Quantum Bayesianism

File:30. Frank Ramsey.jpg, whose interpretation of probability theory closely matches the one adopted by QBism.|256x256px]]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".{{Cite book|title=Complexity, Entropy, and the Physics of Information|last=Jaynes|first=E. T.|publisher=Addison-Wesley|year=1990|editor-last=Zurek|editor-first=W. H.|location=Redwood City, CA|pages=381|chapter=Probability in Quantum Theory}} 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 three categories: those which assert that a probability is an objective property of reality (the propensity school), those who assert that probability is an objective property of the measuring process (frequentists), and those which assert that a probability is a cognitive construct which an agent may use to quantify their ignorance or degree of belief in a proposition (Bayesians). 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 a personalist Bayesian interpretation along the lines of Italian mathematician Bruno de Finetti{{Cite news|url=https://www.quantamagazine.org/20150604-quantum-bayesianism-qbism/|title=A Private View of Quantum Reality|last=Gefter|first=Amanda|work=Quanta|access-date=2017-04-24|language=en-US}} and English philosopher Frank Ramsey.{{cite arXiv|last1=Fuchs|first1=Christopher A.|last2=Schlosshauer|first2=Maximilian|last3=Stacey|first3=Blake C.|date=2014-05-10|title=My Struggles with the Block Universe|eprint=1405.2390|class=quant-ph}}{{Cite book|title=Essays in biography|last=Keynes|first=John Maynard|date=2012-01-01|publisher=Martino Fine Books|isbn=978-1614273264|chapter=F. P. Ramsey|oclc=922625832|author-link=John Maynard Keynes}}

According to QBists, the advantages of adopting this view of probability are twofold. First, for QBists the role of quantum states, such as the wavefunctions of particles, is to efficiently encode probabilities; so quantum states are ultimately degrees of belief themselves. (If one considers any single measurement that is a minimal, informationally complete positive operator-valued measure (POVM), this is especially clear: A quantum state is mathematically equivalent to a single probability distribution, the distribution over the possible outcomes of that measurement.) Regarding quantum states as degrees of belief implies that the event of a quantum state changing when a measurement occurs—the "collapse of the wave function"—is simply the agent updating her beliefs in response to a new experience. Second, it suggests that quantum mechanics can be thought of as a local theory, because the Einstein–Podolsky–Rosen (EPR) criterion of reality can be rejected. The EPR criterion states: "If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of reality corresponding to that quantity."{{Cite book|chapter-url=https://plato.stanford.edu/archives/fall2016/entries/qt-epr/|title=The Stanford Encyclopedia of Philosophy|last=Fine|first=Arthur|date=2016-01-01|publisher=Metaphysics Research Lab, Stanford University|editor-last=Zalta|editor-first=Edward N.|edition=Fall 2016|chapter=The Einstein–Podolsky–Rosen Argument in Quantum Theory}} Arguments that quantum mechanics should be considered a nonlocal theory depend upon this principle, but to a QBist, it is invalid, because a personalist Bayesian considers all probabilities, even those equal to unity, to be degrees of belief.The issue of the interpretation of probabilities equal to unity in quantum theory occurs even for probability distributions over a finite number of alternatives, and thus it is distinct from the issue of events that happen almost surely in measure-theoretic treatments of probability. Therefore, while many interpretations of quantum theory conclude that quantum mechanics is a nonlocal theory, QBists do not.

Christopher Fuchs introduced the term "QBism" and outlined the interpretation in more or less its present form in 2010,{{cite arXiv|last=Fuchs|first=Christopher A.|date=2010-03-26|title=QBism, the Perimeter of Quantum Bayesianism|eprint=1003.5209|class=quant-ph}} carrying further and demanding consistency of ideas broached earlier, notably in publications from 2002.{{Cite journal|last1=Caves|first1=Carlton M.|last2=Fuchs|first2=Christopher A.|last3=Schack|first3=Ruediger|date=2002-01-01|title=Quantum probabilities as Bayesian probabilities|journal=Physical Review A|volume=65|issue=2|pages=022305|arxiv=quant-ph/0106133|doi=10.1103/PhysRevA.65.022305|bibcode=2002PhRvA..65b2305C|s2cid=119515728}}{{cite book|first=C. A. |last=Fuchs |chapter=Quantum Mechanics as Quantum Information (and only a little more |title=Quantum Theory: Reconsideration of Foundations |editor-first=A. |editor-last=Khrennikov |publisher=Växjö University Press |location=Växjö, Sweden |year=2002 |pages=463–543 |arxiv=quant-ph/0205039}} Several subsequent works have expanded and elaborated upon these foundations, notably a Reviews of Modern Physics article by Fuchs and Schack;{{Cite journal|last1=Fuchs|first1=Christopher A.|last2=Schack|first2=Rüdiger|date=2013-01-01|title=Quantum-Bayesian coherence|journal=Reviews of Modern Physics|volume=85|issue=4|pages=1693–1715|arxiv=1301.3274|bibcode=2013RvMP...85.1693F|doi=10.1103/RevModPhys.85.1693|s2cid=18256163}} an American Journal of Physics article by Fuchs, Mermin, and Schack;{{Cite journal|last1=Fuchs|first1=Christopher A.|last2=Mermin|first2=N. David|last3=Schack|first3=Ruediger|date=2014-07-22|title=An introduction to QBism with an application to the locality of quantum mechanics|journal=American Journal of Physics|volume=82|issue=8|pages=749–754|arxiv=1311.5253|doi=10.1119/1.4874855|issn=0002-9505|bibcode=2014AmJPh..82..749F|s2cid=56387090}} and Enrico Fermi Summer School{{Cite web|url=https://www.sif.it/attivita/scuola_fermi|title=International School of Physics "Enrico Fermi"|website=Italian Physical Society|access-date=2017-04-18}} lecture notes by Fuchs and Stacey.{{cite arXiv|last1=Fuchs|first1=Christopher A.|last2=Stacey|first2=Blake C.|date=2016-12-21|title=QBism: Quantum Theory as a Hero's Handbook|eprint=1612.07308|class=quant-ph}}

Prior to the 2010 article, the term "quantum Bayesianism" was used to describe the developments which have since led to QBism in its present form. However, as noted above, QBism subscribes to a particular kind of Bayesianism which does not suit everyone who might apply Bayesian reasoning to quantum theory (see, for example, {{slink|#Other uses of Bayesian probability in quantum physics}} below). Consequently, Fuchs chose to call the interpretation "QBism", pronounced "cubism", preserving the Bayesian spirit via the CamelCase in the first two letters, but distancing it from Bayesianism more broadly. As this neologism is a homophone of Cubism the art movement, it has motivated conceptual comparisons between the two, and media coverage of QBism has been illustrated with art by Picasso and Gris.{{Cite news|url=https://www.faz.net/aktuell/wissen/physik-chemie/philosophische-quantenphysik-ganz-im-auge-des-betrachters-12792104.html|title=Philosophische Quantenphysik : Ganz im Auge des Betrachters|last=von Rauchhaupt|first=Ulf|date=9 February 2014|work=Frankfurter Allgemeine Sonntagszeitung|access-date=2017-04-18|volume=6|page=62|language=de}} However, QBism itself was not influenced or motivated by Cubism and has no lineage to a potential connection between Cubist art and Bohr's views on quantum theory.{{Cite web|url=https://vimeo.com/167261302?from=outro-embed|title=Q3: Quantum Metaphysics Panel|date=13 February 2016|website=Vimeo|access-date=2017-04-18}}

According to QBism, quantum theory is a tool which an agent may use to help manage their expectations, more like probability theory than a conventional physical theory. Quantum theory, QBism claims, is fundamentally a guide for decision making which has been shaped by some aspects of physical reality. Chief among the tenets of QBism are the following:

1. All probabilities, including those equal to zero or one, are valuations that an agent ascribes to their degrees of belief in possible outcomes. As they define and update probabilities, quantum states (density operators), channels (completely positive trace-preserving maps), and measurements (positive operator-valued measures) are also the personal judgements of an agent.
2. The Born rule is normative, not descriptive. 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. Different agents may confer and agree upon the consequences of a measurement, but the outcome is the experience each of them individually has.
4. A measurement apparatus is conceptually an extension of the agent. It should be considered analogous to a sense organ or prosthetic limb—simultaneously a tool and a part of the individual.

