# Operator and Graph Theoretic Techniques for Distinguishing Quantum States via One-Way LOCC
## 3 Operator Structures and One-Way LOCC  
Our initial work [10] identified the importance of certain operator structures for distinguishing various sets of quantum states using one-way LOCC. The following result encompassed our first observation and readily follows from [PERSON]'s result. It suggested deeper operator theoretic connections to the mathematics of one-way LOCC lying in the background. Let \(\Delta:M_{d}(\mathbb{C})\to M_{d}(\mathbb{C})\) be the'map to diagonal' on \(d\times d\) matrices; that is, \(\Delta\) zeros out all off-diagonal entries of a matrix but leaves its diagonal entries unchanged, and so there is an orthonormal basis \(\{|k\rangle\}\) for \(\mathbb{C}^{d}\) such that \(\Delta(\rho)=\sum_{k=1}^{d}|k\rangle\langle k|\rho|k\rangle\langle k|\) is the (von Neumann) measurement map defined by the basis. **Proposition 1**. : _Let \(\{P_{k}\}_{k=1}^{n}\) be a set of \(d\times d\) permutation matrices and let \(\mathcal{S}=\{(I\otimes P_{k})|\Phi\rangle\}\) be the set of corresponding maximally entangled states on \(\mathbb{C}^{d}\otimes\mathbb{C}^{d}\). Then, the following conditions are equivalent:_  
* _The states in_ \(\mathcal{S}\) _are distinguishable by one-way LOCC._
* \(\Delta(P_{j}^{*}P_{i})=0\) _whenever_ \(i\
eq j\)_._  
The null space of the map to diagonal operator has a special structure; it is a linear subspace which is closed under taking adjoints. This observation led us to consider the following notions for the first time in the context of LOCC state distinguishability. First, we recall the basic structure theory for finite-dimensional \(C^{*}\)-algebras, for instance, as exhibited in [14; 15]. Every such algebra is unitarily equivalent to an orthogonal direct sum of ampliated full matrix algebras of the form \(\bigoplus_{k}(M_{m_{k}}(\mathbb{C})\otimes I_{n_{k}})\) for some positive integers \(m_{k},n_{k}\geq 1\). Further, the algebra is unital if it contains the identity operator. **Definition 2**. : _Let \(\mathfrak{A}\) be a unital \(C^{*}\)-algebra. Any linear subspace \(\mathfrak{S}\) contained in \(\mathfrak{A}\) which contains the identity and is closed under taking adjoints is called an operator system._  
Within the setting of such operator structures, the following notion turns out to be key for us. **Definition 3**. : _Let \(\mathcal{H}\) be a Hilbert space and let \(\mathfrak{S}\subseteq B(\mathcal{H})\) be a set of operators on \(\mathcal{H}\) that form an operator system. A vector \(|\psi\rangle\in\mathcal{H}\) is said to be a separating vector for \(\mathfrak{S}\) if \(A|\psi\rangle\
eq 0\) whenever \(A\) is a non-zero element of \(\mathfrak{S}\); in other words, \(A|\psi\rangle=B|\psi\rangle\) with \(A,B\in\mathfrak{S}\) implies \(A=B\)._  
If \(\mathcal{H}\) is finite-dimensional and \(\mathfrak{S}\) is closed under multiplication, and hence a \(C^{*}\)-algebra, then we may use the decomposition above for such algebras to determine the existence of a separating vector as follows. This result was proved in [16]. **Theorem 2**. : _The \(C^{*}\)-algebra \(\bigoplus_{k}(M_{m_{k}}(\mathbb{C})\otimes I_{n_{k}})\) has a separating vector if and only if \(n_{k}\geq m_{k}\) for all \(k\)._  
In the case of the diagonal algebra \(\mathfrak{A}_{\Delta}\), the set of \(d\times d\) diagonal matrices (and so \(\mathfrak{A}_{\Delta}\cong\mathbb{C}^{d}\)), we have \(m_{k}=n_{k}=1\) for all \(1\leq k\leq d\), and hence \(\mathfrak{A}_{\Delta}\) has a separating vector; an example of which can easily be written down: \(|\psi\rangle=\frac{1}{\sqrt{d}}(|0\rangle+\ldots+|d-1\rangle)\). Taken together, these notions and our early results led us to the following general theorem on operator structures and one-way LOCC distinguishability. The first version of the result was proved in [10], and the refined improvement as stated below was established in [11]. **Theorem 3**.