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QBism deals with common questions in the interpretation of quantum theory about the nature of [[wavefunction]] [[Quantum superposition|superposition]], [[Quantum nonlocality|non-locality]], and [[Quantum entanglement|entanglement]].<ref>{{cite journal|last=Timpson|first=Christopher Gordon|year=2008|title=Quantum Bayesianism: A study|url=http://users.ox.ac.uk/~bras2317/qb_s.ps|format=postscript|journal=Studies In History and Philosophy of Science Part B: Studies In History and Philosophy of Modern Physics|volume=39|issue=3|pages=579–609|doi=10.1016/j.shpsb.2008.03.006|via=|accessdate=}}</ref><ref>{{Cite journal|date=2012-07-01|title=Commentary: Quantum mechanics: Fixing the shifty split|url=http://physicstoday.scitation.org/doi/10.1063/PT.3.1618|journal=Physics Today|volume=65|issue=7|pages=8–10|doi=10.1063/PT.3.1618|issn=0031-9228}}</ref><ref>{{Cite journal|date=2012-11-30|title=Measured responses to quantum Bayesianism|url=http://physicstoday.scitation.org/doi/10.1063/PT.3.1803|journal=Physics Today|volume=65|issue=12|pages=12–15|doi=10.1063/PT.3.1803|issn=0031-9228}}</ref> According to QBism, many, but not all, aspects of the quantum formalism are subjective in nature. For example, in this interpretation, a quantum state is not an element of reality—instead it represents the [[Degree of belief|degrees of belief]] an agent has in the outcomes of measurements. As the interpretation of quantum mechanics is important to [[Philosophy of science|philosophers of science]], some compare the idea of degree of belief and its application in QBism with the idea of [[anti-realism]].<ref name=Stairs /> The originators of the interpretation strongly disagree with this characterization, proposing instead that the theory more properly aligns with a kind of realism they call "participatory realism".<ref>{{Cite journal|last=Fuchs|first=Christopher A.|date=2016-01-17|title=On Participatory Realism|url=http://arxiv.org/abs/1601.04360|journal=arXiv:1601.04360 [quant-ph]}}</ref>
QBism deals with common questions in the interpretation of quantum theory about the nature of [[wavefunction]] [[Quantum superposition|superposition]], [[Quantum nonlocality|non-locality]], and [[Quantum entanglement|entanglement]].<ref>{{cite journal|last=Timpson|first=Christopher Gordon|year=2008|title=Quantum Bayesianism: A study|url=http://users.ox.ac.uk/~bras2317/qb_s.ps|format=postscript|journal=Studies In History and Philosophy of Science Part B: Studies In History and Philosophy of Modern Physics|volume=39|issue=3|pages=579–609|doi=10.1016/j.shpsb.2008.03.006|via=|accessdate=}}</ref><ref>{{Cite journal|date=2012-07-01|title=Commentary: Quantum mechanics: Fixing the shifty split|url=http://physicstoday.scitation.org/doi/10.1063/PT.3.1618|journal=Physics Today|volume=65|issue=7|pages=8–10|doi=10.1063/PT.3.1618|issn=0031-9228}}</ref><ref>{{Cite journal|date=2012-11-30|title=Measured responses to quantum Bayesianism|url=http://physicstoday.scitation.org/doi/10.1063/PT.3.1803|journal=Physics Today|volume=65|issue=12|pages=12–15|doi=10.1063/PT.3.1803|issn=0031-9228}}</ref> According to QBism, many, but not all, aspects of the quantum formalism are subjective in nature. For example, in this interpretation, a quantum state is not an element of reality—instead it represents the [[Degree of belief|degrees of belief]] an agent has in the outcomes of measurements. As the interpretation of quantum mechanics is important to [[Philosophy of science|philosophers of science]], some compare the idea of degree of belief and its application in QBism with the idea of [[anti-realism]].<ref name=Stairs /> The originators of the interpretation strongly disagree with this characterization, proposing instead that the theory more properly aligns with a kind of realism they call "participatory realism".<ref>{{Cite journal|last=Fuchs|first=Christopher A.|date=2016-01-17|title=On Participatory Realism|url=http://arxiv.org/abs/1601.04360|journal=arXiv:1601.04360 [quant-ph]}}</ref>