File:Jean Metzinger, 1912, Danseuse au café, Dancer in a café, oil on canvas, 146.1 x 114.3 cm, Albright-Knox Art Gallery, Buffalo, New York.jpg, 1912, Danseuse au café.

One advocate of QBism, physicist David Mermin, describes his rationale for choosing that term over the older and more general "quantum Bayesianism": "I prefer [the] term 'QBist' because [this] view of quantum mechanics differs from others as radically as cubism differs from renaissance painting ..."{{cite arXiv|last=Mermin|first=N. David|date=2013-01-28|title=Annotated Interview with a QBist in the Making|eprint=1301.6551|class=quant-ph}}|391x391px]]

Reactions to the QBist interpretation have ranged from enthusiastic to strongly negative. Some who have criticized QBism claim that it fails to meet the goal of resolving paradoxes in quantum theory. Bacciagaluppi argues that QBism's treatment of measurement outcomes does not ultimately resolve the issue of nonlocality,{{Cite book|last=Bacciagaluppi|first=Guido|date=2014-01-01|publisher=Springer International Publishing|isbn=9783319043814|editor-last=Galavotti|editor-first=Maria Carla|editor-link= Maria Carla Galavotti |series=The Philosophy of Science in a European Perspective|pages=403–416|language=en|doi=10.1007/978-3-319-04382-1_27|editor2-last=Dieks|editor2-first=Dennis|editor3-last=Gonzalez|editor3-first=Wenceslao J.|editor4-last=Hartmann|editor4-first=Stephan|editor5-last=Uebel|editor5-first=Thomas|editor6-last=Weber|editor6-first=Marcel|chapter=A Critic Looks at QBism|title=New Directions in the Philosophy of Science}} and Jaeger finds QBism's supposition that the interpretation of probability is key for the resolution to be unnatural and unconvincing. Norsen{{Cite journal|last=Norsen|first=Travis|year=2014|title=Quantum Solipsism and Non-Locality|url=http://www.ijqf.org/wps/wp-content/uploads/2014/12/Norsen-Bell-paper.pdf|journal=Int. J. Quant. Found.|volume=John Bell Workshop}} has accused QBism of solipsism, and Wallace{{cite arXiv|last=Wallace|first=David|date=2007-12-03|title=The Quantum Measurement Problem: State of Play|eprint=0712.0149|class=quant-ph}} identifies QBism as an instance of instrumentalism; QBists have argued insistently that these characterizations are misunderstandings, and that QBism is neither solipsist nor instrumentalist.{{Cite journal|last1=DeBrota|first1=John B.|last2=Fuchs|first2=Christopher A.|date=2017-05-17|title=Negativity Bounds for Weyl-Heisenberg Quasiprobability Representations|journal=Foundations of Physics|volume=47|issue=8|pages=1009–1030|arxiv=1703.08272|doi=10.1007/s10701-017-0098-z|bibcode=2017FoPh...47.1009D|s2cid=119428587}} A critical article by Nauenberg{{Cite journal|last=Nauenberg|first=Michael|date=2015-03-01|title=Comment on QBism and locality in quantum mechanics|journal=American Journal of Physics|volume=83|issue=3|pages=197–198|arxiv=1502.00123|doi=10.1119/1.4907264|issn=0002-9505|bibcode=2015AmJPh..83..197N|s2cid=117823345}} in the American Journal of Physics prompted a reply by Fuchs, Mermin, and Schack.{{Cite journal|last1=Fuchs|first1=Christopher A.|last2=Mermin|first2=N. David|last3=Schack|first3=Ruediger|date=2015-02-10|title=Reading QBism: A Reply to Nauenberg|journal=American Journal of Physics|volume=83|issue=3|pages=198|arxiv=1502.02841|doi=10.1119/1.4907361|bibcode=2015AmJPh..83..198F}}

Some assert that there may be inconsistencies; for example, Stairs argues that when a probability assignment equals one, it cannot be a degree of belief as QBists say.{{cite journal|last=Stairs|first=Allen|year=2011|title=A loose and separate certainty: Caves, Fuchs and Schack on quantum probability one|url=http://www.terpconnect.umd.edu/~stairs/papers/Loose_and_Separate_Certainty.pdf|journal=Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics|volume=42|issue=3|pages=158–166|doi=10.1016/j.shpsb.2011.02.001|bibcode=2011SHPMP..42..158S}} Further, while also raising concerns about the treatment of probability-one assignments, Timpson suggests that QBism may result in a reduction of explanatory power as compared to other interpretations. Fuchs and Schack replied to these concerns in a later article.{{Cite journal|last1=Fuchs|first1=Christopher A.|last2=Schack|first2=Rüdiger|date=2015-01-01|title=QBism and the Greeks: why a quantum state does not represent an element of physical reality|journal=Physica Scripta|language=en|volume=90|issue=1|pages=015104|arxiv=1412.4211|doi=10.1088/0031-8949/90/1/015104|issn=1402-4896|bibcode=2015PhyS...90a5104F|s2cid=14553716}} Mermin advocated QBism in a 2012 Physics Today article, which prompted considerable discussion. Several further critiques of QBism which arose in response to Mermin's article, and Mermin's replies to these comments, may be found in the Physics Today readers' forum.{{Cite journal|date=2012-11-30|title=Measured responses to quantum Bayesianism|journal=Physics Today|volume=65|issue=12|pages=12–15|doi=10.1063/PT.3.1803|issn=0031-9228|last1=Mermin|first1=N. David|bibcode=2012PhT....65l..12M|doi-access=free}}{{Cite journal|last=Mermin|first=N. David|date=2013-06-28|title=Impressionism, Realism, and the aging of Ashcroft and Mermin|journal=Physics Today|volume=66|issue=7|pages=8|doi=10.1063/PT.3.2024|issn=0031-9228|bibcode=2013PhT....66R...8M}} Section 2 of the Stanford Encyclopedia of Philosophy entry on QBism also contains a summary of objections to the interpretation, and some replies.{{Cite book|chapter-url=https://plato.stanford.edu/entries/quantum-bayesian/|title=Stanford Encyclopedia of Philosophy|last=Healey|first=Richard|publisher=Metaphysics Research Lab, Stanford University|year=2016|editor-last=Zalta|editor-first=Edward N.|chapter=Quantum-Bayesian and Pragmatist Views of Quantum Theory}} Others are opposed to QBism on more general philosophical grounds; for example, Mohrhoff criticizes QBism from the standpoint of Kantian philosophy.{{Cite arXiv|last=Mohrhoff|first=Ulrich|date=2014-09-10|title=QBism: A Critical Appraisal|eprint=1409.3312|class=quant-ph}}

