arXiv subjects

Arthur J. Parzygnat

Publications and source records attributed to Arthur J. Parzygnat.

At least 19 recordsLinked to original sources

arXiv2026

Bayesian inference and retrodiction for faithful states on von Neumann algebras

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.

math.OA↗

arXiv2026

Quantum encodings that preserve persistent homology

Given a data set with a notion of distance, such as a point cloud in Euclidean space, topological data analysis (TDA) uses techniques from algebraic topology and metric geometry to infer the topology of a hypothetical manifold from which the data are sampled. This inference is achieved by calculating topological invariants, some of which are difficult to compute classically. Meanwhile, quantum TDA utilizes quantum processes to extract the invariants used in making such inferences in an attempt to speed up the computations. Because applying transformations to the original classical dataset could alter the associated topological invariants, we investigate which quantum encodings would best preserve the invariants of the original dataset. This line of inquiry is distinct from standard approaches in quantum TDA, whose typical starting point is not from the classical dataset directly, but rather from the associated combinatorial objects, such as simplicial complexes, which typically demand a lot of resources to construct. We take the first step at a more direct approach by focusing on which quantum encodings acting directly on the data are admissible for applying quantum algorithms to extract topological features from classical datasets.

quant-ph↗

arXiv2025

Bipartite quantum states admitting a causal explanation

The statistics of local measurements of joint quantum systems can sometimes be used to distinguish the spatiotemporal structure in which they were measured. We first prove that every bipartite separable density matrix is temporally compatible with direct causal influence for arbitrary finite-dimensional quantum systems and measurements of a tomographically-complete class of observables, which includes all Pauli observables in the case of multi-qubit systems. Equivalently, if a bipartite density matrix is not temporally compatible with direct causal influence, then it must be entangled. We also provide an operational meaning for the two temporal evolutions consistent with such correlations in terms of generalized dephasing channels and pretty good measurements. The two temporal evolutions are Bayesian inverses of each other, which is different from them being Petz recovery maps of each other. Finally, we prove necessary and sufficient conditions for an arbitrary bipartite quantum state to be temporally compatible, thereby providing a temporal analogue of the positive partial transpose criterion valid for quantum systems of any dimension.

quant-ph↗

arXiv2025

Sufficient conditions for localized vibrational modes in one- and two-dimensional discrete lattices

This paper presents a rigorous proof that arbitrarily weak perturbations produce localized vibrational (phonon) modes in one- and two-dimensional discrete lattices, inspired by analogous results for the Schr{ö}dinger and Maxwell equations, and complementing previous explicit solutions for specific perturbations (e.g., decreasing a single mass). In particular, we study monatomic crystals with nearest-neighbor harmonic interactions, corresponding to square lattices of masses and springs, and prove that arbitrary localized perturbations that decrease the net mass lead to localized vibrating modes. The proof employs a straightforward variational method that should be extensible to other discrete lattices, interactions, and perturbations.

cond-mat.other↗

arXiv2025

Operator representation of spatiotemporal quantum correlations

While quantum correlations between two spacelike-separated systems are fully encoded by the bipartite density operator associated with the joint system, there does not exist an analogous operator representing general quantum correlations across space and time. This is in stark contrast to the case of classical random variables, which make no distinction between spacelike and timelike correlations. Despite this, we show that spatiotemporal correlations between light-touch observables (i.e., observables whose eigenvalues are all equal in magnitude) admit a unique operator representation for arbitrary timelike-separated quantum systems. A special case of our result reproduces generalized Pauli observables and pseudo-density matrices, which have, up until now, only been defined for multi-qubit systems. In the case of qutrit systems, we use our results to illustrate an intriguing connection between light-touch observables and symmetric, informationally complete, positive operator-valued measures (SIC-POVMs).

quant-ph↗

arXiv2024

Towards structure-preserving quantum encodings

Harnessing the potential computational advantage of quantum computers for machine learning tasks relies on the uploading of classical data onto quantum computers through what are commonly referred to as quantum encodings. The choice of such encodings may vary substantially from one task to another, and there exist only a few cases where structure has provided insight into their design and implementation, such as symmetry in geometric quantum learning. Here, we propose the perspective that category theory offers a natural mathematical framework for analyzing encodings that respect structure inherent in datasets and learning tasks. We illustrate this with pedagogical examples, which include geometric quantum machine learning, quantum metric learning, topological data analysis, and more. Moreover, our perspective provides a language in which to ask meaningful and mathematically precise questions for the design of quantum encodings and circuits for quantum machine learning tasks.

