Applying Probability Theory to Quantum Modeling

𝗤𝘂𝗮𝗻𝘁𝘂𝗺 𝗣𝗿𝗼𝗯𝗮𝗯𝗶𝗹𝗶𝘁𝘆 × 𝗟𝗟𝗠 𝗜𝗻𝘁𝗲𝗹𝗹𝗶𝗴𝗲𝗻𝗰𝗲 𝖰𝗎𝖺𝗇𝗍𝗎𝗆 𝖺𝗆𝗉𝗅𝗂𝗍𝗎𝖽𝖾𝗌 𝗋𝖾𝖿𝗂𝗇𝖾 𝗅𝖺𝗇𝗀𝗎𝖺𝗀𝖾 𝗉𝗋𝖾𝖽𝗂𝖼𝗍𝗂𝗈𝗇 𝖯𝗁𝖺𝗌𝖾 𝖺𝗅𝗂𝗀𝗇𝗆𝖾𝗇𝗍 𝖾𝗇𝗋𝗂𝖼𝗁𝖾𝗌 𝖼𝗈𝗇𝗍𝖾𝗑𝗍𝗎𝖺𝗅 𝗇𝗎𝖺𝗇𝖼𝖾 Classical probability treats token likelihoods as isolated scalars, but quantum computation reimagines them as amplitude vectors whose phases encode latent context. By mapping transformer outputs onto Hilbert spaces, we unlock interference patterns that selectively amplify coherent meanings while cancelling noise, yielding sharper posteriors with fewer samples. Variational quantum circuits further permit gradient‑based training of unitary operators, allowing language models to entangle distant dependencies without the quadratic memory overhead of classical self‑attention. The result is not simply faster or smaller models, but a fundamentally richer probabilistic grammar where superposition captures ambiguity and measurement collapses it into actionable insight. As qubit counts rise and error rates fall, the convergence of quantum linear algebra and deep semantics promises a new era in which language understanding is limited less by data volume than by our willingness to rethink probability itself. #quantum #ai #llm

Complex joint probabilities as expressions of determinism in quantum mechanics The paper begins with a provocative idea: determinism does not disappear in quantum mechanics; it merely changes form. Hofmann proposes that all the information contained in a quantum state can be represented through complex joint probabilities between observables that cannot be measured simultaneously. In other words, the quantum world does not deny the causal connection between variables; it redefines it within a complex plane, where phases and interferences take the place of classical certainties. Through this approach, classical reality emerges as a limiting case: when measurements have low resolution, the phase oscillations average out, and the system appears to follow a traditional deterministic behavior. Yet, in its deeper structure, quantum determinism is not made of trajectories or points in phase space, but of phase relations that encode how one measurement transforms into another within Hilbert space. The author shows that the laws of transformation between observables can be formulated as complex conditional probabilities, describing how one measurement basis converts into another in a perfectly reversible and deterministic way, without invoking any "underlying realities." Quantum determinism, therefore, does not assign simultaneous values to variables; it preserves their relational coherence without violating the mathematical structure of the theory. When resolution decreases or the environment introduces noise, the phase oscillations fade, and the system approaches a single classical trajectory. In that limit, the world we perceive as "real" and "measurable" emerges. The paper concludes with a profound thesis: quantum determinism does not require realism. What we call "reality" is not independent of measurement; it is a statistical emergence of a deeper order, expressed in terms of complex probabilities. From this perspective, Hilbert space replaces phase space as the true geometry of physical determinism.   By Holger F. Hofmann Hiroshima University University Link https://lnkd.in/duEc997X  

