We study a generalization of relative submajorization that compares pairs of positive operators on representation spaces of some fixed group. A pair equivariantly relatively submajorizes another if there is an equivariant subnormalized channel that takes the components of the first pair to a pair satisfying similar positivity constraints as in the definition of relative submajorization. In the context of the resource theory approach to thermodynamics, this generalization allows one to study transformations by Gibbs-preserving maps that are in addition time-translation symmetric. We find a sufficient condition for the existence of catalytic transformations and a characterization of an asymptotic relaxation of the relation. For classical and certain quantum pairs the characterization is in terms of explicit monotone quantities related to the sandwiched quantum R\'enyi divergences. In the general quantum case the relevant quantities are given only implicitly. Nevertheless, we find a large collection of monotones that provide necessary conditions for asymptotic or catalytic transformations. When applied to time-translation symmetric maps, these give rise to second laws that constrain state transformations allowed by thermal operations even in the presence of catalysts.
翻译:我们研究相对次多数的概括化,比较一些固定组群代表空间的正数操作者。一对相对相对相对的次分类化,如果存在一种等同的次整化渠道,将第一对的成分带给一对符合相对次分化定义中类似的假设性限制的对一对,则另一对相对的次分类化。在对热力学的资源理论方法中,这种概括化使得人们可以研究Gibbs-保藏图的转换,这种转换是附加时间翻译的对称。我们找到了催化变换的充足条件,并确定了关系无症状放松的特征。对于古典和某些量配对来说,定性的描述是明确的单体数量,这与被保护的量子R\'enyi差异有关。在一般量的情况下,有关数量只是隐含的。然而,我们发现大量单体元素的集合为无症状或催化变形提供了必要的条件。当时间转换地图应用到时间转换时,这就产生了限制热力操作所允许的国家变换的催化剂的第二种法律。