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The paper gives a theoretical link between two quantum inference tasks, showing that the Petz recovery map acts as a gradient of the log-likelihood used in maximum-likelihood tomography. That suggests repeated Petz updates can serve as a practical estimation procedure with guaranteed likelihood increase for measurement channels. The results are mathematical, and their value for quantum hardware depends on whether the method scales and performs well on real experimental data.
Connecting Quantum Tomography and Quantum Retrodiction
arXiv:2606.23777v2 Announce Type: replace Abstract: Quantum tomography and quantum retrodiction are traditionally viewed as separate inference tasks. We show that the two are manifestations of the same underlying principle. We prove that the Petz recovery map of a measurement channel is a gradient of the log-likelihood used in maximum-likelihood tomography, and we derive its extension to arbitrary quantum channels. Consequently, for measurement channels, repeated applications of the Petz map monotonically increase the likelihood, naturally yielding a method for quantum tomography.
Abstract:Quantum tomography and quantum retrodiction are traditionally viewed as separate inference tasks. We show that the two are manifestations of the same underlying principle. We prove that the Petz recovery map of a measurement channel is a gradient of the log-likelihood used in maximum-likelihood tomography, and we derive its extension to arbitrary quantum channels. Consequently, for measurement channels, repeated applications of the Petz map monotonically increase the likelihood, naturally yielding a method for quantum tomography.
6+9 pages, 4 figures. Revised version: updated Figs. 2 and 3; added Fig. 4 and supplementary sections on numerical evidence for Conjecture 3 and a thermodynamic reconstruction example; moved the gradient derivation to End Matter
Quantum Physics (quant-ph); Mathematical Physics (math-ph); Statistics Theory (math.ST); Data Analysis, Statistics and Probability (physics.data-an)
arXiv:2606.23777 [quant-ph]
(or arXiv:2606.23777v2 [quant-ph] for this version)
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