Option Pricing with Quantum Mechanical Methods I first encountered a formal treatment of pricing financial derivatives using the framework of quantum mechanics in Baaquie’s book Quantum Finance when it was published. Over the years, the term “quantum finance” has appeared more frequently in literature. I paid limited attention to this line of work until the paper discussed below, which caught my interest by addressing a well-known problem using the language of quantum mechanics. The paper proposes an option pricing model that converts the Fokker–Planck equation into the Schrödinger equation, yielding both the return distribution and a closed-form solution for European options. The model shows that S&P 500 returns follow a Laplace distribution with power-law tails and that quantum methods outperform GBM-based models in explaining return dynamics and put option prices. Findings: -The paper proposes an option pricing model inspired by quantum mechanics to address the long-standing puzzle of overpriced put options. -The authors reformulate the stock return dynamics by transforming the Fokker–Planck equation into a Schrödinger equation. -This framework yields an explicit probability density function for stock returns and a closed-form solution for European option prices. -Empirical results suggest that S&P 500 index returns follow a Laplace distribution with power-law tail behavior rather than a Gaussian distribution. -The quantum-mechanics-based model outperforms traditional GBM-based models in fitting both index returns and observed put option prices. -The findings indicate that high put option prices observed in the market are close to fair value when modeled within this quantum framework. Reference: Minhyuk Jeong, Biao Yang, Xingjia Zhang, Taeyoung Park & Kwangwon Ahn, A quantum model for the overpriced put puzzle, Financial Innovation (2025) 11:130 Join a community of 7,000+ quants—subscribe to the newsletter! https://lnkd.in/gVFDBTCK #options #volatility #quantitativefinance ABSTRACT Put options are known to be priced unusually high in the market, which we refer to as the overpriced put puzzle . This study proposes a quantum model (QM) that can explain such high put option prices as fair prices. Starting from a stochastic differential equation of stock returns, we convert the Fokker–Planck equation into the Schrödinger equation. To model the market force that always draws excess returns back to equilibrium, we specify a diffusion process corresponding to a QM with a delta potential. The results demonstrate that stock returns follow a Laplace distribution and exhibit power law in the tail. We then construct a closed-form solution for European put option pricing, determining that our model better explains the returns of the S&P 500 index and its corresponding put option prices than do geometric Brownian motion-based models. This study has significant implications for investors and risk managers,...

THE SCHRÖDINGER EQUATION Schrödinger’s equation is often given much of the credit for quantum mechanics, yet by itself it means very little without additional postulates. There are independent principles that are crucial for making sense of the theory, one of which was introduced by Max Born. Schrödinger’s equation tells us how the wave function evolves over time, but it was Born who proposed that the square of the wave function’s amplitude is related to the probability of finding a particle in a given region. However, we rarely stop to ask where this idea came from or what inspired it. Interestingly, Born was influenced by Einstein. In the photoelectric effect, when light falls on a surface, electrons are emitted—but only if the light’s frequency is above a certain threshold. Einstein explained this by proposing that light consists of photons, each with energy proportional to its frequency. Classically, the energy of a wave is related to its amplitude. Increasing the intensity of light increases the number of photons, not the energy of each photon. The total energy of the beam is proportional to the square of its amplitude. This insight suggested that amplitude squared is linked to the number of photons—and thus to the likelihood of photoelectric emission. Born extended this idea further. He proposed that the square of the wave function’s amplitude represents the probability of finding a particle in a particular location. This was the key step that connected Schrödinger’s mathematical equation to physical reality, giving rise to the probabilistic interpretation at the heart of quantum mechanics.

ℚuantum VS ℂlassical 𝐆𝐚𝐥𝐭𝐨𝐧 𝐁𝐨𝐚𝐫𝐝 ℂ: one bead, one path, normal distribution from many trials; ℚ: one wave packet, all paths, interference rewrites distribution via probability density of wave function (from solution of 2D Schrödinger equation). ℂ sums impacts into statistics; ℚ sums amplitudes with phase, so same pegs yield fringes, voids, channels instead of a bell curve. ℂ picks one route per run; ℚ traverses peg field coherently, and final pattern is written by constructive and destructive interference. ℂ hides uncertainty in unknown trajectory; ℚ makes all trajectories matter at once, so geometry plus phase replace the classical bell curve. 🔴 Wolfram article, code and simulation: https://lnkd.in/dK6rbVdc Galton board: drop ball through peg rows; each hit sends it left or right into bottom slots; over many drops, slot counts approach a bell curve, the normal distribution: individual bounces are random, aggregate outcome predictable. Replace ball with a quantum particle: it takes all paths at once as a wave, splitting and recombining at each peg; path interference makes the final distribution unlike a bell curve. The linked Wolfram notebook simulates this quantum Galton board by launching a quantum wavepacket downward through a triangular five-row array of hard-wall obstacles and numerically solving the time-dependent Schrödinger equation for the full probability-density evolution.