The matrix representation of affine maps in the higher-dimensional space and the necessary and sufficient condition of a quantum operation which is unital are given.
We answer a question posed by Martin et al. at MFPS XXVI in the affirmative: the free affine monoid over a finite involution group has a quantum representation, in addition to the classical representation already shown to exist in [Martin, K., J. Feng, and S. Krishnan. A Free Object in Quantum Information Theory. Electronic Notes in Theoretical Computer Science 267 (2010), 35–47]. In particular, this implies that in every dimension there are non-trivial classes of quantum channels that have classical representations which were not present in the lower dimensions. En route to establishing this result, we give two characterizations of the diagonal n-qubit channels (one of which is stated in [Bourdon, P.S, and H.T. Williams, Unital quantum operations on the Bloch ball and Bloch region, Phys. Rev. A. 69 (2004), 022314]) and necessary conditions for a Bloch matrix to satisfy.
A linear map between matrix algebras corresponds to the Choi matrix in the tensor product of two matrix algebras whose definition depends on matrix units. Paulsen and Shultz [J. Math. Phys. 54, 072201 (2013)] considered the question if one can replace matrix units by another basis of matrix algebras in the definition of the Choi matrix to retain the correspondence between the complete positivity of maps and the positivity of Choi matrices and gave a sufficient condition on basis under which this is true. In this note, we provide necessary and sufficient conditions to see that the Paulsen–Shultz condition is also necessary.
One of the main goals of any resource theory such as entanglement, quantum thermodynamics, quantum coherence, and asymmetry, is to find necessary and sufficient conditions that determine whether one resource can be converted to another by the set of free operations. Here we find such conditions for a large class of quantum resource theories which we call affine resource theories. Affine resource theories include the resource theories of athermality, asymmetry, and coherence, but not entanglement. Remarkably, the necessary and sufficient conditions can be expressed as a family of inequalities between resource monotones (quantifiers) that are given in terms of the conditional min-entropy. The set of free operations is taken to be (1) the maximal set (i.e., consists of all resource nongenerating quantum channels) or (2) the self-dual set of free operations (i.e., consists of all resource nongenerating maps for which the dual map is also resource nongenerating). As an example, we apply our results to quantum thermodynamics with Gibbs preserving operations, and several other affine resource theories. Finally, we discuss the applications of these results to resource theories that are not affine and, along the way, provide the necessary and sufficient conditions that a quantum resource theory consists of a resource destroying map.