Pub Date : 1900-01-01DOI: 10.1093/oso/9780192896308.003.0005
J. Rau
This chapter introduces the notions of classical and quantum information and discusses simple protocols for their exchange. It defines the entropy as a quantitative measure of information, and investigates its mathematical properties and operational meaning. It discusses the extent to which classical information can be carried by a quantum system and derives a pertinent upper bound, the Holevo bound. One important application of quantum communication is the secure distribution of cryptographic keys; a pertinent protocol, the BB84 protocol, is discussed in detail. Moreover, the chapter explains two protocols where previously shared entanglement plays a key role, superdense coding and teleportation. These are employed to effectively double the classical information carrying capacity of a qubit, or to transmit a quantum state with classical bits, respectively. It is shown that both protocols are optimal.
{"title":"Communication","authors":"J. Rau","doi":"10.1093/oso/9780192896308.003.0005","DOIUrl":"https://doi.org/10.1093/oso/9780192896308.003.0005","url":null,"abstract":"This chapter introduces the notions of classical and quantum information and discusses simple protocols for their exchange. It defines the entropy as a quantitative measure of information, and investigates its mathematical properties and operational meaning. It discusses the extent to which classical information can be carried by a quantum system and derives a pertinent upper bound, the Holevo bound. One important application of quantum communication is the secure distribution of cryptographic keys; a pertinent protocol, the BB84 protocol, is discussed in detail. Moreover, the chapter explains two protocols where previously shared entanglement plays a key role, superdense coding and teleportation. These are employed to effectively double the classical information carrying capacity of a qubit, or to transmit a quantum state with classical bits, respectively. It is shown that both protocols are optimal.","PeriodicalId":445393,"journal":{"name":"Quantum Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131918426","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.1093/oso/9780192896308.003.0002
J. Rau
This chapter explains the approach of ‘operationalism’, which in a physical theory admits only concepts associated with concrete experimental procedures, and lays out its consequences for propositions about measurements, their logical structure, and states. It illustrates these with toy examples where the ability to perform measurements is limited by design. For systems composed of several constituents this chapter introduces the notions of composite and reduced states, statistical independence, and correlations. It examines what it means for multiple systems to be prepared identically, and how this is represented mathematically. The operational requirement that there must be procedures to measure and prepare a state is examined, and the ensuing constraints derived. It is argued that these constraint leave only one alternative to classical probability theory that is consistent, universal, and fully operational, namely, quantum theory.
{"title":"Reasoning About Measurements","authors":"J. Rau","doi":"10.1093/oso/9780192896308.003.0002","DOIUrl":"https://doi.org/10.1093/oso/9780192896308.003.0002","url":null,"abstract":"This chapter explains the approach of ‘operationalism’, which in a physical theory admits only concepts associated with concrete experimental procedures, and lays out its consequences for propositions about measurements, their logical structure, and states. It illustrates these with toy examples where the ability to perform measurements is limited by design. For systems composed of several constituents this chapter introduces the notions of composite and reduced states, statistical independence, and correlations. It examines what it means for multiple systems to be prepared identically, and how this is represented mathematically. The operational requirement that there must be procedures to measure and prepare a state is examined, and the ensuing constraints derived. It is argued that these constraint leave only one alternative to classical probability theory that is consistent, universal, and fully operational, namely, quantum theory.","PeriodicalId":445393,"journal":{"name":"Quantum Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127039238","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.1093/oso/9780192896308.003.0004
J. Rau
This chapter introduces the basic building blocks of quantum computing and a variety of specific algorithms. It begins with a brief review of classical computing and discusses how its key elements – bits, gates, circuits – carry over to the quantum realm. It highlights crucial differences to the classical case, such as the impossibility of copying a qubit. The quantum circuit model is shown to be universal, and a peculiar variant of quantum computing, based on measurements only, is illustrated. That a quantum computer can perform some calculations more efficiently than a classical computer, at least in principle, is exemplified with the Deutsch-Jozsa algorithm. Other examples covered in this chapter are the variational quantum eigensolver, which can be applied to the study of molecules and classical optimization problems; quantum simulation; and entanglement-assisted metrology.
{"title":"Computation","authors":"J. Rau","doi":"10.1093/oso/9780192896308.003.0004","DOIUrl":"https://doi.org/10.1093/oso/9780192896308.003.0004","url":null,"abstract":"This chapter introduces the basic building blocks of quantum computing and a variety of specific algorithms. It begins with a brief review of classical computing and discusses how its key elements – bits, gates, circuits – carry over to the quantum realm. It highlights crucial differences to the classical case, such as the impossibility of copying a qubit. The quantum circuit model is shown to be universal, and a peculiar variant of quantum computing, based on measurements only, is illustrated. That a quantum computer can perform some calculations more efficiently than a classical computer, at least in principle, is exemplified with the Deutsch-Jozsa algorithm. Other examples covered in this chapter are the variational quantum eigensolver, which can be applied to the study of molecules and classical optimization problems; quantum simulation; and entanglement-assisted metrology.","PeriodicalId":445393,"journal":{"name":"Quantum Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127090169","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.1093/oso/9780192896308.003.0003
J. Rau
This chapter introduces the mathematical framework, basic rules, and some key results of quantum theory. After a succinct overview of linear algebra and an introduction to complex Hilbert space, it investigates the correspondence between subspaces of Hilbert space and propositions, their logical structure, and how the pertinent probabilities are calculated. It discusses the mathematical representation of states, observables, and transformations, as well as the rules for calculating expectation values and uncertainties, and for updating states after a measurement. Particular attention is paid to two-level systems, or ‘qubits’, and the connection is made with experimental evidence about binary measurements. The properties of composite systems are discussed in detail, notably the phenomenon of entanglement. The chapter concludes with an investigation of conceptual issues regarding realism, non-contextuality, and locality, as well as the classical limit.
{"title":"Probability in Hilbert Space","authors":"J. Rau","doi":"10.1093/oso/9780192896308.003.0003","DOIUrl":"https://doi.org/10.1093/oso/9780192896308.003.0003","url":null,"abstract":"This chapter introduces the mathematical framework, basic rules, and some key results of quantum theory. After a succinct overview of linear algebra and an introduction to complex Hilbert space, it investigates the correspondence between subspaces of Hilbert space and propositions, their logical structure, and how the pertinent probabilities are calculated. It discusses the mathematical representation of states, observables, and transformations, as well as the rules for calculating expectation values and uncertainties, and for updating states after a measurement. Particular attention is paid to two-level systems, or ‘qubits’, and the connection is made with experimental evidence about binary measurements. The properties of composite systems are discussed in detail, notably the phenomenon of entanglement. The chapter concludes with an investigation of conceptual issues regarding realism, non-contextuality, and locality, as well as the classical limit.","PeriodicalId":445393,"journal":{"name":"Quantum Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128541140","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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