1,774 research outputs found

### On the structure of the adjacency matrix of the line digraph of a regular digraph

We show that the adjacency matrix M of the line digraph of a d-regular
digraph D on n vertices can be written as M=AB, where the matrix A is the
Kronecker product of the all-ones matrix of dimension d with the identity
matrix of dimension n and the matrix B is the direct sum of the adjacency
matrices of the factors in a dicycle factorization of D.Comment: 5 page

### On the Cayley digraphs that are patterns of unitary matrices

A digraph D is the pattern of a matrix M when D has an arc ij if and only if
the ij-th entry of M is nonzero. Study the relationship between unitary
matrices and their patterns is motivated by works in quantum chaology and
quantum computation. In this note, we prove that if a Cayley digraph is a line
digraph then it is the pattern of a unitary matrix. We prove that for any
finite group with two generators there exists a set of generators such that the
Cayley digraph with respect to such a set is a line digraph and hence the
pattern of a unitary matrix

### Some Ulam's reconstruction problems for quantum states

Provided a complete set of putative $k$-body reductions of a multipartite
quantum state, can one determine if a joint state exists? We derive necessary
conditions for this to be true. In contrast to what is known as the quantum
marginal problem, we consider a setting where the labeling of the subsystems is
unknown. The problem can be seen in analogy to Ulam's reconstruction conjecture
in graph theory. The conjecture - still unsolved - claims that every graph on
at least three vertices can uniquely be reconstructed from the set of its
vertex-deleted subgraphs. When considering quantum states, we demonstrate that
the non-existence of joint states can, in some cases, already be inferred from
a set of marginals having the size of just more than half of the parties. We
apply these methods to graph states, where many constraints can be evaluated by
knowing the number of stabilizer elements of certain weights that appear in the
reductions. This perspective links with constraints that were derived in the
context of quantum error-correcting codes and polynomial invariants. Some of
these constraints can be interpreted as monogamy-like relations that limit the
correlations arising from quantum states. Lastly, we provide an answer to
Ulam's reconstruction problem for generic quantum states.Comment: 22 pages, 3 figures, v2: significantly revised final versio

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