arXiv subjects

Cristina Mullican

Publications and source records attributed to Cristina Mullican.

3 recordsLinked to original sources

arXiv2023

Werner states from diagrams

We present two results on multiqubit Werner states, defined to be those states that are invariant under the collective action of any given single-qubit unitary that acts simultaneously on all the qubits. Motivated by the desire to characterize entanglement properties of Werner states, we construct a basis for the real linear vector space of Werner invariant Hermitian operators on the Hilbert space of pure states; it follows that any mixed Werner state can be written as a mixture of these basis operators with unique coefficients. Continuing a study of "polygon diagram" Werner states constructed in earlier work, with a goal to connect diagrams to entanglement properties, we consider a family of multiqubit states that generalize the singlet, and show that their 2-qubit reduced density matrices are separable.

quant-ph↗

arXiv2019

Extending powers of pseudo-Anosovs

Biringer, Johnson, and Minsky showed that a pseudo-Anosov map on a boundary component of an irreducible 3-manifold has a power that partially extends to the interior if and only if the (un)stable laminations of $f$ is an $\mathbb{R}$-projective limit of meridians. We prove that the power required for a pseudo-Anosov map to partially extend is not universally bounded. We construct a family of pseudo-Anosov maps $f_i$ for all $i=1,2...$ on a boundary component of a family of irreducible 3-manifolds $M_i$ such that $f_i^i$ partially extends to the interior of $M_i$ but $f_i^j$ does not for $j<i$.

math.GT↗

arXiv2016

Neighborhood reconstruction and cancellation of graphs

We connect two seemingly unrelated problems in graph theory. Any graph $G$ has an associated neighborhood multiset $\mathscr{N}(G)= \{N(x) \mid x\in V(G)\}$ whose elements are precisely the open vertex-neighborhoods of $G$. In general there exist non-isomorphic graphs $G$ and $H$ for which $\mathscr{N}(G)=\mathscr{N}(H)$. The neighborhood reconstruction problem asks the conditions under which $G$ is uniquely reconstructible from its neighborhood multiset, that is, the conditions under which $\mathscr{N}(G)=\mathscr{N}(H)$ implies $G\cong H$. Such a graph is said to be neighborhood-reconstructible. The cancellation problem for the direct product of graphs seeks the conditions under which $G\times K\cong H\times K$ implies $G\cong H$. Lovasz proved that this is indeed the case if $K$ is not bipartite. A second instance of the cancellation problem asks for conditions on $G$ that assure $G\times K\cong H\times K$ implies $G\cong H$ for any bipartite graph $K$ with $E(K)\neq \emptyset$. A graph $G$ for which this is true is called a cancellation graph. We prove that the neighborhood-reconstructible graphs are precisely the cancellation graphs. We also present some new results on cancellation graphs, which have corresponding implications for neighborhood reconstruction.

math.CO↗