Quantum Bayesianism

British philosopher, mathematician, and economist Frank Ramsey, whose interpretation of probability theory closely matches the one adopted by QBism.[14]

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."[15] 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 a personalist Bayesian interpretation along the lines of Italian mathematician Bruno de Finetti[16] and English philosopher Frank Ramsey.[17][18]

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 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.[19]) 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.[13] 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."[20] 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.[21][22] Therefore, while many interpretations of quantum theory conclude that quantum mechanics is a nonlocal theory, QBists do not.[23]

Fuchs introduced the term "QBism" and outlined the interpretation in more or less its present form in 2010,[24] carrying further and demanding consistency of ideas broached earlier, notably in publications from 2002.[25][26] Several subsequent papers have expanded and elaborated upon these foundations, notably a Reviews of Modern Physics article by Fuchs and Schack;[19] an American Journal of Physics article by Fuchs, Mermin, and Schack;[23] and Enrico Fermi Summer School[27] lecture notes by Fuchs and Stacey.[22]

Prior to the 2010 paper, 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, the Other uses of Bayesian probability in quantum physics section 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,[28] and media coverage of QBism has been illustrated with art by Picasso[7] and Gris.[29] 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.[30]

According to QBism, quantum theory is a tool which an agent may use to help manage his or her expectations, more like probability theory than a conventional physical theory.[13] 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:[31]

1. All probabilities, including those equal to zero or one, are valuations that an agent ascribes to his or her 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 his or her 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.

Jean Metzinger, 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 ..."[28]

Reactions to the QBist interpretation have ranged from enthusiastic[13][28] to strongly negative.[32] 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,[33] and Jaeger finds QBism's supposition that the interpretation of probability is key for the resolution to be unnatural and unconvincing.[12] Norsen[34] has accused QBism of solipsism, and Wallace[35] identifies QBism as an instance of instrumentalism; QBists have argued insistently that these characterizations are misunderstandings, and that QBism is neither solipsist nor instrumentalist.[17][36] A critical article by Nauenberg[32] in the American Journal of Physics prompted a reply by Fuchs, Mermin, and Schack.[37] 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.[38] 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.[1] Fuchs and Schack replied to these concerns in a later article.[39] Mermin advocated QBism in a 2012 Physics Today article,[2] 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.[40][41] Section 2 of the Stanford Encyclopedia of Philosophy entry on QBism also contains a summary of objections to the interpretation, and some replies.[42] Others are opposed to QBism on more general philosophical grounds; for example, Mohrhoff criticizes QBism from the standpoint of Kantian philosophy.[43]

Certain authors find QBism internally self-consistent, but do not subscribe to the interpretation.[44] 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.[45] 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.[46] In addition, some agree with most, but perhaps not all, of the core tenets of QBism; Barnum's position,[47] as well as Appleby's,[48] are examples.

Popularized or semi-popularized media coverage of QBism has appeared in New Scientist,[49] Scientific American,[50] Nature,[51] Science News,[52] the FQXi Community,[53] the Frankfurter Allgemeine Zeitung,[29] Quanta Magazine,[16] Aeon,[54] and Discover.[55] 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.[56][57] Furthermore, Harvard University Press published a popularized treatment of the subject, QBism: The Future of Quantum Physics, in 2016.[13]

The philosophy literature has also discussed QBism from the viewpoints of structural realism and of phenomenology.[58][59][60]

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[82]) to provide a Bayesian understanding of the idea of an "unknown quantum state",[83][84] has found application elsewhere, in topics like quantum key distribution[85] and entanglement detection.[86]

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.[51][87] Although the core tenets of QBism do not demand such a reconstruction, some QBists—Fuchs,[26] 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 studies[88] and in pure mathematics.[89]

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.[90][91] 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.[92] 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.[19][93][94]

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 ${\textstyle d}$-dimensional Hilbert space. If a set of ${\textstyle d^{2}}$ rank-1 projectors ${\displaystyle {\hat {\Pi }}_{i}}$ satisfying

$${\displaystyle \operatorname {tr} {\hat {\Pi }}_{i}{\hat {\Pi }}_{j}={\frac {d\delta _{ij}+1}{d+1}}}$$

exists, then one may form a SIC-POVM ${\textstyle {\hat {H}}_{i}={\frac {1}{d}}{\hat {\Pi }}_{i}}$. An arbitrary quantum state ${\displaystyle {\hat {\rho }}}$ may be written as a linear combination of the SIC projectors

$${\displaystyle {\hat {\rho }}=\sum _{i=1}^{d^{2}}\left[(d+1)P(H_{i})-{\frac {1}{d}}\right]{\hat {\Pi }}_{i},}$$

where ${\textstyle P(H_{i})=\operatorname {tr} {\hat {\rho }}{\hat {H}}_{i}}$ is the Born rule probability for obtaining SIC measurement outcome ${\displaystyle H_{i}}$ implied by the state assignment ${\displaystyle {\hat {\rho }}}$. 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 ${\displaystyle \{{\hat {D}}_{j}\}}$. The urgleichung is the expression obtained from forming the Born rule probabilities, ${\textstyle Q(D_{j})=\operatorname {tr} {\hat {\rho }}{\hat {D}}_{j}}$, for the outcomes of this quantum measurement,

