Coarse-grained entanglement classification through orthogonal arrays

Abstract

Classification of entanglement in multipartite quantum systems is an open problemsolved so far only for bipartite systems and for systems composed of three and four qubits. We propose here a coarse-grained classification of entanglement in systems consisting of N subsystems with an arbitrary number of internal levels each, based on properties of orthogonal arrays with N columns. In particular, we investigate in detail a subset of highly entangled pure states which contains all states defining maximum distance separable codes. To illustrate the methods presented, we analyze systems of four and five qubits, as well as heterogeneous tripartite systems consisting of two qubits and one qutrit or one qubit and two qutrits. I. INTRODUCTIONCharacterization of entanglement in multipartite systems has proven particularly challenging, even for pure states. As one moves from a bipartite to a multipartite scenario involving N parties, the algebraic structure becomes much richer. A pure multipartite quantum state is de-arXiv:1709.05916v4 [math.CO] 28 Jun 2018 := max I:|I|=t J t (I), where |I| denotes the cardinality of the multi-index I. Then, the generalized resolution is defined as(2) Clearly, t < GR < t+1. For two-level OAs, the above quantity is invariant under isomorphisms.However, for multi-level arrays, permutations of symbols within columns can change quantity 2) [23]. Hence the generalized resolution of a multi-level OA is defined as the maximum value of (2) taken over the family of all isomorphic arrays.

Cut-resistant links and multipartite entanglement resistant to particle loss

“…A natural generalization of AME states are the so called k-uniform states, whereby the partial tracing of any subsystems down to k qudits will result in a maximally mixed state [19,20]. Such states can be constructed with help of combinatorial tools like orthogonal arrays [21], which allow for a coarse-grained classification of multipartite entanglement [22]. Another related measure, called the persistence of entanglement [23], is defined by the minimal number of local measurements such that the state becomes completely disentangled for any measurement outcome.…”