POVM Research Articles

In this work, we propose a novel quantum-informed epistemic framework that extends the classical notion of probability by integrating plausibility, credibility, and possibility as distinct yet complementary measures of uncertainty. This enriched quadruple (P, Pl, Cr, Ps) enables a deeper characterization of quantum systems and decision-making processes under partial, noisy, or ambiguous information. Our formalism generalizes the Born rule within a multi-valued logic structure, linking Positive Operator-Valued Measures (POVMs) with data-driven plausibility estimators, agent-based credibility priors, and fuzzy-theoretic possibility functions. We develop a hybrid classical–quantum inference engine that computes a vectorial aggregation of the quadruples, enhancing robustness and semantic expressivity in contexts where classical probability fails to capture non-Kolmogorovian phenomena such as entanglement, contextuality, or decoherence. The approach is validated through three real-world application domains—quantum cybersecurity, quantum AI, and financial computing—where the proposed model outperforms standard probabilistic reasoning in terms of accuracy, resilience to noise, interpretability, and decision stability. Comparative analysis against QBism, Dempster–Shafer, and fuzzy quantum logic further demonstrates the uniqueness of architecture in both operational semantics and practical outcomes. This contribution lays the groundwork for a new theory of epistemic quantum computing capable of modelling and acting under uncertainty beyond traditional paradigms.

Abstract We discuss and experimentally demonstrate the role of quantum coherence in a sequence of two measurements collected at different times using weak measurements. For this purpose, we have realized a weak-sequential measurement protocol with photonic qubits, where the first measurement is carried out as a positive operator-valued measure, whereas the second one is a projective operation. We determine the quasiprobability distributions associated to this procedure using both the commensurate and the Margenau-Hill quasiprobabilities. By tuning the weak measurements, we obtain a quasidistribution that may or may not exhibit negative parts, depending on the suitability of a contextual model for describing the experiment. Our results show how quasidistributions may find application in inspecting quantum monitoring, when part of the initial quantum coherence needs to be preserved.

Abstract A frame is a generalization of a basis of a vector space to a redundant overspanning set whose vectors are linearly dependent. Frames find applications in signal processing and quantum information theory. We present a genetic algorithm that can generate maximally orthogonal frames (MOFs) of arbitrary size n in d-dimensional complex space. First, we formalize the concept of MOF and demonstrate that it depends on the choice of an energy function to weigh the different pairwise overlaps between vectors. Then, we discuss the relation between different energy functions and well-known frame varieties such as tight and Grassmannian frames and complex projective p-designs. Obtaining MOFs poses a global non-convex minimization problem. We discuss the relation with established numerical problems such as the Thomson problem and the problem of finding optimal packings in complex projective space. To tackle the minimization, we design a hybrid genetic algorithm that features local optimization of the parents. To assess the performance of the algorithm, we propose two visualization techniques that allow us to analyze the coherence and uniformity of high-dimensional frames. The genetic algorithm is able to produce highly-symmetric universal frames, such as equiangular tight frames, symmetric, informationally complete, positive operator-valued measurements and maximal sets of mutually unbiased bases, for configurations of up to d = 6 and n = 36, with runtimes of the order of several minutes on a regular desktop computer for the largest configurations.

Quantum simulation, the study of strongly correlated quantum matter using synthetic quantum systems, has been the most successful application of quantum computers to date. It often requires determining observables with high precision, for example when studying critical phenomena near quantum phase transitions. Thus, readout errors must be carefully characterized and mitigated in data postprocessing, using scalable and noise-model agnostic protocols. We present a readout error-mitigation protocol that uses only single-qubit Pauli measurements and avoids experimentally challenging randomized measurements. The proposed approach captures a very broad class of correlated noise models and is scalable to large qubit systems. It is based on a complete and efficient characterization of few-qubit correlated positive operator-valued measures, using overlapping detector tomography. To assess the effectiveness of the protocol, observables are extracted from simulations involving up to 100 qubits employing readout errors obtained from experiments with superconducting qubits.