QBism aims for more than a simple overlay of interpretation. Ideally it would like to rewrite quantum theory solely in terms native to probability theory, so that confusion over the meaning of quantum theory is allayed at the outset. One need not subscribe to a particular reformulation to subscribe to the QBism interpretation of the conventional quantum formalism. The principles which comprise the QBism interpretation are enumerated in the [[Quantum Bayesianism#Core Positions|Core Positions]] section. The most developed QBist reformulation of quantum theory is described in the [[Quantum Bayesianism#The Urgleichung|Urgleichung]] section.
In addition to presenting an interpretation of the existing mathematical structure of quantum theory, QBists have advocated a research program of <nowiki>''reconstructing''</nowiki> quantum theory from basic physical principles whose QBist character is manifest.<ref>{{Cite book|title=QBism: The Future of Quantum Physics|last=von Baeyer|first=Hans Christian|publisher=Harvard University Press|year=2016|isbn=067450464X|location=Cambridge, MA|pages=}}</ref><ref name=":2" /><ref>{{Cite book|url=http://link.springer.com/chapter/10.1007/978-94-017-7303-4_9|title=Some Negative Remarks on Operational Approaches to Quantum Theory|last=Fuchs|first=Christopher A.|last2=Stacey|first2=Blake C.|date=2016-01-01|publisher=Springer Netherlands|year=|isbn=9789401773027|editor-last=Chiribella|editor-first=Giulio|series=Fundamental Theories of Physics|location=|pages=283–305|language=en|doi=10.1007/978-94-017-7303-4_9|editor-last2=Spekkens|editor-first2=Robert W.}}</ref> These reconstruction efforts are still incomplete. The principles which comprise the QBism interpretation are enumerated in the [[Quantum Bayesianism#Core Positions|Core Positions]] section. The most developed QBist reformulation of quantum theory is described in the [[Quantum Bayesianism#The Urgleichung|Urgleichung]] section.


