Optimal protocols for slowly-driven quantum processes
The design of efficient quantum information processing will rely on optimal nonequilibrium transitions of driven quantum systems. Building on a recently-developed geometric framework for computing optimal protocols for classical systems driven in finite-time, we construct a general framework for optimizing the average information entropy for driven quantum systems. Geodesics on the parameter manifold endowed with a positive semi-definite metric correspond to protocols that minimize the average information entropy production in finite-time. We use this framework to explicitly compute the optimal entropy production for a simple two-state quantum system coupled to a heat bath of bosonic oscillators, which has applications to quantum annealing.
pacs:05.70.Ln, 02.40.–k, 05.40.–a, 02.50.Ga, 03.67.–a
Though a unifying set of principles describing all known nonequilibrium phenomena remains undiscovered, many recent developments have illuminated the thermodynamic behavior of small-scale systems. For instance, fluctuation theorems valid far from equilibrium have been developed in the classical setting Evans et al. (1993); Evans and Searles (1994); Gallavotti and Cohen (1995); Crooks (1999); Hatano and Sasa (2001); Jarzynski (1997); Liphardt et al. (2002); Seifert (2005); Sagawa and Ueda (2010); Wang et al. (2002); Carberry et al. (2004); Garnier and Ciliberto (2005); Toyabe et al. (2010) as well as in the quantum regime Deffner and Lutz (2011); Fei and Zhong-Can (2014); Leggio et al. (2013); Chetrite and Mallick (2012); Albash et al. (2013); Crooks (2008); Campisi et al. (2011, 2009); Talkner et al. (2009).
An area of nonequilibrium thermodynamics of particular interest concerns the operation of small-scale information processing systems. The interplay between information as a physical quantity and thermodynamics has a rich history Szilard (1929); Landauer (1961); Bennett (1982).
The physics of information processing is of particular relevance considering the rapid development of information technology and the inevitable approach to computational limits imposed by physical law Frank (2002); Lambson et al. (2011). Optimization schemes for small-scale information processing occurring in finite time will be needed to develop technology capable of approaching those limits Andresen (2011); Chen and Sun (2004).
Current research has uncovered techniques to optimize thermodynamic quantities arising in small-scale systems designed to store and erase classical information Diana et al. (2013); Esposito et al. (2010); Aurell et al. (2012); Zulkowski and DeWeese (2014), including the derivation of a refined second law Aurell et al. (2012). This research couples with the progress made on the general problem of predicting optimal protocols to drive classical systems between stationary states with minimal dissipation Sivak and Crooks (2012); Zulkowski et al. (2012, 2013); Zulkowski and DeWeese (2014); Shenfeld et al. (2009); Brody and Hook (2009); Gomez-Marin et al. (2008); Schmiedl and Seifert (2007); Aurell et al. (2011).
In parallel with classical developments, a greater understanding of optimal processes in the nonequilibrium quantum regime and the efficiency of quantum engines has been achieved Abah and Lutz (2014); Mehta and Polkovnikov (2013); Deffner (2014); Stefanatos (2014); Schmiedl et al. (2009). The success of a recently-proposed linear-response framework for optimal driving of microscopic classical systems Sivak and Crooks (2012) calls out for an extension to quantum systems.
In the geometric formulation of Sivak and Crooks (2012), a generalized inverse diffusion tensor induces a Riemannian manifold structure on the space of parameters, and optimal protocols trace out geodesics of this inverse diffusion tensor. This geometric framework is subsequently developed and exploited in Zulkowski et al. (2012, 2013); Zulkowski and DeWeese (2014).
In this paper, we extend this work to provide a geometric framework for computing control protocols optimizing the average information entropy production Deffner and Lutz (2011). The production of entropy is intimately related to the overall performance of thermodynamic devices by in some sense quantifying irreversibility and providing a bound on the availability of useful work. Entropy production also provides a useful tool in the analysis of nonequilibrium effects.
By twice measuring the density operator of a system interacting with a thermal reservoir at the beginning and end of the protocol, the average information entropy production may be defined and shown to satisfy a fluctuation theorem Deffner and Lutz (2011). While there is still debate about the best way to define thermodynamic quantities along quantum trajectories, this approach provides an avenue for experimental observation Deffner and Lutz (2011). Fortunately, this formalism holds for open quantum systems driven arbitrarily far from equilibrium.