Certain authors find QBism internally self-consistent, but do not subscribe to the interpretation.{{Cite journal|last=Marchildon|first=Louis|date=2015-07-01|title=Why I am not a QBist|journal=Foundations of Physics|language=en|volume=45|issue=7|pages=754–761|arxiv=1403.1146|doi=10.1007/s10701-015-9875-8|issn=0015-9018|bibcode=2015FoPh...45..754M|s2cid=119196825}}
{{Cite web|url=http://fmoldove.blogspot.com/2015/05/interview-with-anti-quantum-zealot-on.html|title=Interview with an anti-Quantum zealot|last=Leifer|first=Matthew|website=Elliptic Composability|access-date=10 March 2017}} For example, Marchildon finds QBism well-defined in a way that, to him, many-worlds interpretations are not, but he ultimately prefers a Bohmian interpretation.{{Cite journal|last=Marchildon|first=Louis|title=Multiplicity in Everett's interpretation of quantum mechanics|journal=Studies in History and Philosophy of Modern Physics|language=en|volume=52|issue=B|pages=274–284|arxiv=1504.04835|doi=10.1016/j.shpsb.2015.08.010|year=2015|bibcode=2015SHPMP..52..274M|s2cid=118398374}} Similarly, Schlosshauer and Claringbold state that QBism is a consistent interpretation of quantum mechanics, but do not offer a verdict on whether it should be preferred.{{Cite book|chapter-url=http://ebooks.cambridge.org/ref/id/CBO9781107706927A085|title=Protective Measurement and Quantum Reality: Towards a New Understanding of Quantum Mechanics|last1=Schlosshauer|first1=Maximilian|last2=Claringbold|first2=Tangereen V. B.|publisher=Cambridge University Press|year=2015|isbn=9781107706927|pages=180–194|language=en|chapter=Entanglement, scaling, and the meaning of the wave function in protective measurement|arxiv=1402.1217|doi=10.1017/cbo9781107706927.014|s2cid=118003617}} In addition, some agree with most, but perhaps not all, of the core tenets of QBism; Barnum's position,{{cite arXiv|last=Barnum|first=Howard N.|date=2010-03-23|title=Quantum Knowledge, Quantum Belief, Quantum Reality: Notes of a QBist Fellow Traveler|eprint=1003.4555|class=quant-ph}} as well as Appleby's,{{Cite journal|last=Appleby|first=D. M.|date=2007-01-01|title=Concerning Dice and Divinity|journal=AIP Conference Proceedings|volume=889|pages=30–39|arxiv=quant-ph/0611261|doi=10.1063/1.2713444|bibcode=2007AIPC..889...30A|s2cid=119529426}} are examples.

Popularized or semi-popularized media coverage of QBism has appeared in New Scientist,See {{Cite news|url=https://www.newscientist.com/article/mg22229680.400-qbism-is-quantum-uncertainty-all-in-the-mind.html|title=QBism: Is quantum uncertainty all in the mind?|last=Chalmers|first=Matthew|date=2014-05-07|work=New Scientist|access-date=2017-04-09|language=en-US}} Mermin criticized some aspects of this coverage; see {{cite arXiv|last=Mermin|first=N. David|date=2014-06-05|title=QBism in the New Scientist|eprint=1406.1573|class=quant-ph}}
See also {{Cite news|url=https://www.newscientist.com/article/2114398-physics-may-be-a-small-but-crucial-fraction-of-our-reality/|title=Physics may be a small but crucial fraction of our reality|last=Webb|first=Richard|date=2016-11-30|work=New Scientist|access-date=2017-04-22|language=en-US}}
See also {{Cite news|url=https://www.newscientist.com/article/mg23631510-200-consciously-quantum-how-you-make-everything-real/|last=Ball|first=Philip|title=Consciously quantum|work=New Scientist|access-date=2017-12-06|date=2017-11-08}} Scientific American,{{Cite journal|last=von Baeyer|first=Hans Christian|title=Quantum Weirdness? It's All in Your Mind|journal=Scientific American|language=en|volume=308|issue=6|pages=46–51|doi=10.1038/scientificamerican0613-46|pmid=23729070|year=2013|bibcode=2013SciAm.308f..46V}} Nature,{{Cite journal|last=Ball|first=Philip|date=2013-09-12|title=Physics: Quantum quest|journal=Nature|language=en|volume=501|issue=7466|pages=154–156|doi=10.1038/501154a|pmid=24025823|bibcode=2013Natur.501..154B|doi-access=free}} Science News,{{Cite news|url=https://www.sciencenews.org/blog/context/qbists-tackle-quantum-problems-adding-subjective-aspect-science|title='QBists' tackle quantum problems by adding a subjective aspect to science|last=Siegfried|first=Tom|date=2014-01-30|work=Science News|access-date=2017-04-20|language=en}} the FQXi Community,{{Cite web|url=http://fqxi.org/community/articles/display/218|title=Painting a QBist Picture of Reality|last=Waldrop|first=M. Mitchell|website=fqxi.org|language=en-US|access-date=2017-04-20}} the Frankfurter Allgemeine Zeitung, Quanta Magazine, Aeon,{{Cite news|url=https://aeon.co/essays/materialism-alone-cannot-explain-the-riddle-of-consciousness|title=Materialism alone cannot explain the riddle of consciousness|last=Frank|first=Adam|date=2017-03-13|work=Aeon|access-date=2017-04-22|editor-last=Powell|editor-first=Corey S.|language=en|author-link=Adam Frank}} Discover,{{Cite news|url=http://discovermagazine.com/2017/may-2017/the-war-over-reality|title=The War Over Reality|last=Folger|first=Tim|author-link=Tim Folger |date=May 2017|work=Discover Magazine|access-date=2017-05-10}} Nautilus Quarterly,{{cite web|url=https://nautil.us/my-quantum-leap-14132/ |title=My Quantum Leap |first=Bob |last=Henderson |website=Nautilus Quarterly |date=2022-02-23 |access-date=2022-02-23}} and Big Think.{{cite web|url=https://bigthink.com/13-8/qbism-quantum-physics/ |title=QBism: The most radical interpretation of quantum mechanics ever |first=Adam |last=Frank |author-link=Adam Frank |date=2023-09-07 |access-date=2023-09-21 |website=Big Think}} In 2018, two popular-science books about the interpretation of quantum mechanics, Ball's Beyond Weird and Ananthaswamy's Through Two Doors at Once, devoted sections to QBism.{{Cite book|title=Beyond Weird: Why Everything You Thought You Knew About Quantum Physics is Different|last=Ball|first=Philip|author-link=Philip Ball |publisher=Penguin Random House|year=2018|isbn=9781847924575|oclc=1031304139 |location=London}}{{Cite book|title=Through Two Doors at Once: The Elegant Experiment That Captures the Enigma of Our Quantum Reality|last=Ananthaswamy|first=Anil|author-link=Anil Ananthaswamy |publisher=Penguin Random House|year=2018|isbn=9781101986097|oclc=1089112651 |location=New York}} Furthermore, Harvard University Press published a popularized treatment of the subject, QBism: The Future of Quantum Physics, in 2016.

The philosophy literature has also discussed QBism from the viewpoints of structural realism and of phenomenology.{{cite journal|first=Dean |last=Rickles |author-link=Dean Rickles |url=https://www.ingentaconnect.com/content/imp/mm/2019/00000017/00000002/art00005 |title=Johntology: Participatory Realism and its Problems |journal=Mind and Matter |volume=17 |issue=2 |year=2019 |pages=205–211}}{{cite book|first=Michel |last=Bitbol |author-link=Michel Bitbol |chapter=A Phenomenological Ontology for Physics: Merleau-Ponty and QBism |title=Phenomenological Approaches to Physics |series=Synthese Library (Studies in Epistemology, Logic, Methodology, and Philosophy of Science) |volume=429 |publisher=Springer |year=2020 |isbn=978-3-030-46972-6 |oclc=1193285104 |doi=10.1007/978-3-030-46973-3_11 |editor-first1=Harald |editor-last1=Wiltsche |editor-first2=Philipp |editor-last2=Berghofer |pages=227–242|s2cid=226714879 |chapter-url=http://philsci-archive.pitt.edu/19512/1/Quantum_Merleau_Michel_HW_PB.pdf }}{{cite book|first=Laura |last=de La Tremblaye |chapter=QBism from a Phenomenological Point of View: Husserl and QBism |title=Phenomenological Approaches to Physics |series=Synthese Library (Studies in Epistemology, Logic, Methodology, and Philosophy of Science) |volume=429 |publisher=Springer |year=2020 |isbn=978-3-030-46972-6 |oclc=1193285104 |doi=10.1007/978-3-030-46973-3_12 |editor-first1=Harald |editor-last1=Wiltsche |editor-first2=Philipp |editor-last2=Berghofer |pages=243–260|s2cid=226670546 }} Ballentine argues that "the initial assumption of QBism is not valid" because the inferential probability of Bayesian theory used by QBism is not applicable to quantum mechanics.{{Cite journal |last=Ballentine |first=Leslie |date=2020-06-01 |title=Reviews of quantum foundations |url=https://pubs.aip.org/physicstoday/article/73/6/11/909666/Reviews-of-quantum-foundations |journal=Physics Today |language=en |volume=73 |issue=6 |pages=11–12 |doi=10.1063/PT.3.4488 |bibcode=2020PhT....73f..11B |s2cid=219759324 |issn=0031-9228|doi-access=free }}

