Multiparty pure entangled states and local measurement with LHV models and efficient classical sampling via the PEPS formalism

The difficulty of simulating quantum systems on classical computers has attracted considerable attention over many years. While the origin of this difficulty is still to be fully understood, the entanglement of quantum states and dynamics plays a role in many cases. In the context of many-body physics, overcoming this problem relies on using ansatz states that one hopes are simple enough to compute with, yet sophisticated enough to capture important phenomena. In this work we consider one such approach — the so-called PEPS (‘projected entangled pair states’) formalism for a spin lattice [1]. In the PEPS formalism, an underlying lattice of pairwise-entangled ‘virtual’ particles is ‘projected’ at each site to give the final multiparticle entangled ansatz state. The use of entangled ‘virtual bonds’ brings complexity to the description, while helping to break the multiparty entanglement down into two-particle form that may be more tractable. Nevertheless, PEPS states can still be very complex. The cluster state of measurement based quantum computation is an example of a PEPS state [2], so sampling the outcomes of local measurements on PEPS states can be classically difficult. In this work [3] we use the PEPS formalism for the simulation of quantum systems, but in the context of generalised entanglement that has arisen in the foundations of quantum theory [4]. A quantum state of two or more particles is said to be (quantum) entangled if it cannot be written as a probabilistic mixture of products of local operators drawn from the set of single particle quantum states. However, in some contexts (particularly involving restricted measurements), one may consider allowing the local operators to be drawn from a non-quantum set of operators (the dual space of the restricted measurements) other than the set single particle quantum states. In such situations states that are entangled in the quantum setting may become separable from such a generalised perspective [5]. If an operator has a separable decomposition w.r.t to a set S, we say that the operator is S-separable. The above observation naturally leads to the following question: if we treat the quantum entangled virtual bonds in the PEPS formalism as separable states with respect to some non-quantum state space, then is it possible to exploit this separability in order to improve the current classical simulation techniques or provide alternative classical descriptions (such as local hidden variable models) for some PEPS states? We suppose that the many-particle quantum state consists of d level particles arranged on a lattice, and that the lattice sites are all of degree v. The PEPS formalism allocates a ‘virtual’ quantum particle of D-levels to either end of each edge on the lattice, such that the two particles corresponding to each edge are in a maximally entangled state |D〉. The ansatz for the state (of the d-level particles) is then obtained by applying a linear transformation A taking the v virtual particles at each site (which live on C⊗v D ) into a real particle at that site (living on Cd). Collectively this gives a final (unnormalised) quantum state for the whole lattice from which we may try to calculate properties of