We begin by constructing a general positive semi-definite tensor on the space of control parameters for the quantum system interacting with the thermal reservoir. We assume the dynamics of the system are described by a master equation of Lindblad form, arising from an adiabatic, rotating-wave approximation in a sense we make concrete below Albash et al. (2012).
With the general tensor in hand, we compute optimal protocols for a simple two-state system coupled to a thermal bath of bosonic oscillators. The system Hamiltonian may be interpreted as describing a spin- particle coupled to a time-dependent magnetic field with components in the and -directions. This system has applications in quantum annealing Albash et al. (2012); Kadowaki and Nishimori (1998). For this simple system, we demonstrate the existence of null directions of the metric tensor, which correspond to directions in parameter space in which there is no overall change in information entropy. We derive an approximate expression for the optimal overall entropy production.
Ii The quantum tensor
Our model consists of two distinct components: the system under our control and a large collection of quantum degrees of freedom acting as the thermal reservoir. Together, the system and the bath degrees of freedom evolve unitarily according to the von Neumann equation , where
The operator is the system Hamiltonian acting in the Hilbert space of system states while is the Hamiltonian for the bath degrees of freedom. The interaction term consists of a (weak) coupling and Hermitian operators and operating on the system and bath Hilbert spaces respectively.
We are only interested in the time evolution of the density operator of the system, denoted , which can be obtained from by tracing over the bath degrees of freedom. We follow the construction of Albash et al. (2012), which obtains a quantum Markovian master equation governing via an adiabatic, rotating-wave approximation. Specifically, the derivation of Eq. (II) in Albash et al. (2012) utilizes the so-called “standard adiabatic approximation”
where denotes the total evolution time, is defined to be the minimum ground state energy gap of , and
Here, is a dimensionless measure of time. Though it is possible to compute higher-order terms in Albash et al. (2012), we will assume that Eq. (II) adequately approximates the time-evolution of the quantum system over finite, but sufficiently long time scales.
Assuming a weak coupling between the system and bath degrees of freedom, we have a master equation in Lindblad form:
Here, consists of the system Hamiltonian and the Lamb shift Hamiltonian, which arises through the coupling of the system with the thermal reservoir.
Assume that the system Hamiltonian has time-dependent eigenvalues with time-dependent eigenkets . Then the operators are defined by
where both and are Hermitian and can be computed from the spectral-density matrix Albash et al. (2012). To ease notation we will suppress the time dependence of the Lindblad operators .
It is crucial to note that the time dependence of the terms defining arises only through the time dependence of the spectrum and eigenkets of the system Hamiltonian. Therefore, the time dependence of the Lindblad operator stems from the control parameters . If time appeared explicitly in the terms defining , we could not interpret the approximation developed in this section as giving rise to a semi-definite metric on the space of control parameters.
In what follows, we denote the control parameter protocol by . We assume the protocol to be sufficiently smooth to be twice-differentiable. The framework in the classical setting is versatile and can handle situations in which jump discontinuities are present Zulkowski and DeWeese (2014). Jump discontinuities at the end points of the protocol commonly arise in optimal finite-time driving processes of classical systems Schmiedl and Seifert (2007); Gomez-Marin et al. (2008); Aurell et al. (2011); Seifert (2012); Esposito et al. (2010); Aurell et al. (2012). For simplicity we only admit twice-differentiable protocols though one could in principle extend this approach to piecewise-continuous .
By definition Deffner and Lutz (2011), the average information entropy is given by
is the equilibrium distribution defined by the instantaneous control parameters and is related to the thermal bath temperature by where is Boltzmann’s constant. Using Eq. (II), we see
where represents the duration of the protocol. This equation follows from the cyclic property of the trace:
The trace term involving and the commutator vanishes because we may permute into the commutator with . As for the term, it too can be permuted into the commutator and , naturally. A little more work goes into showing that . Since by definition, it is sufficient to show that .
From Eq. (7), we need only establish that . First, note that
where we have suppressed the time dependence for brevity. This further implies that
establishing our claim.
We see from Eq. (II) that the average information entropy is proportional to . This seems reasonable since if , then there would be no coupling between the system and the bath. The system would then evolve in time unitarily and the average entropy would vanish.
