Detecting correlated errors in SPAM tomography

TL;DR

This paper investigates SPAM tomography where neither states nor measurement devices are known, identifying gauge parameters, and introduces gauge-invariant quantities for detecting correlated errors in quantum systems.

Contribution

It introduces gauge-invariant quantities for SPAM tomography that can detect correlated errors without prior knowledge of states or devices.

Findings

Identifies at most d^2(d^2-1) gauge parameters in d-dimensional systems.

Derives gauge-invariant quantities accessible experimentally.

Provides conditions to detect correlations between SPAM errors.

Abstract

Whereas in standard quantum state tomography one estimates an unknown state by performing various measurements with known devices, and whereas in detector tomography one estimates the POVM elements of a measurement device by subjecting to it various known states, we consider here the case of SPAM (state preparation and measurement) tomography where neither the states nor the measurement device are assumed known. For $d$-dimensional systems measured by $d$-outcome detectors, we find there are at most $d^2(d^2-1)$ "gauge" parameters that can never be determined by any such experiment, irrespective of the number of unknown states and unknown devices. For the case $d=2$ we find new gauge-invariant quantities that can be accessed directly experimentally and that can be used to detect and describe SPAM errors. In particular, we identify conditions whose violations detect the presence of correlations between SPAM errors. From the perspective of SPAM tomography, standard quantum state tomography and detector tomography are protocols that fix the gauge parameters through the assumption that some set of fiducial measurements is known or that some set of fiducial states is known, respectively.