The states generated by a multiport beam splitter usually display genuine multipartite entanglement between the many spatial modes. Here we investigate the different classes of multipartite entangled states that arise in this practical situation, working within the paradigm of stochastic local operations with classical communication. We highlight three scenarios, one where the multipartite entanglement classes follow a total number hierarchy, another where the various classes follow a nonclassicality degree hierarchy, and a third one that is a combination of the previous two. Moreover, the multipartite entanglement of higher-dimensional versions of Dicke states relates naturally to our results.
The states generated by a multiport beam splitter usually display genuine multipartite entanglement between the many spatial modes. Here we investigate the different classes of multipartite entangled states that arise in this practical situation, working within the paradigm of stochastic local operations with classical communication. We highlight three scenarios, one where the multipartite entanglement classes follow a total number hierarchy, another where the various classes follow a nonclassicality degree hierarchy, and a third one that is a combination of the previous two. Moreover, the multipartite entanglement of higher-dimensional versions of Dicke states relates naturally to our results.
Abstract Multipartite entanglement plays a central role in optical quantum technologies. One way to entangle two photons is to prepare them in orthogonal internal states, for example, in two polarisations, and then send them through a balanced beam splitter. Post-selecting on the cases where there is one photon in each output port results in a maximally entangled state. This idea can be extended to schemes for the post-selected generation of larger entangled states. Typically, switching between different types of entangled states requires different arrangements of beam splitters and so a new experimental setup. Here, we demonstrate a simple and versatile scheme to generate different types of genuine tripartite entangled states with only one experimental setup. We send three photons through a three-port splitter and vary their internal states before post-selecting on certain output distributions. This results in the generation of tripartite W, G and GHZ states. We obtain fidelities of up to ( 87.3 ± 1.1 ) % with regard to the respective ideal states, confirming a successful generation of genuine tripartite entanglement.
We develop a novel method in classifying the multipartite entanglement state of 2×N ×N under stochastic local operations and classical communication (SLOCC). In this method, all inequivalent classes of a genuine entangled state can be assorted directly without knowing the classification information of lower dimension ones for any given dimension N. It also gives a nature explanation for the non-local parameters remaining in the entanglement classes when N ≥ 4. Entanglement is at the heart of the quantum information theory (QIT) and is now thought as a physical resource to realize quantum information tasks, such as quantum cryptography [1, 2], superdense coding [3, 4], and quantum computation [5]. The study of entanglement improves our further understanding of quantum non-locality [6] and also is of particular interest in QIT. One such case is the classification of multipartite entanglement. Two quantum states can be employed to implement the same tasks in QIT and are thought to be equivalent while they are mutually convertible by Stochastic Local Operations and Classical Communication (SLOCC) [7]. Nevertheless, in practice the classification of multipartite entanglement and in high dimensions in the Hilbert space seems to be a formidable task [8]. Matrix decomposition keeps to be a useful tool as in the two-partite case [9, 10]. A widely adopted philosophy in dealing with this issue is first to know the classification of the state in lower dimension (or less partite) and then extend to the higher dimension case [11, 12] (or more partite [13, 14]) in an inductive way. However, nontrivial aspect emerges as the dimension increases, i.e. some non-local parameters may nest in the entangled states [15, 16]. In literature, in
The nonclassicality of single-mode quantum states is studied in relation to the entanglement created by a beam splitter. It is shown that properly defined quantifications---based on the quantum superposition principle---of the amounts of nonclassicality and entanglement are strictly related to each other. This can be generalized to the amount of genuine multipartite entanglement, created from a nonclassical state by an N splitter. As a consequence, a single-mode state of a given amount of nonclassicality is fully equivalent, as a resource, to exactly the same amount of entanglement. This relation is also considered in the context of multipartite entanglement and multimode nonclassicality.
The main concern of this paper is how to define proper measures of multipartite entanglement for mixed quantum states. Since the structure of partial separability and multipartite entanglement is getting complicated if the number of subsystems exceeds two, one cannot expect the existence of an ultimate scalar entanglement measure, which grasps even a small part of the rich hierarchical structure of multipartite entanglement, and some higher-order structure characterizing that is needed. In this paper we make some steps in this direction. First, we reveal the lattice-theoretic structure of the partial separability classification, introduced earlier [Sz. Szalay and Z. K\"ok\'enyesi, Phys. Rev. A 86, 032341 (2012)]. It turns out that, mathematically, the structure of the entanglement classes is the up-set lattice of the structure of the different kinds of partial separability, which is the down-set lattice of the lattice of the partitions of the subsystems. It also turns out that, physically, this structure is related to the local operations and classical communication convertibility: If a state from a class can be mapped into another one, then that class can be found higher in the hierarchy. Second, we introduce the notion of multipartite monotonicity, expressing that a given set of entanglement monotones, while measuring the different kinds of entanglement, shows also the same hierarchical structure as the entanglement classes. Then we construct such hierarchies of entanglement measures and propose a physically well-motivated one, being the direct multipartite generalization of the entanglement of formation based on the entanglement entropy, motivated by the notion of statistical distinguishability. The multipartite monotonicity shown by this set of measures motivates us to consider the measures to be the different manifestations of some ``unified'' notion of entanglement.
Entanglement shared among multiple parties presents complex challenges for the characterisation of different types of entanglement. One of the most fundamental insights is the fact that some mixed states can feature entanglement across every possible cut of a multipartite system yet can be produced via a mixture of states separable with respect to different partitions. To distinguish states that genuinely cannot be produced from mixing such partition-separable states, the term genuine multipartite entanglement was coined. All these considerations originate in a paradigm where only a single copy of the state is distributed and locally acted upon. In contrast, advances in quantum technologies prompt the question of how this picture changes when multiple copies of the same state become locally accessible. Here we show that multiple copies unlock genuine multipartite entanglement from partially separable states, i.e., mixtures of the partition-separable states, even from undistillable ensembles, and even more than two copies can be required to observe this effect. With these findings, we characterise the notion of genuine multipartite entanglement in the paradigm of multiple copies and conjecture a strict hierarchy of activatable states and an asymptotic collapse of the hierarchy.
Abstract This chapter takes a closer look at entanglement in multipartite systems. An examination of pure tripartite systems serves a starting point from which we discover the equivalence classes of Greenberger-Horne-Zeilinger (GHZ) states and W states. We then continue with the mixed-state case and discuss the notions of full separability, partition-separability and biseparability versus genuine tripartite entanglement, before stating the GHZ theorem in the formulation following Mermin. For systems of three or more parties we then formulate the definitions for k-separability and genuine multipartite entanglement (GME), as well as for k-producibility and entanglement depth, followed by prominent examples for GME states such as the generalized GHZ states and Dicke states. We give an overview of the problem of detecting GME using various linear and non-linear GME witnesses, lifted witnesses, PPT mixers, before discussing challenges for the characterization and quantification of multipartite entanglement. We close by analysing the phenomenon of entangled entanglement.