Abstract The Jordan product plays a pivotal role in quantum information theory, particularly within the framework of quantum compatibility. This research introduces the concept of the triple Jordan product and establishes its fundamental properties. Furthermore, for qubit systems, we characterize Jordan compatibility, Lie compatibility of three positive operator-valued measures and elucidate the interrelationships among Lie compatibility, joint compatibility, and Jordan compatibility.

Abstract Einstein–Podolsky–Rosen (EPR) steering, as an asymmetric form of nonlocal correlations, is a crucial resource for one-sided device-independent quantum information tasks. Although many methods have been proposed to certify steering, efficient detection of EPR steerable states remains an important and difficult issue, and it is desirable to study it from as many angles as possible. In this work, a class of finite tight frames, equiangular tight frames, are applied to explore the problem of certifying EPR steering, where each frame can be used to construct a positive operator-valued measure. More specifically, based on two different forms of correlation matrices derived from equiangular tight frames, two steerability criteria that are readily computable for arbitrary dimensional bipartite systems are proposed. These criteria are illustrated via several detailed examples and their relative advantage is exhibited in certain cases.

Abstract We construct relational observables in group field theory (GFT) in terms of covariant positive operator-valued measures (POVMs), using techniques developed in the context of quantum reference frames (QRFs). We focus on matter QRFs; this can be generalized to other types of frames within the same POVM-based framework. The resulting family of relational observables provides a covariant framework to extract localized observables from GFT, which is typically defined in a perspective-neutral way. Then, we compare this formalism with previous proposals for relational observables in GFT. We find that our QRF-based relational observables overcome the intrinsic limitations of previous proposals while reproducing the same continuum limit results concerning expectation values of the number and volume operators on coherent states. Nonetheless, there can be important differences for more complex operators, as well as for other types of GFT states. Finally, we also use a specific class of POVMs to show how to project states and operators from the more general perspective-neutral GFT Fock space to a perspective-dependent one where a scalar matter field plays the role of a relational clock.

Abstract Antidistinguishability, a key concept in quantum information, describes quantum states that can be excluded by null measurement outcomes, thus enabling indirect state certification through incompatible measurements. This work systematically explores the fundamental properties of quantum state antidistinguishability. First, we present several essential characteristics of antidistinguishable quantum states. Leveraging the Bloch representation, we establish a rigorous correspondence between quantum states and positive-operator-valued measures (POVMs). Through this approach, we prove that in qubit systems, real quantum states can be antidistinguished only by real POVMs, while complex states necessitate complex POVMs. However, the scenario becomes significantly more intricate in high-dimensional quantum systems. Finally, we derive the general forms for a three-state antidistinguishable set through real and complex POVMs, respectively, achieving a complete solution to the antidistinguishability problem within qubit systems.

Abstract Generative models aim to learn the probability distributions underlying data, enabling the generation of new, realistic samples. Quantum-inspired generative models, such as Born machines based on the matrix product state (MPS) framework, have demonstrated remarkable capabilities in unsupervised learning tasks. This study advances the Born machine paradigm by introducing trainable token embeddings through positive operator-valued measurements (POVMs), replacing the traditional approach of static tensor indices. Key technical innovations include encoding tokens as quantum measurement operators with trainable parameters and leveraging QR decomposition to adjust the physical dimensions of the MPS. This approach maximizes the utilization of operator space and enhances the model’s expressiveness. Empirical results on RNA data demonstrate that the proposed method significantly reduces negative log-likelihood (NLL) compared to one-hot embeddings, with higher physical dimensions further enhancing single-site probabilities and multi-site correlations. The model also outperforms GPT-2 in single-site estimation and achieves competitive correlation modeling, showcasing the potential of trainable POVM embeddings for complex data correlations in quantum-inspired sequence modeling.

We develop a relativistic framework of integral quantization applied to the motion of spinless particles in the four-dimensional Minkowski spacetime. The proposed scheme is based on coherent states generated by the action of the Heisenberg–Weyl group and has been motivated by the Hamiltonian description of the geodesic motion in General Relativity. We believe that this formulation should also allow for a generalization to the motion of test particles in curved spacetimes. A key element in our construction is the use of suitably defined positive operator-valued measures. We show that this approach can be used to quantize the one-dimensional nonrelativistic harmonic oscillator, recovering the standard Hamiltonian as obtained by the canonical quantization. A direct application of our model, including a computation of transition amplitudes between states characterized by fixed positions and momenta, is postponed to a forthcoming article.