quant-ph↗

arXiv2024

Quantum Mutual Information in Time

While the quantum mutual information is a fundamental measure of quantum information, it is only defined for spacelike-separated quantum systems. Such a limitation is not present in the theory of classical information, where the mutual information between two random variables is well-defined irrespective of whether or not the variables are separated in space or separated in time. Motivated by this disparity between the classical and quantum mutual information, we employ the pseudo-density matrix formalism to define a simple extension of quantum mutual information into the time domain. As in the spatial case, we show that such a notion of quantum mutual information in time serves as a natural measure of correlation between timelike-separated systems, while also highlighting ways in which quantum correlations distinguish between space and time. We also show how such quantum mutual information is time-symmetric with respect to quantum Bayesian inversion, and then we conclude by showing how mutual information in time yields a Holevo bound for the amount of classical information that may be extracted from sequential measurements on an ensemble of quantum states.

quant-ph↗

arXiv2024

Time-symmetric correlations for open quantum systems

Two-time expectation values of sequential measurements of dichotomic observables are known to be time symmetric for closed quantum systems. Namely, if a system evolves unitarily between sequential measurements of dichotomic observables $\mathscr{O}_{A}$ followed by $\mathscr{O}_{B}$, then it necessarily follows that $\langle\mathscr{O}_{A}\,,\mathscr{O}_{B}\rangle=\langle\mathscr{O}_{B}\,,\mathscr{O}_{A}\rangle$, where $\langle\mathscr{O}_{A}\,,\mathscr{O}_{B}\rangle$ is the two-time expectation value corresponding to the product of the measurement outcomes of $\mathscr{O}_{A}$ followed by $\mathscr{O}_{B}$, and $\langle\mathscr{O}_{B}\,,\mathscr{O}_{A}\rangle$ is the two-time expectation value associated with the time reversal of the unitary dynamics, where a measurement of $\mathscr{O}_{B}$ precedes a measurement of $\mathscr{O}_{A}$. In this work, we show that a quantum Bayes' rule implies a time symmetry for two-time expectation values associated with open quantum systems, which evolve according to a general quantum channel between measurements. Such results are in contrast with the view that processes associated with open quantum systems -- which may lose information to their environment -- are not reversible in any operational sense. We give an example of such time-symmetric correlations for the amplitude-damping channel, and we propose an experimental protocol for the potential verification of the theoretical predictions associated with our results.

quant-ph↗

arXiv2024

Reversing information flow: retrodiction in semicartesian categories

In statistical inference, retrodiction is the act of inferring potential causes in the past based on knowledge of the effects in the present and the dynamics leading to the present. Retrodiction is applicable even when the dynamics is not reversible, and it agrees with the reverse dynamics when it exists, so that retrodiction may be viewed as an extension of inversion, i.e., time-reversal. Recently, an axiomatic definition of retrodiction has been made in a way that is applicable to both classical and quantum probability using ideas from category theory. Almost simultaneously, a framework for information flow in in terms of semicartesian categories has been proposed in the setting of categorical probability theory. Here, we formulate a general definition of retrodiction to add to the information flow axioms in semicartesian categories, thus providing an abstract framework for retrodiction beyond classical and quantum probability theory. More precisely, we extend Bayesian inference, and more generally Jeffrey's probability kinematics, to arbitrary semicartesian categories.

math.CT↗

arXiv2023

SVD Entanglement Entropy

In this paper, we introduce a new quantity called SVD entanglement entropy. This is a generalization of entanglement entropy in that it depends on two different states, as in pre- and post-selection processes. This SVD entanglement entropy takes non-negative real values and is bounded by the logarithm of the Hilbert space dimensions. The SVD entanglement entropy can be interpreted as the average number of Bell pairs distillable from intermediates states. We observe that the SVD entanglement entropy gets enhanced when the two states are in the different quantum phases in an explicit example of the transverse-field Ising model. Moreover, we calculate the Rényi SVD entropy in various field theories and examine holographic calculations using the AdS/CFT correspondence.

hep-th↗

arXiv2023

Virtual quantum broadcasting

The quantum no-broadcasting theorem states that it is impossible to produce perfect copies of an arbitrary quantum state, even if the copies are allowed to be correlated. Here we show that, although quantum broadcasting cannot be achieved by any physical process, it can be achieved by a virtual process, described by a Hermitian-preserving trace-preserving map. This virtual process is canonical: it is the only map that broadcasts all quantum states, is covariant under unitary evolution, is invariant under permutations of the copies, and reduces to the classical broadcasting map when subjected to decoherence. We show that the optimal physical approximation to the canonical broadcasting map is the optimal universal quantum cloning, and we also show that virtual broadcasting can be achieved by a virtual measure-and-prepare protocol, where a virtual measurement is performed, and, depending on the outcomes, two copies of a virtual quantum state are generated. Finally, we use canonical virtual broadcasting to prove a uniqueness result for quantum states over time.