QBist foundational research stimulated interest in symmetric, informationally-complete, positive operator-valued measures ([[SIC-POVM|SIC-POVMs)]], which now have applications in quantum theory outside of foundational studies.<ref>{{Cite journal|last=Scott|first=A. J.|date=2006-01-01|title=Tight informationally complete quantum measurements|url=http://stacks.iop.org/0305-4470/39/i=43/a=009|journal=Journal of Physics A: Mathematical and General|language=en|volume=39|issue=43|pages=13507|doi=10.1088/0305-4470/39/43/009|issn=0305-4470}}</ref><ref>{{Cite web|url=http://www.oalib.com/paper/3573714|title=Discrete phase space and minimum-uncertainty states - Open Access Library|last=Wootters|first=William K.|last2=Sussman|first2=Daniel M.|date=|website=www.oalib.com|archive-url=|archive-date=|dead-url=|access-date=2017-03-07}}</ref><ref name=":0">{{Cite journal|last=Fuchs|first=Christopher A.|last2=Schlosshauer|first2=Maximilian|last3=Stacey|first3=Blake C.|date=2014-05-10|title=My Struggles with the Block Universe|url=http://arxiv.org/abs/1405.2390|journal=arXiv:1405.2390 [physics, physics:quant-ph]}}</ref><ref>{{Cite journal|last=Appleby|first=D. M.|last2=Bengtsson|first2=Ingemar|last3=Brierley|first3=Stephen|last4=Grassl|first4=Markus|last5=Gross|first5=David|last6=Larsson|first6=Jan-\AAke|date=2012-05-01|title=The Monomial Representations of the Clifford Group|url=http://dl.acm.org/citation.cfm?id=2230996.2230999|journal=Quantum Info. Comput.|volume=12|issue=5–6|pages=404–431|issn=1533-7146}}</ref> Likewise, a quantum version of the [[de Finetti's theorem|de Finetti theorem]], introduced by Caves, Fuchs, and Schack (independently reproving a result found using different means by Störmer<ref>{{cite journal|last=Störmer|first=E.|year=1969|title=Symmetric states of infinite tensor products of C*-algebras|journal=J. Funct. Anal.|volume=3|pages=48–68|doi=10.1016/0022-1236(69)90050-0}}</ref>) to provide a Bayesian understanding of the idea of an "unknown quantum state",<ref>{{Cite journal|last=Caves|first=Carlton M.|last2=Fuchs|first2=Christopher A.|last3=Schack|first3=Ruediger|date=2002-08-20|year=|title=Unknown quantum states: The quantum de Finetti representation|url=http://aip.scitation.org/doi/10.1063/1.1494475|journal=Journal of Mathematical Physics|volume=43|issue=9|pages=4537–4559|doi=10.1063/1.1494475|issn=0022-2488|via=}}</ref><ref>{{cite web|url=http://math.ucr.edu/home/baez/week251.html|title=This Week's Finds in Mathematical Physics (Week 251)|year=2007|author=[[John C. Baez|J. Baez]]|accessdate=29 April 2012}}</ref> has found application elsewhere, in topics like [[quantum key distribution]]<ref>{{Cite journal|last=Renner|first=Renato|date=2005-12-30|title=Security of Quantum Key Distribution|url=http://arxiv.org/abs/quant-ph/0512258|journal=arXiv:quant-ph/0512258}}</ref> and [[quantum entanglement|entanglement]] detection.<ref>{{Cite journal|last=Doherty|first=Andrew C.|last2=Parrilo|first2=Pablo A.|last3=Spedalieri|first3=Federico M.|date=2005-01-01|year=|title=Detecting multipartite entanglement|url=http://link.aps.org/doi/10.1103/PhysRevA.71.032333|journal=Physical Review A|volume=71|issue=3|pages=|doi=10.1103/PhysRevA.71.032333|via=}}</ref>
QBist foundational research stimulated interest in symmetric, informationally-complete, positive operator-valued measures ([[SIC-POVM|SIC-POVMs)]], which now have applications in quantum theory outside of foundational studies.<ref>{{Cite journal|last=Scott|first=A. J.|date=2006-01-01|title=Tight informationally complete quantum measurements|url=http://stacks.iop.org/0305-4470/39/i=43/a=009|journal=Journal of Physics A: Mathematical and General|language=en|volume=39|issue=43|pages=13507|doi=10.1088/0305-4470/39/43/009|issn=0305-4470}}</ref><ref>{{Cite web|url=http://www.oalib.com/paper/3573714|title=Discrete phase space and minimum-uncertainty states - Open Access Library|last=Wootters|first=William K.|last2=Sussman|first2=Daniel M.|date=|website=www.oalib.com|archive-url=|archive-date=|dead-url=|access-date=2017-03-07}}</ref><ref name=":0">{{Cite journal|last=Fuchs|first=Christopher A.|last2=Schlosshauer|first2=Maximilian|last3=Stacey|first3=Blake C.|date=2014-05-10|title=My Struggles with the Block Universe|url=http://arxiv.org/abs/1405.2390|journal=arXiv:1405.2390 [physics, physics:quant-ph]}}</ref><ref>{{Cite journal|last=Appleby|first=D. M.|last2=Bengtsson|first2=Ingemar|last3=Brierley|first3=Stephen|last4=Grassl|first4=Markus|last5=Gross|first5=David|last6=Larsson|first6=Jan-\AAke|date=2012-05-01|title=The Monomial Representations of the Clifford Group|url=http://dl.acm.org/citation.cfm?id=2230996.2230999|journal=Quantum Info. Comput.|volume=12|issue=5–6|pages=404–431|issn=1533-7146}}</ref> Likewise, a quantum version of the [[de Finetti's theorem|de Finetti theorem]], introduced by Caves, Fuchs, and Schack (independently reproving a result found using different means by Störmer<ref>{{cite journal|last=Störmer|first=E.|year=1969|title=Symmetric states of infinite tensor products of C*-algebras|journal=J. Funct. Anal.|volume=3|pages=48–68|doi=10.1016/0022-1236(69)90050-0}}</ref>) to provide a Bayesian understanding of the idea of an "unknown quantum state",<ref>{{Cite journal|last=Caves|first=Carlton M.|last2=Fuchs|first2=Christopher A.|last3=Schack|first3=Ruediger|date=2002-08-20|year=|title=Unknown quantum states: The quantum de Finetti representation|url=http://aip.scitation.org/doi/10.1063/1.1494475|journal=Journal of Mathematical Physics|volume=43|issue=9|pages=4537–4559|doi=10.1063/1.1494475|issn=0022-2488|via=}}</ref><ref>{{cite web|url=http://math.ucr.edu/home/baez/week251.html|title=This Week's Finds in Mathematical Physics (Week 251)|year=2007|author=[[John C. Baez|J. Baez]]|accessdate=29 April 2012}}</ref> has found application elsewhere, in topics like [[quantum key distribution]]<ref>{{Cite journal|last=Renner|first=Renato|date=2005-12-30|title=Security of Quantum Key Distribution|url=http://arxiv.org/abs/quant-ph/0512258|journal=arXiv:quant-ph/0512258}}</ref> and [[quantum entanglement|entanglement]] detection.<ref>{{Cite journal|last=Doherty|first=Andrew C.|last2=Parrilo|first2=Pablo A.|last3=Spedalieri|first3=Federico M.|date=2005-01-01|year=|title=Detecting multipartite entanglement|url=http://link.aps.org/doi/10.1103/PhysRevA.71.032333|journal=Physical Review A|volume=71|issue=3|pages=|doi=10.1103/PhysRevA.71.032333|via=}}</ref>

