# Operator and Graph Theoretic Techniques for Distinguishing Quantum States via One-Way LOCC
## 3 Operator Structures and One-Way LOCC
### Application: States from the Stabilizer Formalism for Quantum Error Correction  
In [11], we developed connections between quantum error correction [18; 19; 20; 21; 22; 23] and the study of one-way LOCC, including the fact that every one-way LOCC protocol naturally defines a quantum error correcting code defined by the distinguishable states. This alsoled to new derivations of some known results and new examples of distinguishable states. Here, we present one of the applications from that paper.  
Sets of unitary operators with the features discussed above are plentiful in one of the foundational areas of quantum error correction, the'stabilizer formalism' [21], which gives a toolbox for generating and identifying codes from the Pauli group.  
Let \(\mathcal{P}_{n}\) be the \(n\)-qubit Pauli group, that is, the unitary subgroup on \((\mathbb{C}^{2})^{\otimes n}\) with generating set as follows:  
\[\mathcal{P}_{n}:=\left\langle\pm iI;X_{j},Y_{j},Z_{j}:1\leq j\leq n\right\rangle,\]  
where \(X_{1}=X\otimes I\otimes\cdots\otimes I=X\otimes I^{\otimes(n-1)}\), etc.  
The _Clifford group_ is the normalizer subgroup of \(\mathcal{P}_{n}\) inside the group of \(n\)-qubit unitary operators. It is known that if \(G\) is a subgroup of \(\mathcal{P}_{n}\), with \(S_{0}=\{g_{1},\ldots g_{m}\}\), a minimal generating set for a maximal abelian subgroup \(S\) of \(G\), then there exists a unitary \(U\) in the Clifford group such that \(U^{*}g_{j}U=Z_{j}\), for all \(1\leq j\leq m\). This allows us to focus on the generating Pauli operators for deriving more general results.  
In [11], we proved the following. For succinctness, we use terminology from the stabilizer formalism in the theorem hypotheses without giving precise details here, as they are not necessary to appreciate the result.  
Let \(\{U_{i}\}\subseteq\mathcal{P}_{n}\) be a complete set of \(4^{k}\)-encoded logical Pauli operators for a stabilizer \(k\)-qubit code on an \(n\)-qubit Hilbert space. Then, the set of states \(\mathcal{S}=\{(I\otimes U_{i})|\Phi\rangle\}\) is distinguishable by one-way LOCC if and only if \(k\leq\frac{n}{2}\).  
The basic idea behind this proof as an application of the result above is as follows: The \(4^{k}\) element set, \(\mathcal{P}_{n,k}=\langle X_{j},Z_{j}:1\leq j\leq k\rangle/\{\pm iI\}\), forms a set of encoded operations for the code subspace (up to unitary equivalence). However, \(\mathfrak{S}_{0}:=\operatorname{span}\left(\mathcal{P}_{n,k}\right)= \operatorname{Alg}\left(\mathcal{P}_{n,k}\right)=M_{2^{k}}\otimes I_{2^{n-k}}\). Hence, from the theorem above, the states \(\mathcal{S}\) are distinguishable by one-way LOCC if and only if \(\mathfrak{S}_{0}\) has a separating vector if \(2^{k}\leq 2^{n-k}\), or equivalently \(2k\leq n\).  
The upper bound in this result (\(2k=n\)) gives sets that saturate a known bound [5; 24] for the size of one-way distinguishable sets of maximally entangled states on \(\mathbb{C}^{d}\otimes\mathbb{C}^{d}\) (\(d=2^{n}\)). For \(2k<n\), this produces (non-trivial) distinguishable sets, which is significant as it is known [8] that many sets defined from \(\mathcal{P}_{n}\) with less than \(2^{n}\) operators (here \(4^{k}<2^{n}\)) are not distinguishable even with positive partial transpose operations, and hence not with one-way LOCC.