This observation allows us to drastically simplify the mathematics since the expressions inside the integral need only be kept to -th order in . In other words, we may compute the evolution of using only the von Neumann equation
We wish to approximate Eq (II) when the protocol duration is large in the sense of Eq. (2). To achieve this end we utilize the so-called derivative truncation method Zulkowski et al. (2012, 2013); Zulkowski and DeWeese (2014), which assumes a specific form for the density operator in terms of the equilibrium system density operator and the first order derivative of the protocol :
The Einstein summation convention is assumed here and throughout for the index and the operator is Hermitian and traceless.
Substituting Eq. (15) into the von Neumann equation Eq. (14) and ignoring derivative terms of order higher than first, we obtain equations for the unknown operators where indexes the finite set of control parameters.
These equations are most conveniently expressed in terms of the operator basis where is an eigenket of with eigenvalue . The convenience of this choice arises in the time independence of the eigenvalues , which can be illustrated by the following argument:
As evolves according to Eq. (14), we must have
for all positive integers . Since the coefficients of the characteristic polynomial of can be expressed in terms of combinations of traces of powers of , it follows that the spectrum is time independent; i.e.; .
It follows immediately that
by the Spectral Theorem. Using the eigenket basis affords us a simple expression for : we need only compute the time-derivative of the projection operators since the eigenvalues are time-independent. However, the (time-dependent) energy eigenkets are more convenient for practical applications and so we express all relevant quantities in terms of this basis.
Using the derivative truncation approximation, we can deduce the approximate eigenkets of in terms of the energy eigenkets:
Furthermore, . This follows immediate from the approximation Eq. (15) and the definition of eigenkets.
The construction of the approximate eigenkets and eigenvalues forces the diagonal entries of in the energy eigenket basis to vanish. This is consistent with the requirement of positivity as both and are positive in Eq. (15), but the term involving is sensitive to the rate of change of which could be negative.
and so we obtain the expansion of in the energy eigenket operator basis:
with and .
Furthermore, from Eq. (II) we have
By constructing an explicit expression for we may now construct a quadratic functional approximating :
which follows from .
We compute the trace in the basis in which is diagonal. Since , we have
From this follows a useful expression for the entropy production
after symmetrization in the and indices. We may write this expression as
where the components of the matrix depend on and the energy eigenvalues and eigenkets. We include the explicit components of in the appendix.
Eq. (II) approximates the average information entropy produced during a finite-time driving protocol of a quantum system weakly coupled to a large thermal bath using only quantities directly calculable from the time-dependent system Hamiltonian and the Lindblad operators.
We will now use Eq. (II) to explicitly compute the average information entropy produced by driving a simple two-state model quantum system.
Iii Two-state model system
We apply Eq. (II) to a simple two-state system with system Hamiltonian
We further assume that the system is coupled to a thermal bath of bosonic oscillators so that the full Hamiltonian is
where , and is a small coupling constant.
It is convenient to work in an eigenbasis of represented by the column vectors
In this basis,
It is most convenient to express the control parameters in polar form: . In this form, it is not hard to show that the eigenvalues of are with eigenvectors
For simplicity, we assume and . A straightforward calculation demonstrates that
are the operators defining for this model system.
Choosing the bath oscillator frequencies so that and for and setting , we have from Eq. (II)
While this functional is non-negative as expected from the fluctuation theorem of Deffner and Lutz (2011), we see that it vanishes if is held constant. If is allowed to vary while is fixed, no information entropy is generated on average. In terms of Riemannian geometry, this means that the metric tensor possesses null directions.
We apply the Euler-Lagrange equation to obtain the optimal entropy production:
Upon integration, we find
Therefore, the optimal average entropy is given by
If we choose , then
Eq. (III) exhibits the expected behavior of the optimal average entropy production. Fig. 1 illustrates the dependence of on the constants and . We see that has a profile described by . The quantities and are related to the bosonic frequencies of the thermal bath Albash et al. (2012) and are thereby related to the noise of the quantum system. The bosonic frequencies consequently have a relatively simple influence on the overall average entropy production in the finite-time long duration limit via Eq. (III).
