Jordan products of quantum channels and their compatibility

Given two quantum channels, we examine the task of determining whether they are compatible—meaning that one can perform both channels simultaneously but, in the future, choose exactly one channel whose output is desired (while forfeiting the output of the other channel). Here, we present several results concerning this task. First, we show it is equivalent to the quantum state marginal problem, i.e., every quantum state marginal problem can be recast as the compatibility of two channels, and vice versa. Second, we show that compatible measure-and-prepare channels (i.e., entanglement-breaking channels) do not necessarily have a measure-and-prepare compatibilizing channel. Third, we extend the notion of the Jordan product of matrices to quantum channels and present sufficient conditions for channel compatibility. These Jordan products and their generalizations might be of independent interest. Last, we formulate the different notions of compatibility as semidefinite programs and numerically test when families of partially dephasing-depolarizing channels are compatible.

Maps Preserving k-Jordan Products on Operator Algebras

For any positive integer k, the k-Jordan product of a , b in a ring R is defined by { a , b } k = { { a , b } k − 1 , b } 1 , where { a , b } 0 = a and { a , b } 1 = a b + b a . A map f on R is k-Jordan zero-product preserving if { f ( a ) , f ( b ) } k = 0 whenever { a , b } k = 0 for a , b ∈ R ; it is strong k-Jordan product preserving if { f ( a ) , f ( b ) } k = { a , b } k for all a , b ∈ R . In this paper, strong k-Jordan product preserving nonlinear maps on general rings and k-Jordan zero-product preserving additive maps on standard operator algebras are characterized, generalizing some known results.

Derivation Requirements on Prime Near-Rings for Commutative Rings

Near-ring is an extension of ring without having to fulfill a commutative of the addition operations and left distributive of the addition and multiplication operations It has been found that some theorems related to a prime near-rings are commutative rings involving the derivation of the Lie products and the derivation of the Jordan product. The contribution of this paper is developing the previous theorem by inserting derivations to the Lie products and the Jordan product. Keywords: Derivation, Prime Near-Ring, Lie Products and Jordan Products.