How much entanglement is needed for quantum error correction?

TL;DR

This paper investigates the relationship between entanglement and error correction capability in quantum codes, revealing that high entanglement is not always necessary and depends on code type and measures used.

Contribution

It characterizes a tradeoff between code distance and geometric entanglement, showing that some codes require high entanglement while others can have low entanglement regardless of distance.

Findings

Geometric entanglement grows at least linearly with code distance for certain code families.

For some codes, the overlap with product states decreases exponentially with code distance.

Existence of codes with low entanglement states despite high error correction capability.

Abstract

It is commonly believed that logical states of quantum error-correcting codes have to be highly entangled such that codes capable of correcting more errors require more entanglement to encode a qubit. Here, we show that the validity of this belief depends on the specific code and the choice of entanglement measure. To this end, we characterize a tradeoff between the code distance $d$ quantifying the number of correctable errors, and the geometric entanglement measure of logical states quantifying their maximal overlap with product states or more general ``topologically trivial" states. The maximum overlap is shown to be exponentially small in $d$ for three families of codes: (1) low-density parity check codes with commuting check operators, (2) stabilizer codes, and (3) codes with a constant encoding rate. Equivalently, the geometric entanglement of any logical state of these codes grows at least linearly with $d$. On the opposite side, we also show that this distance-entanglement tradeoff does not hold in general. For any constant $d$ and $k$ (number of logical qubits), we show there exists a family of codes such that the geometric entanglement of some logical states approaches zero in the limit of large code length.