arXiv subjects

Xing M. Wang

Publications and source records attributed to Xing M. Wang.

15 recordsLinked to original sources

arXiv2026

Quantum Measurement Without Collapse or Many Worlds: The Branched Hilbert Subspace Interpretation

We propose the Branched Hilbert Subspace Interpretation (BHSI) as an alternative perspective on quantum measurement. BHSI describes measurement as a unitary branching of the local Hilbert space into decoherent, independent, and unitarily evolving subspaces, while updating observer states (through their equipment) by causally engaging and disengaging operators. Unlike the Copenhagen Interpretation (CI), BHSI avoids wave function collapse while maintaining the Born rule through the branch weights associated with the initial system state. Unlike the Many-Worlds Interpretation (MWI), BHSI sidesteps parallel worlds by entangling branches with the local environment within a single world. We compare BHSI features with those of CI, MWI, and Bohmian Mechanics (BM). We investigate its implications for the double-slit experiment, Bell tests, Wigner and his friend, black hole radiation, and the delayed-choice quantum eraser. We examine quantum teleportation, demonstrating that locally controlled decoherence and recoherence processes (CDRP) can be observed. Specifically, we suggest experiments using modern Stern-Gerlach interferometers (SGI) to visualize the CDRP, measure branch weights that encode the Born rule, and predict the electromagnetic (EM) phase shift resulting from the independent unitary evolution of decoherent branches. BHSI thus provides a minimalist alternative to interpretations based on collapse or many-worlds.

quant-ph↗

arXiv2026

Probability Bracket Notation: Multivariable Systems and Static Bayesian Networks

We extend Probability Bracket Notation (PBN), inspired by the Dirac notation in quantum mechanics, to multivariable probability systems and static Bayesian networks (BNs). By defining probability distributions and conditional expectations in a unified, basis-independent algebraic form, PBN provides a systematic way to represent and manipulate dependencies among random variables. Using the well-known Student BN as an illustrative probabilistic graphical model, we demonstrate prediction, bottom-up and top-down inference, and expectation calculations within the PBN framework. We show that, for a large N-node binary BN, after a one-time preprocessing, inference along a d-separable chain with k intermediate nodes requires O(k2^k) operations, compared to O(N2^N) for direct computation from the full joint distribution. We further extend PBN to networks with continuous variables, including linear Gaussian models, and introduce a hybrid Healthcare BN that combines discrete and continuous variables. In this model, discrete-display nodes serve as proxies for continuous parents, enabling user-specific predictions. Overall, PBN provides an operator-based framework that unifies representation and computation, with potential applications in education, data analytics, and machine learning.

cs.AI↗

arXiv2026

Einstein's Electron and Local Unitary Branching: Boundaries of Islands of Coherence and Quantum Nonlocality

The Branched Hilbert Subspace Interpretation (BHSI) aims to provide a unitary account of quantum measurement while maintaining a single-world ontology. The framework reexamines scenarios such as Einstein 1927 electron-diffraction thought experiment by treating measurement as a finite dynamical process of information recording, comprising a sequence of unitary operations: branching, engaging, and disengaging. This perspective motivates a testable proposal: a dual-layer experiment in which the particle transit time between layers is shorter than the sensor response time, enabling a direct probe of measurement timing and potentially uncommitted outcomes. We introduce the Island of Coherence (IOC) as an operationally isolated quantum system, mathematically described by a Local Hilbert Subspace (LHS), which coexists with the background spacetime and within which unitary branching occurs. Historically, the first quantization already implies this dual structure. Applying the Gleason and Busch theorems to local unitary branching, the Born rule follows from the amplitudes given in the initial state. Moreover, quantum nonlocality (e.g., in Bell tests or tunneling) arises naturally from the inner-product structure of the LHS, which possesses no intrinsic spacetime metric. BHSI thus provides a coherent framework in which relativistic causality and quantum correlations remain structurally compatible.

quant-ph↗

arXiv2026

Stern-Gerlach Interferometers with Dual Sensing: Probing Recoherence and Lifecycles of Islands of Coherence

