nLab perfect tensor

Idea

In the context of tensor network states and holographic entanglement entropy, a rank $n$ tensor $T \in V^{ \otimes_n }$ for $V$ a self-dual object (hence equipped with a non-degenerate inner product $V\otimes V \to \mathbf{1}$) is called perfect if all of its tensor/hom-adjuncts

$$V^{\otimes_{k}} \longrightarrow V^{\otimes_{n-k}}$$

for $k \leq n/2$ are isometries.

Applications

Holographic entanglement entropy

In the HaPPY code tensor network the fact that all vertices carry a perfect tensor leads to the Ryu-Takayanagi formula for the corresponding entanglement entropies (see at holographic entanglement entropy for more).

References

• Fernando Pastawski, Beni Yoshida, Daniel Harlow, John Preskill, Section 2 of: Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence, JHEP 06 (2015) 149 (arXiv:1503.06237)