# Operator and Graph Theoretic Techniques for Distinguishing Quantum States via One-Way LOCC
## 4 Graph Theory and Distinguishing Product States  
The following section contains a brief review of the main results and some applications from [13], in which we used graph theory to study the problem of distinguishing sets of product states via one-way LOCC. Our graph-theoretic work in LOCC is ongoing, and here we give a new graph-theoretic perspective and proof for the case that [PERSON] only has access to a single-qubit Hilbert space. We shall write \(G=(V,E)\) for a _simple graph_ with vertex set \(V\) and edge set \(E\). For \(v,w\in V\), we write \(v\sim w\) if the edge \(\{v,w\}\in E\). The _complement_ of \(G\) is the graph \(\overline{G}=(V,\overline{E})\), where the edge set \(\overline{E}\) consists of all two-element sets from \(V\) that are not in \(E\). Another graph \(G^{\prime}\) is a _subgraph_ of \(G\), written \(G^{\prime}\leq G\), if \(V^{\prime}\subseteq V\) and \(E^{\prime}\subseteq E\) with \(v,w\in V^{\prime}\) whenever \(\{v,w\}\in E^{\prime}\). Given a graph \(G=(V,E)\), a function \(\phi:V\to\mathbb{C}^{d}\backslash\{0\}\) is an _orthogonal representation_ of \(G\) if for all vertices \(v_{i}\
eq v_{j}\in V\),  
\[v_{i}\
ot\sim v_{j}\iff\langle\phi(v_{i}),\phi(v_{j})\rangle=0.\]  
Orthogonal representations have been discussed in graph theory, for instance, [27; 28]. Note the biconditional in the definition, which is stronger than conditions for graph colouring. This allows us to uniquely define the graph associated with a function \(\phi\). We introduced a graph-theoretic perspective to distinguishing product states as follows. Suppose we are given a set of product states \(\{|\psi_{k}^{A}\rangle\otimes|\psi_{k}^{B}\rangle\}_{k=1}^{r}\) on \(\mathcal{H}_{A}\otimes\mathcal{H}_{B}\). The graph of these states from [PERSON]'s perspective is the unique graph \(G_{A}\) with vertex set \(V=\{1,2,\ldots,r\}\) such that the map \(k\mapsto|\psi_{k}^{A}\rangle\) is an orthogonal representation of \(G_{A}\). Likewise, the graph of the states from [PERSON]'s perspective is the graph \(G_{B}\) with vertex set \(V\) such that \(k\mapsto|\psi_{k}^{B}\rangle\) is an orthogonal representation of \(G_{B}\). Observe that by construction, the set of product states are mutually orthogonal precisely when [PERSON]'s graph is a subgraph of the complement of [PERSON]'s graph, that is, \(G_{A}\leq\overline{G_{B}}\). The following concepts from graph theory are central for us. Given a graph \(G=(V,E)\), a set of graphs \(\{G_{i}=(V_{i},E_{i})\}\)_covers_\(G\) if \(V=\cup_{i}V_{i}\) and \(E=\cup_{i}E_{i}\). A collection of graphs \(\{G_{i}\}\) is a _clique cover_ for \(G\) if \(\{G_{i}\}\) covers \(G\) and if each of the \(G_{i}\) is a complete graph (i.e., a clique). The clique cover number \(\operatorname{cc}(G)\) is the smallest possible number of subgraphs contained in a clique cover of \(G\). A clique cover can be thought of as a collection of (notnecessarily disjoint) induced subgraphs of \(G\), each of which is a complete graph (there is an edge between every pair of vertices) with the condition that every edge is contained in at least one of the cliques. The following is one of our main results from [13], and gives a characterization of when product states are one-way LOCC distinguishable in terms of the underlying [PERSON] and [PERSON] graph structures and a related decomposition of [PERSON]'s Hilbert space. We note this is a corrected version of the theorem from [13]. The revision was made to condition (3), as the earlier version gave a condition that was sufficient but not necessary. Theorem 4.2: _Given a set of product states in \(\mathcal{H}_{A}\otimes\mathcal{H}_{B}\), let \(G_{A}\) and \(G_{B}\) be the graphs of the states from [PERSON] and [PERSON]'s perspectives, respectively. Let \(\phi:V_{A}\rightarrow\mathcal{H}_{A}\) be the association of vertices with [PERSON]'s states and assume that the set \(\{\phi(v):v\in V\}\) spans \(\mathcal{H}_{A}\)._  
_Then, the states are distinguishable with one-way LOCC with [PERSON] measuring first if and only if there exists_  
1.