Inference is a versatile tool that underlies scientific discovery, machine learning, and everyday decision-making: it describes how an agent updates a probability distribution as partial information is acquired from multiple measurements, reducing ignorance about a system's latent state. We define an inferential efficiency as the ratio of information gain to cumulative memory erasure cost, with inefficiency arising from unexploited correlations between the measured system and memories, and/or between memories and environment (noise). Using this efficiency, we benchmark two limiting measurement paradigms: sequential, in which the same memory is exploited iteratively, and parallel, in which many memories are exploited simultaneously. In both cases, the minimal erasure cost reflects correlations across memories: temporal in sequential, spatial in parallel. Remarkably, when all system-memory correlations are exploited for inference, both paradigms attain the same minimal erasure cost, even in the presence of noise. Conversely, the parallel paradigm performs better in the presence of unexploited correlations, stemming from hidden memories' degrees of freedom. This approach provides a quantitative, physically grounded criterion to compare inference strategies, determine their efficiency, and link target information gains to their minimal entropic cost.
Inference is a versatile tool that underlies scientific discovery, machine learning, and everyday decision-making: it describes how an agent updates a probability distribution as partial information is acquired from multiple measurements, reducing ignorance about a system's latent state. We define an inferential efficiency as the ratio of information gain to cumulative memory erasure cost, with inefficiency arising from unexploited correlations between the measured system and memories, and/or between memories and environment (noise). Using this efficiency, we benchmark two limiting measurement paradigms: sequential, in which the same memory is exploited iteratively, and parallel, in which many memories are exploited simultaneously. In both cases, the minimal erasure cost reflects correlations across memories: temporal in sequential, spatial in parallel. Remarkably, when all system-memory correlations are exploited for inference, both paradigms attain the same minimal erasure cost, even in the presence of noise. Conversely, the parallel paradigm performs better in the presence of unexploited correlations, stemming from hidden memories' degrees of freedom. This approach provides a quantitative, physically grounded criterion to compare inference strategies, determine their efficiency, and link target information gains to their minimal entropic cost.
This paper investigates the potential for accelerating probability inference using quantum entanglement. We propose a novel approach where probability models are mapped onto entangled quantum bit (qubit) systems, enabling parallel exploration of all possible probabilistic branches. The core claim is that this entanglement-assisted method can significantly reduce the computational time required for complex probabilistic models, particularly those with a large number of possible outcomes. The mechanism leverages the inherent correlations within entangled qubits to perform parallel computations. We introduce a theoretical framework outlining this process and discuss the potential advantages over classical sequential methods. The exploration of this area promises to revolutionize probabilistic computing, opening new avenues for tackling computationally intensive problems in fields like machine learning, statistical physics, and decision theory. The efficiency gains stem from the ability to simultaneously evaluate multiple probability branches, a capability fundamentally unavailable to classical computers. The ultimate goal is to demonstrate a quantifiable speedup in inference processes.
This paper investigates the application of Bayesian inference for optimizing quantum measurement schemes. Traditional quantum measurement strategies often rely on heuristic approaches or specific system constraints, potentially leading to suboptimal outcomes. We propose a framework where the quantum measurement process is modeled as a Bayesian inference problem. This allows us to systematically explore the measurement landscape, incorporating prior knowledge and iteratively refining the measurement strategy based on observed outcomes. The core of this approach lies in formulating the measurement problem as a probabilistic model and utilizing Bayesian updating to determine the optimal measurement basis or parameters. We demonstrate, through theoretical analysis and conceptual design, that Bayesian optimization can lead to improved measurement accuracy and efficiency compared to conventional methods. The key benefits include reduced measurement errors, enhanced signal-to-noise ratios, and potentially the ability to extract more information from quantum systems. This work presents a novel methodology for quantum measurement, bridging the gap between classical Bayesian optimization and the unique properties of quantum mechanics.
