# Universal efficiency at optimal work with Bayesian statistics

###### Abstract

If the work per cycle of a quantum heat engine is averaged over an appropriate prior distribution for an external parameter , the work becomes optimal at Curzon-Ahlborn efficiency. More general priors of the form yield optimal work at an efficiency which stays close to CA value, in particular near equilibrium the efficiency scales as one-half of the Carnot value. This feature is analogous to the one recently observed in literature for certain models of finite-time thermodynamics. Further, the use of Bayes’ theorem implies that the work estimated with posterior probabilities also bears close analogy with the classical formula. These findings suggest that the notion of prior information can be used to reveal thermodynamic features in quantum systems, thus pointing to a new connection between thermodynamic behavior and the concept of information.

###### pacs:

05.70.-a, 03.65.-w, 05.70.Ln, 02.50.CwThe connection between thermodynamics and the concept of information is one of the most subtle analogies in our physical theories. It has played a central role in the exorcism of Maxwell’s demon Leffbook (). It is also crucial to how we may understand and exploit quantum information Vedral2008 (); Ueda2009 (); Jacobs2008 (). To make nanodevices Joachim2000 (); Serreli2007 () that are functional and useful, we need to understand their performance with regard to heat dissipation and optimal information processing. To model such systems, standard thermodynamic processes and heat cycles have been generalised using quantum systems as the working media Hastapoulus76 ()-AJM2008 (). It is well accepted that the maximal efficiency, , where is the hot (cold) bath temperature, is only obtained by a reversible heat engine which however involves infinitely slow processes. For heat cycles running in a finite time, the concept of power output becomes meaningful. Curzon and Ahlborn Curzon1975 (), first of all displayed an elegant formula in the so called endoreversible approximation for efficiency at maximum power, . The appearance of an optimal efficiency in different models with a value close to Curzon-Ahlborn (CA) value, has raised the issue of its universality that has captured the imagination of workers in this area since many years Leff (). More recently AJM2008 (),Broeck2005 ()-Segal2010 (), a universal form for optimal efficiency, has been discussed within finite-time thermodynamics in the near-equilibrium regime (small value of ) as given by .

On the other hand, in recent years, Bayesian methods of statistical inference have gained popularity in physics Dose (). In Bayesian probability theory, the central role is played by the concept of prior information. It represents state of our knowledge about a system before any experimental data is acquired. The assignment of a unique distribution to a given prior information is a non-trivial issue but may be argued on the basis of maximum entropy principle and certain requirements of invariance Jaynes1968 (); Jaynes2003 (). Using Bayes’ theorem Jeffreys1939 () one can then update the prior probabilities based on the new information gathered from the data. Recently, Bayesian methods have been applied to modular structure of networks Hofman2008 (), inference of density of states Habeck2007 (), the interpretation of quantum probabilities Caves2002 () and other inverse problems Lemm2000 ().

In this letter, a Bayesian approach is used to show that the efficiency at optimal work for a quantum heat engine is related to CA value, after the work per cycle is averaged over the prior distribution of an external parameter. In contrast to the finite-time models Esposito2009 (), the heat cycle considered here is performed infinitely slowly. The application of Bayes’ theorem gives the optimal efficiency exactly at CA value for a whole class of priors and for arbitrary bath temperatures. The present analysis thus provides a novel argument for the emergence of thermodynamic behavior in quantum heat engines from a Bayesian perspective.

As a model of a heat engine, consider a quantum system with Hamiltonian , where energy eigenvalues . The factor depends on the energy level as well as other fixed parameters/constants of the system; is a controllable external parameter equivalent to say, the applied magnetic field for a spin-1/2 system. Other examples of this class are 1-d quantum harmonic oscillator ( equivalent to frequency) and a particle in 1-d box ( inversely proportional to the square of box-width). Initially, the quantum system is in thermal state at temperature with its eigenvalues given by the canonical probabilities . The quantum analogue of a classical Otto cycle between two heat baths at temperatures and involves the following steps Kieu2004 (): (i) the system is detached from the hot bath and made to undergo the first quantum adiabatic process, during which the system hamiltonian changes to , where , without any transitions between the levels and so the system continues to occupy its initial state. For , this process is the analogue of an adiabatic expansion. The work done by the system in this stage is defined as the change in mean energy ; (ii) the system with modified energy spectrum is brought in thermal contact with the cold bath and it achieves a thermal state . The modified canonical probabilities now correspond to temperature . On average, heat rejected to the bath in this stage is defined as ; (iii) the system is now detached from the cold bath and made to undergo a second quantum adiabatic process (compression) during which the hamiltonian changes back to . Work done on the system in this stage is ; (iv) finally, the system is brought in thermal contact with the hot bath again. Heat is absorbed by the system in this stage whence it recovers its initial state and its temperature attains back the value . The total work done on average in a cycle is calculated to be

(1) | |||||

(2) |

Similarly, heat exchanged with hot bath in stage (iv) is given by Heat exchanged by the system with the cold bath is . The efficiency of the engine , is given by

(3) |

For convenience, we express , using Eq. (3). Consider an ensemble of such systems where now the value of parameter may vary from system to system. If the ensemble corresponds to an actual preparation according to a certain probability distribution , then the state of the system can be expressed as . Each system in the ensemble is made to perform the quantum heat cycle described above, with a fixed efficiency . We wish to study the optimal characteristics of the average work, in particular the efficiency at which the work becomes optimal. Clearly, choice of the probability distribution is expected to play a significant role in the conclusions. In the following, we analyse this problem by choosing a distribution according to the prior information available and show that the efficiency at optimal work is closely associated with CA value.