File:Being Bayesian in a Quantum World 2005 conference photo at University of Konstanz.jpg conference Being Bayesian in a Quantum World. |360x360px]]

The views of many physicists (Bohr, Heisenberg, Rosenfeld, von Weizsäcker, Peres, etc.) are often grouped together as the "Copenhagen interpretation" of quantum mechanics. Several authors have deprecated this terminology, claiming that it is historically misleading and obscures differences between physicists that are as important as their similarities.{{Cite journal|last=Stacey|first=Blake C.|date=2016-05-28|title=Von Neumann Was Not a Quantum Bayesian|journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences|volume=374|issue=2068|pages=20150235|arxiv=1412.2409|doi=10.1098/rsta.2015.0235|pmid=27091166|issn=1364-503X|bibcode=2016RSPTA.37450235S|s2cid=16829387}}{{Cite journal|last=Peres|first=Asher|date=2002-03-01|title=Karl Popper and the Copenhagen interpretation|journal=Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics|volume=33|issue=1|pages=23–34|arxiv=quant-ph/9910078|doi=10.1016/S1355-2198(01)00034-X|bibcode=2002SHPMP..33...23P}}
{{Cite book|last=Żukowski|first=Marek|title=Quantum [Un]Speakables II |chapter=Bell's Theorem Tells Us Not What Quantum Mechanics is, but What Quantum Mechanics is Not |date=2017-01-01|publisher=Springer International Publishing|isbn=9783319389851|editor-last=Bertlmann|editor-first=Reinhold|series=The Frontiers Collection|pages=175–185|language=en|arxiv=1501.05640|doi=10.1007/978-3-319-38987-5_10|s2cid=119214547|editor2-last=Zeilinger|editor2-first=Anton}}
{{Cite journal|last=Camilleri|first=Kristian|date=2009-02-01|title=Constructing the Myth of the Copenhagen Interpretation|url=https://muse.jhu.edu/article/259149|journal=Perspectives on Science|volume=17|issue=1|pages=26–57|issn=1530-9274|doi=10.1162/posc.2009.17.1.26|s2cid=57559199}} QBism shares many characteristics in common with the ideas often labeled as "the Copenhagen interpretation", 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, for instance, would be a substantial misrepresentation.{{Cite book|last=Mermin|first=N. David|title=Quantum [Un]Speakables II |chapter=Why QBism is Not the Copenhagen Interpretation and What John Bell Might Have Thought of It |date=2017-01-01|publisher=Springer International Publishing|isbn=9783319389851|editor-last=Bertlmann|editor-first=Reinhold|series=The Frontiers Collection|pages=83–93|language=en|arxiv=1409.2454|doi=10.1007/978-3-319-38987-5_4|s2cid=118458259|editor2-last=Zeilinger|editor2-first=Anton}}{{cite journal|last=Fuchs|first=Christopher A.|year=2017|title=Notwithstanding Bohr, the Reasons for QBism |journal=Mind and Matter| volume=15 |pages=245–300 |arxiv=1705.03483 |bibcode=2017arXiv170503483F}}

QBism takes probabilities to be personal judgments of the individual agent who is using quantum mechanics. This contrasts with older Copenhagen-type views, which hold that probabilities are given by quantum states that are in turn fixed by objective facts about preparation procedures.{{Cite journal|last=Peres|first=Asher|author-link=Asher Peres|date=1984-07-01|title=What is a state vector?|journal=American Journal of Physics|volume=52|issue=7|pages=644–650|doi=10.1119/1.13586|issn=0002-9505|bibcode=1984AmJPh..52..644P}}
{{Cite journal|last1=Caves|first1=Carlton M.|last2=Fuchs|first2=Christopher A.|last3=Schack|first3=Rüdiger|date=2007-06-01|title=Subjective probability and quantum certainty|journal=Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics|series=Probabilities in quantum mechanics|volume=38|issue=2|pages=255–274|arxiv=quant-ph/0608190|doi=10.1016/j.shpsb.2006.10.007|bibcode=2007SHPMP..38..255C|s2cid=119549678}} 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. QBism claims that these points on which it differs from previous Copenhagen-type interpretations resolve the obscurities that many critics have found in the latter, by changing the role that quantum theory plays (even though QBism does not yet provide a specific underlying ontology). Specifically, QBism posits that quantum theory is a normative tool which an agent may use to better navigate reality, rather than a set of mechanics governing it.

Approaches to quantum theory, like QBism,{{Cite journal |last1=Harrigan |first1=Nicholas |last2=Spekkens |first2=Robert W. |date=2010-02-01 |title=Einstein, Incompleteness, and the Epistemic View of Quantum States |journal=Foundations of Physics |language=en |volume=40 |issue=2 |pages=125–157 |arxiv=0706.2661 |doi=10.1007/s10701-009-9347-0 |issn=0015-9018 |bibcode=2010FoPh...40..125H |s2cid=32755624}} which treat quantum states as expressions of information, knowledge, belief, or expectation are called "epistemic" interpretations.{{Cite book|title=What is Quantum Information?|last=Cabello|first=Adán|publisher=Cambridge University Press|year=2017|isbn=9781107142114|editor-last=Lombardi|editor-first=Olimpia|editor-link=Olimpia Lombardi|pages=138–143|chapter=Interpretations of quantum theory: A map of madness|arxiv=1509.04711|editor2-last=Fortin|editor2-first=Sebastian|editor3-last=Holik|editor3-first=Federico|editor4-last=López|editor4-first=Cristian|bibcode=2015arXiv150904711C|doi=10.1017/9781316494233.009|s2cid=118419619}} 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. Furthermore, not all authors who advocate views of this type propose an answer to the question of what the information represented in quantum states concerns. In the words of the paper that introduced the Spekkens Toy Model:

if a quantum state is a state of knowledge, and it is not knowledge of local and noncontextual hidden variables, then what is it knowledge about? We do not at present have a good answer to this question. We shall therefore remain completely agnostic about the nature of the reality to which the knowledge represented by quantum states pertains. This is not to say that the question is not important. Rather, we see the epistemic approach as an unfinished project, and this question as the central obstacle to its completion. Nonetheless, we argue that even in the absence of an answer to this question, a case can be made for the epistemic view. The key is that one can hope to identify phenomena that are characteristic of states of incomplete knowledge regardless of what this knowledge is about.{{Cite journal|last=Spekkens|first=Robert W.|date=2007-01-01|title=Evidence for the epistemic view of quantum states: A toy theory|journal=Physical Review A|volume=75|issue=3|pages=032110|arxiv=quant-ph/0401052|doi=10.1103/PhysRevA.75.032110|bibcode=2007PhRvA..75c2110S|s2cid=117284016}}

