Bayes' rule plays a crucial role in logical inference in information and physical sciences alike.Its extension into the quantum regime has been the subject of several recent works.These quantum versions of Bayes' rule have been expressed in the language of Hilbert spaces.In this paper, we derive the expression for the Petz recovery map within any quasiprobability representation, with explicit formulas for the two canonical choices of "normal quasiprobability representations" (which include discrete-Wigner representations) and of representations based on symmetric informationally complete positive operator-valued measures (SIC-POVMs).By using the same mathematical syntax of (quasi)stochastic matrices acting on (quasi)stochastic vectors, the core difference in logical inference between classical and quantum theory is found in the manipulation of the reference prior rather than in the representation of the channel.
Bayes' rule plays a crucial role in logical inference in information and physical sciences alike.Its extension into the quantum regime has been the subject of several recent works.These quantum versions of Bayes' rule have been expressed in the language of Hilbert spaces.In this paper, we derive the expression for the Petz recovery map within any quasiprobability representation, with explicit formulas for the two canonical choices of "normal quasiprobability representations" (which include discrete-Wigner representations) and of representations based on symmetric informationally complete positive operator-valued measures (SIC-POVMs).By using the same mathematical syntax of (quasi)stochastic matrices acting on (quasi)stochastic vectors, the core difference in logical inference between classical and quantum theory is found in the manipulation of the reference prior rather than in the representation of the channel.
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.
Wigner functions provide a way to do quantum physics using quasiprobabilities, that is, ``probability'' distributions that can go negative. Informationally complete positive operator-valued measures, a much younger subject than phase-space formulations of quantum mechanics, are less familiar but provide wholly probabilistic representations of quantum theory. In this paper, we show that the Born rule links these two classes of structure and discuss the art of interconverting between them. In particular, we demonstrate that the operator bases corresponding to minimal discrete Wigner functions (Wigner bases) are orthogonalizations of minimal informationally complete measurements (MICs). By not imposing a particular discrete phase-space structure at the outset, we push Wigner functions to their limits in a suitably quantified sense, revealing an extremality condition satisfied by the symmetric informationally complete measurements (SICs). This contributes another member to the set of optimality criteria that SICs are known to satisfy. Finally, we speculate that astute choices of MICs from the orthogonalization preimages of Wigner bases may in general give quantum measurements that are conceptually illuminating for the quasiprobability representations defined by those Wigner bases.
Bayes' rule is widely regarded as the canonical prescription for belief updating. We show, however, that one can arbitrarily preserve pre-specified beliefs while appearing to perform Bayesian updates via "prior hacking": engineering a reference prior distribution such that, for a fixed channel and evidence, the update matches a chosen target distribution. We prove that this is generically possible in both classical and quantum settings whenever Bayesian inversions are well-defined (with the Petz recovery map as the quantum analogue to Bayes' rule), and provide constructive algorithms for doing so. We further establish a duality between prior hacking and Schrödinger bridge problems (a key object in statistical physics with applications in generative modelling), yielding in the quantum setting a unique, inference-consistent selection among candidate bridges. This formally establishes the Bayes-like updating that Schrödinger bridges are performing with respect to the process as opposed to the reference prior, both in classical and quantum settings.
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.
Quasiprobability representations are well-established tools in quantum information science, with applications ranging from the classical simulability of quantum computation to quantum process tomography, quantum error correction, and quantum sensing. While traditional quasiprobability representations typically employ real-valued distributions, recent developments highlight the usefulness of complex-valued ones -- most notably, via the family of Kirkwood--Dirac quasiprobability distributions. Building on the framework of Schmid et al. [Quantum 8, 1283 (2024)], we extend the analysis to encompass complex-valued quasiprobability representations that need not preserve the identity channel. Additionally, we also extend previous results to consider mappings towards infinite-dimensional spaces. We show that, for each system, every such representation can be expressed as the composition of two maps that are completely characterized by their action on states and on the identity (equivalently, on effects) for that system. Our results apply to all complex-valued quasiprobability representations of any finite-dimensional, tomographically-local generalized probabilistic theory, with finite-dimensional quantum theory serving as a paradigmatic example. In the quantum case, the maps' action on states and effects corresponds to choices of frames and dual frames for the representation. This work offers a unified mathematical framework for analyzing complex-valued quasiprobability representations in generalized probabilistic theories.
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.