In a multiqubit system, when a detector's resolution is insufficient to distinguish between qubits, individual qubit measurement becomes impractical. This limitation results in coarse-grained (CG) measurements that dilute system information and, more critically, hinder accurate assessment of qubit correlations, making quantum state tomography (QST) infeasible. To address this issue, we propose a scheme to utilize CG measurements for QST by integrating two-qubit gates and optimizing their corresponding positive operator-valued measures (POVMs) using parametrized quantum circuits. Specifically, we focus on constructing generalized symmetric informationally complete POVMs by maximizing the von Neumann entropy of the Gram matrix of these CG POVMs. Our numerical simulations demonstrate the effectiveness of this approach for QST using a two-qubit model and further verify the scalability of our scheme toward N-qubit systems. Published by the American Physical Society 2025

Stationary quantum information sources emit sequences of correlated qudits-that is, structured quantum stochastic processes. If an observer performs identical measurements on a qudit sequence, the outcomes are a realization of a classical stochastic process. We introduce quantum-information-theoretic properties for separable qudit sequences that serve as bounds on the classical information properties of subsequent measured processes. For sources driven by hidden Markov dynamics, we describe how an observer can temporarily or permanently synchronize to the source's internal state using specific positive operator-valued measures or adaptive measurement protocols. We introduce a method for approximating an information source with an independent and identically distributed, Markov, or larger memory model through tomographic reconstruction. We identify broad classes of separable processes based on their quantum information properties and the complexity of measurements required to synchronize to and accurately reconstruct them.

Self-testing positive operator-valued measures and certifying randomness

Abstract The Husimi function (Q-function) of a quantum state is the distribution function of the density operator in the coherent state representation. It is widely used in theoretical research, such as in quantum optics. The Wehrl entropy is the Shannon entropy of the Husimi function, and is non-zero even for pure states. This entropy has been extensively studied in mathematical physics. Recent research also suggests a significant connection between the Wehrl entropy and many-body quantum entanglement in spin systems. We investigate the statistical interpretation of the Husimi function and the Wehrl entropy, taking the system of N spin-1/2 particles as an example. Due to the completeness of coherent states, the Husimi function and Wehrl entropy can be explained via the positive operator-valued measurement (POVM) theory, although the coherent states are not a set of orthonormal basis. Here, with the help of the Bayes’ theorem, we provide an alternative probabilistic interpretation for the Husimi function and the Wehrl entropy. This interpretation is based on direct measurements of the system, and thus does not require the introduction of an ancillary system as in the POVM theory. Moreover, under this interpretation the classical correspondences of the Husimi function and the Wehrl entropy are just phase-space probability distribution function of N classical tops, and its associated entropy, respectively. Therefore, this explanation contributes to a better understanding of the relationship between the Husimi function, Wehrl entropy, and classical-quantum correspondence. The generalization of this statistical interpretation to continuous-variable systems is also discussed.

We offer new results and new directions in the study of operator-valued kernels and their factorizations. Our approach provides both more explicit realizations and new results, as well as new applications. These include: (i) an explicit covariance analysis for Hilbert space-valued Gaussian processes, (ii) optimization results for quantum gates (from quantum information), (iii) new results for positive operator-valued measures, and (iv) a new approach/result in inverse problems for quantum measurements.

We study the evolution of a quantum many-body system driven by two competing measurements, which induces a topological entanglement transition between two distinct area law phases. We employ a positive operator-valued measurement with variable coupling between the system and detector within free fermion dynamics. This approach allows us to continuously track the universal properties of the transition between projective and continuous monitoring. Our findings suggest that the percolation universality of the transition in the projective limit is unstable when the system-detector coupling is reduced.