Leifer and Spekkens propose a way of treating quantum probabilities as Bayesian probabilities, thereby considering quantum states as epistemic, which they state is "closely aligned in its philosophical starting point" with QBism.{{cite journal|last2=Spekkens|first2=Robert W.|year=2013|title=Towards a Formulation of Quantum Theory as a Causally Neutral Theory of Bayesian Inference|journal=Phys. Rev. A|volume=88|issue=5|pages=052130|arxiv=1107.5849|doi=10.1103/PhysRevA.88.052130|last1=Leifer|first1=Matthew S.|bibcode=2013PhRvA..88e2130L|s2cid=43563970}} However, they remain deliberately agnostic about what physical properties or entities quantum states are information (or beliefs) about, as opposed to QBism, which offers an answer to that question. Another approach, advocated by Bub and Pitowsky, argues that quantum states are information about propositions within event spaces that form non-Boolean lattices.{{Cite book|chapter-url=https://philpapers.org/rec/BUBTDA|title=Many Worlds?: Everett, Quantum Theory & Reality|last1=Bub|first1=Jeffrey|last2=Pitowsky|first2=Itamar|date=2010-01-01|publisher=Oxford University Press|editor-last=Saunders|editor-first=Simon|pages=433–459|chapter=Two dogmas about quantum mechanics|arxiv=0712.4258|editor2-last=Barrett|editor2-first=Jonathan|editor3-last=Kent|editor3-first=Adrian|editor4-last=Wallace|editor4-first=David|bibcode=2007arXiv0712.4258B}} On occasion, the proposals of Bub and Pitowsky are also called "quantum Bayesianism".{{Cite journal|last=Duwell|first=Armond|title=Uncomfortable bedfellows: Objective quantum Bayesianism and the von Neumann–Lüders projection postulate|journal=Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics|volume=42|issue=3|pages=167–175|doi=10.1016/j.shpsb.2011.04.003|bibcode=2011SHPMP..42..167D|year=2011}}

Zeilinger and Brukner have also proposed an interpretation of quantum mechanics in which "information" is a fundamental concept, and in which quantum states are epistemic quantities.{{cite journal |first1=Časlav |last1=Brukner |first2=Anton |last2=Zeilinger |title=Conceptual inadequacy of the Shannon information in quantum measurements |journal=Physical Review A |year=2001 |volume=63 |issue=2 |pages=022113 |doi=10.1103/PhysRevA.63.022113 |arxiv=quant-ph/0006087 |bibcode=2001PhRvA..63b2113B |s2cid=119381924}}
{{cite journal |first1=Časlav |last1=Brukner |first2=Anton |last2=Zeilinger |title=Information Invariance and Quantum Probabilities |journal=Foundations of Physics |year=2009 |volume=39 |issue=7 |pages=677–689 |doi=10.1007/s10701-009-9316-7 |arxiv=0905.0653 |bibcode=2009FoPh...39..677B |s2cid=73599204}} Unlike QBism, the Brukner–Zeilinger interpretation treats some probabilities as objectively fixed. In the Brukner–Zeilinger interpretation, a quantum state represents the information that a hypothetical observer in possession of all possible data would have. Put another way, a quantum state belongs in their interpretation to an optimally informed agent, whereas in QBism, any agent can formulate a state to encode her own expectations.{{cite journal |first=Andrei |last=Khrennikov |title=Reflections on Zeilinger–Brukner information interpretation of quantum mechanics |journal=Foundations of Physics |year=2016 |volume=46 |issue=7 |pages=836–844 |doi=10.1007/s10701-016-0005-z |arxiv=1512.07976 |bibcode=2016FoPh...46..836K |s2cid=119267791}} Despite this difference, in Cabello's classification, the proposals of Zeilinger and Brukner are also designated as "participatory realism", as QBism and the Copenhagen-type interpretations are.

Bayesian, or epistemic, interpretations of quantum probabilities were proposed in the early 1990s by Baez and Youssef.{{cite journal |first=Saul |last=Youssef |title=A Reformulation of Quantum Mechanics |journal= Modern Physics Letters A |volume=6 |issue=3 |pages=225–236 |year=1991 |url=https://physics.bu.edu/~youssef/quantum/mpl.pdf |doi=10.1142/S0217732391000191 |bibcode=1991MPLA....6..225Y }}
{{cite journal |first=Saul |last=Youssef |title=Quantum Mechanics as Bayesian Complex Probability Theory |journal=Modern Physics Letters A |volume=9 |issue=28 |pages=2571–2586 |year=1994 |arxiv=hep-th/9307019 |doi=10.1142/S0217732394002422 |bibcode=1994MPLA....9.2571Y |s2cid=18506337 }}

R. F. Streater argued that "[t]he first quantum Bayesian was von Neumann", basing that claim on von Neumann's textbook The Mathematical Foundations of Quantum Mechanics.{{cite book |title=Lost Causes in and beyond Physics |url=https://archive.org/details/lostcausesphysic00stre |url-access=limited |last=Streater |first=R. F. |publisher=Springer |year=2007 |isbn=978-3-540-36581-5 |page=[https://archive.org/details/lostcausesphysic00stre/page/n76 70]}} Blake Stacey disagrees, arguing that the views expressed in that book on the nature of quantum states and the interpretation of probability are not compatible with QBism, or indeed, with any position that might be called quantum Bayesianism.

Comparisons have also been made between QBism and the relational quantum mechanics (RQM) espoused by Carlo Rovelli and others.{{Cite book|last=Brukner|first=Časlav|title=Quantum [Un]Speakables II |date=2017-01-01|publisher=Springer International Publishing|isbn=9783319389851|editor-last=Bertlmann|editor-first=Reinhold|series=The Frontiers Collection|pages=95–117|language=en|chapter=On the Quantum Measurement Problem|arxiv=1507.05255|doi=10.1007/978-3-319-38987-5_5|s2cid=116892322|editor2-last=Zeilinger|editor2-first=Anton}}
{{cite arXiv|last=Marlow|first=Thomas|date=2006-03-07|title=Relationalism vs. Bayesianism|eprint=gr-qc/0603015}}
{{cite journal|last=Pusey |first=Matthew F. |title=An inconsistent friend |journal=Nature Physics |volume=14 |issue=10 |pages=977–978 |date=2018-09-18 |doi=10.1038/s41567-018-0293-7|bibcode=2018NatPh..14..977P |s2cid=126294105 }}{{Cite journal|last=Pienaar|first=Jacques|date=2021|title=QBism and Relational Quantum Mechanics compared|url=http://dx.doi.org/10.1007/s10701-021-00501-5|journal=Foundations of Physics|volume=51|issue=5|page=96|doi=10.1007/s10701-021-00501-5|arxiv=2108.13977|bibcode=2021FoPh...51...96P|s2cid=237363865|issn=0015-9018}} In both QBism and RQM, quantum states are not intrinsic properties of physical systems.{{Cite journal|last1=Cabello|first1=Adán|last2=Gu|first2=Mile|last3=Gühne|first3=Otfried|last4=Larsson|first4=Jan-Åke|last5=Wiesner|first5=Karoline|date=2016-01-01|title=Thermodynamical cost of some interpretations of quantum theory|journal=Physical Review A|volume=94|issue=5|pages=052127|arxiv=1509.03641|doi=10.1103/PhysRevA.94.052127|bibcode=2016PhRvA..94e2127C|s2cid=601271}} Both QBism and RQM deny the existence of an absolute, universal wavefunction. Furthermore, both QBism and RQM insist that quantum mechanics is a fundamentally local theory.{{Cite journal|last1=Smerlak|first1=Matteo|last2=Rovelli|first2=Carlo|date=2007-02-26|title=Relational EPR|journal=Foundations of Physics|volume=37|issue=3|pages=427–445|arxiv=quant-ph/0604064|bibcode=2007FoPh...37..427S|doi=10.1007/s10701-007-9105-0|s2cid=11816650|issn=0015-9018}} In addition, Rovelli, like several QBist authors, advocates reconstructing quantum theory from physical principles in order to bring clarity to the subject of quantum foundations.{{Cite journal|last=Rovelli|first=Carlo|date=1996-08-01|title=Relational quantum mechanics|journal=International Journal of Theoretical Physics|language=en|volume=35|issue=8|pages=1637–1678|arxiv=quant-ph/9609002|doi=10.1007/BF02302261|issn=0020-7748|bibcode=1996IJTP...35.1637R|s2cid=16325959}} (The QBist approaches to doing so are different from Rovelli's, and are described below.) One important distinction between the two interpretations is their philosophy of probability: RQM does not adopt the Ramsey–de Finetti school of personalist Bayesianism. Moreover, RQM does not insist that a measurement outcome is necessarily an agent's experience.

