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A central result in the foundations of quantum mechanics is the Kochen-Specker theorem. In short, it states that quantum mechanics is in conflict with classical models in which the result of a measurement does not depend on which other compatible measurements are jointly performed. Here compatible measurements are those that can be implemented simultaneously or, more generally, those that are jointly measurable. This conflict is generically called quantum contextuality. In this review, an introduction to this subject and its current status is presented. Several proofs of the Kochen-Specker theorem and different notions of contextuality are reviewed. How to experimentally test some of these notions is explained, and connections between contextuality and nonlocality or graph theory are discussed. Finally, some applications of contextuality in quantum information processing are reviewed., Funding Agencies|Project Qdisc [US-15097]; FEDER funds; Agencia Estatal de Investigacion [PID2020-113738GB-I00]; QuantERA grant SECRET by MINECO [PCI2019-111885-2]; Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) [447948357, 440958198]; FQXi Fund (Silicon Valley Community Foundation); Sino-German Center for Research Promotion [M0294]; Austrian Science Fund (FWF) [ZK 3, F7113]; ERC [683107/TempoQ]

A central result in the foundations of quantum mechanics is the Kochen-Specker theorem. In short, it states that quantum mechanics is in conflict with classical models in which the result of a measurement does not depend on which other compatible measurements are jointly performed. Here compatible measurements are those that can be implemented simultaneously or, more generally, those that are jointly measurable. This conflict is generically called quantum contextuality. In this review, an introduction to this subject and its current status is presented. Several proofs of the Kochen-Specker theorem and different notions of contextuality are reviewed. How to experimentally test some of these notions is explained, and connections between contextuality and nonlocality or graph theory are discussed. Finally, some applications of contextuality in quantum information processing are reviewed., Funding Agencies|Project Qdisc [US-15097]; FEDER funds; Agencia Estatal de Investigacion [PID2020-113738GB-I00]; QuantERA grant SECRET by MINECO [PCI2019-111885-2]; Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) [447948357, 440958198]; FQXi Fund (Silicon Valley Community Foundation); Sino-German Center for Research Promotion [M0294]; Austrian Science Fund (FWF) [ZK 3, F7113]; ERC [683107/TempoQ]

A central result in the foundations of quantum mechanics is the Kochen-Specker theorem. In short, it states that quantum mechanics is in conflict with classical models in which the result of a measurement does not depend on which other compatible measurements are jointly performed. Here compatible measurements are those that can be implemented simultaneously or, more generally, those that are jointly measurable. This conflict is generically called quantum contextuality. In this review, an introduction to this subject and its current status is presented. Several proofs of the Kochen-Specker theorem and different notions of contextuality are reviewed. How to experimentally test some of these notions is explained, and connections between contextuality and nonlocality or graph theory are discussed. Finally, some applications of contextuality in quantum information processing are reviewed., Funding Agencies|Project Qdisc [US-15097]; FEDER funds; Agencia Estatal de Investigacion [PID2020-113738GB-I00]; QuantERA grant SECRET by MINECO [PCI2019-111885-2]; Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) [447948357, 440958198]; FQXi Fund (Silicon Valley Community Foundation); Sino-German Center for Research Promotion [M0294]; Austrian Science Fund (FWF) [ZK 3, F7113]; ERC [683107/TempoQ]

A central result in the foundations of quantum mechanics is the Kochen-Specker theorem. In short, it states that quantum mechanics is in conflict with classical models in which the result of a measurement does not depend on which other compatible measurements are jointly performed. Here compatible measurements are those that can be implemented simultaneously or, more generally, those that are jointly measurable. This conflict is generically called quantum contextuality. In this review, an introduction to this subject and its current status is presented. Several proofs of the Kochen-Specker theorem and different notions of contextuality are reviewed. How to experimentally test some of these notions is explained, and connections between contextuality and nonlocality or graph theory are discussed. Finally, some applications of contextuality in quantum information processing are reviewed., Funding Agencies|Project Qdisc [US-15097]; FEDER funds; Agencia Estatal de Investigacion [PID2020-113738GB-I00]; QuantERA grant SECRET by MINECO [PCI2019-111885-2]; Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) [447948357, 440958198]; FQXi Fund (Silicon Valley Community Foundation); Sino-German Center for Research Promotion [M0294]; Austrian Science Fund (FWF) [ZK 3, F7113]; ERC [683107/TempoQ]

A central result in the foundations of quantum mechanics is the Kochen-Specker theorem. In short, it states that quantum mechanics is in conflict with classical models in which the result of a measurement does not depend on which other compatible measurements are jointly performed. Here compatible measurements are those that can be implemented simultaneously or, more generally, those that are jointly measurable. This conflict is generically called quantum contextuality. In this review, an introduction to this subject and its current status is presented. Several proofs of the Kochen-Specker theorem and different notions of contextuality are reviewed. How to experimentally test some of these notions is explained, and connections between contextuality and nonlocality or graph theory are discussed. Finally, some applications of contextuality in quantum information processing are reviewed., Funding Agencies|Project Qdisc [US-15097]; FEDER funds; Agencia Estatal de Investigacion [PID2020-113738GB-I00]; QuantERA grant SECRET by MINECO [PCI2019-111885-2]; Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) [447948357, 440958198]; FQXi Fund (Silicon Valley Community Foundation); Sino-German Center for Research Promotion [M0294]; Austrian Science Fund (FWF) [ZK 3, F7113]; ERC [683107/TempoQ]

We show that the phenomenon of quantum contextuality can be used to certify lower bounds on the dimension accessed by the measurement devices. To prove this, we derive bounds for different dimensions and scenarios of the simplest noncontextuality inequalities. Some of the resulting dimension witnesses work independently of the prepared quantum state. Our constructions are robust against noise and imperfections, and we show that a recent experiment can be viewed as an implementation of a state-independent quantum dimension witness.

We show that the phenomenon of quantum contextuality can be used to certify lower bounds on the dimension accessed by the measurement devices. To prove this, we derive bounds for different dimensions and scenarios of the simplest noncontextuality inequalities. Some of the resulting dimension witnesses work independently of the prepared quantum state. Our constructions are robust against noise and imperfections, and we show that a recent experiment can be viewed as an implementation of a state-independent quantum dimension witness.

We show that the phenomenon of quantum contextuality can be used to certify lower bounds on the dimension accessed by the measurement devices. To prove this, we derive bounds for different dimensions and scenarios of the simplest noncontextuality inequalities. Some of the resulting dimension witnesses work independently of the prepared quantum state. Our constructions are robust against noise and imperfections, and we show that a recent experiment can be viewed as an implementation of a state-independent quantum dimension witness.

We show that the phenomenon of quantum contextuality can be used to certify lower bounds on the dimension accessed by the measurement devices. To prove this, we derive bounds for different dimensions and scenarios of the simplest noncontextuality inequalities. Some of the resulting dimension witnesses work independently of the prepared quantum state. Our constructions are robust against noise and imperfections, and we show that a recent experiment can be viewed as an implementation of a state-independent quantum dimension witness.

We show that the phenomenon of quantum contextuality can be used to certify lower bounds on the dimension accessed by the measurement devices. To prove this, we derive bounds for different dimensions and scenarios of the simplest noncontextuality inequalities. Some of the resulting dimension witnesses work independently of the prepared quantum state. Our constructions are robust against noise and imperfections, and we show that a recent experiment can be viewed as an implementation of a state-independent quantum dimension witness.