The Branched Hilbert Subspace Interpretation (BHSI) addresses the quantum measurement problem by preserving unitary quantum evolution within a single world. Its central concept is the Island of Coherence (IOC), an operationally isolated, inseparable quantum system mathematically described by a Local Hilbert Space (LHS), which carries no intrinsic spacetime metric and coexists with the spacetime in which the IOC is embedded, a dual structure implicated by the first quantization. This paper advances BHSI on both experimental and conceptual frontiers. Experimentally, we propose a three-stage dual-sensing Stern-Gerlach interferometer (SGI) designed to probe the fuzzy spatiotemporal boundaries associated with IOC transitions. Stage 1 targets uncommitted timing events, manifested as sensor-detector mismatches; Stage 2 investigates conditional recoherence, a signature of local, time-extended branching; and Stage 3 employs controlled electromagnetic phase shifts to discriminate between unitary and retrocausal recoherence mechanisms. Conceptually, we introduce the lifecycle of IOCs, describing how coherent domains emerge, persist, and fragment across scales. We draw structural analogies between fuzzy IOC boundaries and phenomenological bag models in quantum field theory, and between primordial global Hilbert space fragmentation and Hilbert space fragmentation in many-body systems. Altogether, BHSI offers a consistent and experimentally testable approach to resolving the quantum measurement problem.

quant-ph↗

arXiv2025

From Dirac Notation to Probability Bracket Notation: Time Evolution and Path Integral under Wick Rotations

In this work, we advance the development of the Probability Bracket Notation (PBN), a formalism inspired by Dirac's notation in quantum mechanics, to provide a unified framework for probability modeling. We demonstrate that under a Special Wick Rotation (SWR), an imaginary-time map, the Schrödinger equation, the transition amplitude, and its associated path integral in Dirac notation transform into the master equation, the transition probability, and its Euclidean path integral in the PBN, from which we can reproduce the master equation, representing induced micro-diffusion processes. By extending this approach through a General Wick Rotation (GWR) and employing an anti-Hermitian wave-number operator, we perform parallel derivations of path integrals in both the Dirac and PBN frameworks. This leads to the formulation of the Euclidean Lagrangian for induced diffusions and the strong-damping harmonic oscillator (described by the Smoluchowski diffusion equation). Our findings highlight the versatility of the PBN in bridging quantum mechanics and stochastic processes, offering a coherent notation system for analyzing time evolution and path integrals across these domains.

math-ph↗

arXiv2025

Entropy Gain and Information Loss by Measurements

When the von Neumann entropy (VNE) of a system increases due to measurements, certain information is lost, some of which may be recoverable. We define information retrievability (IR) and information loss (IL) as functions of the density matrix through VNE to illustrate the relationship between gain and loss. We demonstrate that when a pure, unbiased m-qubit state collapses into a maximally mixed state, it experiences the maximal loss of information and the highest gain in entropy, equivalent to the m-bit classical Shannon entropy. We analyze the VNE, IR, and IL of single qubits, entangled photon pairs in Bell tests, three-qubit systems in quantum teleportation, multiple-qubit systems of GHZ and W states, and two-qubit Werner mixed states, emphasizing their IL dependence on parameters such as polarization bias and qubit count. Data exchange between two observers in Bell tests can recover some of the lost quantum information and eliminate the associated quantum entropy, even years later. The need to recover knowledge explains why no spooky action occurs at a distance. We show that measuring the Bell, GHZ, and marginally entangled Werner states yields the same minimum entropy gain (ln2) and equal minimal information loss (50 percent).

quant-ph↗

arXiv2025

Probability Bracket Notation: Markov Sequence Projector of Visible and Hidden Markov Models in Dynamic Bayesian Networks