QBism should be distinguished from other applications of Bayesian inference in quantum physics, and from quantum analogues of Bayesian inference. For example, some in the field of computer science have introduced a kind of quantum Bayesian network, which they argue could have applications in "medical diagnosis, monitoring of processes, and genetics".{{Cite journal|last=Tucci|first=Robert R.|date=1995-01-30|title=Quantum bayesian nets|journal=International Journal of Modern Physics B|volume=09|issue=3|pages=295–337|arxiv=quant-ph/9706039|doi=10.1142/S0217979295000148|issn=0217-9792|bibcode=1995IJMPB...9..295T|s2cid=18217167}}{{cite journal|last1=Moreira|first1=Catarina|last2=Wichert|first2=Andreas|date=2016|title=Quantum-Like Bayesian Networks for Modeling Decision Making|journal=Frontiers in Psychology |volume=7|pages=11|doi=10.3389/fpsyg.2016.00011|pmid=26858669|pmc=4726808|doi-access=free}} Bayesian inference has also been applied in quantum theory for updating probability densities over quantum states,{{Cite journal|title=Principles of quantum inference |doi=10.1016/0003-4916(91)90182-8|last=Jones|first=K. R. W.|language=en|volume=207|issue=1 |journal=Annals of Physics|pages=140–170|year=1991 |bibcode=1991AnPhy.207..140J }} and MaxEnt methods have been used in similar ways.{{Cite web|url=https://math.ucr.edu/home/baez/bayes.html |title=Bayesian Probability Theory and Quantum Mechanics|last=Baez|first=John|date=2003-09-12|access-date=2017-04-18}}{{Cite journal|title=Reconstruction of Quantum States of Spin Systems: From Quantum Bayesian Inference to Quantum Tomography |doi=10.1006/aphy.1998.5802|last1=Bužek|first1=V.|last2=Derka|first2=R.|language=en|last3=Adam|first3=G.|last4=Knight|first4=P. L.|volume=266|issue=2 |journal=Annals of Physics|pages=454–496|year=1998 |bibcode=1998AnPhy.266..454B }} Bayesian methods for quantum state and process tomography are an active area of research.{{Cite journal|last1=Granade|first1=Christopher|last2=Combes|first2=Joshua|last3=Cory|first3=D. G.|date=2016-01-01|title=Practical Bayesian tomography|journal=New Journal of Physics|language=en|volume=18|issue=3|pages=033024|arxiv=1509.03770|doi=10.1088/1367-2630/18/3/033024|issn=1367-2630|bibcode=2016NJPh...18c3024G|s2cid=88521187}}

Conceptual concerns about the interpretation of quantum mechanics and the meaning of probability have motivated technical work. A quantum version of the de Finetti theorem, introduced by Caves, Fuchs, and Schack (independently reproving a result found using different means by Størmer{{cite journal|last=Størmer|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|hdl=10852/45014|hdl-access=free}}) to provide a Bayesian understanding of the idea of an "unknown quantum state",{{Cite journal|last1=Caves|first1=Carlton M.|last2=Fuchs|first2=Christopher A.|last3=Schack|first3=Ruediger|date=2002-08-20|title=Unknown quantum states: The quantum de Finetti representation|journal=Journal of Mathematical Physics|volume=43|issue=9|pages=4537–4559|arxiv=quant-ph/0104088|bibcode=2002JMP....43.4537C|doi=10.1063/1.1494475|s2cid=17416262|issn=0022-2488}}{{cite web|url=http://math.ucr.edu/home/baez/week251.html|title=This Week's Finds in Mathematical Physics (Week 251)|first=J. |last=Baez|year=2007|access-date=2017-04-18|author-link=John C. Baez}} has found application elsewhere, in topics like quantum key distribution{{cite arXiv|last=Renner|first=Renato|date=2005-12-30|title=Security of Quantum Key Distribution|eprint=quant-ph/0512258}} and entanglement detection.{{Cite journal|last1=Doherty|first1=Andrew C.|last2=Parrilo|first2=Pablo A.|last3=Spedalieri|first3=Federico M.|date=2005-01-01|title=Detecting multipartite entanglement|url=http://espace.library.uq.edu.au/view/UQ:76023/UQ76023.pdf|journal=Physical Review A|volume=71|issue=3|pages=032333|arxiv=quant-ph/0407143|bibcode=2005PhRvA..71c2333D|doi=10.1103/PhysRevA.71.032333|s2cid=44241800}}

Adherents of several interpretations of quantum mechanics, QBism included, have been motivated to reconstruct quantum theory. The goal of these research efforts has been to identify a new set of axioms or postulates from which the mathematical structure of quantum theory can be derived, in the hope that with such a reformulation, the features of nature which made quantum theory the way it is might be more easily identified.{{cite book|title=Quantum Theory: Informational Foundations and Foils|volume=181|last1=Chiribella|first1=Giulio|last2=Spekkens|first2=Rob W.|publisher=Springer|year=2016|pages=1–18|chapter=Introduction |doi=10.1007/978-94-017-7303-4|series=Fundamental Theories of Physics|isbn=978-94-017-7302-7|arxiv=1208.4123|bibcode=2016qtif.book.....C |s2cid=118699215}} Although the core tenets of QBism do not demand such a reconstruction, some QBists—Fuchs, in particular—have argued that the task should be pursued.

One topic prominent in the reconstruction effort is the set of mathematical structures known as symmetric, informationally-complete, positive operator-valued measures (SIC-POVMs). QBist foundational research stimulated interest in these structures, which now have applications in quantum theory outside of foundational studiesTechnical references on SIC-POVMs include the following:

{{Cite journal|last=Scott|first=A. J.|date=2006-01-01|title=Tight informationally complete quantum measurements|journal=Journal of Physics A: Mathematical and General|language=en|volume=39|issue=43|pages=13507–13530|arxiv=quant-ph/0604049|bibcode=2006JPhA...3913507S|doi=10.1088/0305-4470/39/43/009|s2cid=33144766|issn=0305-4470}}

{{Cite arXiv|last1=Wootters|first1=William K.|author-link=William Wootters|last2=Sussman|first2=Daniel M.|title=Discrete phase space and minimum-uncertainty states|eprint=0704.1277|class=quant-ph|year=2007}}

{{Cite journal|last1=Appleby|first1=D. M.|last2=Bengtsson|first2=Ingemar|last3=Brierley|first3=Stephen|last4=Grassl|first4=Markus|last5=Gross|first5=David|last6=Larsson|first6=Jan-Åke|date=2012-05-01|title=The Monomial Representations of the Clifford Group|url=http://dl.acm.org/citation.cfm?id=2230996.2230999|journal=Quantum Information & Computation|volume=12|issue=5–6|pages=404–431|doi=10.26421/QIC12.5-6-3|arxiv=1102.1268|bibcode=2011arXiv1102.1268A|s2cid=1250951|issn=1533-7146}}

