Causal memory and structural complexity of correlated quantum sequences
Abstract
What makes one quantum process more complex than another? We adopt an operational approach inspired by classical complexity science, which characterises structure through the memory required for causal generation. We develop a fully-quantum extension of computational mechanics for quantum space series: Correlated sequences of quantum outputs that do not return to interact with their generator. For stationary, memoryful quantum space series admitting finite-dimensional, primitive causal models, we show that any memory-minimal causal generator can be chosen to produce outputs isometrically, without a discarded environment. The resulting canonical matrix product state representation uniquely determines the irreducible memory required for generation. In particular, the minimum memory entropy is exactly the entanglement entropy across the past-future bipartition, giving entanglement a direct operational interpretation as irreducible memory for causal generation. The same structure determines the minimum memory dimension through the corresponding Schmidt rank. Within this class, we further show that the fundamental entropy rate vanishes, that causal generation is symmetric under time reversal, and that all memoryful processes possess infinite quantum Markov order with respect to arbitrary sharp measurements. These results establish an operational framework for quantifying intrinsic quantum structure, revealing deep connections between causal generation, complexity, memory, entropy, and entanglement.
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BibTeX
@misc{elliott2026causal,
title = {{Causal memory and structural complexity of correlated quantum sequences}},
author = {Thomas J. Elliott},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06519},
eprint = {2610.06519},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06519},
}
RIS
TY - UNPB
TI - Causal memory and structural complexity of correlated quantum sequences
AU - Thomas J. Elliott
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.06519
UR - https://arxiv.org/abs/2610.06519
AB - What makes one quantum process more complex than another? We adopt an operational approach inspired by classical complexity science, which characterises structure through the memory required for causal generation. We develop a fully-quantum extension of computational mechanics for quantum space series: Correlated sequences of quantum outputs that do not return to interact with their generator. For stationary, memoryful quantum space series admitting finite-dimensional, primitive causal models, we show that any memory-minimal causal generator can be chosen to produce outputs isometrically, without a discarded environment. The resulting canonical matrix product state representation uniquely determines the irreducible memory required for generation. In particular, the minimum memory entropy is exactly the entanglement entropy across the past-future bipartition, giving entanglement a direct operational interpretation as irreducible memory for causal generation. The same structure determines the minimum memory dimension through the corresponding Schmidt rank. Within this class, we further show that the fundamental entropy rate vanishes, that causal generation is symmetric under time reversal, and that all memoryful processes possess infinite quantum Markov order with respect to arbitrary sharp measurements. These results establish an operational framework for quantifying intrinsic quantum structure, revealing deep connections between causal generation, complexity, memory, entropy, and entanglement.
ER -