# Operator and Graph Theoretic Techniques for Distinguishing Quantum States via One-Way LOCC
## 2 One-Way LOCC and Operator Relations  
We will use the traditional quantum information notation throughout the paper. In particular, we use the Dirac bra-ket notation for vectors, which labels a given fixed orthonormal basis for \(\mathbb{C}^{d}\), with \(d\geq 1\) fixed, as \(\{|i\rangle:0\leq i\leq d-1\}\), and the corresponding dual vectors as \(|i\rangle^{*}=\langle i|\). For \(n\geq 1\), the \(n\)-qudit Hilbert space is the tensor product \((\mathbb{C}^{d})^{\otimes n}\), which has an orthonormal basis given by \(|i_{1}i_{2}\cdots i_{n}\rangle:=|i_{1}\rangle\otimes|i_{2}\rangle\otimes \ldots\otimes|i_{n}\rangle\). We also denote the set of complex \(m\times m\) matrices, for a fixed \(m\geq 1\), by \(M_{m}(\mathbb{C})\). Given a finite-dimensional Hilbert space \(\mathcal{H}\), we will write \(B(\mathcal{H})\) for the algebra of bounded (continuous) linear operators on \(\mathcal{H}\), which can be identified with \(M_{m}(\mathbb{C})\) via matrix representations when \(\dim\mathcal{H}=m\). The Pauli operators play an important role in many of our applications, and are given as matrix representations on the single-qubit basis \(\{|0\rangle,|1\rangle\}\) for \(\mathcal{H}=\mathbb{C}^{2}\) by:  
\[X= \begin{pmatrix}0&1\\ 1&0\end{pmatrix} ,\quad Y= \begin{pmatrix}0&-i\\ i&0\end{pmatrix} ,\quad Z= \begin{pmatrix}1&0\\ 0&-1\end{pmatrix} ;\]  
where the operators are described by \(X|0\rangle=|1\rangle\), \(X|1\rangle=|0\rangle\), etc. The basic set up for the LOCC framework is as follows: multiple parties share a set of quantum states, on which each party can perform local quantum operations. They can then transmit their results only using _classical_ information in prescribed directions. The key problem we have focussed on in our work is to distinguish quantum states amongst a set of known states, where two parties, called [PERSON] (\(A\)) and [PERSON] (\(B\)), can perform quantum measurements on their individual subsystems, and then communicate classically. Further, as general LOCC operations are very difficult to characterize mathematically, we have largely restricted ourselves to the case of _one-way_ LOCC, where the communication is limited to one predetermined direction, generally from \(A\) to \(B\). This still captures many key examples and settings (though not all). Hence, the bipartite case we consider makes the following assumptions:  
* Two parties \(A\), \(B\) are separated physically. * They control their (finite-dimensional) subsystem Hilbert spaces \(\mathcal{H}_{A}\), \(\mathcal{H}_{B}\); for simplicity, we often assume \(\mathcal{H}_{A}=\mathcal{H}_{B}=\mathbb{C}^{d}\) for some fixed \(d\geq 2\). * The state of the composite system \(\mathcal{H}_{A}\otimes\mathcal{H}_{B}\) is assumed to be a pure state amongst a known set of states \(\mathcal{S}=\{|\psi_{i}\rangle\}_{i}\subseteq\mathcal{H}_{A}\otimes\mathcal{ H}_{B}\). * The goal of \(A\) and \(B\) is then to identify the particular \(i\) using only one-way LOCC measurements. The mathematical description of measurements defined by one-way LOCC protocols is given as follows [7]. Definition 1: A one-way LOCC measurement, with \(A\) going first, is a set of positive operators \(\mathbb{M}=\{A_{k}\otimes B_{k,j}\}_{k,j}\) on \(\mathcal{H}_{A}\otimes\mathcal{H}_{B}\) such that  
\[\sum_{k}A_{k}=I_{A}\quad\text{and}\quad\sum_{j}B_{k,j}=I_{B}\quad\forall k.\]  
Each of the sets \(\{A_{k}\}_{k}\), \(\{B_{k}j\}_{j}\) form what is called a _positive operator valued measure_ (POVM) on \(\mathcal{H}_{A}\) and \(\mathcal{H}_{B}\), respectively. If outcome \((k,j)\) is obtained, for any \(k\) and a particular \(j\), the conclusion is that the prepared state was the state identified with the pair \(k,j\). Without loss of generality, one can further assume each \(A_{k}\) is a scalar multiple of a (pure) rank one projection.