When different quantum networks describe the same state

One state, different descriptions

A quantum many-body state can require an enormous collection of numbers to describe directly. Tensor networks organize those numbers through smaller building blocks called tensors, connected by internal links. Contracting the network means summing over the shared internal indices to obtain the coefficients of the quantum state.

But the internal description is not unique. In a one-dimensional matrix product state, an invertible change on an internal bond can cancel against its inverse on the neighboring tensor, leaving the state unchanged. A fundamental theorem asks the converse: if two descriptions produce the same states, what transformation must connect their building blocks?

Delcamp and Kjær’s preprint, submitted on November 9, 2026, introduces a two-dimensional framework and proves such a theorem under specific conditions. The result is mathematical, rather than an experimental measurement or a numerical performance benchmark.

Adding structure between the bonds

The authors begin by rewriting one-dimensional matrix product states in algebraic language. They then use categorification: replacing familiar mathematical objects with richer category-theoretic counterparts.

In their construction, the analogue of the bond space is a 2-vector space, defined here as a finite, semisimple, complex-linear category. Concretely, the resulting networks have labels attached to the regions between their bonds. Each bond space splits into blocks identified by the labels on its two sides, and tensors respect the matching of neighboring region labels.

The lattice geometry is encoded separately in a free monoidal category, a structure that records how elementary pieces can be joined. A network representation is a module category over that structure: it specifies how the geometrical building instructions act on the categorical bond data.

These abstractions allow different entanglement structures within one framework. With only one region-label type, the construction recovers ordinary projected entangled pair states, or PEPS. Other choices include networks built from GHZ-type entanglement resources around plaquettes.

Why the boundary matters

The theorem concerns uniform networks, meaning the same defining tensor is repeated throughout a patch. Their edges remain open and are closed using specially constructed boundary data. Those data themselves form a one-dimensional tensor network carrying region labels.

The allowed patches are called shapes: connected skew Young diagrams, obtained by removing one upper-left-aligned arrangement of rows of boxes from another. Rectangles are included, but not every connected square-lattice patch is allowed.

Several assumptions are essential. Both representations must be non-degenerate: the matrices of bond-space blocks have no entirely zero row or column. Boundary data must have full support, with a nonzero component in every nonzero bond block.

For one representation, the incoming boundary vectors must also be cyclic: acting with the local tensor and varying its physical index must span the relevant outgoing bond space. The outgoing boundary covectors must satisfy the dual, cocyclic, condition, spanning the corresponding space of linear functionals. Finally, the second representation’s two relevant composite bond spaces must be no larger than the first’s.

What the theorem proves

Under these assumptions, suppose the representations generate exactly the same states on every allowed shape, using their respective boundary data. The authors prove that the representations are equivalent as module categories, with an equivalence that preserves those boundary data and is unique up to isomorphism.

Concretely, the transformation combines invertible maps on horizontal and vertical bond spaces with a consistent bijection between region labels. Joining these maps while matching shared labels produces a matrix product operator that intertwines the two descriptions.

The proof constructs transformations by leaving selected bonds open and closing the others with boundary data. Cyclicity and cocyclicity ensure that these partially closed networks span the necessary spaces. Equality of the completed states then makes the transformations well defined and invertible. Further arguments show that they respect region labels and fit together consistently across larger patches.

A stronger condition, not a ground-state guarantee

The paper also defines 2-injectivity. It requires every region-pair bond block to be nonzero and the local tensors to span the full space of compatible maps. This guarantees cyclicity and cocyclicity for any full-support boundary data. With a single region-label type, it reduces to ordinary injectivity.

The authors intend this condition to describe states that are unique ground states of local Hamiltonians. That physical property is not established here as a general consequence.

Where the result stops

The theorem is not an unrestricted equivalence test for arbitrary PEPS. It assumes exact equality across every allowed shape, not approximate agreement or equality on one finite sample. Periodic boundaries and analogous theorems for other geometries remain proposed extensions.

A broader classification of phases protected by higher or non-invertible symmetries motivates the framework, but is explicitly outside this paper. Its demonstrated contribution is a new language for two-dimensional networks and a conditional theorem identifying when apparently different descriptions are mathematically equivalent.