Revision as of 01:17, 13 March 2017

Part of a series of articles about
Quantum mechanics
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Fundamentals
Experiments
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Scientists

Quantum Bayesianism most often refers to an interpretation of quantum theory, also called QBism, which gives a "subjective Bayesian account of quantum probability."[1] Rooted in the prior work of Carlton Caves, Christopher Fuchs, and Ruediger Schack during the early 2000s, QBism itself is primarily associated with Fuchs and Schack and has more recently been advocated by David Mermin.[2] QBism draws from the fields of quantum information and Bayesian probability, claiming to correct, clarify, and extend the Copenhagen interpretation that is commonly presented in textbooks.[3]

The term "Quantum Bayesianism" may sometimes refer more generically to the use of a Bayesian or personalist (aka "subjective") treatment of the probabilities that appear in quantum theory. QBism, in particular, has been referred to as "the radical Bayesian interpretation".[4] It attempts, on a philosophical level, to provide an understanding of quantum theory, and on a more technical level, to derive as much of quantum theory from informational considerations as possible. The remainder of this article focuses primarily on QBism.

QBism deals with common questions in the interpretation of quantum theory about the nature of wavefunction superposition, non-locality, and entanglement.[5][6][7] According to QBism, many, but not all, aspects of the quantum formalism are subjective in nature. For example, in this interpretation, a quantum state is not an element of reality—instead it represents the degrees of belief an agent has in the outcomes of measurements. As the interpretation of quantum mechanics is important to philosophers of science, some compare the idea of degree of belief and its application in QBism with the idea of anti-realism.[1] The originators of the interpretation strongly disagree with this characterization, proposing instead that the theory more properly aligns with a kind of realism they call "participatory realism".[8]

In addition to presenting an interpretation of the existing mathematical structure of quantum theory, QBists have advocated a research program of ''reconstructing'' quantum theory from basic physical principles whose QBist character is manifest.[9][10][11] These reconstruction efforts are still incomplete. The principles which comprise the QBism interpretation are enumerated in the Core Positions section. The most developed QBist reformulation of quantum theory is described in the Urgleichung section.

QBist foundational research stimulated interest in symmetric, informationally-complete, positive operator-valued measures (SIC-POVMs), which now have applications in quantum theory outside of foundational studies.[12][13][14][15] Likewise, a quantum version of the de Finetti theorem, introduced by Caves, Fuchs, and Schack (independently reproving a result found using different means by Störmer[16]) to provide a Bayesian understanding of the idea of an "unknown quantum state",[17][18] has found application elsewhere, in topics like quantum key distribution[19] and entanglement detection.[20]

History and Development

E.T. Jaynes, a promoter of the use of Bayesian probability in statistical physics, once suggested that quantum theory is "[a] peculiar mixture describing in part realities of Nature, in part incomplete human information about Nature—all scrambled up by Heisenberg and Bohr into an omelette that nobody has seen how to unscramble."[21] The present form of QBism developed out of efforts to separate these parts using the tools of quantum information theory and personalist Bayesian probability theory.

