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Amaral B, Terra Cunha M. On geometrical aspects of the graph approach to contextuality. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2024; 382:20230008. [PMID: 38281724 DOI: 10.1098/rsta.2023.0008] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/31/2023] [Accepted: 12/23/2023] [Indexed: 01/30/2024]
Abstract
The connection between contextuality and graph theory has paved the way for numerous advancements in the field. One notable development is the realization that sets of probability distributions in many contextuality scenarios can be effectively described using well-established convex sets from graph theory. This geometric approach allows for a beautiful characterization of these sets. The application of geometry is not limited to the description of contextuality sets alone; it also plays a crucial role in defining contextuality quantifiers based on geometric distances. These quantifiers are particularly significant in the context of the resource theory of contextuality, which emerged following the recognition of contextuality as a valuable resource for quantum computation. In this paper, we provide a comprehensive review of the geometric aspects of contextuality. Additionally, we use this geometry to define several quantifiers, offering the advantage of applicability to other approaches to contextuality where previously defined quantifiers may not be suitable. This article is part of the theme issue 'Quantum contextuality, causality and freedom of choice'.
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Affiliation(s)
- Barbara Amaral
- Departamento de Matemática, Universidade Federal de Minas Gerais, Caixa Postal 702, 30123-970, Belo Horizonte, Minas Gerais, Brazil
- Departamento de Matemática, Universidade Federal de Ouro Preto, Ouro Preto, Minas Gerais, Brazil
- Departamento de Física e Matemática, CAP - Universidade Federal de São João del-Rei, 36.420-000, Ouro Branco, Minas Gerais, Brazil
- International Institute of Physics, Federal University of Rio Grande do Norte, 59078-970, PO Box 1613, Natal, Brazil
- Department of Mathematical Physics, Institute of Physics, University of São Paulo, R. do Matão 1371, São Paulo 05508-090, Brazil
| | - Marcelo Terra Cunha
- Departamento de Matemática, Universidade Federal de Minas Gerais, Caixa Postal 702, 30123-970, Belo Horizonte, Minas Gerais, Brazil
- Departamento de Matemática Aplicada, IMECC-Unicamp, 13084-970, Campinas, São Paulo, Brazil
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Lima P, Cardoso W, Pádua S. Integrated photonic circuits for contextuality tests via sequential measurements in three-level quantum systems. OPTICS EXPRESS 2024; 32:5550-5566. [PMID: 38439278 DOI: 10.1364/oe.504966] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/05/2023] [Accepted: 12/27/2023] [Indexed: 03/06/2024]
Abstract
In this paper, we present a protocol to obtain photonic circuits that can be used in the implementation of contextuality tests on qutrit systems. The use of photonic integrated circuits offers several advantages for performing this type of task. These include scalability, accuracy, robustness, high-speed and efficient quantum measurements, precise control over the phase properties of photons by using electrically driven heaters to induce a thermo-optic phase shift and resistance to noise. We relate the average values that appear in the inequalities with the probability of photon counting in the circuit outputs and present a realizable configuration for the desired device, taking into account state-dependent and state-independent contextuality tests.
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Karvonen M. Neither Contextuality nor Nonlocality Admits Catalysts. PHYSICAL REVIEW LETTERS 2021; 127:160402. [PMID: 34723585 DOI: 10.1103/physrevlett.127.160402] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/17/2021] [Accepted: 08/24/2021] [Indexed: 06/13/2023]
Abstract
We show that the resource theory of contextuality does not admit catalysts, i.e., there are no correlations that can enable an otherwise impossible resource conversion and still be recovered afterward. As a corollary, we observe that the same holds for nonlocality. As entanglement allows for catalysts, this adds a further example to the list of "anomalies of entanglement," showing that nonlocality and entanglement behave differently as resources. We also show that catalysis remains impossible even if, instead of classical randomness, we allow some more powerful behaviors to be used freely in the free transformations of the resource theory.