For simplicity, we now consider a two-level system as our working medium, with and , so that the initial energy levels are and . The work over a cycle in this case is

(4) |

where Boltzmann’s constant is put equal to unity. The average work with the initial state for a given , can be expressed as

(5) |

A central issue in Bayesian probability is to assign a unique prior distribution corresponding to a given prior information. If the only prior information about the continuous parameter is that it takes positive real values but otherwise we have complete ignorance about it, then Jeffreys has suggested the prior distribution Jeffreys (); Jaynes1968 (), or in a finite range, , where and are the minimal and the maximal energy splitting achievable for the two-level system. For the above choice, we obtain

(6) |

It can be seen that the average work vanishes for and . In between these values of , the average work exhibits a maximum. We look for the efficiency at which this work becomes maximal for the given range , by imposing the condition . Here we consider the limit of which gives

(7) |

The solution of this equation has been plotted against in Fig. 1. Interestingly, in the asymptotic limit of , the above expression reduces to

(8) |

which yields the efficiency at optimal work as , exactly the CA value. More significantly, the conclusion also holds in general i.e. for a working system with spectrum and with Jeffreys’ prior. It is to be noted that in the asymptotic limits, the expression for average work (Eq. (6)) diverges. However, the limits are taken after the derivative of work is set equal to zero in order to obtain well-defined expressions for the efficiency.

It is conceivable that other choices of the prior may yield similar results. To study consequences of deviations from the above choice, we consider a class of prior distributions, , defined in the range , where and . Upon optimisation of the average work as defined in Eq. (5) over , we get

(9) |

In the limit becoming very large, the above integrals can be evaluated using the standard results stint (). Then the above equation is simplified to

(10) |

where . Now as , the above equation reduces to Eq. (8) and so CA value is also a limiting value for this model. Interestingly, even for other allowed values of , the solution of Eq. (10) depends only on the ratio , apart from the parameter . In particular, Laplace and Bayes have advocated a uniform prior to quantify the state of complete ignorance. For this case, we set . Then the above equation becomes , which has only one real solution given as

(11) |

This solution along with other numerical solutions of (10) for general are shown in Fig. 2. Remarkably, these curves stay very close to the CA value. However, at this point it is not possible to say in general what prior information may be quantified by the parameter . The curves in Fig. 2 are also closely similar to those observed in finite-time models at optimal power Esposito2009 (). It is seen here that in the near-equilibrium regime, all the curves merge into each other and approach the CA value which is approximately in this limit. This can be shown as follows: taking to be close to unity in the near-equilibrium case, is close to zero. The efficiency being bounded from above by the Carnot value is thus small too. On using these facts in the expansion of Eq. (10), we get

(12) |

Thus we recover the linear term mentioned earlier. For general values of , the CA value is a lower bound for the efficiency at optimal work when gammaamin ().

So far we have observed that the use of Jeffreys’ prior implies that efficiency at optimal work approaches CA value for arbitrary bath temperatures. Further, the efficiency also approaches a universal form for a class of priors, for nearly equal bath temperatures. In the following, we show that application of Bayes’ theorem can restore the efficiency back to the exact CA value even for the latter choice. Bayes’ theorem gives a prescription to convert the prior probabilities into posterior probabilities. Note that during the first quantum adiabatic process on the two-level system, the energy levels change from to , but the system continues to occupy its initial state. The respective occupation probabilities are now interpreted as conditional probabilities, given by and . If the system is found in the up () state, the work done in this step is and the posterior probabilities are given by

(13) |

The average work for this process is now given by . On the other hand, if the system is found in the down () state, the work is zero. Similarly, for the second quantum adiabatic process, the work performed can be either or 0 and the avarage work for that process can be similarly calculated using the respective posterior probabilities. Now choosing the prior and with the system being in up state, the average work for the total cycle , in the limit of large is given by

(14) |

where . So using posterior probabilities, a well defined expression for average work is obtained even if the prior is non-normalisable in the asymptotic limit. More generally, given the value of external parameter and assuming canonical probabilities to find the system in th state, we infer the probability about the value of , if the system is actually found in the th state. Remarkably, the work given by eq. (14) attains optimal value exactly at the CA efficiency, regardless of the value of in the prior. Furthermore, the average work shows the same dependence on efficiency as found for the classical Otto cycle in Leff ().

In conclusion, we have argued the emergence of CA value as the efficiency at optimal work in quantum heat engines within a Bayesian framework. This effect of incorporating Bayesian probabilities leading to classical thermodynamic behavior in quantum systems has not been addressed before and may shed new light on the connection between information and thermodynamics. Due to current interest in small scale engines, the observation of similar curves (Fig. 2) as obtained in some recently proposed models of these engines, points to an interesting link between finite time models and our model based on the idea of prior information. Addressing these issues would hopefully lead to a broader perspective on the performance characteristics of small engines and also help to understand the limits of their performance based on principles of information.

## Acknowledements

The author expresses his sincere thanks to Arvind, Pranaw Rungta and Lingaraj Sahu for interest in the work and useful discussions.

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