Watanabe Gentaro, Venkatesh B Prasanna, Talkner Peter, Campisi Michele, Hänggi Peter
Asia Pacific Center for Theoretical Physics (APCTP), San 31, Hyoja-dong, Nam-gu, Pohang, Gyeongbuk 790-784, Korea and Department of Physics, POSTECH, San 31, Hyoja-dong, Nam-gu, Pohang, Gyeongbuk 790-784, Korea.
Asia Pacific Center for Theoretical Physics (APCTP), San 31, Hyoja-dong, Nam-gu, Pohang, Gyeongbuk 790-784, Korea.
Phys Rev E Stat Nonlin Soft Matter Phys. 2014 Mar;89(3):032114. doi: 10.1103/PhysRevE.89.032114. Epub 2014 Mar 12.
Generalized measurements of an observable performed on a quantum system during a force protocol are investigated and conditions that guarantee the validity of the Jarzynski equality and the Crooks relation are formulated. In agreement with previous studies by M. Campisi, P. Talkner, and P. Hänggi [Phys. Rev. Lett. 105, 140601 (2010); Phys. Rev. E 83, 041114 (2011)], we find that these fluctuation relations are satisfied for projective measurements; however, for generalized measurements special conditions on the operators determining the measurements need to be met. For the Jarzynski equality to hold, the measurement operators of the forward protocol must be normalized in a particular way. The Crooks relation additionally entails that the backward and forward measurement operators depend on each other. Yet, quite some freedom is left as to how the two sets of operators are interrelated. This ambiguity is removed if one considers selective measurements, which are specified by a joint probability density function of work and measurement results of the considered observable. We find that the respective forward and backward joint probabilities satisfy the Crooks relation only if the measurement operators of the forward and backward protocols are the time-reversed adjoints of each other. In this case, the work probability density function conditioned on the measurement result satisfies a modified Crooks relation. The modification appears as a protocol-dependent factor that can be expressed by the information gained by the measurements during the forward and backward protocols. Finally, detailed fluctuation theorems with an arbitrary number of intervening measurements are obtained.
研究了在力协议期间对量子系统执行的可观测量的广义测量，并制定了保证Jarzynski等式和Crooks关系有效性的条件。与M. Campisi、P. Talkner和P. Hänggi之前的研究一致[《物理评论快报》105, 140601 (2010); 《物理评论E》83, 041114 (2011)]，我们发现这些涨落关系对于投影测量是满足的；然而，对于广义测量，需要满足关于确定测量的算符的特殊条件。为了使Jarzynski等式成立，正向协议的测量算符必须以特定方式归一化。Crooks关系还要求反向和正向测量算符相互依赖。然而，关于这两组算符如何相互关联仍有相当大的自由度。如果考虑选择性测量，这种模糊性就会消除，选择性测量由所考虑可观测量的功和测量结果的联合概率密度函数指定。我们发现，只有当正向和反向协议的测量算符彼此是时间反转的伴随算符时，相应的正向和反向联合概率才满足Crooks关系。在这种情况下，以测量结果为条件的功概率密度函数满足一个修正的Crooks关系。这种修正表现为一个依赖于协议的因子，它可以由正向和反向协议期间测量获得的信息来表示。最后，得到了具有任意数量中间测量的详细涨落定理。
