We present a simple, dimension-independent criterion which guarantees that some quantum channel Φ is divisible, i.e., that there exists a non-trivial factorization Φ = Φ1Φ2. The idea is to first define an “elementary” channel Φ2 and then to analyze when ΦΦ2−1 is completely positive. The sufficient criterion obtained this way—which even yields an explicit factorization of Φ—is that one has to find orthogonal unit vectors x, x⊥ such that 〈x⊥|KΦKΦ⊥|x〉=〈x|KΦKΦ⊥|x〉={0} where KΦ is the Kraus subspace of Φ and KΦ⊥ is its orthogonal complement. Of course, using linearity this criterion can be reduced to finitely many equalities. Generically, this division even lowers the Kraus rank which is why repeated application—if possible—results in a factorization of Φ into in some sense “simple” channels. Finally, be aware that our techniques are not limited to the particular elementary channel we chose.
We develop a quantum message-passing framework for factor graphs over finite abelian groups. Our starting point is the task of discriminating between a collection of quantum states indexed by the elements of a finite abelian group G whose overlaps respect the structure of a group-covariant pure-state channel (PSC). For such channels, we show that the Gram matrix constructed from the output states is diagonalized by the character basis of the dual group G. Hence, the channel is characterized, up to isometric equivalence, by its character-indexed eigen list.
Based on this representation, we analyze the induced classical-quantum channels associated with check, equality, homomorphism, marginalization, and automorphism factors. For each factor, we derive explicit update rules showing that if the incoming messages are heralded mixtures of group-covariant PSCs, then the outgoing message remains in the same class. This provides a closed quantum message-passing framework for tree-structured factor graphs assembled from these primitives.
The framework applies directly to several standard code families over finite abelian groups, including polar codes, LDPC codes, and convolutional and turbo codes. It recovers the previously studied q-ary formulation as the special case (G=Zq), while extending the belief propagation with quantum messages (BPQM) framework introduced by Renes to non-cyclic alphabets and more general factor-graph constraints described by homomorphisms between products of abelian groups.
The estimation of the decoherence process of an open quantum system is of both theoretical significance and experimental appealing. Practically, the decoherence can be easily estimated if the coherence evolution satisfies some simple relations. We introduce a framework for studying evolution equation of coherence. Based on this framework, we prove a simple factorization relation (FR) for the l1 norm of coherence, and identified the sets of quantum channels for which this FR holds. By using this FR, we further determine condition on the transformation matrix of the quantum channel which can support permanently freezing of the l1 norm of coherence. We finally reveal the universality of this FR by showing that it holds for many other related coherence and quantum correlation measures.