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Abstract The prospect of realizing highly entangled states on quantum processors with fundamentally different hardware geometries raises the question: to what extent does a state of a quantum spin system have an intrinsic geometry? In this paper, we propose that both states and dynamics of a spin system have a canonically associatedcoarse geometry, in the sense of Roe, on the set of sites in the thermodynamic limit. For a state$$\phi $$ on an (abstract) spin system with an infinite collection of sitesX, we define a universal coarse structure$$\mathcal {E}_{\phi }$$ on the setXwith the property that a state has decay of correlations with respect to a coarse structure$$\mathcal {E}$$ onXif and only if$$\mathcal {E}_{\phi }\subseteq \mathcal {E}$$ . We show that under mild assumptions, the coarsely connected completion$$(\mathcal {E}_{\phi })_{con}$$ is stable under quasi-local perturbations of the state$$\phi $$ . We also develop in parallel a dynamical coarse structure for arbitrary quantum channels, and prove a similar stability result. We show that several order parameters of a state only depend on the coarse structure of an underlying spatial metric, and we establish a basic compatibility between the dynamical coarse structure associated with a quantum circuit$$\alpha $$ and the coarse structure of the state$$\psi \circ \alpha $$ where$$\psi $$ is any product state.
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“Fine synergies” describe motor adaptation in people with drop foot in a way that supplements traditional “coarse synergies”
Synergy analysis via dimensionality reduction is a standard approach in biomechanics to capture the dominant features of limb kinematics or muscle activation signals, which can be called “coarse synergies.” Here we demonstrate that the less dominant features of these signals, which are often explicitly disregarded or considered noise, can nevertheless exhibit “fine synergies” that reveal subtle, yet functionally important, adaptations. To find the coarse synergies, we applied non-negative matrix factorization (NMF) to unilateral EMG data from eight muscles of the involved leg in ten people with drop-foot (DF), and of the right leg of 16 unimpaired (control) participants. We then extracted the fine synergies for each group by removing the coarse synergies (i.e., first two factors explaining 85% of variance) from the data and applying Principal Component Analysis (PCA) to those residuals. Surprisingly, the time histories and structure of the coarse EMG synergies showed few differences between DF and controls—even though the kinematics of drop-foot gait is evidently different from unimpaired gait. In contrast, the structure of the fine EMG synergies (as per their PCA loadings) showed significant differences between groups. In particular, loadings forTibialis Anterior,Peroneus Longus,Gastrocnemius Lateralis,BicepsandRectus Femoris,Vastus MedialisandLateralismuscles differed between groups ( ). We conclude that the multiple differences found in the structure of the fine synergies extracted from EMG in people with drop-foot vs. unimpaired controls—not visible in the coarse synergies—likely reflect differences in their motor strategies. Coarse synergies, in contrast, seem to mostly reflect the gross features of EMG in bipedal gait that must be met by all participants—and thus show few differences between groups. However, drawing insights into the clinical origin of these differences requires well-controlled clinical trials. We propose that fine synergies should not be disregarded in biomechanical analysis, as they may be more informative of the disruption and adaptation of muscle coordination strategies in participants due to drop-foot, age and/or other gait impairments.
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- Award ID(s):
- 2113096
- PAR ID:
- 10565896
- Publisher / Repository:
- Frontiers
- Date Published:
- Journal Name:
- Frontiers in Sports and Active Living
- Volume:
- 5
- ISSN:
- 2624-9367
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.3389/fspor.2023.1080170