{{Cite journal|last1=Hou |first1=Zhibo |last2=Tang |first2=Jun-Feng |last3=Shang |first3=Jiangwei |last4=Zhu |first4=Huangjun |last5=Li |first5=Jian |last6=Yuan |first6=Yuan |last7=Wu |first7=Kang-Da |last8=Xiang |first8=Guo-Yong |last9=Li |first9=Chuan-Feng |date=2018-04-12 |title=Deterministic realization of collective measurements via photonic quantum walks|journal=Nature Communications |language=En |volume=9 |issue=1 |pages=1414 |arxiv=1710.10045 |bibcode=2018NatCo...9.1414H |doi=10.1038/s41467-018-03849-x |pmid=29650977 |pmc=5897416 |issn=2041-1723}} and in pure mathematics.{{Cite journal|last1=Appleby|first1=Marcus|last2=Flammia|first2=Steven|last3=McConnell|first3=Gary|last4=Yard|first4=Jon|date=2017-04-24|title=SICs and Algebraic Number Theory|journal=Foundations of Physics|language=en|volume=47|issue=8|pages=1042–1059|arxiv=1701.05200|bibcode=2017FoPh...47.1042A|doi=10.1007/s10701-017-0090-7|s2cid=119334103|issn=0015-9018}}

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.{{Cite journal|last1=Fuchs|first1=Christopher A.|last2=Schack|first2=Rüdiger|date=2010-01-08|title=A Quantum-Bayesian Route to Quantum-State Space|journal=Foundations of Physics|language=en|volume=41|issue=3|pages=345–356|arxiv=0912.4252|doi=10.1007/s10701-009-9404-8|issn=0015-9018|bibcode=2011FoPh...41..345F|s2cid=119277535}}{{Cite journal|last1=Appleby|first1=D. M.|last2=Ericsson|first2=Åsa|last3=Fuchs|first3=Christopher A.|date=2010-04-27|title=Properties of QBist State Spaces|journal=Foundations of Physics|language=en|volume=41|issue=3|pages=564–579|arxiv=0910.2750|doi=10.1007/s10701-010-9458-7|issn=0015-9018|bibcode=2011FoPh...41..564A|s2cid=119296426}} 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.{{Cite journal|last=Rosado|first=José Ignacio|date=2011-01-28|title=Representation of Quantum States as Points in a Probability Simplex Associated to a SIC-POVM|journal=Foundations of Physics|language=en|volume=41|issue=7|pages=1200–1213|arxiv=1007.0715|doi=10.1007/s10701-011-9540-9|issn=0015-9018|bibcode=2011FoPh...41.1200R|s2cid=119102347}} 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. Fuchs, Schack, and others have taken to calling this restatement of the Born rule the urgleichung, from the German for "primal equation" (see Ur- prefix), because of the central role it plays in their reconstruction of quantum theory.{{cite journal|last1=Appleby|first1=Marcus|last2=Fuchs|first2=Christopher A.|last3=Stacey|first3=Blake C.|last4=Zhu|first4=Huangjun|date=2016-12-09|title=Introducing the Qplex: A Novel Arena for Quantum Theory|arxiv=1612.03234|doi=10.1140/epjd/e2017-80024-y|volume=71|issue=7|journal=The European Physical Journal D|page=197|bibcode=2017EPJD...71..197A|s2cid=119240836}}{{Cite journal|last1=Słomczyński|first1=Wojciech|last2=Szymusiak|first2=Anna|date=2020-09-30|title=Morphophoric POVMs, generalised qplexes, and 2-designs|url=https://quantum-journal.org/papers/q-2020-09-30-338/|journal=Quantum|language=en|volume=4|pages=338|arxiv=1911.12456|bibcode=2020Quant...4..338S|doi=10.22331/q-2020-09-30-338|s2cid=221663304|issn=2521-327X}}

The following discussion presumes some familiarity with the mathematics of quantum information theory, and in particular, the modeling of measurement procedures by POVMs. Consider a quantum system to which is associated a $d$-dimensional Hilbert space. If a set of $d^2$ rank-1 projectors $\backslash hat\{\backslash Pi\}\_i$ satisfying$$\backslash operatorname\{tr\}\backslash hat\{\backslash Pi\}\_i\backslash hat\{\backslash Pi\}\_j=\backslash frac\{d\backslash delta\_\{ij\}+1\}\{d+1\}$$exists, then one may form a SIC-POVM $\backslash hat\{H\}\_i=\backslash frac\{1\}\{d\}\backslash hat\{\backslash Pi\}\_i$. An arbitrary quantum state $\backslash hat\{\backslash rho\}$ may be written as a linear combination of the SIC projectors$$\backslash hat\{\backslash rho\}=\backslash sum\_\{i=1\}^\{d^2\}\; \backslash left[(d+1)P(H\_i)-\backslash frac\; 1\; d\; \backslash right]\; \backslash hat\{\backslash Pi\}\_i,$$where $P(H\_i)=\backslash operatorname\{tr\}\backslash hat\{\backslash rho\}\; \backslash hat\{H\}\_i$ is the Born rule probability for obtaining SIC measurement outcome $H\_i$ implied by the state assignment $\backslash hat\{\backslash rho\}$. We follow the convention that operators have hats while experiences (that is, measurement outcomes) do not. Now consider an arbitrary quantum measurement, denoted by the POVM $\backslash \{\backslash hat\{D\}\_j\backslash \}$. The urgleichung is the expression obtained from forming the Born rule probabilities, $Q(D\_j)=\backslash operatorname\{tr\}\backslash hat\{\backslash rho\}\; \backslash hat\{D\}\_j$, for the outcomes of this quantum measurement, $$Q(D\_j)=\backslash sum\_\{i=1\}^\{d^2\}\backslash left[(d+1)P(H\_i)-\backslash frac\{1\}\{d\}\backslash right]P(D\_j\backslash mid\; H\_i),$$where $P(D\_j\backslash mid\; H\_i)\backslash equiv\backslash operatorname\{tr\}\backslash hat\{\backslash Pi\}\_i\backslash hat\{D\}\_j$ is the Born rule probability for obtaining outcome $D\_j$ implied by the state assignment $\backslash hat\{\backslash Pi\}\_i$. The $P(D\_j\backslash mid\; H\_i)$ term may be understood to be a conditional probability in a cascaded measurement scenario: Imagine that an agent plans to perform two measurements, first a SIC measurement and then the $\backslash \{D\_j\backslash \}$ measurement. After obtaining an outcome from the SIC measurement, the agent will update her state assignment to a new quantum state $\backslash hat\{\backslash rho\}\text{'}$ before performing the second measurement. If she uses the Lüders rule{{Cite book|title=Compendium of Quantum Physics|url=https://archive.org/details/compendiumquantu00gree|url-access=limited|last1=Busch|first1=Paul|author-link1=Paul Busch (physicist) |last2=Lahti|first2=Pekka|date=2009-01-01|publisher=Springer Berlin Heidelberg|isbn=9783540706229|editor-last=Greenberger|editor-first=Daniel|pages=[https://archive.org/details/compendiumquantu00gree/page/n370 356]–358|language=en|chapter=Lüders Rule|doi=10.1007/978-3-540-70626-7_110|editor2-last=Hentschel|editor2-first=Klaus|editor3-last=Weinert|editor3-first=Friedel}} for state update and obtains outcome $H\_i$ from the SIC measurement, then $\backslash hat\{\backslash rho\}\text{'}=\backslash hat\{\backslash Pi\}\_i$. Thus the probability for obtaining outcome $D\_j$ for the second measurement conditioned on obtaining outcome $H\_i$ for the SIC measurement is $P(D\_j\backslash mid\; H\_i)$.