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Regula B, Takagi R. One-Shot Manipulation of Dynamical Quantum Resources. PHYSICAL REVIEW LETTERS 2021; 127:060402. [PMID: 34420334 DOI: 10.1103/physrevlett.127.060402] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/29/2021] [Accepted: 06/21/2021] [Indexed: 06/13/2023]
Abstract
We develop a unified framework to characterize one-shot transformations of dynamical quantum resources in terms of resource quantifiers, establishing universal conditions for exact and approximate transformations in general resource theories. Our framework encompasses all dynamical resources represented as quantum channels, including those with a specific structure-such as boxes, assemblages, and measurements-thus immediately applying in a vast range of physical settings. For the particularly important manipulation tasks of distillation and dilution, we show that our conditions become necessary and sufficient for broad classes of important theories, enabling an exact characterization of these tasks and establishing a precise connection between operational problems and resource monotones based on entropic divergences. We exemplify our results by considering explicit applications to quantum communication, where we obtain exact expressions for one-shot quantum capacity and simulation cost assisted by no-signaling, separability-preserving, and positive partial transpose-preserving codes; as well as to nonlocality, contextuality, and measurement incompatibility, where we present operational applications of a number of relevant resource measures.
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Affiliation(s)
- Bartosz Regula
- School of Physical and Mathematical Sciences, Nanyang Technological University, 637371, Singapore
| | - Ryuji Takagi
- School of Physical and Mathematical Sciences, Nanyang Technological University, 637371, Singapore
- Center for Theoretical Physics and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
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Amaral B. Resource theory of contextuality. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2019; 377:20190010. [PMID: 31522637 PMCID: PMC6754716 DOI: 10.1098/rsta.2019.0010] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 06/10/2019] [Indexed: 06/10/2023]
Abstract
In addition to the important role of contextuality in foundations of quantum theory, this intrinsically quantum property has been identified as a potential resource for quantum advantage in different tasks. It is thus of fundamental importance to study contextuality from the point of view of resource theories, which provide a powerful framework for the formal treatment of a property as an operational resource. In this contribution, we review recent developments towards a resource theory of contextuality and connections with operational applications of this property. This article is part of the theme issue 'Contextuality and probability in quantum mechanics and beyond'.
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Affiliation(s)
- Barbara Amaral
- Departamento de Física e Matemática, CAP - Universidade Federal de São João del-Rei, 36.420-000 Ouro Branco, MG, Brazil
- Department of Mathematical Physics, Institute of Physics, University of São Paulo, R. do Matao 1371, São Paulo 05508-090, SP, Brazil
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Skrzypczyk P, Linden N. Robustness of Measurement, Discrimination Games, and Accessible Information. PHYSICAL REVIEW LETTERS 2019; 122:140403. [PMID: 31050470 DOI: 10.1103/physrevlett.122.140403] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/29/2018] [Indexed: 06/09/2023]
Abstract
We introduce a resource theory of measurement informativeness. This allows us to define an associated quantifier, which we call the robustness of measurement. It describes how much "noise" must be added to a measurement before it becomes completely uninformative. We show that this geometric quantifier has operational significance in terms of the advantage the measurement provides over guessing at random in a suitably chosen state discrimination game and that it is the single-shot generalization of the accessible information of a certain quantum-to-classical channel. Using this insight, we further show that the recently introduced robustness of asymmetry or coherence is the single-shot generalization of the accessible information of an ensemble. Finally, we discuss more generally the connection between robustness-based measures, discrimination problems, and information-theoretic quantities.
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Affiliation(s)
- Paul Skrzypczyk
- H. H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol, BS8 1TL, United Kingdom
| | - Noah Linden
- School of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW, United Kingdom
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Mansfield S, Kashefi E. Quantum Advantage from Sequential-Transformation Contextuality. PHYSICAL REVIEW LETTERS 2018; 121:230401. [PMID: 30576205 DOI: 10.1103/physrevlett.121.230401] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/30/2018] [Indexed: 06/09/2023]
Abstract
We introduce a notion of contextuality for transformations in sequential contexts, distinct from the Bell-Kochen-Specker and Spekkens notions of contextuality. Within a transformation-based model for quantum computation we show that strong sequential-transformation contextuality is necessary and sufficient for deterministic computation of nonlinear functions if classical components are restricted to mod2 linearity and matching constraints apply to any underlying ontology. For probabilistic computation, sequential-transformation contextuality is necessary and sufficient for advantage in this task and the degree of advantage quantifiably relates to the degree of contextuality.
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Affiliation(s)
- Shane Mansfield
- Sorbonne Université, CNRS, Laboratoire d'Informatique de Paris 6, F-75005 Paris, France
| | - Elham Kashefi
- Sorbonne Université, CNRS, Laboratoire d'Informatique de Paris 6, F-75005 Paris, France
- School of Informatics, University of Edinburgh, 10 Crichton Street, Edinburgh EH8 9AB, United Kingdom
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