Combining Determinism and Indeterminism
- URL: http://arxiv.org/abs/2009.03996v4
- Date: Thu, 19 Nov 2020 18:22:48 GMT
- Title: Combining Determinism and Indeterminism
- Authors: Michael Stephen Fiske
- Abstract summary: We show that the bi-immune symmetric group is dense in Sym$(mathbbN)$ with respect to the pointwise convergence topology.
The complete structure of the bi-immune symmetric group and its subgroups generated by one or more bi-immune rearrangements is unknown.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: Our goal is to construct mathematical operations that combine indeterminism
measured from quantum randomness with computational determinism so that
non-mechanistic behavior is preserved in the computation. Formally, some
results about operations applied to computably enumerable (c.e.) and bi-immune
sets are proven here, where the objective is for the operations to preserve
bi-immunity. While developing rearrangement operations on the natural numbers,
we discovered that the bi-immune rearrangements generate an uncountable
subgroup of the infinite symmetric group (Sym$(\mathbb{N})$) on the natural
numbers $\mathbb{N}$.
This new uncountable subgroup is called the bi-immune symmetric group. We
show that the bi-immune symmetric group contains the finitary symmetric group
on the natural numbers, and consequently is highly transitive. Furthermore, the
bi-immune symmetric group is dense in Sym$(\mathbb{N})$ with respect to the
pointwise convergence topology. The complete structure of the bi-immune
symmetric group and its subgroups generated by one or more bi-immune
rearrangements is unknown.
Related papers
- Quantum Cellular Automata on Symmetric Subalgebras [6.158725838873227]
We investigate quantum cellular automata on one-dimensional spin systems defined over a subalgebra of the full local operator algebra.
For systems where each site carries a regular representation of $G$, we establish a complete classification of such subalgebra QCAs.
arXiv Detail & Related papers (2024-11-28T17:22:50Z) - Symmetry-restricted quantum circuits are still well-behaved [45.89137831674385]
We show that quantum circuits restricted by a symmetry inherit the properties of the whole special unitary group $SU(2n)$.
It extends prior work on symmetric states to the operators and shows that the operator space follows the same structure as the state space.
arXiv Detail & Related papers (2024-02-26T06:23:39Z) - Pseudorandomness from Subset States [0.34476492531570746]
We show it is possible to obtain quantum pseudorandomness and pseudoentanglement from random subset states.
We show that the trace distance is negligibly small, as long as the subsets are of an appropriate size.
arXiv Detail & Related papers (2023-12-14T18:36:16Z) - DHR bimodules of quasi-local algebras and symmetric quantum cellular
automata [0.0]
We show that for the double spin flip action $mathbbZ/2mathbbZtimes mathbbZ/2mathbbZZcurvearrowright mathbbC2otimes mathbbC2$, the group of symmetric QCA modulo symmetric finite depth circuits in 1D contains a copy of $S_3$, hence is non-abelian.
arXiv Detail & Related papers (2023-03-31T18:33:07Z) - Deep Learning Symmetries and Their Lie Groups, Algebras, and Subalgebras
from First Principles [55.41644538483948]
We design a deep-learning algorithm for the discovery and identification of the continuous group of symmetries present in a labeled dataset.
We use fully connected neural networks to model the transformations symmetry and the corresponding generators.
Our study also opens the door for using a machine learning approach in the mathematical study of Lie groups and their properties.
arXiv Detail & Related papers (2023-01-13T16:25:25Z) - The Differential Structure of Generators of GNS-symmetric Quantum Markov
Semigroups [0.0]
We show that the generator of a GNS-symmetric quantum Markov semigroup can be written as the square of a derivation.
This generalizes a result of Cipriani and Sauvageot for tracially symmetric semigroups.
Compared to the tracially symmetric case, the derivations in the general case satisfy a twisted product rule, reflecting the non-triviality of their modular group.
arXiv Detail & Related papers (2022-07-19T12:59:40Z) - A Practical Method for Constructing Equivariant Multilayer Perceptrons
for Arbitrary Matrix Groups [115.58550697886987]
We provide a completely general algorithm for solving for the equivariant layers of matrix groups.
In addition to recovering solutions from other works as special cases, we construct multilayer perceptrons equivariant to multiple groups that have never been tackled before.
Our approach outperforms non-equivariant baselines, with applications to particle physics and dynamical systems.
arXiv Detail & Related papers (2021-04-19T17:21:54Z) - Abelian Neural Networks [48.52497085313911]
We first construct a neural network architecture for Abelian group operations and derive a universal approximation property.
We extend it to Abelian semigroup operations using the characterization of associative symmetrics.
We train our models over fixed word embeddings and demonstrate improved performance over the original word2vec.
arXiv Detail & Related papers (2021-02-24T11:52:21Z) - Constraints on Maximal Entanglement Under Groups of Permutations [73.21730086814223]
Sets of entanglements are inherently equal, lying in the same orbit under the group action.
We introduce new, generalized relationships for the maxima of those entanglement by exploiting the normalizer and normal subgroups of the physical symmetry group.
arXiv Detail & Related papers (2020-11-30T02:21:22Z) - Joint measurability meets Birkhoff-von Neumann's theorem [77.34726150561087]
We prove that joint measurability arises as a mathematical feature of DNTs in this context, needed to establish a characterisation similar to Birkhoff-von Neumann's.
We also show that DNTs emerge naturally from a particular instance of a joint measurability problem, remarking its relevance in general operator theory.
arXiv Detail & Related papers (2018-09-19T18:57:45Z)
This list is automatically generated from the titles and abstracts of the papers in this site.
This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.