An Analog of the 2-Wasserstein Metric in Non-Commutative Probability Under Which the Fermionic Fokker-Planck Equation is Gradient Flow for the Entropy
An Analog of the 2-Wasserstein Metric in Non-Commutative Probability Under Which the Fermionic Fokker-Planck Equation is Gradient Flow for the Entropy
复制标题
DOI：
10.1007/s00220-014-2124-8
复制
发表时间：
2014-11-01
影响因子：
2.4
通讯作者：
Maas, Jan
中科院分区：
文献类型：
--
作者：
Carlen, Eric A.;Maas, Jan
设Clifford代数over，它是由n个自伴随算子Q (j), j = 1，一个欧式破碎的垂直棒，n满足正则反对易关系，Q (i) Q (j) + Q (j) Q (i) = 2 (ij) i生成的von Neumann代数，设tau表示归一化迹on。在量子力学中，这种代数是由n个费米子自由度产生的可观测值的代数。表示所有正算子的集合使tau(rho) = 1；这些是非交换概率空间中概率密度的非交换类似物。费米子Fokker-Planck方程是经典Fokker-Planck方程的量子力学模拟，它与经典Fokker-Planck方程有很多共同之处，比如相同的最佳超收缩性。在本文中，我们构造了一个黎曼度规，我们证明了它是经典2-瓦瑟斯坦度规的自然类比，我们证明了，与经典情况类似，费米子的福克-普朗克方程是这个度规中相对熵相对于基态的梯度流。我们推导了一些结果，例如这个度规的尖锐Talagrand不等式，并证明了一些与这个度规有关的结果。提出了几个悬而未决的问题。
Let denote the Clifford algebra over , which is the von Neumann algebra generated by n self-adjoint operators Q (j) , j = 1,aEuro broken vertical bar,n satisfying the canonical anticommutation relations, Q (i) Q (j) + Q (j) Q (i) = 2 delta (ij) I, and let tau denote the normalized trace on . This algebra arises in quantum mechanics as the algebra of observables generated by n fermionic degrees of freedom. Let denote the set of all positive operators such that tau(rho) = 1; these are the non-commutative analogs of probability densities in the non-commutative probability space . The fermionic Fokker-Planck equation is a quantum-mechanical analog of the classical Fokker-Planck equation with which it has much in common, such as the same optimal hypercontractivity properties. In this paper we construct a Riemannian metric on that we show to be a natural analog of the classical 2-Wasserstein metric, and we show that, in analogy with the classical case, the fermionic Fokker-Planck equation is gradient flow in this metric for the relative entropy with respect to the ground state. We derive a number of consequences of this, such as a sharp Talagrand inequality for this metric, and we prove a number of results pertaining to this metric. Several open problems are raised.