Finite-Function-Encoding Quantum States
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 funct...
| Main Authors: | , , , , , , |
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| Format: | Article |
| Language: | English |
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Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften
2022-05-01
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| Series: | Quantum |
| Online Access: | https://quantum-journal.org/papers/q-2022-05-09-708/pdf/ |
| _version_ | 1818200625057366016 |
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| author | Paul Appel Alexander J. Heilman Ezekiel W. Wertz David W. Lyons Marcus Huber Matej Pivoluska Giuseppe Vitagliano |
| author_facet | Paul Appel Alexander J. Heilman Ezekiel W. Wertz David W. Lyons Marcus Huber Matej Pivoluska Giuseppe Vitagliano |
| author_sort | Paul Appel |
| collection | DOAJ |
| description | 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. \cite{2017PhRvA..95e2340S} 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. |
| first_indexed | 2024-12-12T02:40:38Z |
| format | Article |
| id | doaj.art-09c2ea7fc9374fee9c57ecd0dbb1ea46 |
| institution | Directory Open Access Journal |
| issn | 2521-327X |
| language | English |
| last_indexed | 2024-12-12T02:40:38Z |
| publishDate | 2022-05-01 |
| publisher | Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften |
| record_format | Article |
| series | Quantum |
| spelling | doaj.art-09c2ea7fc9374fee9c57ecd0dbb1ea462022-12-22T00:41:11ZengVerein zur Förderung des Open Access Publizierens in den QuantenwissenschaftenQuantum2521-327X2022-05-01670810.22331/q-2022-05-09-70810.22331/q-2022-05-09-708Finite-Function-Encoding Quantum StatesPaul AppelAlexander J. HeilmanEzekiel W. WertzDavid W. LyonsMarcus HuberMatej PivoluskaGiuseppe VitaglianoWe 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. \cite{2017PhRvA..95e2340S} 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.https://quantum-journal.org/papers/q-2022-05-09-708/pdf/ |
| spellingShingle | Paul Appel Alexander J. Heilman Ezekiel W. Wertz David W. Lyons Marcus Huber Matej Pivoluska Giuseppe Vitagliano Finite-Function-Encoding Quantum States Quantum |
| title | Finite-Function-Encoding Quantum States |
| title_full | Finite-Function-Encoding Quantum States |
| title_fullStr | Finite-Function-Encoding Quantum States |
| title_full_unstemmed | Finite-Function-Encoding Quantum States |
| title_short | Finite-Function-Encoding Quantum States |
| title_sort | finite function encoding quantum states |
| url | https://quantum-journal.org/papers/q-2022-05-09-708/pdf/ |
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