Statistical Process Tensors

• Statistical process tensors are a unified framework that generalizes classical and quantum stochastic processes using multilinear algebra to capture multi-point correlations.
• They encode full moment and cumulant hierarchies, enabling analysis of high-dimensional, non-Gaussian dynamics through symmetry and invariant decompositions.
• Computational methods, such as matrix product operator representations, facilitate efficient simulation and dimensionality reduction in non-Markovian quantum systems.

A statistical process tensor generalizes the notion of stochastic processes—both classical and quantum—into the multilinear algebraic setting, providing a unified mathematical structure for representing, analyzing, and computing multi-point correlations and higher-order statistical information. This framework is central in fields such as quantum stochastic dynamics, high-dimensional signal processing, spatial statistics, and machine learning, enabling the systematic study of random, correlated, and entangled processes in spaces of arbitrary order and structure (O'Donovan et al., 19 Feb 2025, Mathews, 2018, Pandey et al., 2024, Cochin et al., 6 Mar 2026).

1. Core Definitions and Characterizations

Consider an order-$N$ process where each realization can be encoded as a multilinear map or a tensor in a Hilbert or vector space. In classical settings, V-valued random variables $X_1,...,X_n$ generate the raw process tensor

$$T_n = \mathbb{E}[X_1 \otimes \cdots \otimes X_n] \in V^{\otimes n}$$

as an aggregator of all joint moments (Mathews, 2018). The quantum process tensor extends this to open or monitored quantum systems. Letting $S$ denote the system, $E$ its environment, and $ρ_0$ their joint initial state, controlled by a sequence of completely-positive trace-preserving (CPTP) maps $𝓜_j$, the output state at the end of interventions is

$ρ_{\mathrm{out}} = \mathrm{Tr}_E[U \circ 𝓜_k \circ \cdots \circ 𝓜_1 \circ U(ρ_0)]$

The process tensor $\Upsilon$ is the Choi–Jamiołkowski state on the "butterfly" Hilbert space $$B = \bigotimes_{j=1}^k (H_{\mathrm{in}}^j \otimes H_{\mathrm{out}}^j)$$, with (O'Donovan et al., 19 Feb 2025): $X_1,...,X_n$0 In high-dimensional stochastic modeling, an order-$X_1,...,X_n$1 complex random tensor

$X_1,...,X_n$2

has statistical features expressed via its mean, auto-covariance, and cumulant tensors. For a proper (circularly symmetric) tensor, the pseudo-covariance vanishes (Pandey et al., 2024).

2. Moment and Cumulant Structure

A unifying feature of statistical process tensors is the full moment and cumulant hierarchy:

• The $X_1,...,X_n$3th-order moment tensor of a random process is

$X_1,...,X_n$4

• Cumulant tensors $X_1,...,X_n$5 are computed via Möbius inversion on set partitions, generalizing scalar cumulants to the tensor setting.
• In quantum process tensors, the projected process ensemble (PPE)—the set $X_1,...,X_n$6 arising from orthonormal local projections—admits a hierarchy of moments:

$X_1,...,X_n$7

with cumulant tensors

$X_1,...,X_n$8

encoding fluctuations and higher-order entanglement (O'Donovan et al., 19 Feb 2025).

3. Symmetry, Schur–Weyl Decomposition, and Statistical Invariants

Statistical process tensors admit structural decompositions revealing invariants under various symmetry groups:

• The Schur–Weyl decomposition expresses $X_1,...,X_n$9 as a direct sum of symmetry-adapted components labeled by partitions $$T_n = \mathbb{E}[X_1 \otimes \cdots \otimes X_n] \in V^{\otimes n}$$0:

$$T_n = \mathbb{E}[X_1 \otimes \cdots \otimes X_n] \in V^{\otimes n}$$1

where $$T_n = \mathbb{E}[X_1 \otimes \cdots \otimes X_n] \in V^{\otimes n}$$2 are Schur functors (irreducible GL($$T_n = \mathbb{E}[X_1 \otimes \cdots \otimes X_n] \in V^{\otimes n}$$3) modules) and $$T_n = \mathbb{E}[X_1 \otimes \cdots \otimes X_n] \in V^{\otimes n}$$4 are irreducible $$T_n = \mathbb{E}[X_1 \otimes \cdots \otimes X_n] \in V^{\otimes n}$$5-modules (Mathews, 2018).