quant-ph↗

arXiv2023

On dynamical measures of quantum information

In this work, we use the theory of quantum states over time to define an entropy $S(ρ,\mathcal{E})$ associated with quantum processes $(ρ,\mathcal{E})$, where $ρ$ is a state and $\mathcal{E}$ is a quantum channel responsible for the dynamical evolution of $ρ$. The entropy $S(ρ,\mathcal{E})$ is a generalization of the von Neumann entropy in the sense that $S(ρ,\mathrm{id})=S(ρ)$ (where $\mathrm{id}$ denotes the identity channel), and is a dynamical analogue of the quantum joint entropy for bipartite states. Such an entropy is then used to define dynamical formulations of the quantum conditional entropy and quantum mutual information, and we show such information measures satisfy many desirable properties, such as a quantum entropic Bayes' rule. We also use our entropy function to quantify the information loss/gain associated with the dynamical evolution of quantum systems, which enables us to formulate a precise notion of information conservation for quantum processes.

quant-ph↗

arXiv2023

Axioms for retrodiction: achieving time-reversal symmetry with a prior

We propose a category-theoretic definition of retrodiction and use it to exhibit a time-reversal symmetry for all quantum channels. We do this by introducing retrodiction families and functors, which capture many intuitive properties that retrodiction should satisfy and are general enough to encompass both classical and quantum theories alike. Classical Bayesian inversion and all rotated and averaged Petz recovery maps define retrodiction families in our sense. However, averaged rotated Petz recovery maps, including the universal recovery map of Junge-Renner-Sutter-Wilde-Winter, do not define retrodiction functors, since they fail to satisfy some compositionality properties. Among all the examples we found of retrodiction families, the original Petz recovery map is the only one that defines a retrodiction functor. In addition, retrodiction functors exhibit an inferential time-reversal symmetry consistent with the standard formulation of quantum theory. The existence of such a retrodiction functor seems to be in stark contrast to the many no-go results on time-reversal symmetry for quantum channels. One of the main reasons is because such works defined time-reversal symmetry on the category of quantum channels alone, whereas we define it on the category of quantum channels and quantum states. This fact further illustrates the importance of a prior in time-reversal symmetry.

quant-ph↗

arXiv2023

From time-reversal symmetry to quantum Bayes' rules

Bayes' rule $\mathbb{P}(B|A)\mathbb{P}(A)=\mathbb{P}(A|B)\mathbb{P}(B)$ is one of the simplest yet most profound, ubiquitous, and far-reaching results of classical probability theory, with applications in any field utilizing statistical inference. Many attempts have been made to extend this rule to quantum systems, the significance of which we are only beginning to understand. In this work, we develop a systematic framework for defining Bayes' rule in the quantum setting, and we show that a vast majority of the proposed quantum Bayes' rules appearing in the literature are all instances of our definition. Moreover, our Bayes' rule is based upon a simple relationship between the notions of state over time and a time-reversal symmetry map, both of which are introduced here.

quant-ph↗

arXiv2023

Bayesian inversion and the Tomita-Takesaki modular group

We show that conditional expectations, optimal hypotheses, disintegrations, and adjoints of unital completely positive maps, are all instances of Bayesian inverses. We study the existence of the latter by means of the Tomita-Takesaki modular group and we provide extensions of a theorem of Takesaki as well as a theorem of Accardi and Cecchini to the setting of not necessarily faithful states on finite-dimensional $C^*$-algebras.

math.OA↗

arXiv2022

On quantum states over time

In 2017, D. Horsman, C. Heunen, M. Pusey, J. Barrett, and R. Spekkens proved that there is no physically reasonable assignment that takes a quantum channel and an initial state and produces a joint state on the tensor product of the input and output spaces. The interpretation was that there is a clear distinction between space and time in the quantum setting that is not visible classically, where in the latter, one can freely use Bayes' theorem to go between joint states and marginals with noisy channels. In this paper, we prove that there actually is such a physically reasonable assignment, bypassing the no-go result of Horsman et al., and we illustrate that this is achievable by restricting the domain of their assignment to a domain which represents the given data more faithfully.

quant-ph↗

arXiv2021

The information loss of a stochastic map

We provide a stochastic extension of the Baez-Fritz-Leinster characterization of the Shannon information loss associated with a measure-preserving function. This recovers the conditional entropy and a closely related information-theoretic measure that we call conditional information loss. Although not functorial, these information measures are semi-functorial, a concept we introduce that is definable in any Markov category. We also introduce the notion of an entropic Bayes' rule for information measures, and we provide a characterization of conditional entropy in terms of this rule.

cs.IT↗

arXiv2021

Conditional Distributions for Quantum Systems

Conditional distributions, as defined by the Markov category framework, are studied in the setting of matrix algebras (quantum systems). Their construction as linear unital maps are obtained via a categorical Bayesian inversion procedure. Simple criteria establishing when such linear maps are positive are obtained. Several examples are provided, including the standard EPR scenario, where the EPR correlations are reproduced in a purely compositional (categorical) manner. A comparison between the Bayes map, the Petz recovery map, and the Leifer-Spekkens acausal belief propagation is provided, illustrating some similarities and key differences.

quant-ph↗