This thesis concerns the foundations of inference – probability theory,entropic inference, information geometry, etc. – and its application to the Entropic Dynamics (ED) approach to Quantum Mechanics (QM) [21, 22, 41, 53, 56–61, 150–153, 165, 195, 196, 268]. The first half of this thesis, chapters 2-6, concern the development of the inference framework. We begin in chapter 2 by discussing de- ductive inference, which involves formal logic and it’s role in access- ing the truth of propositions. We eventually discover that deductive inference is incomplete, in that it can’t address situations in which we have incomplete information. This necessitates a theory of inductive inference (probability theory), which is developed in chapter 3. Prob- ability theory is derived as a framework for manipulating degrees of belief of propositions, in a way which is consistent with its deduc- tive counterpart [47]. In chapter 4 we review the construction of en- tropic inference as a means for updating our beliefs in the presence of new information. The entropy functional is designed through the pro- cess of eliminative induction by imposing a principle of minimal updating (PMU) and various constraints [47, 238, 242, 266, 267]. Chapter 5 con- siders the design of another entropic functional, the total correlation and all its variants, for the purposes of ranking join distributions with respect to their correlations [46]. Finally, in chapter 6, we discuss the application of a special case of the correlation functionals from chap- ter 5, the mutual information, to problems in experimental physics and machine learning [42–45]. The second half of this thesis, chapters 7-9, concerns the ED ap- proach to QM. In particular, chapters 8 and 9 involve the inclusion of particles with spin 1/2 into the framework [41, 61]. These devel- opments are the main contribution of this thesis to the body of work in the ED approach. The problem is defined as an application of in- ference to the dynamics of quantum particles which have definite yet unknown positions and follow continuous trajectories. Through the method of maximum entropy developed in chapter 4, we can deter- mine the transition probability that these particles will move from one location to another. Geometric algebra (GA) [90, 140] is chosen as the preferred representation for the algebra of spin, which is then introduced through constraints in the maximum entropy method. A quantum mechanics is subsequently developed by constructing an epistemic phase space of probabilities
This paper introduces a novel approach to Bayesian inference that draws inspiration from the principles of quantum mechanics, specifically superposition and decoherence. Traditional Bayesian inference often struggles with exploring the vast and complex inference landscape, leading to slow convergence and potential inaccuracies. Our method utilizes a quantum-inspired representation of probability distributions as quantum states, enabling the simultaneous consideration of multiple hypotheses within a superposition. Controlled decoherence protocols are then employed to guide the inference process, effectively selecting the most probable hypothesis while minimizing computational cost. We demonstrate, through theoretical analysis and mathematical formulations, that this approach can dramatically improve the exploration of the inference landscape, resulting in faster convergence and enhanced accuracy compared to conventional Bayesian methods. The core of this work lies in the mapping of Bayesian model parameters to quantum states and the implementation of controlled decoherence, offering a fundamentally new perspective on probabilistic reasoning. This research aims to provide a framework for tackling complex Bayesian problems efficiently and effectively, potentially unlocking new possibilities in various fields, including machine learning, statistics, and scientific discovery.
In reinforcement learning, an agent interacts sequentially with an environment to maximize a reward, receiving only partial, probabilistic feedback. This creates a fundamental exploration-exploitation trade-off: the agent must explore to learn the hidden dynamics while exploiting this knowledge to maximize its target objective. While extensively studied classically, applying this framework to quantum systems requires dealing with hidden quantum states that evolve via unknown dynamics. We formalize this problem via a framework where the environment maintains a hidden quantum memory evolving via unknown quantum channels, and the agent intervenes sequentially using quantum instruments. For this setting, we adapt an optimistic maximum-likelihood estimation algorithm. We extend the analysis to continuous action spaces, allowing us to model general positive operator-valued measures (POVMs). By controlling the propagation of estimation errors through quantum channels and instruments, we prove that the cumulative regret of our strategy scales as O(K) over K episodes. Furthermore, via a reduction to the multi-armed quantum bandit problem, we establish information-theoretic lower bounds demonstrating that this sublinear scaling is strictly optimal up to polylogarithmic factors. As a physical application, we consider state-agnostic work extraction. When extracting free energy from a sequence of non-i.i.d. quantum states correlated by a hidden memory, any lack of knowledge about the source leads to thermodynamic dissipation. In our setting, the mathematical regret exactly quantifies this cumulative dissipation. Using our adaptive algorithm, the agent uses past energy outcomes to improve its extraction protocol on the fly, achieving sublinear cumulative dissipation, and, consequently, an asymptotically zero dissipation rate.
Causal discovery, the process of inferring cause-and-effect relationships from observational data, is a fundamental challenge across science and engineering. Traditional approaches often rely on assumptions about functional forms, linearity, or acyclicity, which may not hold in complex systems. This paper explores the "entropic geometry" framework as a powerful, non-parametric approach to causal discovery. By representing causal structures within the convex cone of attainable marginal entropies, this method can uncover intricate, non-convex constraints that are inaccessible to standard probabilistic techniques, particularly in scenarios involving hidden variables or quantum entanglement. We review the theoretical underpinnings of entropic geometry, illustrating how entropic inequalities precisely characterize the compatibility of observed data with various causal models. Furthermore, we discuss its versatility in expanding the set of known entropic constraints for complex scenarios, including multi-party interactions and systems with shared randomness. The paper highlights how this geometric perspective provides a rigorous and expressive language for understanding causal relationships, bridging concepts from information theory, statistical physics, and machine learning. We discuss the implications for empirical causal inference, computational challenges, and future research directions, positioning entropic geometry as a critical tool for advancing causal understanding in diverse scientific disciplines.