[Paper Review] State retrieval beyond Bayes' retrodiction

Jacopo Surace, Matteo Scandi|arXiv (Cornell University)|Jan 24, 2022

Quantum Mechanics and Applications4 citations

TL;DR

This paper proposes a general framework for optimal state retrieval in irreversible dynamics by introducing state retrieval maps that satisfy physical principles, surpassing Bayes' retrodiction and the Petz recovery map in accuracy. It derives an optimal retrieval map via a maximisation principle, showing improved performance in both classical and quantum regimes, with numerical evidence suggesting that adding an involutivity condition recovers the Bayes-inspired reverse as a unique solution.

ABSTRACT

In the context of irreversible dynamics, associating to a physical process its intuitive reverse can result to be a quite ambiguous task. It is a standard choice to define the reverse process using Bayes' theorem, but, in general, this choice is not optimal. In this work we explore whether it is possible to characterise an optimal reverse map building from the concept of state retrieval maps. In doing so, we propose a set of principles that state retrieval maps should satisfy. We find out that the Bayes inspired reverse is just one case in a whole class of possible choices, which can be optimised to give a map retrieving the initial state more precisely than the Bayes rule. Our analysis has the advantage of naturally extending to the quantum regime. In fact, we find a class of reverse transformations containing the Petz recovery map as a particular case, corroborating its interpretation as quantum analogue of the Bayes retrieval. Finally, we present numerical evidences that by adding a single extra axiom one can isolate the usual reverse process derived from Bayes' theorem.

Motivation & Objective

• To address the arbitrariness in defining reverse maps for irreversible dynamics, especially when standard inverse operations fail.
• To develop a physically motivated framework for state retrieval that generalizes Bayes' theorem and the Petz recovery map.
• To identify optimal retrieval maps that minimize retrieval error using a principled maximisation approach.
• To explore whether the Bayes-inspired reverse can be uniquely recovered by adding an additional physical principle, such as involutivity.
• To extend the framework naturally to the quantum regime, providing a coherent interpretation of the Petz map as a special case.

Proposed method

• Define state retrieval maps as physical transformations that perfectly retrieve a fiducial state π and satisfy detailed balance with respect to π.
• Introduce five physical desiderata: (1) physicality, (2) agreement with inverse on invertible maps, (3) perfect retrieval of π, (4) detailed balance of Φ̃Φ w.r.t. π, and (5) positive eigenvalues of Φ̃Φ.
• Propose a maximisation principle over the space of maps satisfying the five principles to identify the optimal retrieval map.
• Derive the optimal retrieval map in terms of a parametrised family of maps, with the Bayes rule and Petz map corresponding to the identity in the parameter space.
• Use numerical simulations to compare retrieval performance of the optimal map against Bayes and Petz maps, showing consistent improvement.
• Investigate whether adding an involutivity condition (reversing the reverse gives the original map) uniquely recovers the Bayes-inspired reverse, supported by numerical evidence.

Experimental results

Research questions

• RQ1 Can a principled, physically motivated framework for state retrieval be constructed that generalizes Bayes' retrodiction and the Petz recovery map?
• RQ2 Does a maximisation principle over the space of physically valid retrieval maps yield a map that outperforms both Bayes' rule and the Petz map in state retrieval accuracy?
• RQ3 Is the Bayes-inspired reverse the unique solution when an additional principle—namely, that the reverse of the reverse is the original map—is imposed?
• RQ4 How does the proposed framework naturally extend to the quantum regime, and what is the role of the Petz map within this framework?
• RQ5 Can the set of state retrieval maps be fully characterised using geometric and algebraic tools such as the vertices of the set U(Φπ, π)?

Key findings

• The optimal retrieval map, derived via a maximisation principle over physically valid maps, consistently outperforms both the Bayes-inspired reverse and the Petz recovery map in state retrieval accuracy, both on average and for individual states.
• The Bayes-inspired reverse map for classical stochastic maps is recovered numerically when an additional involutivity principle is imposed, suggesting it may be uniquely characterised by this condition.
• The Petz recovery map is shown to be a special case of the proposed class of state retrieval maps, reinforcing its interpretation as a quantum analogue of Bayes' retrodiction.
• The framework naturally extends to the quantum regime, with the same maximisation principle yielding a quantum retrieval map that outperforms the Petz map in tested examples.
• Numerical evidence indicates that the only map satisfying the involutivity condition and the five physical principles is the Bayes-inspired reverse, though an analytical proof remains open.
• The set of admissible retrieval maps is constrained by the geometry of U(Φπ, π), and only the identity map satisfies the positivity conditions in tested cases, suggesting strong uniqueness under the full set of principles.