Ideal stochastic process modeling with post-quantum quasiprobabilistic theories

TL;DR

The paper addresses the challenge of achieving ideal memory efficiency in stochastic process modeling by introducing negative-machine (n-machine) generators that permit quasiprobabilities. By using Rényi-2 collision entropy, it demonstrates that the memory of an ideal model can reach the half-excess entropy $\mathbf{E}_{\frac{1}{2}}$, revealing negativity as a resource for nonclassical memory advantage. A constructive protocol splits classical states and injects negativity to produce a quasi-realization that reproduces the same process while reducing memory, with explicit saturation demonstrated for the Perturbed Coin Process and the Simple Nonunifilar Source. The work situates these n-machines within generalized probabilistic theories of HMMs and contrasts them with q-machines via quasiprobability representations, suggesting a principled path toward memory-efficient stochastic simulators and deeper links between negativity and memory crypticity. Overall, the results illuminate the role of post-quantum resources in stochastic modeling and open avenues for connecting memory efficiency to thermodynamics and open-system dynamics.

Abstract

In stochastic modeling, the excess entropy -- the mutual information shared between a process's past and future -- represents the fundamental lower bound of the memory needed to simulate its dynamics. However, this bound cannot be saturated by either classical machines or their enhanced quantum counterparts. Simulating a process fundamentally requires us to store more information in the present than is shared between the past and the future. Here, we consider a generalization of hidden Markov models beyond classical and quantum models, referred to as n-machines, that allow for negative quasiprobabilities. We show that under the collision entropy measure of information, the minimal memory of such models can equal the excess entropy. Our results suggest that negativity can be a useful resource for achieving nonclassical memory advantage.

Figures (15)

• Figure 1: Hierarchy of different models generating the same stochastic process. As discussed in the main text, the sets of n-machine (circled in red) belong to a subset of GPT HMM.
• Figure 2: A hidden quantum Markov model (bottom figure) can be constructed from a classical HMM (top figure). In this illustration, two states $\sigma_0,\sigma_1$ of the HMM are encoded into quantum states $|\sigma_0\rangle,|\sigma_1\rangle$ (which may be non-orthogonal) in which the action of the unitary $U$ of the hidden quantum Markov model (HQMM) is defined (see Eq. \ref{['eq: q machine unitary']}). The hidden memory state of the HQMM is illustrated by the top wire, which propagates the unitaries over time. At time step $t$, an ancilla $|0\rangle$ is fed into unitary $U$ along with the memory state, then a subsystem at the output of $U$ is measured to output the realization $x_t$.
• Figure 3: Illustration of the simulation/generation task as synchronization of two identical and independent machines' internal states.
• Figure 4: Simple illustration of the protocol. Starting from the $\epsilon$-machine (top diagram), the n-machine (bottom diagram) is constructed by copying the causal states. The transition arrows in red are the extended transitions allowing quasiprobabilities with their net effects still emulating the previous transitions (in blue) of the $\epsilon$-machine.
• Figure 5: $\epsilon$-machine representation of Perturbed Coin Process. The nodes represent the internal states of the model and the edges labelled $x|p$ represent the transition made between states with probability $p$ while emitting symbol $x$. The same notation is used throughout in subsequent diagrams.