Bayes or Heisenberg: Who(se) Rules?

TL;DR

This work proposes a unified TB framework that recasts quantum measurement and evolution as probabilistic state updates over a probabilistic state vector, enabling scalable neural implementations. It introduces the Heisenberg–Bayes POVM (HB-POVM) and shows that, without postselection, probabilistic HB-POVM inference is equivalent to Bayesian updates in a generative hidden Markov model (gHMM); with postselection, HB-POVM remains tractable while gHMM inference becomes intractable. Pro-bits and unitary-stochastic gates enable efficient, tensorized neural representations of quantum-like reasoning, with neural HB-POVM updates naturally incorporating skip connections as logit priors. The TB algorithms connect perception, memory, and symbolic reasoning, offering a biologically plausible pathway to integrate symbolic labels into continuous perceptual processing, and drawing connections to transformer LLMs through attention-like measurement processes. Overall, the paper advances a probabilistic, neural-compatible framing of quantum cognition that preserves essential quantum-like features (e.g., order effects in PVM) while providing tractable, interpretable mechanisms for brain-inspired computation and AI, with potential implications for memory, attention, and symbol grounding in large-scale models.

Abstract

Although quantum systems are generally described by quantum state vectors, we show that in certain cases their measurement processes can be reformulated as probabilistic equations expressed in terms of probabilistic state vectors. These probabilistic representations can, in turn, be approximated by the neural network dynamics of the Tensor Brain (TB) model. The Tensor Brain is a recently proposed framework for modeling perception and memory in the brain, providing a biologically inspired mechanism for efficiently integrating generated symbolic representations into reasoning processes.

Figures (4)

• Figure 1: Overall architecture. The scene input at the bottom is processed by several processing layers denoted by $\vec{\nu}$. $\vec{\nu}$ denotes low-level cortical areas such as primary sensory cortices, early sensory regions, and maybe even subcortical regions. $\vec{\nu}$ feeds into the representation layer $X$, while the top layer $Y$ encodes symbolic indices. The system’s evolution is governed by a neural network (shown on the right), with $\mathbf{h}$ denoting the hidden layer within the network. $\mathbf{g}^+(\cdot)$ symbolizes embodiment (see Section \ref{['sec:exti2']}).
• Figure 2: Top: The probabilistic PVM. The random variable $Y$ is part of the Markov chain defined by the random variables $X, X', \ldots$. $X$ and $X'$ are unmeasured and remain in probabilistic states (indicated by green circles). In postselection, the binary variable $Z=1$ filters the selected outcomes. Bottom left: The probabilistic HB-POVM represented as a valid probabilistic model. Under postselection, dependencies exist from $X$ to $Y$ and from $Y$ to $Z$. Bottom center and right: The generalized HMM (gHMM). Figure (c) shows the collider structure, while (d) illustrates the resulting model.
• Figure 3: Illustrative example with $n=2$. Left: Correct polynomial interpolation. Right: Incorrect interpolation. While the outputs are correct for $\gamma_1, \gamma_2 \in \{0, 1\}$, the interpolation becomes biased for $\gamma_1, \gamma_2 \in (0, 1)$. The right-hand plot demonstrates a form of overfitting, suggesting that regularized solutions are generally preferable. $f^{\textit{NN}^{\textit{evol}} }(\cdot)$ needs to have large capacity but should be regularized for smooth interpolation.
• Figure 4: Order effects and interference in probabilistic quantum models. Top row: The PVM—both quantum and probabilistic—exhibits order effects, since exchanging $Y$ and $Y'$ yields different outcomes. In the quantum case, this corresponds to measurements in different bases. Middle row: The quantum PVM displays interference, whereas the probabilistic PVM does not. Here, the cluster variable $C$ is modeled as a child of $Y$. Bottom row: The HB-POVM shows no order effect, and the probabilistic HB-POVM exhibits no interference. $\textbf{B}'$ might be different from $\textbf{B}$.

Theorems & Definitions (2)

• Remark 1
• Conjecture 1