Optimal logical Bell measurements on stabilizer codes with linear optics

TL;DR

This paper establishes a theoretical framework for logical Bell measurements on stabilizer codes using linear optics, providing bounds and criteria for success, applicable to various quantum codes, enhancing quantum communication and error correction.

Contribution

It demonstrates that logical Bell measurements can be mapped to physical measurements, deriving success bounds and criteria, and applies this to multiple stabilizer codes with optimal schemes.

Findings

Logical BMs can be reduced to physical BMs on qubit pairs.

A general upper bound on success probability is established.

Schemes achieving the upper bound are demonstrated for multiple codes.

Abstract

Bell measurements (BMs) are ubiquitous in quantum information and technology. They are basic elements for quantum commmunication, computation, and error correction. In particular, when performed on logical qubits encoded in physical photonic qubits, they allow for a read-out of stabilizer syndrome information to enhance loss tolerance in qubit-state transmission and fusion. However, even in an ideal setting without photon loss, BMs cannot be done perfectly based on the simplest experimental toolbox of linear optics. Here we demonstrate that any logical BM on stabilizer codes can always be mapped onto a single physical BM perfomed on any qubit pair from the two codes. As a necessary condition for the success of a logical BM, this provides a general upper bound on its success probability, especially ruling out the possibility that the stabilizer information obtainable from only partially succeeding, physical linear-optics BMs could be combined into the full logical stabilizer information. We formulate sufficient criteria to find schemes for which a single successful BM on the physical level will always allow to obtain the full logical information by suitably adapting the subsequent physical measurements. Our approach based on stabilizer group theory is generally applicable to any stabilizer code, which we demonstrate for quantum parity, five-qubit, standard and rotated planar surface, tree, and seven-qubit Steane codes. Our schemes attain the general upper bound for all these codes, while this bound had previously only been reached for the quantum parity code.