# Operator and Graph Theoretic Techniques for Distinguishing Quantum States via One-Way LOCC
## 4 Graph Theory and Distinguishing Product States
#### Single-Qubit Sender and Graph Theory
Moreover, we can define orthogonal (one-dimensional) subspaces of \(\mathcal{H}_{A}\) by \(\mathcal{H}_{i}=\text{span}\left\{\phi_{A}(v):v\in V(G_{i})\right\}\), for \(i=1,2\). The states are thus distinguishable via one-way LOCC with [PERSON] first performing a measurement defined by the Hilbert space decomposition \(\mathcal{H}_{A}=\mathcal{H}_{1}\oplus\mathcal{H}_{2}\), then sending the outcome (\(j=1\) or \(2\)) to [PERSON], who then performs a measurement defined by the orthogonal states \(\{\phi_{B}(v):v\in V(G_{j})\}\) to determine the state. If \(V_{0}\) is non-empty, let \(v_{1}\in V(G_{1})\setminus V_{0}\), and choose \(v_{2}\in V(G_{2})\) such that \(v_{1}\) and \(v_{2}\) have no edge in \(G\) (and hence also in \(G_{A}\)) connecting them. Note that such a vertex exists in \(V(G_{2})\) as otherwise \(G=G_{1}\cup G_{2}\) would be a single clique and so \(\operatorname{cc}\left(G\right)=1\). Additionally, it is necessary that \(v_{2}\in V(G_{2})\setminus V_{0}\), as \(v_{1}\) connects with all vertices in \(V_{0}\). It follows that \(|\psi_{i}\rangle=\phi_{A}(v_{i})\), \(i=1,2\), are orthogonal and hence form an orthonormal basis for \(\mathcal{H}_{A}=\mathbb{C}^{2}\). Furthermore, \(\langle\psi_{i}|\phi_{A}(w_{i})\rangle\
eq 0\) for all \(w_{i}\in V(G_{i})\) and \(i=1,2\). This sets up a one-way LOCC protocol as follows: [PERSON] measures on the basis of \(\{|\psi_{1}\rangle,|\psi_{2}\rangle\}\) for \(\mathcal{H}_{A}\), and communicates the outcome to [PERSON]. As \(G\leq\overline{G_{B}}\) and \(G_{j}\) is a clique, the states \(\{\phi_{B}(v):v\in V(G_{j})\}\) are mutually orthogonal in \(\mathcal{H}_{B}\), and so [PERSON] can determine the state by performing a measurement defined by these states and the projection onto the orthogonal complement of their span. This completes the proof. It is fairly straightforward to give examples that satisfy \(G_{A}=\overline{G_{B}}\) and \(\operatorname{cc}(G_{A})\leq 2\). For instance, the set \(\{|00\rangle,|01\rangle,|1+\rangle,|1-\rangle\}\), where \(|+\rangle,|-\rangle\) in this case is any qubit basis different than the standard basis. Here, [PERSON] would measure on the standard basis, and then [PERSON] would measure on the basis suggested by [PERSON]'s outcome communicated to him. A simple example of a distinguishable set for which \(G_{A}\) is a proper subgraph of \(\overline{G_{B}}\) is given by the standard two-qubit basis \(\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\}\) (left as an easy exercise: \(\operatorname{cc}(G_{A})=2<4=\operatorname{cc}(\overline{G_{B}})\)). We finish by presenting a 'nice' indistinguishable example, in that \(G_{A}=\overline{G_{B}}\), but nevertheless the states fail to be distinguishable due to the failure of the clique cover condition. **Example 4**. : _Consider the (unnormalized) states \(|\psi_{i}\rangle\in\mathbb{C}^{2}\otimes\mathbb{C}^{3}\), for \(1\leq i\leq 5\), defined as follows:_  
\[|\psi_{1}\rangle =|1\rangle\otimes|1\rangle\] \[|\psi_{2}\rangle =|0\rangle\otimes(|0\rangle+|1\rangle)\] \[|\psi_{3}\rangle =|0\rangle\otimes(|0\rangle-|1\rangle)\] \[|\psi_{4}\rangle =(|0\rangle+|1\rangle)\otimes|2\rangle\] \[|\psi_{5}\rangle =(|0\rangle-|1\rangle)\otimes|2\rangle\]  
_These states can be distinguished with full LOCC operations, with [PERSON] measuring first followed by [PERSON] and then once more by [PERSON]. We show that [PERSON]'s initial measurement is necessary and that one-way distinguishability is impossible with [PERSON] going first._  
_Observe here that \(G_{A}=\overline{G}_{B}\). Moreover, the complement of [PERSON]'s graph has clique cover number \(\mathsf{cc}(\overline{G_{B}})=4>2\), with a clique cover of minimal size given by the vertex sets:_  
\[\Big{\{}\{1,5\},\ \{1,4\},\ \{2,3,4\},\ \{2,3,5\}\Big{\}}\]  
_Hence, the set of states is not distinguishable via one-way LOCC with [PERSON] going first._