Positive contraction mappings for classical and quantum Schrödinger systems
Positive contraction mappings for classical and quantum Schrödinger systems
复制标题
经典和量子薛定谔系统的正收缩映射
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
M. Pavon
中科院分区：
文献类型：
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作者：
T. Georgiou;M. Pavon
经典的薛定谔桥在路径空间中为扩散过程寻找最可能的概率定律，该定律在时间上的两个端点处匹配边缘;可能性由所寻求的定律和先验之间的相对熵量化。Jamison证明了新的定律是通过对先验进行乘法函数变换得到的。这种变换的特点是在端点概率测度的空间上的自同构，这已经被Fortet，Beurling和其他人研究过。对于在离散时间和空间中演化的过程以及在非交换概率空间中定义的过程，也可以提出类似的问题。本文件建立在早期的工作Pavon和Ticozzi，并开始建立解决方案，薛定谔系统的马尔可夫链。我们的方法是基于希尔伯特度量，并表明薛定谔桥的解决方案是由压缩映射的不动点。我们以类似的方式处理……的转向。
The classical Schrodinger bridge seeks the most likely probability law for a diffusion process, in path space, that matches marginals at two end points in time; the likelihood is quantified by the relative entropy between the sought law and a prior. Jamison proved that the new law is obtained through a multiplicative functional transformation of the prior. This transformation is characterised by an automorphism on the space of endpoints probability measures, which has been studied by Fortet, Beurling, and others. A similar question can be raised for processes evolving in a discrete time and space as well as for processes defined over non-commutative probability spaces. The present paper builds on earlier work by Pavon and Ticozzi and begins by establishing solutions to Schrodinger systems for Markov chains. Our approach is based on the Hilbert metric and shows that the solution to the Schrodinger bridge is provided by the fixed point of a contractive map. We approach, in a similar manner, the steering of ...