[Paper Review] A tensor version of the quantum Wielandt theorem

Mateusz Michałek, Tim Seynnaeve+1 more|arXiv (Cornell University)|Nov 13, 2018

Advanced Operator Algebra Research4 references4 citations

TL;DR

This paper establishes a tensor version of the quantum Wielandt theorem for projected entangled pair states (PEPS), proving that injectivity in 2D and higher-dimensional tensor networks is well-defined with finite blocking. Using nonconstructive Noetherian arguments from algebraic geometry, it shows the existence of a finite grid size that guarantees injectivity, resolving a long-standing open problem in quantum many-body physics and tensor networks.

ABSTRACT

We prove boundedness results for the injectivity regions for PEPS. Our result is a higher-dimensional generalization of the quantum Wielandt inequality.

Motivation & Objective

• To resolve the long-standing open question of whether a quantum Wielandt-type theorem exists for projected entangled pair states (PEPS) in two or more dimensions.
• To establish that injectivity in PEPS is well-defined, meaning there exists a finite blocking size after which the virtual-to-physical map becomes injective.
• To generalize the quantum Wielandt inequality—previously known only for matrix product states (MPS) in one dimension—to higher-dimensional tensor networks.
• To provide a nonconstructive proof of bounded injectivity regions using tools from algebraic geometry, specifically Hilbert's Basis Theorem and Zariski closed sets.
• To demonstrate that the existence of a finite parent Hamiltonian with local support for every injective PEPS is mathematically justified.

Proposed method

• Uses a grid-based framework to model PEPS on n-dimensional lattices with virtual and physical indices.
• Defines an injective region as a grid where the tensor contraction map from virtual to physical indices is injective.
• Applies the Hilbert Basis Theorem by showing that the set of non-injective configurations forms a Zariski closed subset in the parameter space.
• Leverages the fact that the rank condition for injectivity is equivalent to the vanishing of certain minors (polynomial conditions), making the set of non-injective tensors algebraic.
• Uses a descending chain argument on grids of increasing size, showing that the chain of non-injective sets stabilizes due to Noetherianity.
• Reformulates the result as the existence of a finite collection of grids such that any injective PEPS becomes injective on at least one of them, regardless of physical dimension d.

Experimental results

Research questions

• RQ1 Does a quantum Wielandt-type theorem exist for projected entangled pair states (PEPS) in two or more dimensions?
• RQ2 Is the notion of injectivity in PEPS well-defined, in the sense that there exists a finite blocking size after which injectivity is guaranteed?
• RQ3 Can the existence of a finite-support parent Hamiltonian for every injective PEPS be established using algebraic geometry?
• RQ4 What is the structure of the set of non-injective PEPS configurations in the parameter space?
• RQ5 Can the injectivity of a PEPS be decided algorithmically using the proposed framework?

Key findings

• The paper proves that for any n-dimensional PEPS with bond dimension D and physical dimension d, there exists a finite grid size such that injectivity is achieved if and only if the tensor network becomes injective on one of finitely many fixed grid shapes.
• The existence of such a finite blocking size is guaranteed by the Noetherian property of polynomial rings, via Hilbert's Basis Theorem, applied to the algebraic variety of non-injective configurations.
• The bound on the grid size is nonconstructive, meaning the exact size is not computed, but its existence is proven using algebraic geometry.
• The result generalizes the quantum Wielandt theorem from matrix product states (MPS) to higher-dimensional tensor networks (PEPS), extending the injectivity result to 2D and beyond.
• The proof implies that the injectivity of a PEPS can be decided algorithmically, as the set of non-injective tensors is Zariski closed and thus decidable via polynomial ideal membership.
• The result holds uniformly across all physical dimensions d, with the finite collection of grids depending only on n and D, not on d.