Quantum Contextuality

Mladen Pavicic

Center of Excellence CEMS, Photonics and Quantum Optics Unit, Ruder Bošković Institute and Institute of Physics, Zagreb, Croatia

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Quantum contextual sets have been recognized as resources for universal quantum computation, quantum steering and quantum communication. Therefore, we focus on engineering the sets that support those resources and on determining their structures and properties. Such engineering and subsequent implementation rely on discrimination between statistics of measurement data of quantum states and those of their classical counterparts. The discriminators considered are inequalities defined for hypergraphs whose structure and generation are determined by their basic properties. The generation is inherently random but with the predetermined quantum probabilities of obtainable data. Two kinds of statistics of the data are defined for the hypergraphs and six kinds of inequalities. One kind of statistics, often applied in the literature, turn out to be inappropriate and two kinds of inequalities turn out not to be noncontextuality inequalities. Results are obtained by making use of universal automated algorithms which generate hypergraphs with both odd and even numbers of hyperedges in any odd and even dimensional space – in this paper, from the smallest contextual set with just three hyperedges and three vertices to arbitrarily many contextual sets in up to 8-dimensional spaces. Higher dimensions are computationally demanding although feasible.

Classical computers are binary devices while quantum ones are non-binary ones. Their discriminators are hypergraphs which determines how states supporting a computation are arranged. In quantum computers stabilizer operations initialized by superpositions of states rely on quantum gates that exhibit contextuality via contextual hypergraphs. Quantum gates are described by edges of a hypergraph.

It turns out that contextual non-binary hypergraphs are essential for designing quantum computation and communication and that their structure and implementation rely on a differentiation from their classical non-contextual binary counterparts independently of their possible coordinatization. Alternatively we can generate arbitrarily many contextual sets from simplest possible vector components and then make use of their structure by implementing the hypergraphs with the help of YES-NO measurements so as to collect data from each gate/edge and then postselect them.

This results in collecting data from the same ports/vertices belonging to different gates and eventually establish relations between vertices/vectors and edges/gates that yield several noncontextuality inequalities which serve us as alternative discriminators between contextual and noncontextual sets. The protocol consists in automated generation of hypergraphs from which contextual ones are filtered out to implement and carry out computations.

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[1] Mirko Navara and Karl Svozil, "Form of contextuality predicting probabilistic equivalence between two sets of three mutually noncommuting observables", Physical Review A 109 2, 022222 (2024).

[2] David Amaro-Alcalá, Barry C. Sanders, and Hubert de Guise, "Benchmarking of universal qutrit gates", Physical Review A 109 1, 012621 (2024).

[3] Mladen Pavičić, "Non-Kochen–Specker Contextuality", Entropy 25 8, 1117 (2023).

[4] Mladen Pavičić and Norman D. Megill, "Automated generation of arbitrarily many Kochen-Specker and other contextual sets in odd-dimensional Hilbert spaces", Physical Review A 106 6, L060203 (2022).

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