Optimal two-qubit tomography based on local and global measurements: Maximal robustness against errors as described by condition numbers

TL;DR

This paper develops an optimal two-qubit tomography protocol that maximizes robustness against errors by minimizing the condition number, using a combination of local and nonlocal measurements, and provides feasible experimental setups.

Contribution

It introduces a new tomographic protocol with minimal condition number for two-qubit states, improving robustness over standard methods, and extends the approach to multiqubit and multilevel systems.

Findings

Achieves the lowest possible condition number of 1 for robustness.

Proposes experimentally feasible measurement setups using linear optics.

Extends the method to multiqubit and qudit systems.

Abstract

We present an error analysis of various tomographic protocols based on the linear inversion for the reconstruction of an unknown two-qubit state. We solve the problem of finding a tomographic protocol which is the most robust against errors in terms of the lowest value (i.e., equal to 1) of a condition number, as required by the Gastinel-Kahan theorem. In contrast, standard tomographic protocols, including those based on mutually unbiased bases, are nonoptimal for determining all 16 elements of an unknown two-qubit density matrix. Our method is based on the measurements of the 16 generalized Pauli operators, where twelve of them can be locally measured, and the other four require nonlocal Bell measurements. Our method corresponds to selectively measuring, one by one, all of the real and imaginary elements of an unknown two-qubit density matrix. We describe two experimentally feasible setups of this protocol for the optimal reconstruction of two photons in an unknown polarization state using conventional detectors and linear-optical elements. Moreover, we define the operators for the optimal reconstruction of the states of multiqubit or multilevel (qudit) systems.