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Category:Interpretations of quantum mechanics In physics and the philosophy of physics, Quantum Bayesianism usually refers to an interpretation of quantum mechanics also known as QBism. This interpretation takes an agent's action and experience as the central concerns of the theory and uses a subjective Bayesian account of probabilities to understand the Born rule as a normative addition to good decision-making. Rooted in the prior work of Carlton Caves, Christopher Fuchs, and Rüdiger Schack during the early 2000s, QBism itself is primarily associated with Fuchs and Schack and has more recently been adopted by David Mermin. QBism draws from the fields of quantum information and Bayesian probability and aims to eliminate the interpretational conundrums that have beset quantum theory. The QBist interpretation is historically derivative of the views of the various physicists that are often grouped together as "the" Copenhagen interpretation, but is itself distinct from them. Theodor Hänsch has characterized Qbism as sharpening the Copenhagen interpretation and making it more consistent.

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".

QBism deals with common questions in the interpretation of quantum theory about the nature of wavefunction superposition, nonlocality, and entanglement. 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. 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. For this reason, some philosophers of science have deemed QBism a form of anti-realism. The originators of the interpretation disagree with this characterization, proposing instead that the theory more properly aligns with a kind of realism they call "participatory realism", wherein reality consists of more than can be captured by any putative third-person account of it.

In addition to presenting an interpretation of the existing mathematical structure of quantum theory, some QBists have advocated a research program of reconstructing quantum theory from basic physical principles whose QBist character is manifest, as described in the Reconstructing quantum theory section below. The QBist interpretation itself, as described in the Core positions section, however, does not depend on any particular reconstruction.

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 and in pure mathematics. Likewise, a quantum version of the de Finetti theorem, introduced by Caves, Fuchs, and Schack (independently reproving a result found using different means by Størmer) to provide a Bayesian understanding of the idea of an "unknown quantum state", has found application elsewhere, in topics like quantum key distribution and entanglement detection.

History and development

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

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." 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 and English philosopher Frank Ramsey.

According to QBists, the advantages of adopting this view of probability are twofold. First, it suggests that the Einstein-Podolsky-Rosen (EPR) criterion of reality should be rejected because it identifies "probability one" assignments with elements of reality preexisting the quantum measurement outcomes. A personalist Bayesian considers all probabilities, even those equal to unity, to be degrees of belief. Therefore, QBists do not conclude, as many although not all other interpretations of quantum theory do, that quantum mechanics is a nonlocal theory. Second, for QBists the role of quantum states such as the wavefunctions of a particles, in physics is to efficiently encode probabilities; so quantum states, including wavefunctions, are 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.

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

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 homonym of Cubism the art movement, it has motivated conceptual comparisons between the two. However, QBism itself was not influenced or motivated by Cubism and has no lineage to a potential connection between Cubism and Bohr's views on quantum theory.

Core positions

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. 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 as:

1. All probabilities, including probability-1 assignments, 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 rule is normative, not descriptive or prescriptive. 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.

Reception and criticism

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 ..." – David Mermin

Reactions to the QBist interpretation have ranged from delight to outrage. 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 and Jaeger finds QBism's supposition that the interpretation of probability is key for the resolution to be unnatural and unconvincing. Norsen has accused QBism of solipsism, and Wallace identifies QBism as an instance of instrumentalism; QBists have argued strongly that these characterizations are misunderstandings, and that QBism is neither solipsist nor instrumentalist. A critical article by Nauenberg in the American Journal of Physics prompted a reply by Fuchs, Mermin, and Schack; Some assert that there may be inconsistencies; for example, Stairs argues that probability-1 assignments cannot be degrees of belief as QBists say. Further, while also raising concerns about the treatment of probability-1 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. Several further critiques of QBism which arose in response to Mermin's Physics Today article, 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 also contains a list of objections and replies to the interpretation. Others are opposed to QBism on more general philosophical grounds; for example, Mohrhoff criticizes QBism from the standpoint of Kantian philosophy.