The POVM theorem is a central result in Bohmian mechanics, grounding the measurement formalism of standard quantum mechanics in a statistical analysis based on the quantum equilibrium hypothesis (the Born rule for Bohmian particle positions). It states that the outcome statistics of an experiment are described by a positive operator-valued measure (POVM) acting on the Hilbert space of the measured system. In light of recent debates about the scope and status of this result, we provide a systematic presentation of the POVM theorem and its underlying assumptions with a focus on their conceptual foundations and physical justifications. We conclude with a brief discussion of the scope of the POVM theorem-especially the sense in which it does (and does not) place limits on what is "measurable" in Bohmian mechanics.

Abstract We point out that the Gaussian wave-packet formalism can serve as a concrete realization of the joint measurement of position and momentum, which is an essential element in understanding Heisenberg’s original philosophy of the uncertainty principle, in line with the universal framework of error, disturbance, and their uncertainty relations developed by Lee and Tsutsui. We show that our joint measurement in the Gaussian phase space, being a Positive-Operator-Valued-Measure (POVM) measurement, smoothly interpolates between the projective measurements of position and momentum. We, for the first time, have obtained the Lee–Tsutsui (LT) error and the refined Lee error for the position-momentum measurement. We find that the LT uncertainty relation becomes trivial, $0=0$, in the limiting case of projective measurement of either position or momentum. Remarkably, in contrast to the LT relation, the refined Lee uncertainty relation, which assesses errors for local representability, provides a constant lower bound unaffected by these limits and is invariably saturated, for a pure Gaussian initial state. The obtained lower bound is in agreement with Heisenberg’s value.

The emergence of classical behavior in quantum theory is often ascribed to the interaction of a quantum system with its environment, which can be interpreted as environmental monitoring of the system. As a result, off-diagonal elements of the density matrix of the system are damped in the basis of a preferred observable, often taken to be the position, leading to the phenomenon of decoherence. This effect can be modeled dynamically in terms of a Lindblad equation driven by the position operator. Here the question of decoherence resulting from a monitoring of position and momentum, i.e., a phase-space measurement, by the environment is addressed. There is no standard quantum observable corresponding to the detection of phase-space points, which is forbidden by Heisenberg's uncertainty principle. This issue is addressed by use of a coherent-state-based positive operator-valued measure for modeling phase-space monitoring by the environment. In this scheme, decoherence in phase space implies the diagonalization of the density matrix in both position and momentum representations. This is shown to be linked to a Lindblad equation where position and momentum appear as two independent Lindblad operators.

Quantum illumination (QI) is an entanglement-based protocol for improving LiDAR/radar detection of unresolved targets beyond what a classical LiDAR/radar of the same average transmitted energy can do. Originally proposed by Seth Lloyd as a discrete-variable quantum LiDAR, it was soon shown that his proposal offered no quantum advantage over its best classical competitor. Continuous-variable, specifically Gaussian-state, QI has been shown to offer a true quantum advantage, both in theory and in table-top experiments. Moreover, despite its considerable drawbacks, the microwave version of Gaussian-state QI continues to attract research attention. A recent QI study by Armanpreet Pannu, Amr Helmy, and Hesham El Gamal (PHE), however, has: (i) combined the entangled state from Lloyd’s QI with the channel models from Gaussian-state QI; (ii) proposed a new positive operator-valued measurement for that composite setup; and (iii) claimed that, unlike Gaussian-state QI, PHE QI achieves the Nair–Gu lower bound on QI target-detection error probability at all noise brightnesses. PHE’s analysis was asymptotic, i.e., it presumed infinite-dimensional entanglement. The current paper works out the finite-dimensional performance of PHE QI. It shows that there is a threshold value for the entangled-state dimensionality below which there is no quantum advantage, and above which the Nair–Gu bound is approached asymptotically. Moreover, with both systems operating with error-probability exponents 1 dB lower than the Nair–Gu bound, PHE QI requires enormously higher entangled-state dimensionality than does Gaussian-state QI to achieve useful error probabilities in both high-brightness (100 photons/mode) and moderate-brightness (1 photon/mode) noise. Furthermore, neither system has an appreciable quantum advantage in low-brightness (much less than 1 photon/mode) noise.