Quantum Error Correction Difficulties: "Measurements destroy quantum information" Intuition

Question

Context: In section 10.1.1, Nielsen and Chuang describe the difficulties QEC faces compared to classical error correction. Particularly that

Measurements destroy quantum information.

Shor states in his "Scheme for reducing decoherence in quantum computer memory" paper

Our encoding scheme stores information in the entanglement between qubits, so that no information is stored in any one specific qubit

Question: I was wondering, if one could generally claim that in QEC, due to the collapse of the wave function, it is impossible to store error correcting properties in a single qubit itself but rather in the relationship ("use entanglement as a resource"?) with the actual qubit and ancillaries? Would that be a valid statement?

Answer 1

Not really, this statement does not capture the intuition for the use of entanglement, as it is not due to the measurement problem.

It is more akin to the idea of redundancy in classical error correction, where we encode the information into a larger set of bits. This helps against noise that is usually assumed to act locally, on individual bits.

In quantum error correction, in the overwhelming majority of constructions, it is similarly assumed that noise is local, i.e., it will act on individual qubits with high probability and on multiple qubits at the same time with much lower probability. The more qubits local noise has to act on to change the information the harder it will be to corrupt our stored information. With entanglement, we spread the information to multiple data qubits, and the local operators that can change the quantum information in a single qubit become instead multi-qubit logical operators. These high weight operators are much harder (lower probability) for local noise to create.

When noise is not too strong, it will mostly create smaller errors (smaller than logical operators), and in good coding constructions we can correct them. As regards to the challenges with the measurement issue, that is related to how we actually implement error correction. You can build your intuition based on the similarity to calculating parity checks in classical error correction. When using parity checks, you don't actually need the information encoded, you are using extra information about the system to use for recovery. QEC schemes have to allow for syndrome measurements that determine whether the state is noisy without destroying the information. This is typically done by transferring the value of parity check operators to an ancillary system and measure it out from there (technically it's a phase kickback, or operator eigenvalue measurement).