . Then we can use these finite time information of ( ) G u R to deduce the full information of ( ) G u R . The reason for the validity of this method is where the number of fitting parameters is 2D 2 with D being the dimension of the system. This method was used to measure the quantum work distributions [37][38][39]. Note that the Jarzynski equality and the discovery in this work, equation (2), are both exact relations and in principle the equalities only hold for infinitely many trials. But in practice one can only perform finite number of trials in Jarzynski equality. Usually the convergence can be achieved with small number of trials for small system. To achieve convergence for bigger system in Jarzynski equality, more trials are required.

Physical model study
Here we use a specific model to verify our finding, equation (2), the relations between the generating function of work and the family of Rényi divergences. is the excitation spectrum. We now consider a sudden quench process in the 1D quantum Ising model with a transverse field. At t=0, the quantum Ising model is prepared in a thermal equilibrium state with density matrix Tr e H i i . Then we suddenly change the transverse field from l i to l f . The characteristic function of work distribution in this sudden quench process can be exactly calculated as (see appendix B for detailed derivation) With this exact solution we are now ready to verify our results about the generating function of dissipated work and the family of Rényi divergences.
In the sudden quench process, the generating function of work is given by f . The Jarzynski equality for free energy change is 0, 5 2 to get the information of Rényi divergence.
In figure 4(a), we present the real (black squares) and imaginary (black circles) of the characteristic function of the work distribution in a sudden quench of a single spin from H i to H f as described above. The solid red line and solid blue line are respectively the real part and imaginary part of the characteristic function fitted by equation (62). We then make use of the fitted function to calculate the family of Rényi divergences through equation (49), which is marked by the red circles in figure 4(b). The solid red line presents the exact solution of the family of Rényi divergences. We can see that the family of Rényi divergences obtained from the characteristic function of work distributions agree well with the exact solution.

Conclusions
We have established an exact relation which connects a fundamental quantity in thermodynamics, the dissipated work in a system driven arbitrarily far from equilibrium, to a fundamental concept in information theory, the family of Rényi divergences. We find that the generating function of the dissipated work under an arbitrary timedependent driving is related to the family of Rényi-divergences between the forward process and its time reversed process. Since Rényi divergences is a quantitative measure of distinguishability between two states, the Rényi divergence between the forward state and the backward state appears in equation (2) measures the breaking of time reversal symmetry. The full statistics of dissipated work in an arbitrary driving process is determined by the family of Rényi divergences. Moreover, for quantum system, one can extract the family of Rényi divergences from Ramsey interference of a single spin. In this work we studied the case that the system is isolated from the bath in the time-dependent driving process, it would be interesting to study whether the results still hold when the system and bath are coupled in the course of driven. In addition, in the recent formulation of quantum thermodynamics, the family of Rényi divergences plays a central role, it would be intriguing to find the connections of this work and the new formulation of quantum thermodynamics.
In this appendix, we prove that the real part and imaginary part of the characteristic function of quantum work distribution satisfy the Cauchy-Riemann condition. The characteristic function of quantum work distribution is Its complex conjugate is given by The derivatives of the characteristic function with respect to the u R and u I respectively are given by  The derivatives of the complex conjugate of the characteristic function with respect to the u R and u I respectively are  The real part and imaginary part of the characteristic function of work distribution is . A 8 I From equations (A3)-(A8), it is straightforward to check that G R (u) and G I (u) satisfy the Cauchy-Riemann condition, Here periodic boundary condition is applied. The Hamiltonian is invariant under a π rotation along the z axis to all the spins, which is described by where we define a q q = -