Experimentally exploring compressed sensing quantum tomography

Abstract

In the light of the progress in quantum technologies, the task of verifying the correct functioning of processes and obtaining accurate tomographic information about quantum states becomes increasingly important. Compressed sensing, a machinery derived from the theory of signal processing, has emerged as a feasible tool to perform robust and significantly more resource-economical quantum state tomography for intermediate-sized quantum systems. In this work, we provide a comprehensive analysis of compressed sensing tomography in the regime in which tomographically complete data is available with reliable statistics from experimental observations of a multi-mode photonic architecture. Due to the fact that the data is known with high statistical significance, we are in a position to systematically explore the quality of reconstruction depending on the number of employed measurement settings, randomly selected from the complete set of data, and on different model assumptions. We present and test a complete prescription to perform efficient compressed sensing and are able to reliably use notions of model selection and cross-validation to account for experimental imperfections and finite counting statistics. Thus, we establish compressed sensing as an effective tool for quantum state tomography, specifically suited for photonic systems.

Adaptive compressive tomography: A numerical study

“…To start off, the a priori assumption about r requires additional verification before it can be applied to CS, since the resulting estimator accuracy hinges on the validity of this assumption. Next, as one has no means of validating the final estimator self-consistently, the usual solution is to compare the estimator with some assumed target state [9,17,18]. In the presence of experimental errors, there is simply no guarantee whether such a comparison is actually trustworthy.…”

A comparative study of estimation methods in quantum tomography

“…More recently, quantum tomography has become a crucial validation tool current quantum technology applications [38,58,29,67]. The experimental challenges have stimulated research in new directions such as compressed sensing [35,54,27,70,20,5,3], estimation of permutationally invariant states [73], adaptive and selflearning tomography [55,34,60,26,39,63], incomplete tomography [74], minimax bounds [25,4], Bayesian estimation [15,33], and confidence regions [7,18,71,24,53]. Since 'full tomography' becomes impossible for systems composed of even a f < l a t e x i t s h a 1 _ b a s e 6 4 = " 7 j z U X Q l i 6…”

Objective compressive quantum process tomography

“…One attempt to directly reduce the measurement cost of QPT with non-IC measurements and entropy methods was reported [29]. Simultaneously, the method of compressed sensing [30][31][32][33][34][35][36] was applied to QPT [37][38][39] to reconstruct lowrank or sparse quantum processes with a small set of specialized compressive measurements. However, this concept only works under the assumption that ρ Φ should either possess a rank no larger than some known integer r, or be sparse in some known basis of known sparsity, all of which demand reliable evidence.…”