Finite-Function-Encoding Quantum States

J 2022

Finite-Function-Encoding Quantum States

APPEL, Paul, Alexander J. HEILMAN, Ezekiel W. WERTZ, David W. LYONS, Marcus HUBER et. al.

Basic information

Original name

Finite-Function-Encoding Quantum States

Authors

APPEL, Paul (276 Germany), Alexander J. HEILMAN, Ezekiel W. WERTZ, David W. LYONS, Marcus HUBER, Matej PIVOLUSKA (703 Slovakia, guarantor, belonging to the institution) and Giuseppe VITAGLIANO

Edition

QUANTUM, WIEN, VEREIN FORDERUNG OPEN ACCESS PUBLIZIERENS QUANTENWISSENSCHAF, 2022, 2521-327X

Other information

Language

English

Type of outcome

Článek v odborném periodiku

Field of Study

10306 Optics

Country of publisher

Austria

Confidentiality degree

není předmětem státního či obchodního tajemství

References:

URL

Impact factor

Impact factor: 6.400

RIV identification code

RIV/00216224:14610/22:00125965

Organization unit

Institute of Computer Science

DOI

http://dx.doi.org/10.22331/q-2022-05-09-708

UT WoS

000799374500001

Keywords in English

non-locality; entanglement; high-dimensional quantum states

Abstract

V originále

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.

Links

MUNI/G/1596/2019, interní kód MU

Name: Development of algorithms for application of quantum computers in electronic-structure calculations in solid-state physics and chemistry (Acronym: Qubits4PhysChem)

Investor: Masaryk University, INTERDISCIPLINARY - Interdisciplinary research projects

Citovat

APPEL, Paul, Alexander J. HEILMAN, Ezekiel W. WERTZ, David W. LYONS, Marcus HUBER, Matej PIVOLUSKA and Giuseppe VITAGLIANO. Finite-Function-Encoding Quantum States. 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.

@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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