Using the formalism of Sivak and Crooks (2012); Zulkowski et al. (2012, 2013); Zulkowski and DeWeese (2014), developed for classical systems, we were able to construct a general approximation of the average information entropy of a quantum system driven in finite-time in terms of a quadratic functional of velocities in parameter space. This functional can be interpreted as endowing parameter space with a semi-definite metric in which optimal protocols are equivalent to geodesics. For a simple two-state driven quantum system weakly coupled to a thermal bosonic bath, we were able to derive an approximate expression for the average information entropy. This expression has the characteristic dependence with a coefficient compactly expressed in terms of quantities related to the bosonic bath oscillator frequencies.
Interestingly, in the simple two-state example studied here, the quadratic functional approximation for we derived possesses null directions when expressed in terms of metric geometry on the space of control parameters. In fact, for this model system, changing results in a shift of energies. If alone is changed, then it turns out that the system density matrix evolves exactly without a change in average information entropy. Consequently, our solution contains a null direction, which makes the quantum tensor not positive definite but positive semi-definite. It seems likely that this behavior is generic, though a proof is lacking.
It is gratifying that this framework allowed us to obtain a general expression for the approximate entropy production in driven non-equilibrium quantum systems, as well as a closed-form solution for the minimum entropy production possible for a specific system with relevance to quantum annealing. We are encouraged by the success of this approach for this simple system and we hope that this program will lead to further insight into the optimization of quantum systems out of equilibrium.
M.R.D. gratefully acknowledges support from the McKnight Foundation and the Hellman Family Faculty Fund. M.R.D. and P.R.Z. were partly supported by the National Science Foundation through Grant No. IIS-1219199. This material is based upon work supported in part by the US Army Research Laboratory and the US Army Research Office under Contract No. W911NF-13-1-0390.
We record here the components of appearing in Eq. (25) for the convenience of the reader.
- Evans et al. (1993) D. J. Evans, E. G. D. Cohen, and G. P. Morriss, Phys. Rev. Lett. 71, 2401 (1993).
- Evans and Searles (1994) D. J. Evans and D. J. Searles, Phys. Rev. E 50, 1645 (1994).
- Gallavotti and Cohen (1995) G. Gallavotti and E. G. D. Cohen, Phys. Rev. Lett. 74, 2694 (1995).
- Crooks (1999) G. E. Crooks, Phys. Rev. E 60, 2721 (1999).
- Hatano and Sasa (2001) T. Hatano and S. Sasa, Phys. Rev. Lett. 86, 3463 (2001).
- Jarzynski (1997) C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997).
- Liphardt et al. (2002) J. T. Liphardt, S. Dumont, S. B. Smith, I. Tinoco Jr, and C. Bustamante, Science 296, 1832 (2002).
- Seifert (2005) U. Seifert, Phys. Rev. Lett. 95, 040602 (2005).
- Sagawa and Ueda (2010) T. Sagawa and M. Ueda, Phys. Rev. Lett. 104, 090602 (2010).
- Wang et al. (2002) G. M. Wang, E. M. Sevick, E. Mittag, D. J. Searles, and D. J. Evans, Phys. Rev. Lett. 89, 050601 (2002).
- Carberry et al. (2004) D. M. Carberry, J. C. Reid, G. M. Wang, E. M. Sevick, D. J. Searles, and D. J. Evans, Phys. Rev. Lett. 92, 140601 (2004).
- Garnier and Ciliberto (2005) N. Garnier and S. Ciliberto, Phys. Rev. E 71, 060101 (2005).
- Toyabe et al. (2010) S. Toyabe, T. Sagawa, M. Ueda, E. Muneyuki, and M. Sano, Nat. Phys. 6, 988 (2010).
- Deffner and Lutz (2011) S. Deffner and E. Lutz, Phys. Rev. Lett. 107, 140404 (5 pp.) (2011).
- Fei and Zhong-Can (2014) L. Fei and O. Zhong-Can, Chin. Phys. B 23 (2014).
- Leggio et al. (2013) B. Leggio, A. Napoli, H.-P. Breuer, and A. Messina, Phys. Rev. E 87 (2013).
- Chetrite and Mallick (2012) R. Chetrite and K. Mallick, J. Stat. Phys. 148 (2012).
- Albash et al. (2013) T. Albash, D. A. Lidar, M. Marvian, and P. Zanardi, Phys. Rev. E 88 (2013).