There are many interpretations of probability theory. Broadly speaking, these interpretations fall into one of two categories: those which assert that a probability is an objective property of reality and those which assert that a probability is a subjective, mental construct which an agent may use to quantify their ignorance or degree of belief in a proposition. QBism begins by asserting that all probabilities, even those appearing in quantum theory, are most properly viewed as members of the latter category. Specifically, QBism adopts the personalist Bayesian position of Italian mathematician Bruno de Finetti.[22]

The advantages of adopting this view of probability are twofold in QBism. First, it suggests that the Einstein Podolsky Rosen (EPR) criterion of reality[23] should be rejected because it identifies "probability one" assignments with elements of reality preexisting the quantum measurement outcomes.[24] A personalist Bayesian considers all probabilities, even those equal to unity, to be degrees of belief. Therefore, a QBist does not conclude, as many interpretations of quantum theory do, that quantum mechanics is a nonlocal theory. Second, if probabilities are degrees of belief, and one can calculate probabilities from a wavefunction, then wavefunctions must be degrees of belief themselves. With respect to a minimal, informationally complete POVM, this point is made especially clear: A quantum state is then conceptually no more than a single probability distribution over the possible outcomes.

Fuchs introduced the term QBism and outlined the interpretation in more or less its present form in 2010,[25] carrying further and demanding consistency of the ideas first broached in reference [26] with Caves and Schack and in reference [27]. Several subsequent papers, notably references [10], [28], and [24], have expanded and elaborated these foundations.

Core Positions

  1. All probabilities, including probability-1 assignments, are valuations that an agent ascribes to his or her degrees of belief in possible outcomes. As quantum states imply probability assignments, they too are degrees of belief.
  2. The Born rule is normative, not descriptive or prescriptive. It is a relation to which an agent should strive to adhere in their probability and quantum state assignments.
  3. Quantum measurement outcomes are personal experiences for the agent gambling on them.
  4. A measurement apparatus is conceptually an extention of the agent, analogous to a prosthetic limb.

The Urgleichung

The most extensively explored QBist reformulation of quantum theory involves the use of SIC-POVMs to rewrite quantum states (either pure or mixed) as a set of probabilities defined over the outcomes of a "Bureau of Standards" measurement.[29][30][31][14] That is, if one expresses a density matrix as a probability distribution over the outcomes of a SIC-POVM experiment, one can reproduce all the statistical predictions implied by the density matrix from the SIC-POVM probabilities instead. The Born rule then takes the role of relating one valid probability distribution to another, rather than of deriving probabilities from something apparently more fundamental.

Reception

Reactions to the QBism interpretation have ranged from delight[32] to outrage[33]. Some who have criticized QBism claim that it fails to meet the goal of resolving paradoxes in quantum theory. Mohrhoff, for example, criticized QBism from the standpoint of Kantian philosophy.[34] Others may have misunderstood the claims of QBism. See Nauenberg's article [35] which prompted the reply [36]. Still others find QBism internally self-consistent, but do not subscribe to the interpretation.[37][38] Several critiques of QBism which arose in response to Mermin's Physics Today article[39] and his replies to these comments may be found in the Physics Today readers' forum. Section 2 of the Stanford Encyclopedia of Philosophy entry on QBism contains a list of objections and replies to the interpretation. A curated list of various other published criticisms of QBism can be found on page 2278 of reference [14].

Nobel laureate Theodor Hänsch has promoted the QBism interpretation.[40]

Relation to Other Interpretations

The Copenhagen Interpretation

QBism shares many characteristics in common with the Copenhagen interpretation of quantum mechanics, but the differences are important; to conflate them or to regard QBism as a minor modification of the points of view of Bohr or Heisenberg would be a substantial misrepresentation.[41] QBism takes probabilities to be personal judgments of the individual agent who is using quantum mechanics while the Copenhagen view holds that probabilities are given by something ontologically prior, namely a wavefunction. QBism considers a measurement to be any action that an agent takes to elicit a response from the world and the outcome of that measurement to be the experience the world's response induces back on that agent. As a consequence, communication between agents is the only means by which different agents can attempt to compare their internal experiences. Most variants of the Copenhagen interpretation, however, hold that the outcomes of experiments are agent-independent pieces of reality for anyone to access. Although not yet claiming to provide an overt underlying ontology, QBism claims that such changes resolve the obscurities that many critics have found in the Copenhagen interpretation by supplanting the role that quantum theory plays. Rather than a mechanics of reality, QBism claims that quantum theory is a normative tool which an agent may use to better navigate reality.[24]