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

Relation to other interpretations

Group photo from the 2005 University of Konstanz conference Being Bayesian in a Quantum World.

Copenhagen interpretations

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. QBism shares many characteristics in common with the ideas often labaled 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.

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 these points on which it differs from Copenhagen resolve the obscurities that many critics have found in the Copenhagen interpretation, by changing 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.

Other epistemic interpretations

Approaches to quantum theory, like QBism, 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. 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.

QBism answers the question "what is a quantum state information about?" with the reply, "It encodes expectations about potential future experiences." By contrast, for example, Bub and Pitowsky argue that quantum states are information about propositions within event spaces that form non-Boolean lattices.

Von Neumann's views

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. 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.

Relational quantum mechanics

Comparisons have also been made between QBism and the relational quantum mechanics espoused by Carlo Rovelli and others.

Other uses of Bayesian probability in quantum physics

QBism should be distinguished from other applications of Bayesian inference or quantum analogues of Bayesian inference, in quantum physics. 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". (A Bayesian framework is also used for neural networks.) Bayesian inference has also been applied in quantum theory for updating probability densities over quantum states, and MaxEnt methods have been used in similar ways.

Reconstructing quantum theory

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. 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.

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. 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. Fuchs, Schack and others have taken to calling this restatment 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.

Consider a $d$-dimensional quantum system. If a set of ${\textstyle d^{2}}$ rank-1 projectors ${\hat {\Pi }}_{i}$ satisfying

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

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

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

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

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

where $P(D_{j}|H_{i})\equiv {\text{tr}}{\hat {\Pi }}_{i}{\hat {D}}_{j}$ is the Born rule probability for obtaining outcome $D_{j}$ implied by the state assignment ${\hat {\Pi }}_{i}$. The $P(D_{j}|H_{i})$ 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 $\{D_{j}\}$ measurement. After obtaining an outcome from the SIC measurement, the agent will update her state assignment to a new quantum state ${\hat {\rho }}'$ before performing the second measurement. If she uses the Lüders rule for state update and obtains outcome $H_{i}$ from the SIC measurement, then ${\textstyle {\hat {\rho }}'={\hat {\Pi }}_{i}}$. Thus the probability for obtaining outcome $D_{j}$ for the second measurement conditioned on obtaining outcome $H_{i}$ for the SIC measurement is $P(D_{j}|H_{i})$.

Note that the urgleichung is structurally very similar to the law of total probability, which is the expression

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

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, many QBists 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.

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 $P(D_{j})$ 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 ${\hat {\rho }}$ with SIC representation ${\textstyle P(H_{i})}$. After some time $t$, quantum theory instructs us to use the unitarily rotated state ${\textstyle {\hat {U}}{\hat {\rho }}{\hat {U}}^{\dagger }}$ with SIC representation ${\textstyle P_{t}(H_{i})={\text{tr}}\left[({\hat {U}}{\hat {\rho }}{\hat {U}}^{\dagger }){\hat {H}}_{i}\right]={\text{tr}}\left[{\hat {\rho }}({\hat {U}}^{\dagger }{\hat {H}}_{i}{\hat {U}})\right]}$. 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 ${\hat {\rho }}$. Then the Schrödinger equation is completely captured in the urgleichung for this measurement:

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

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. Alternative reconstruction efforts are in the beginning stages.

See also

• Bayes factor
• Bayesian inference
• Bayesian probability
• Credible intervals
• Degree of belief
• Doxastic logic
• Philosophy of science
• Probability interpretation
• Probability theory
• Quantum information
• Interpretations of Quantum Mechanics
• Quantum probability
• SIC-POVM
• Statistical inference

External links

• 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
• Quantum-Bayesian and Pragmatist Views of Quantum Theory – Stanford Encyclopedia of Philosophy article
• Painting a QBist Picture of Reality – Foundational Questions Institute article
• 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

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