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@article{1859041,
author = {Appel, Paul and Heilman, Alexander J. and Wertz, Ezekiel W. and Lyons, David W. and Huber, Marcus and Pivoluska, Matej and Vitagliano, Giuseppe},
article_location = {WIEN},
article_number = {708},
doi = {http://dx.doi.org/10.22331/q-2022-05-09-708},
keywords = {non-locality; entanglement; high-dimensional quantum states},
language = {eng},
issn = {2521-327X},
journal = {QUANTUM},
title = {Finite-Function-Encoding Quantum States},
url = {https://quantum-journal.org/papers/q-2022-05-09-708/},
volume = {6},
year = {2022}
}
TY - JOUR
ID - 1859041
AU - Appel, Paul - Heilman, Alexander J. - Wertz, Ezekiel W. - Lyons, David W. - Huber, Marcus - Pivoluska, Matej - Vitagliano, Giuseppe
PY - 2022
TI - Finite-Function-Encoding Quantum States
JF - QUANTUM
VL - 6
IS - 708
SP - 1-34
EP - 1-34
PB - VEREIN FORDERUNG OPEN ACCESS PUBLIZIERENS QUANTENWISSENSCHAF
SN - 2521327X
KW - non-locality
KW - entanglement
KW - high-dimensional quantum states
UR - https://quantum-journal.org/papers/q-2022-05-09-708/
N2 - We introduce finite-function-encoding (FFE) states which encode arbitrary d-valued logic functions, i.e., multivariate functions over the ring of integers modulo d, and investigate some of their structural properties. We also point out some differences between polynomial and non-polynomial function encoding states: The former can be associated to graphical objects, that we dub tensor-edge hypergraphs (TEH), which are a generalization of hypergraphs with a tensor attached to each hyperedge encoding the coefficients of the different monomials. To complete the framework, we also introduce a notion of finite-function-encoding Pauli (FP) operators, which correspond to elements of what is known as the generalized symmetric group in mathematics. First, using this machinery, we study the stabilizer group associated to FFE states and observe how qudit hypergraph states introduced in Ref. [1] admit stabilizers of a particularly simpler form. Afterwards, we investigate the classification of FFE states under local unitaries (LU), and, after showing the complexity of this problem, we focus on the case of bipartite states and especially on the classification under local FP operations (LFP). We find all LU and LFP classes for two qutrits and two ququarts and study several other special classes, pointing out the relation between maximally entangled FFE states and complex Butson-type Hadamard matrices. Our investigation showcases also the relation between the properties of FFE states, especially their LU classification, and the theory of finite rings over the integers.
ER -
APPEL, Paul, Alexander J. HEILMAN, Ezekiel W. WERTZ, David W. LYONS, Marcus HUBER, Matej PIVOLUSKA and Giuseppe VITAGLIANO. Finite-Function-Encoding Quantum States. \textit{QUANTUM}. WIEN: VEREIN FORDERUNG OPEN ACCESS PUBLIZIERENS QUANTENWISSENSCHAF, 2022, vol.~6, No~708, p.~1-34. ISSN~2521-327X. Available from: https://dx.doi.org/10.22331/q-2022-05-09-708.
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