- Crooks (2008) G. Crooks, J. Stat. Mech.:Theor. Exp. 2008 (2008).
- Campisi et al. (2011) M. Campisi, P. Haenggi, and P. Talkner, Rev. Mod. Phys. 83 (2011).
- Campisi et al. (2009) M. Campisi, P. Talkner, and P. Haenggi, Phys. Rev. Lett. 102 (2009).
- Talkner et al. (2009) P. Talkner, M. Campisi, and P. Haenggi, J. Stat. Mech.:Theor. Exp. (2009).
- Szilard (1929) L. Szilard, Z. Phys. 53, 840 (1929).
- Landauer (1961) R. Landauer, IBM J. Res. Dev. 5, 183 (1961).
- Bennett (1982) C. Bennett, Int. J. Theor. Phys. 21, 905 (1982).
- Frank (2002) M. Frank, Comput. Sci. Eng. 4, 16 (2002).
- Lambson et al. (2011) B. Lambson, D. Carlton, and J. Bokor, Phys. Rev. Lett. 107, 010604 (2011).
- Andresen (2011) B. Andresen, Angew. Chem. Int. Ed. 50, 2690 (2011).
- Chen and Sun (2004) L. Chen and F. Sun, eds., Advances in Finite Time Thermodynamics: Analysis and Optimization, (Nova Science, New York, 2004).
- Diana et al. (2013) G. Diana, G. B. Bagci, and M. Esposito, Phys. Rev. E 87, 012111 (2013).
- Esposito et al. (2010) M. Esposito, R. Kawai, K. Lindenberg, and C. Van den Broeck, Europhys. Lett. 89, 20003 (2010).
- Aurell et al. (2012) E. Aurell, K. Gawedzki, C. Mejia-Monasterio, R. Mohayaee, and P. Muratore-Ginanneschi, J. Stat. Phys. 147, 487 (2012).
- Zulkowski and DeWeese (2014) P. R. Zulkowski and M. R. DeWeese, Phys. Rev. E 89, 052140 (2014).
- Sivak and Crooks (2012) D. A. Sivak and G. E. Crooks, Phys. Rev. Lett. 108, 190602 (2012).
- Zulkowski et al. (2012) P. R. Zulkowski, D. A. Sivak, G. E. Crooks, and M. R. DeWeese, Phys. Rev. E 86, 041148 (2012).
- Zulkowski et al. (2013) P. R. Zulkowski, D. A. Sivak, and M. R. DeWeese, PLoS ONE 8, e82754 (2013).
- Shenfeld et al. (2009) D. K. Shenfeld, H. Xu, M. P. Eastwood, R. O. Dror, and D. E. Shaw, Phys. Rev. E 80, 046705 (2009).
- Brody and Hook (2009) D. C. Brody and D. W. Hook, J. Phys. A 42, 023001 (2009).
- Gomez-Marin et al. (2008) A. Gomez-Marin, T. Schmiedl, and U. Seifert, J. Chem. Phys. 129, 024114 (2008).
- Schmiedl and Seifert (2007) T. Schmiedl and U. Seifert, Phys. Rev. Lett. 98, 108301 (2007).
- Aurell et al. (2011) E. Aurell, C. Mejía-Monasterio, and P. Muratore-Ginanneschi, Phys. Rev. Lett. 106, 250601 (4) (2011).
- Abah and Lutz (2014) O. Abah and E. Lutz, EPL 106 (2014).
- Mehta and Polkovnikov (2013) P. Mehta and A. Polkovnikov, Annals of Physics 332, 110 (2013).
- Deffner (2014) S. Deffner, J. Phys. B 47 (2014).
- Stefanatos (2014) D. Stefanatos, Phys. Rev. E 90 (2014).
- Schmiedl et al. (2009) T. Schmiedl, E. Dieterich, P.-S. Dieterich, and U. Seifert, J. Stat. Mech.:Theor. Exp. (2009).
- Albash et al. (2012) T. Albash, S. Boixo, D. A. Lidar, and P. Zanardi, New Journal of Physics 14, 123016 (2012).
- Kadowaki and Nishimori (1998) T. Kadowaki and H. Nishimori, Phys. Rev. E 58, 5355 (1998).
- Seifert (2012) U. Seifert, Reports on Progress in Physics 75, 126001 (2012).