# Entanglement of Subspaces and Error Correcting Codes

## TL;DR

This paper introduces a new measure called entanglement of subspaces to quantify bipartite entanglement, explores its properties, and demonstrates its significance in quantum error correction codes, including analysis of Shor's nine-qubits code.

## Contribution

It defines entanglement of subspaces, proves its additivity for maximally entangled subspaces, and links these concepts to the structure of quantum error correction codes.

## Key findings

- Maximally entangled subspaces are additive.
- The subspace of logical codewords in non-degenerate codes is maximally entangled.
- Analysis of Shor's nine-qubits code using orthogonal subspaces.

## Abstract

We introduce the notion of entanglement of subspaces as a measure that quantify the entanglement of bipartite states in a randomly selected subspace. We discuss its properties and in particular we show that for maximally entangled subspaces it is additive. Furthermore, we show that maximally entangled subspaces can play an important role in the study of quantum error correction codes. We discuss both degenerate and non-degenerate codes and show that the subspace spanned by the logical codewords of a non-degenerate code is a 2k-totally (maximally) entangled subspace. As for non-degenerate codes, we provide a mathematical definition in terms of subspaces and, as an example, we analyze Shor's nine qubits code in terms of 22 mutually orthogonal subspaces.