Code to accompany Classification of joint quantum measurements based on entanglement cost of localization

Jef Pauwels, Alejandro Pozas-Kerstjens, Flavio del Santo, and Nicolas Gisin

This is a repository containing the computational appendix of the article "Classification of joint quantum measurements based on entanglement cost of localization. Jef Pauwels, Alejandro Pozas-Kerstjens, Flavio del Santo, and Nicolas Gisin. Phys. Rev. X 15, 021013 (2025) (arXiv:2408.00831)." It provides the codes for analytically solving the initial levels in the hierarchies for localizing joint two-qubit measurements, and for exploring higher levels and the three-qubit scenario. It also contains all the measurements found by numerical searches, for the third level of the hierarchy in the two-qubit scenario, and for the first two levels in the three-qubit scenario.

The code is written in Mathematica and MATLAB.

Mathematica packages required:

• QI for quantum information tools.

MATLAB libraries required:

• QETLAB for quantum information tools.
• YALMIP for setting up semidefinite programs.
• Mosek (or any other SDP solver) for solving semidefinite programs.

Files:

• AllOneEbitSolutions.nb: Notebook that analytically finds all solutions to Equation 4 in the manuscript, giving all the two-qubit measurements that can be localized with one ebit in the finite-consumption Vaidman scheme.

• AllThreeEbitSolutions.nb: Notebook that analytically finds all solutions to Equation 5 in the manuscript, giving all the two-qubit measurements that can be localized with three ebits in the finite-consumption Vaidman scheme.

• findUniqueBases.m: Auxiliary script to identify representatives of measurements according to Definition 2 in the manuscript.

• numsearch: MATLAB files that perform the numerical searches on the cases that are too complicated to solve analytically. This folder includes the search codes (N2q9e for the third level for two-qubit measurements, N3q2e for the first level for three-qubit measurements, and N3q17e for the second level for three-qubit measurements), the functions that are minimized (fmin2ebits, fmin9ebits, and fmin17ebits), functions to check equivalence of bases (localunitarilyequiv2 for two-qubit bases and localunitarilyequiv3 for three-qubit bases), and auxiliary functions (BasesFixed, BasesRandom, PartialTrace, sigma, and vector).

• results: Folder containing the results as .mat files. In each of them knownM contains all the corresponding measurements.

If you would like to cite this work, please use the following format:

J. Pauwels, A. Pozas-Kerstjens, and N. Gisin, Classification of joint quantum measurements based on entanglement cost of localization, Phys. Rev. X 15, 021013 (2025), arXiv:2408.00831

@article{pauwels2025localizable,
  author = {Pauwels, Jef and Pozas-Kerstjens, Alejandro and del Santo, Flavio and Gisin, Nicolas},
  title = {Classification of joint quantum measurements based on entanglement cost of localization},
  journal = {Phys. Rev. X},
  volume = {15},
  issue = {2},
  pages = {021013},
  numpages = {20},
  year = {2025},
  month = {Apr},
  publisher = {American Physical Society},
  doi = {10.1103/PhysRevX.15.021013},
  url = {https://link.aps.org/doi/10.1103/PhysRevX.15.021013},
  archivePrefix = {arXiv},
  eprint = {2408.00831},
  year = {2025}
}