Note that the urgleichung is structurally very similar to the law of total probability, which is the expression$$P(D\_j)=\backslash sum\_\{i=1\}^\{d^2\}P(H\_i)P(D\_j\backslash mid\; H\_i).$$They functionally differ only by a dimension-dependent affine transformation of the SIC probability vector. As QBism says that quantum theory is an empirically-motivated normative addition to probability theory, Fuchs and others find the appearance of a structure in quantum theory analogous to one in probability theory to be an indication that a reformulation featuring the urgleichung prominently may help to reveal the properties of nature which made quantum theory so successful.

The urgleichung does not replace the law of total probability. Rather, the urgleichung and the law of total probability apply in different scenarios because $P(D\_j)$ and $Q(D\_j)$ refer to different situations. $P(D\_j)$ is the probability that an agent assigns for obtaining outcome $D\_j$ on her second of two planned measurements, that is, for obtaining outcome $D\_j$ after first making the SIC measurement and obtaining one of the $H\_i$ outcomes. $Q(D\_j)$, on the other hand, is the probability an agent assigns for obtaining outcome $D\_j$ when she does not plan to first make the SIC measurement. The law of total probability is a consequence of coherence within the operational context of performing the two measurements as described. The urgleichung, in contrast, is a relation between different contexts which finds its justification in the predictive success of quantum physics.

The SIC representation of quantum states also provides a reformulation of quantum dynamics. Consider a quantum state $\backslash hat\{\backslash rho\}$ with SIC representation $P(H\_i)$. The time evolution of this state is found by applying a unitary operator $\backslash hat\{U\}$ to form the new state $\backslash hat\{U\}\backslash hat\{\backslash rho\}\backslash hat\{U\}^\backslash dagger$, which has the SIC representation

$$P\_t(H\_i)=\backslash operatorname\{tr\}\backslash left[(\backslash hat\{U\}\backslash hat\{\backslash rho\}\backslash hat\{U\}^\backslash dagger)\; \backslash hat\{H\}\_i\backslash right]=\backslash operatorname\{tr\}\backslash left[\backslash hat\{\backslash rho\}(\backslash hat\{U\}^\backslash dagger\; \backslash hat\{H\}\_i\; \backslash hat\{U\})\; \backslash right].$$

The second equality is written in the Heisenberg picture of quantum dynamics, with respect to which the time evolution of a quantum system is captured by the probabilities associated with a rotated SIC measurement $\backslash \{D\_j\backslash \}=\backslash \{\backslash hat\{U\}^\backslash dagger\; \backslash hat\{H\}\_j\backslash hat\{U\}\backslash \}$ of the original quantum state $\backslash hat\{\backslash rho\}$. Then the Schrödinger equation is completely captured in the urgleichung for this measurement:$$P\_t(H\_j)=\backslash sum\_\{i=1\}^\{d^2\}\backslash left[(d+1)P(H\_i)-\backslash frac\; 1\; d\; \backslash right]P(D\_j\backslash mid\; H\_i).$$In these terms, the Schrödinger equation is an instance of the Born rule applied to the passing of time; an agent uses it to relate how she will gamble on informationally complete measurements potentially performed at different times.

Those QBists who find this approach promising are pursuing a complete reconstruction of quantum theory featuring the urgleichung as the key postulate. (The urgleichung has also been discussed in the context of category theory.{{Cite journal|last=van de Wetering|first=John|title=Quantum theory is a quasi-stochastic process theory|journal=Electronic Proceedings in Theoretical Computer Science|volume=266|issue=2018|pages=179–196|arxiv=1704.08525|doi=10.4204/EPTCS.266.12|year=2018|s2cid=53635011}}) Comparisons between this approach and others not associated with QBism (or indeed with any particular interpretation) can be found in a book chapter by Fuchs and Stacey{{Cite book|title=Quantum Theory: Informational Foundations and Foils|last1=Fuchs|first1=Christopher A.|last2=Stacey|first2=Blake C.|date=2016-01-01|publisher=Springer Netherlands|isbn=9789401773027|editor-last=Chiribella|editor-first=Giulio|series=Fundamental Theories of Physics|volume=181 |pages=283–305|language=en|chapter=Some Negative Remarks on Operational Approaches to Quantum Theory|arxiv=1401.7254|doi=10.1007/978-94-017-7303-4_9|s2cid=116428784|editor2-last=Spekkens|editor2-first=Robert W.}} and an article by Appleby et al. As of 2017, alternative QBist reconstruction efforts are in the beginning stages.{{Cite web|url=http://fqxi.org/grants/large/awardees/view/__details/2016/chiribella|title=The Observer Observed: a Bayesian Route to the Reconstruction of Quantum Theory|last1=Chiribella|first1=Giulio|last2=Cabello|first2=Adán|website=FQXi: Foundational Questions Institute|access-date=2017-04-18|last3=Kleinmann|first3=Matthias}}

{{Portal|Physics}}

{{cols|colwidth=18em}}

• Bayes factor
• Bayesian inference
• Credible intervals
• Degree of belief
• Doxastic logic
• Philosophy of science
• Quantum logic
• Quantum probability
• Statistical inference

{{colend}}

{{Reflist}}

• [http://physics.bu.edu/~youssef/quantum/quantum_refs.html Exotic Probability Theories and Quantum Mechanics: References]
• [https://arxiv.org/abs/quant-ph/0105039 Notes on a Paulian Idea: Foundational, Historical, Anecdotal and Forward-Looking Thoughts on the Quantum] – Cerro Grande Fire Series, Volume 1
• [https://arxiv.org/abs/1405.2390 My Struggles with the Block Universe] – Cerro Grande Fire Series, Volume 2
• [http://www.abc.net.au/radionational/programs/philosopherszone/a-universe-alive-with-possibility/6802576 Why the multiverse is all about you] – [http://www.abc.net.au/radionational/programs/philosopherszone/ The Philosopher's Zone] interview with Fuchs
• [https://www.quantamagazine.org/20150604-quantum-bayesianism-qbism/ A Private View of Quantum Reality] – Quanta Magazine interview with Fuchs
• [https://theconversation.com/qbism-quantum-mechanics-is-not-a-description-of-objective-reality-it-reveals-a-world-of-genuine-free-will-200487 Rüdiger Schack on QBism] in The Conversation
• [http://stias.ac.za/research/projects/participatory-realism/ Participatory Realism] – 2017 conference at the [http://stias.ac.za/ Stellenbosch Institute for Advanced Study]
• [https://web.archive.org/web/20090511060206/http://www.uni-konstanz.de/ppm/events/bbqw2005/ Being Bayesian in a Quantum World] – 2005 conference at the University of Konstanz
• {{Cite news|url=http://www.investigacionyciencia.es/revistas/investigacion-y-ciencia/la-red-de-la-memoria-712/el-puzle-de-la-teora-cuntica-15554|title=El puzle de la teoría cuántica: ¿Es posible zanjar científicamente el debate sobre la naturaleza del mundo cuántico?|last=Cabello|first=Adán|date=September 2017|work=Investigación y Ciencia}}
• {{cite AV media|url=https://www.youtube.com/watch?v=95fKJF5frtE |title=Some Tenets of QBism |people = Fuchs, Christopher (presenter); Stacey, Blake (editor); Thisdell, Bill (editor)|date=2018-04-25 |publisher=YouTube |access-date=2018-05-17}}
• {{cite arXiv|eprint=1810.13401 |title=FAQBism |last1=DeBrota |first1=John B. |last2=Stacey |first2=Blake C. |date=2018-10-31|class=quant-ph }}

{{Quantum mechanics topics|state=expanded}}

{{Quantum information}}

{{Authority control}}

Category:Interpretations of quantum mechanics

Category:Bayesian statistics