Von Neumann's Views

In the book Lost Causes in and beyond Physics, R. F. Streater writes, "[t]he first quantum Bayesian was von Neumann. In Die mathematischen Grundlagen der Quantenmechanik, he describes the measurement process of say the spin polarization of an electron source ...".[42] Not everyone agrees.[43]

Relational Quantum Mechanics

Comparisons have also been made between QBism and the relational quantum mechanics espoused by Carlo Rovelli and others.[44][45][46][14]

Other Epistemic Views

Approaches to quantum mechanics, like QBism,[47] which treat quantum states as expressions of information, knowledge, belief, or expectation are called "epistemic" interpretations. These approaches differ from each other in what they consider quantum states to be information or expectations 'about', as well as in the technical features of the mathematics they employ.[48]

Other Uses of Bayesian Probability in Quantum Physics

QBism should be distinguished from other applications of Bayesian probability in quantum physics.[10][49] For example, quantum computer science uses Bayesian networks, which find applications in "medical diagnosis, monitoring of processes, and genetics".[50][51] (A Bayesian framework is also used for neural networks.[52]) Bayesian inference has also been applied in quantum theory for updating probability densities over quantum states,[53] and MaxEnt methods have been used in similar ways.[49][54]

In his book Physics from Fisher Information, Roy Frieden attempts to derive physics from Fisher information.[55] His claims have been disputed by Streater.[56] Although seemingly related, Frieden's approach is actually quite different from QBism. QBism does not attempt to derive physics, or, specifically, quantum theory, from any aspects of probability theory. Rather, QBism proposes that the personalist Bayesian interpretation of probability theory allows one to understand the proper functional role of quantum theory.