Special projections $$T_n = \mathbb{E}[X_1 \otimes \cdots \otimes X_n] \in V^{\otimes n}$$6 onto these components extract quantities such as fully symmetric (raw moments), alternating (top-level joint cumulants), and mixed-symmetry (pairwise covariances). This enables a coordinate-free analysis of statistical content: means, variances, and higher cumulants are all special cases of projected process tensors.

4. Quantum Extensions: Chaoticity and Spatiotemporal Correlations

In quantum many-body systems, statistical process tensors serve as vehicles for diagnosing dynamical complexity. The hierarchy of PPE moments recovers known chaos quantifiers:

• The Alicki–Fannes quantum dynamical entropy (QDE),

$$T_n = \mathbb{E}[X_1 \otimes \cdots \otimes X_n] \in V^{\otimes n}$$7

quantifies unpredictability of conditional outputs.

• Butterfly-flutter fidelity,

$$T_n = \mathbb{E}[X_1 \otimes \cdots \otimes X_n] \in V^{\otimes n}$$8

measures the mean overlap of conditional output states; it tends to zero in strongly chaotic regimes (O'Donovan et al., 19 Feb 2025).

Higher moments and cumulants directly reveal non-Gaussian, multipartite spatiotemporal entanglement. In the limit of quantum chaos, the $$T_n = \mathbb{E}[X_1 \otimes \cdots \otimes X_n] \in V^{\otimes n}$$9th moment approaches the projector onto the symmetric subspace, while for localized/integrable dynamics, this structure remains low-rank.

5. Computational Constructions and High-Dimensional Regimes

The practical evaluation of statistical process tensors in high-dimensional systems entails severe computational challenges:

• In non-Markovian open quantum systems, the process tensor is realized as a matrix product operator (MPO) capturing all environmental multitime correlations. Efficient construction exploits time-translational invariance (TTI), reducing both memory and computation from $S$0 to $S$1 in the system Hilbert space dimension $S$2 (Cochin et al., 6 Mar 2026).
• The MPO form admits simulation of auxiliary quantities—e.g., multi-time quantum correlations, steady states, and qubit readout statistics in circuit QED—with scalability to large environments and long memory depths.

In classical spatial statistics and morphometry, the high-order process tensor is compressed via Schur projections or via assumptions such as covariance separability, yielding interpretable “shape fingerprints” and dimensionality reduction for applications like 4D CT imaging (Mathews, 2018).

6. Statistical Analysis and Asymptotic Laws

Statistical process tensors inherit spectral and probabilistic properties from random matrix and random tensor theory. For Hermitian process tensors (covariance/auto-correlation tensors), the spectral decomposition

$S$3

generalizes eigenstructure analysis; in the high-dimensional limit, eigenvalues cluster according to Wigner’s semicircle law. For non-symmetric tensors, the spectrum of singular values follows Marčenko–Pastur laws, and the detection of spiked low-rank signals demonstrates BBP-type phase transitions—as the signal-to-noise ratio traverses a threshold, statistical inference shifts from random to informative (Pandey et al., 2024).

7. Applications and Unified Perspective

Statistical process tensors provide a common language for:

• Quantum dynamics: they fully characterize the evolution under generalized, multi-time interventions, encoding all spatiotemporal correlations and non-Markovian memory (O'Donovan et al., 19 Feb 2025, Cochin et al., 6 Mar 2026).
• Spatial processes: raw, central, and joint moment tensors capture morphological variability and population structure (Mathews, 2018).
• Signal processing and machine learning: tensor-based models underlie denoising, component analysis, and spectral learning, as well as the characterization of high-dimensional covariance/separability and phase transition phenomena (Pandey et al., 2024).

This structure unifies traditional moment analysis, group-invariant decompositions, random multiway analysis, and quantum process characterization, forming the foundation of contemporary approaches to analyzing complex, high-dimensional, non-Gaussian, and strongly correlated dynamics.