The Petz recovery map is considered one of the key candidates for the quantum analogue of Bayesian inference and Jeffrey's conditionalization. Since, there seems to be a natural connection between Bayesian inference and the notion of state tomography, it is natural to ask if the Petz recovery can be used for this latter task. In this paper, we discuss such recent results on iterated Petz recovery and relate them to Bayesian approaches to quantum state tomography. We highlight the limitations of direct Petz iteration and show how an extended Petz construction, by lifting the inference problem to a classical distribution over candidate quantum states, recovers the structure of Bayesian and maximum likelihood tomography. This provides perspective on the modifications or nuances required for a Petz approach to quantum retrodiction to perform quantum state tomography.
The Petz recovery map is considered one of the key candidates for the quantum analogue of Bayesian inference and Jeffrey's conditionalization. Since, there seems to be a natural connection between Bayesian inference and the notion of state tomography, it is natural to ask if the Petz recovery can be used for this latter task. In this paper, we discuss such recent results on iterated Petz recovery and relate them to Bayesian approaches to quantum state tomography. We highlight the limitations of direct Petz iteration and show how an extended Petz construction, by lifting the inference problem to a classical distribution over candidate quantum states, recovers the structure of Bayesian and maximum likelihood tomography. This provides perspective on the modifications or nuances required for a Petz approach to quantum retrodiction to perform quantum state tomography.
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.
Retrodiction is the act of inferring a cause from its effects, the most common example of which is Bayesian inference. Retrodiction can be defined by its structural process-theoretic properties, which are mathematically captured by category theory. This categorical definition of retrodiction has recently been shown to potentially isolate the Petz recovery map as a unique universal candidate for quantum Bayesian inference. This paper extends these results to the infinite-dimensional setting on von Neumann algebras. In the process, we provide a pedagogical review of the Petz recovery map in infinite dimensions and its relation to the more commonly used expression in the finite-dimensional setting. We formalize the open question as to whether these categorical axioms for retrodiction do in fact uniquely characterize the Petz recovery map. If such a characterization holds, this would show that Bayesian inversion and the Petz recovery map are structural necessities and not simply useful algorithms for classical and quantum inference.
Bayes' rule is central to classical statistical inference: given a prior and a stochastic model, it determines the corresponding retrodictive conditional probabilities. In quantum theory, however, the question of whether an equally general rule of retrodiction exists and, if so, whether it is uniquely determined remains nontrivial. Here we prove that a retrodiction assignment defined for all priors and channels, which takes each predicted output state back to its prior, must coincide with the Petz transpose map if it is involutive, respects sequential and parallel composition, and agrees with unitary inversion. The theorem holds for channels between finite-dimensional full matrix algebras, with the input and output spaces restricted to the supports of the corresponding reference states. In particular, affinity under probabilistic mixtures and the characteristic square-root dependence on the reference states follow from these requirements rather than being postulated. Without involutivity, the remaining requirements characterize a family of doubly rotated Petz maps. Our result thus explains why the Petz transpose map arises in seemingly different areas of quantum theory: it is the unique quantum analogue of Bayes' rule compatible with these consistency requirements.
Reversibility lies at the heart of physics and information theory, linking entropy, efficiency and information recovery. Yet, definitions of a “reverse process” have been context-specific, relying on physical assumptions rather than a general framework. This thesis develops a universal recipe for reverse processes using Bayesian inference, formalized through Bayes’ rule and the Petz recovery map. We first show how these tools recover conventional notions of reversal while yielding new insights, and justify the Petz transpose as the quantum analog of Bayes’ rule via quasiprobability representations. We then compare Bayesian reversibility with dilation-based approaches, proving their equivalence through the role of correlations. This motivates the notion of tabletop time-reversible processes, special cases where correlations can be neglected with respect to reversal. Beyond the core results, we apply these methods to fluctuation relations, quantum thermal operations, and the Loschmidt paradox, and reflect on implications for the philosophy of time. In sum, Bayesian inference provides a regime-independent foundation for defining and analyzing (ir)reversibility across physics and information theory.
According to Quantum Darwinism, system-environment interactions both einselect particular system properties and encode them redundantly in many independent subsets of the environment, called fragments. This redundancy implies that an observer can recover the einselected information by accessing just one such fragment. However, the protocol by which such reconstruction should occur is often left unspecified. Considering a system Γ interacting with a multipartite environment Ξ, we investigate whether, and under what conditions, the einselected state of Γ can be recovered from environmental fragments using the Petz recovery map. We show that the fidelity between the system's initial state and the state reconstructed via Petz recovery develops a plateau as a function of the fragment size. Our results are supported by both analytical arguments and numerical simulations of large but tractable models.
Quantum state vector reconstruction is a fundamental problem in quantum information theory, crucial for tasks like quantum state tomography and quantum error correction. Traditional methods often struggle with noisy measurements and lack a principled framework for uncertainty quantification. This paper introduces a novel algorithm, Quantum State Vector Reconstruction via Bayesian Inference, that leverages Bayesian inference to reconstruct a quantum state vector from a set of measurements, enhancing accuracy and offering a more robust approach compared to existing methods. We detail the core mechanism, emphasizing the integration of Bayesian estimation for uncertainty quantification and refinement of the reconstruction process. The algorithm incorporates a probabilistic approach to estimate the state vector distribution, improving reconstruction by incorporating uncertainty estimates directly into the reconstruction process. This results in a more reliable and accurate state vector reconstruction, particularly when dealing with noisy measurements. This work addresses the limitations of conventional methods by providing a principled Bayesian framework for state vector reconstruction, offering a significant advancement in this field.