See also

External links

References

  1. ^ a b Stairs, Allen (2011). "A loose and separate certainty: Caves, Fuchs and Schack on quantum probability one" (pdf). Studies In History and Philosophy of Science Part B: Studies In History and Philosophy of Modern Physics. 42 (3): 158–166. doi:10.1016/j.shpsb.2011.02.001.
  2. ^ Mermin, N. David (2014-03-27). "Physics: QBism puts the scientist back into science". Nature. 507 (7493): 421–423. doi:10.1038/507421a.
  3. ^ Tammaro, Elliott (2014-08-09). "Why Current Interpretations of Quantum Mechanics are Deficient". arXiv:1408.2093 [quant-ph].
  4. ^ Jaeger, Gregg (2009). "3.7. The radical Bayesian interpretation". Entanglement, information, and the interpretation of quantum mechanics (Online-Ausg. ed.). Berlin: Springer. pp. 170–179. ISBN 978-3-540-92127-1.
  5. ^ Timpson, Christopher Gordon (2008). "Quantum Bayesianism: A study" (postscript). Studies In History and Philosophy of Science Part B: Studies In History and Philosophy of Modern Physics. 39 (3): 579–609. doi:10.1016/j.shpsb.2008.03.006.
  6. ^ "Commentary: Quantum mechanics: Fixing the shifty split". Physics Today. 65 (7): 8–10. 2012-07-01. doi:10.1063/PT.3.1618. ISSN 0031-9228.
  7. ^ "Measured responses to quantum Bayesianism". Physics Today. 65 (12): 12–15. 2012-11-30. doi:10.1063/PT.3.1803. ISSN 0031-9228.
  8. ^ Fuchs, Christopher A. (2016-01-17). "On Participatory Realism". arXiv:1601.04360 [quant-ph].
  9. ^ von Baeyer, Hans Christian (2016). QBism: The Future of Quantum Physics. Cambridge, MA: Harvard University Press. ISBN 067450464X.
  10. ^ a b c Fuchs, Christopher A.; Schack, Ruediger (2013-01-01). "Quantum-Bayesian coherence". Reviews of Modern Physics. 85 (4): 1693–1715. doi:10.1103/RevModPhys.85.1693.
  11. ^ Fuchs, Christopher A.; Stacey, Blake C. (2016-01-01). Chiribella, Giulio; Spekkens, Robert W. (eds.). Some Negative Remarks on Operational Approaches to Quantum Theory. Fundamental Theories of Physics. Springer Netherlands. pp. 283–305. doi:10.1007/978-94-017-7303-4_9. ISBN 9789401773027.
  12. ^ Scott, A. J. (2006-01-01). "Tight informationally complete quantum measurements". Journal of Physics A: Mathematical and General. 39 (43): 13507. doi:10.1088/0305-4470/39/43/009. ISSN 0305-4470.
  13. ^ Wootters, William K.; Sussman, Daniel M. "Discrete phase space and minimum-uncertainty states - Open Access Library". www.oalib.com. Retrieved 2017-03-07. {{cite web}}: Cite has empty unknown parameter: |dead-url= (help)
  14. ^ a b c d Fuchs, Christopher A.; Schlosshauer, Maximilian; Stacey, Blake C. (2014-05-10). "My Struggles with the Block Universe". arXiv:1405.2390 [physics, physics:quant-ph].
  15. ^ Appleby, D. M.; Bengtsson, Ingemar; Brierley, Stephen; Grassl, Markus; Gross, David; Larsson, Jan-\AAke (2012-05-01). "The Monomial Representations of the Clifford Group". Quantum Info. Comput. 12 (5–6): 404–431. ISSN 1533-7146.
  16. ^ Störmer, E. (1969). "Symmetric states of infinite tensor products of C*-algebras". J. Funct. Anal. 3: 48–68. doi:10.1016/0022-1236(69)90050-0.
  17. ^ Caves, Carlton M.; Fuchs, Christopher A.; Schack, Ruediger (2002-08-20). "Unknown quantum states: The quantum de Finetti representation". Journal of Mathematical Physics. 43 (9): 4537–4559. doi:10.1063/1.1494475. ISSN 0022-2488.
  18. ^ J. Baez (2007). "This Week's Finds in Mathematical Physics (Week 251)". Retrieved 29 April 2012.
  19. ^ Renner, Renato (2005-12-30). "Security of Quantum Key Distribution". arXiv:quant-ph/0512258.
  20. ^ Doherty, Andrew C.; Parrilo, Pablo A.; Spedalieri, Federico M. (2005-01-01). "Detecting multipartite entanglement". Physical Review A. 71 (3). doi:10.1103/PhysRevA.71.032333.
  21. ^ Jaynes, E.T., 1990, Complexity, Entropy, and the Physics of Information, W. H. Zurek (Eds.), Addison-Wesley.
  22. ^ Gefter, Amanda. "A Private View of Quantum Reality". WIRED. Retrieved 2017-03-03.
  23. ^ Fine, Arthur (2016-01-01). Zalta, Edward N. (ed.). The Stanford Encyclopedia of Philosophy (Fall 2016 ed.). Metaphysics Research Lab, Stanford University.
  24. ^ a b c Fuchs, Christopher A.; Stacey, Blake C. (2016-12-21). "QBism: Quantum Theory as a Hero's Handbook". arXiv:1612.07308 [quant-ph].
  25. ^ Fuchs, Christopher A. (2010-03-26). "QBism, the Perimeter of Quantum Bayesianism". arXiv:1003.5209 [quant-ph].
  26. ^ Caves, Carlton M.; Fuchs, Christopher A.; Schack, Ruediger (2002-01-01). "Quantum probabilities as Bayesian probabilities". Physical Review A. 65 (2). doi:10.1103/PhysRevA.65.022305.
  27. ^ C. A. Fuchs, "Quantum Mechanics as Quantum Information (and only a little more),'' in Quantum Theory: Reconsideration of Foundations,|| edited by A. Khrennikov (Växjö University Press, Växjö, Sweden, 2002), pp.~463--543. https://arxiv.org/abs/quant-ph/0205039.
  28. ^ Fuchs, Christopher A.; Mermin, N. David; Schack, Ruediger (2014-07-22). "An introduction to QBism with an application to the locality of quantum mechanics". American Journal of Physics. 82 (8): 749–754. doi:10.1119/1.4874855. ISSN 0002-9505.
  29. ^ Fuchs, Christopher A.; Schack, Rüdiger (2010-01-08). "A Quantum-Bayesian Route to Quantum-State Space". Foundations of Physics. 41 (3): 345–356. doi:10.1007/s10701-009-9404-8. ISSN 0015-9018.
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