With the symbolic framework of Probability Bracket Notation (PBN), the Markov Sequence Projector (MSP) is introduced to expand the evolution formula of Homogeneous Markov Chains (HMCs). The well-known weather example, a Visible Markov Model (VMM), illustrates that the full joint probability of a VMM corresponds to a specifically projected Markov state sequence in the expanded evolution formula. In a Hidden Markov Model (HMM), the probability basis (P-basis) of the hidden Markov state sequence and the P-basis of the observation sequence exist in the sequential event space. The full joint probability of an HMM is the product of the (unknown) projected hidden sequence of Markov states and their transformations into the observation P-bases. The Viterbi algorithm is applied to the famous Weather-Stone HMM example to determine the most likely weather-state sequence given the observed stone-state sequence. Our results are verified using the Elvira software package. Using the PBN, we unify the evolution formulas for Markov models like VMMs, HMMs, and factorial HMMs (with discrete time). We briefly investigated the extended HMM, addressing the feedback issue, and the continuous-time VMM and HMM (with discrete or continuous states). All these models are subclasses of Dynamic Bayesian Networks (DBNs) essential for Machine Learning (ML) and Artificial Intelligence (AI).

cs.AI↗

arXiv2025

Probability Bracket Notation for Probability Modeling

Following the Dirac Notation in Quantum Mechanics (QM), we propose the Bracket Notation (PBN) by defining a probability-bra (P-bra), P-ket, P-bracket, P-identity, etc. Using the PBN, many formulae, such as normalizations and expectations in systems of one or more random variables, can now be written in abstract basis-independent expressions, which are easy to expand by inserting a proper P-identity. The time evolution of homogeneous Markov processes can also be formatted in such a way. Our system P-kets are identified with probability vectors, and our system P-bra is comparable to the Doi state function or the Peliti standard bra. In the Heisenberg picture of the PBN, a random variable becomes a stochastic process, and the Chapman-Kolmogorov equations are obtained by inserting a time-dependent P-identity. Also, some QM expressions in the Dirac notation are naturally transformed into probability expressions in PBN by a special Wick rotation. Potential applications show the usefulness of the PBN beyond the constrained domain and range of Hermitian operators on Hilbert Spaces in QM all the way to IT.

math.PR↗

arXiv2024

Probability Bracket Notation, Wick-Matsubara Relation, Density Operators, and Microscopic Probability Modeling

Following the Dirac vector bracket notation (VBN), we proposed the probability bracket notation (PBN) in our previous paper. We mentioned that under the special Wick rotation (imaginary time), a stationary Schrodinger equation in the Hilbert space transforms into the master equation of a microscopic probabilistic process (MPP) in the probability space. In this article, we first study the MPP of the system of a single particle, we show that the energy expectation of the MPP eventually approaches the lowest energy level in its initial condition and its von Neumann entropy finally vanishes. Then we explore the MPP for the quantum system of identical particles in the Fock space, we recover the expected occupation number of particles and the grand partition function in quantum statistics by connecting time with temperature (the Wick-Matsubara relation). We also reproduce the internal energy of an ideal gas in thermodynamics by using the relation. To address the entropy issue and relate the PBN with research topics of statistics in the literature, we express the density operators and the von Neumann entropy in the probability space by using the PBN. The Wick-Matsubara relation plus the PBN might provide a new way of microscopic probability modeling.

math.PR↗

arXiv2024

Probability Bracket Notation, Markov Chains, Stochastic Processes, and Microscopic Probabilistic Processes

Inspired by the Dirac vector probability notation (VPN), we propose the Probability Bracket Notation (PBN), a new set of symbols defined similarly (but not identically) as in the VPN. Applying the PBN to fundamental definitions and theorems for discrete and continuous random variables, we show that the PBN could play a similar role in the probability space as the VBN in the Hilbert vector. Our system P-kets are identified with the probability vectors in Markov chains (MC). The master equation of homogeneous MC in the Schrodinger pictures can be basis-independent. Our system P-bra is linked to the Doi state function and the Peliti standard bra. Transformed from the Schrodinger picture to the Heisenberg picture, the time dependence of the system P-ket of a homogeneous MC (HMC) is shifted to the observable as a stochastic process. Using the correlations established by the special Wick rotation (SWR), the microscopic probabilistic processes (MPPs) are investigated for single and many-particle systems. The expected occupation number of particles in quantum statistics is reproduced by associating time with temperature (the Wick-Matsubara relation).