$${\displaystyle Q(D_{j})=\sum _{i=1}^{d^{2}}\left[(d+1)P(H_{i})-{\frac {1}{d}}\right]P(D_{j}\mid H_{i}),}$$

where ${\displaystyle P(D_{j}\mid H_{i})\equiv \operatorname {tr} {\hat {\Pi }}_{i}{\hat {D}}_{j}}$ is the Born rule probability for obtaining outcome ${\displaystyle D_{j}}$ implied by the state assignment ${\displaystyle {\hat {\Pi }}_{i}}$. The ${\displaystyle P(D_{j}\mid H_{i})}$ 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 ${\displaystyle \{D_{j}\}}$ measurement. After obtaining an outcome from the SIC measurement, the agent will update her state assignment to a new quantum state ${\displaystyle {\hat {\rho }}'}$ before performing the second measurement. If she uses the Lüders rule[95] for state update and obtains outcome ${\displaystyle H_{i}}$ from the SIC measurement, then ${\textstyle {\hat {\rho }}'={\hat {\Pi }}_{i}}$. Thus the probability for obtaining outcome ${\displaystyle D_{j}}$ for the second measurement conditioned on obtaining outcome ${\displaystyle H_{i}}$ for the SIC measurement is ${\displaystyle P(D_{j}\mid H_{i})}$. Note that the urgleichung is structurally very similar to the law of total probability, which is the expression

$${\displaystyle P(D_{j})=\sum _{i=1}^{d^{2}}P(H_{i})P(D_{j}\mid H_{i}).}$$

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.[19][22]

It is important to recognize that the urgleichung does not replace the law of total probability. Rather, the urgleichung and the law of total probability apply in different scenarios because ${\displaystyle P(D_{j})}$ and ${\displaystyle Q(D_{j})}$ refer to different situations. ${\displaystyle P(D_{j})}$ is the probability that an agent assigns for obtaining outcome ${\displaystyle D_{j}}$ on her second of two planned measurements, that is, for obtaining outcome ${\displaystyle D_{j}}$ after first making the SIC measurement and obtaining one of the ${\displaystyle H_{i}}$ outcomes. ${\displaystyle Q(D_{j})}$, on the other hand, is the probability an agent assigns for obtaining outcome ${\displaystyle D_{j}}$ 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 ${\displaystyle {\hat {\rho }}}$ with SIC representation ${\textstyle P(H_{i})}$. The time evolution of this state is found by applying a unitary operator ${\displaystyle {\hat {U}}}$ to form the new state ${\textstyle {\hat {U}}{\hat {\rho }}{\hat {U}}^{\dagger }}$, which has the SIC representation

$${\displaystyle P_{t}(H_{i})=\operatorname {tr} \left[({\hat {U}}{\hat {\rho }}{\hat {U}}^{\dagger }){\hat {H}}_{i}\right]=\operatorname {tr} \left[{\hat {\rho }}({\hat {U}}^{\dagger }{\hat {H}}_{i}{\hat {U}})\right].}$$

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 ${\textstyle \{D_{j}\}=\{{\hat {U}}^{\dagger }{\hat {H}}_{j}{\hat {U}}\}}$ of the original quantum state ${\displaystyle {\hat {\rho }}}$. Then the Schrödinger equation is completely captured in the urgleichung for this measurement:

$${\displaystyle P_{t}(H_{j})=\sum _{i=1}^{d^{2}}\left[(d+1)P(H_{i})-{\frac {1}{d}}\right]P(D_{j}\mid H_{i}).}$$

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.[93] (The urgleichung has also been discussed in the context of category theory.[96]) 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[97] and an article by Appleby et al.[93] As of 2017, alternative QBist reconstruction efforts are in the beginning stages.[98]

• Exotic Probability Theories and Quantum Mechanics: References
• Notes on a Paulian Idea: Foundational, Historical, Anecdotal and Forward-Looking Thoughts on the Quantum – Cerro Grande Fire Series, Volume 1
• My Struggles with the Block Universe – Cerro Grande Fire Series, Volume 2
• Why the multiverse is all about you – The Philosopher's Zone interview with Fuchs
• A Private View of Quantum Reality – Quanta Magazine interview with Fuchs
• Rüdiger Schack on quantum Bayesianism – Machine Intelligence Research Institute interview with Schack
• Participatory Realism – 2017 conference at the Stellenbosch Institute for Advanced Study
• Being Bayesian in a Quantum World – 2005 conference at the University of Konstanz
• Cabello, Adán (September 2017). "El puzle de la teoría cuántica: ¿Es posible zanjar científicamente el debate sobre la naturaleza del mundo cuántico?". Investigación y Ciencia.
• Fuchs, Christopher (presenter); Stacey, Blake (editor); Thisdell, Bill (editor) (2018-04-25). Some Tenets of QBism. YouTube. Retrieved 2018-05-17.
• DeBrota, John B.; Stacey, Blake C. (2018-10-31). "FAQBism". arXiv:1810.13401 [quant-ph].