[Paper Review] Families of pure PEPS with efficiently simulatable local hidden variable models for most measurements

Hussain Anwar, Sania Jevtic+2 more|arXiv (Cornell University)|Dec 11, 2014

Quantum Computing Algorithms and Architecture2 references3 citations

TL;DR

This paper constructs families of pure, multiparty entangled PEPS (Projected Entangled Pair States) that admit local hidden variable models and are efficiently simulatable classically for any restricted set of measurements whose dual strictly contains a pure quantum state. The key contribution is a general construction using generalized probabilistic theories and PEPS formalism, showing that highly entangled quantum states can be classically simulatable under measurement restrictions, even when universal for measurement-based quantum computation under unrestricted measurements.

ABSTRACT

An important problem in quantum information theory is to understand what makes entangled quantum systems non-local or hard to simulate efficiently. In this work we consider situations in which various parties have access to a restricted set of measurements on their particles, and construct entangled quantum states that are essentially classical for those measurements. In particular, given any set of local measurements on a large enough Hilbert space whose dual strictly contains (i.e. contains an open neighborhood of) a pure state, we use the PEPS formalism and ideas from generalized probabilistic theories to construct pure multiparty entangled states that have (a) local hidden variable models, and (b) can be efficiently simulated classically. We believe that the examples we construct cannot be efficiently classically simulated using previous techniques. Without the restriction on the measurements, the states that we construct are non-local, and in some proof-of-principle cases are universal for measurement based quantum computation.

Motivation & Objective

• To understand the conditions under which highly entangled quantum states can be efficiently simulated classically despite non-locality under unrestricted measurements.
• To identify when a restricted set of local measurements allows for local hidden variable models and efficient classical simulation of entangled quantum states.
• To generalize existing results (e.g., Gottesman-Knill, Bell's theorem) beyond Pauli bases to arbitrary measurement sets with suitable duality conditions.
• To construct explicit families of pure entangled PEPS that are classically simulatable for a broad class of measurement restrictions.
• To explore the interplay between entanglement structure, measurement restrictions, and computational universality in measurement-based quantum computation.

Proposed method

• Uses the PEPS formalism to construct pure, multiparty entangled quantum states on lattices with local entanglement structure.
• Applies the concept of generalized separability with respect to a set of measurements M, where the dual set R strictly contains a pure quantum state.
• Defines R-positive and strictly R-positive operators to ensure valid probabilities under Born’s rule for the restricted measurements.
• Employs a tensor network construction where virtual bonds are entangled in a way that ensures the resulting physical state is R-separable under M.
• Uses continuity and perturbation arguments to ensure that a pure state |ψ⟩ lies in the interior of the dual set R, enabling efficient classical sampling.
• Demonstrates that the resulting states are universal for measurement-based quantum computation when unrestricted measurements are allowed, despite being classically simulatable under M.

Experimental results

Research questions

• RQ1 Under what conditions on a set of restricted local measurements M can highly entangled quantum states still admit local hidden variable models?
• RQ2 Can pure entangled PEPS be constructed such that they are efficiently simulatable classically for a given measurement set M, even when non-local under unrestricted measurements?
• RQ3 How does the duality between measurement sets M and their dual R influence the existence of generalized separable decompositions?
• RQ4 What is the role of the virtual bond dimension and lattice connectivity in enabling classical simulatability while preserving long-range entanglement?
• RQ5 Can such constructions be extended beyond the d ≥ 2^v constraint to allow for broader classes of measurement sets?

Key findings

• For any set of local measurements M whose dual R strictly contains a pure quantum state, the paper constructs families of pure, multiparty entangled PEPS that admit local hidden variable models.
• The constructed states are efficiently simulatable classically for measurements in M, leveraging the generalized separability and R-positivity framework.
• The states remain non-local and can be universal for measurement-based quantum computation when unrestricted measurements are applied, demonstrating a sharp contrast with classical simulatability under M.
• Entanglement length is infinite in some cases, as shown via entanglement swapping using Bell basis measurements, even though Bell measurements are included in M.
• The construction is translationally invariant in many cases, indicating that quantum entanglement is not confined to specific particles or regions.
• The method generalizes beyond Pauli bases and applies to arbitrary measurement sets M, provided the dual R strictly contains a pure state, enabling broad applicability.