cs.OH↗

arXiv2011

Dirac Notation, Fock Space and Riemann Metric Tensor in Information Retrieval Models

Using Dirac Notation as a powerful tool, we investigate the three classical Information Retrieval (IR) models and some their extensions. We show that almost all such models can be described by vectors in Occupation Number Representations (ONR) of Fock spaces with various specifications on, e.g., occupation number, inner product or term-term interactions. As important cases of study, Concept Fock Space (CFS) is introduced for Boolean model; the basic formulas for Singular Value Decomposition (SVD) of Latent Semantic Indexing (LSI) Model are manipulated in terms of Dirac notation. And, based on SVD, a Riemannian metric tensor is introduced, which not only can be used to calculate the relevance of documents to a query, but also may be used to measure the closeness of documents in data clustering.

cs.IR↗

arXiv2011

Probability Bracket Notation, Term Vector Space, Concept Fock Space and Induced Probabilistic IR Models

After a brief introduction to Probability Bracket Notation (PBN) for discrete random variables in time-independent probability spaces, we apply both PBN and Dirac notation to investigate probabilistic modeling for information retrieval (IR). We derive the expressions of relevance of document to query (RDQ) for various probabilistic models, induced by Term Vector Space (TVS) and by Concept Fock Space (CFS). The inference network model (INM) formula is symmetric and can be used to evaluate relevance of document to document (RDD); the CFS-induced models contain ingredients of all three classical IR models. The relevance formulas are tested and compared on different scenarios against a famous textbook example.

cs.IR↗

arXiv2009

Induced Hilbert Space, Markov Chain, Diffusion Map and Fock Space in Thermophysics

In this article, we continue to explore Probability Bracket Notation (PBN), proposed in our previous article. Using both Dirac vector bracket notation (VBN) and PBN, we define induced Hilbert space and induced sample space, and propose that there exists an equivalence relation between a Hilbert space and a sample space constructed from the same base observable(s). Then we investigate Markov transition matrices and their eigenvectors to make diffusion maps with two examples: a simple graph theory example, to serve as a prototype of bidirectional transition operator; a famous text document example in IR literature, to serve as a tutorial of diffusion map in text document space. We show that the sample space of the Markov chain and the Hilbert space spanned by the eigenvectors of the transition matrix are not equivalent. At the end, we apply our PBN and equivalence proposal to Thermophysics by associating sample (phase) space with the Hilbert space of a single particle and the Fock space of many-particle systems.

cs.OH↗

arXiv2009

Probability Bracket Notation: the Unified Expressions of Conditional Expectation and Conditional Probability in Quantum Modeling

After a brief introduction to Probability Bracket Notation (PBN), indicator operator and conditional density operator (CDO), we investigate probability spaces associated with various quantum systems: system with one observable (discrete or continuous), system with two commutative observables (independent or dependent) and a system of indistinguishable non-interacting many-particles. In each case, we derive unified expressions of conditional expectation (CE), conditional probability (CP), and absolute probability (AP): they have the same format for discrete or continuous spectrum; they are defined in both Hilbert space (using Dirac notation) and probability space (using PBN); and they may be useful to deal with CE of non-commutative observables.

math.PR↗

arXiv2009

Probability Bracket Notation: Probability Space, Conditional Expectation and Introductory Martingales

In this paper, we continue to explore the consistence and usability of Probability Bracket Notation (PBN) proposed in our previous articles. After a brief review of PBN with dimensional analysis, we investigate probability spaces in terms of PBN by introducing probability spaces associated with random variables (R.V) or associated with stochastic processes (S.P). Next, we express several important properties of conditional expectation (CE) and some their proofs in PBN. Then, we introduce martingales based on sequence of R.V or based on filtration in PBN. In the process, we see PBN can be used to investigate some probability problems, which otherwise might need explicit usage of Measure theory. Whenever applicable, we use dimensional analysis to validate our formulas and use graphs for visualization of concepts in PBN. We hope this study shows that PBN, stimulated by and adapted from Dirac notation in Quantum Mechanics (QM), may have the potential to be a useful tool in probability modeling, at least for those who are already familiar with Dirac notation in QM.

math.PR↗