In physics and the philosophy of physics, **quantum Bayesianism** is a collection of related approaches to the interpretation of quantum mechanics, of which the most prominent is **QBism** (pronounced "cubism"). QBism is an interpretation that takes an agent's actions and experiences as the central concerns of the theory. QBism deals with common questions in the interpretation of quantum theory about the nature of wavefunction superposition, quantum measurement, and entanglement.^{ [1] }^{ [2] } 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 about the possible outcomes of measurements. For this reason, some philosophers of science have deemed QBism a form of anti-realism.^{ [3] }^{ [4] } 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.^{ [5] }^{ [6] }

- History and development
- Core positions
- Reception and criticism
- Relation to other interpretations
- Copenhagen interpretations
- Other epistemic interpretations
- Von Neumann's views
- Relational quantum mechanics
- Other uses of Bayesian probability in quantum physics
- Technical developments and reconstructing quantum theory
- See also
- References
- External links

This interpretation is distinguished by its use of a subjective Bayesian account of probabilities to understand the quantum mechanical 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.^{ [7] } 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,^{ [8] }^{ [9] } but is itself distinct from them.^{ [9] }^{ [10] } Theodor Hänsch has characterized QBism as sharpening those older views and making them more consistent.^{ [11] }

More generally, any work that uses a Bayesian or personalist (a.k.a. "subjective") treatment of the probabilities that appear in quantum theory is also sometimes called *quantum Bayesian*. QBism, in particular, has been referred to as "the radical Bayesian interpretation".^{ [12] }

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. The ultimate goal of this research is to identify what aspects of the ontology of the physical world make quantum theory a good tool for agents to use.^{ [13] } However, the QBist interpretation itself, as described in the * Core positions * section, does not depend on any particular reconstruction.

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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 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 ^{ [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] }

- 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.
- 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.
- 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.
- 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.

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] }

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.^{ [14] }^{ [61] } 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.^{ [10] }^{ [31] }

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.^{ [13] }^{ [62] } 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.^{ [10] } 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.^{ [22] }^{ [42] }

Approaches to quantum theory, like QBism,^{ [63] } which treat quantum states as expressions of information, knowledge, belief, or expectation are called "epistemic" interpretations.^{ [6] } 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.

^{ [64] }

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.^{ [65] } 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.^{ [65] } Another approach, advocated by Bub and Pitowsky, argues that quantum states are information about propositions within event spaces that form non-Boolean lattices.^{ [66] } On occasion, the proposals of Bub and Pitowsky are also called "quantum Bayesianism".^{ [67] }

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.^{ [68] } 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.^{ [69] } 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.^{ [6] }

Bayesian, or epistemic, interpretations of quantum probabilities were proposed in the early 1990s by Baez and Youssef.^{ [70] }^{ [71] }

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.*^{ [72] } 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.^{ [14] }

Comparisons have also been made between QBism and the relational quantum mechanics (RQM) espoused by Carlo Rovelli and others.^{ [73] } In both QBism and RQM, quantum states are not intrinsic properties of physical systems.^{ [74] } 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.^{ [23] }^{ [75] } 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.^{ [76] } (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.^{ [6] }^{ [17] } Moreover, RQM does not insist that a measurement outcome is necessarily an agent's experience.^{ [17] }

QBism should be distinguished from other applications of Bayesian inference in quantum physics, and from quantum analogues of Bayesian inference.^{ [19] }^{ [70] } 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".^{ [77] }^{ [78] } Bayesian inference has also been applied in quantum theory for updating probability densities over quantum states,^{ [79] } and MaxEnt methods have been used in similar ways.^{ [70] }^{ [80] } Bayesian methods for quantum state and process tomography are an active area of research.^{ [81] }

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 -dimensional Hilbert space. If a set of rank-1 projectors satisfying

exists, then one may form a SIC-POVM . An arbitrary quantum state may be written as a linear combination of the SIC projectors

where is the Born rule probability for obtaining SIC measurement outcome implied by the state assignment . 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 . The urgleichung is the expression obtained from forming the Born rule probabilities, , for the outcomes of this quantum measurement,

where is the Born rule probability for obtaining outcome implied by the state assignment . The 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 measurement. After obtaining an outcome from the SIC measurement, the agent will update her state assignment to a new quantum state before performing the second measurement. If she uses the Lüders rule^{ [95] } for state update and obtains outcome from the SIC measurement, then . Thus the probability for obtaining outcome for the second measurement conditioned on obtaining outcome for the SIC measurement is . Note that the urgleichung is structurally very similar to the law of total probability, which is the expression

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 and refer to different situations. is the probability that an agent assigns for obtaining outcome on her second of two planned measurements, that is, for obtaining outcome after first making the SIC measurement and obtaining one of the outcomes. , on the other hand, is the probability an agent assigns for obtaining outcome *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 with SIC representation . The time evolution of this state is found by applying a unitary operator to form the new state , which has the SIC representation

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 of the original quantum state . Then the Schrödinger equation is completely captured in the urgleichung for this measurement:

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] }

The **Copenhagen interpretation** is a collection of views about the meaning of quantum mechanics principally attributed to Niels Bohr and Werner Heisenberg. It is one of the oldest of numerous proposed interpretations of quantum mechanics, as features of it date to the development of quantum mechanics during 1925–1927, and it remains one of the most commonly taught.

The **many-worlds interpretation** (**MWI**) is an interpretation of quantum mechanics that asserts that the universal wavefunction is objectively real, and that there is no wavefunction collapse. This implies that all possible outcomes of quantum measurements are physically realized in some "world" or universe. In contrast to some other interpretations, such as the Copenhagen interpretation, the evolution of reality as a whole in MWI is rigidly deterministic. Many-worlds is also called the **relative state formulation** or the **Everett interpretation**, after physicist Hugh Everett, who first proposed it in 1957. Bryce DeWitt popularized the formulation and named it *many-worlds* in the 1960s and 1970s.

The **Einstein–Podolsky–Rosen paradox** is a thought experiment proposed by physicists Albert Einstein, Boris Podolsky and Nathan Rosen (EPR), with which they argued that the description of physical reality provided by quantum mechanics was incomplete. In a 1935 paper titled "Can Quantum-Mechanical Description of Physical Reality be Considered Complete?", they argued for the existence of "elements of reality" that were not part of quantum theory, and speculated that it should be possible to construct a theory containing them. Resolutions of the paradox have important implications for the interpretation of quantum mechanics.

**Quantum entanglement** is a physical phenomenon that occurs when a group of particles is generated, interact, or share spatial proximity in a way such that the quantum state of each particle of the group cannot be described independently of the state of the others, including when the particles are separated by a large distance. The topic of quantum entanglement is at the heart of the disparity between classical and quantum physics: entanglement is a primary feature of quantum mechanics lacking in classical mechanics.

The **de Broglie–Bohm theory**, also known as the *pilot wave theory*, **Bohmian mechanics**, **Bohm's interpretation**, and the **causal interpretation**, is an interpretation of quantum mechanics. In addition to the wavefunction, it also postulates an actual configuration of particles exists even when unobserved. The evolution over time of the configuration of all particles is defined by a guiding equation. The evolution of the wave function over time is given by the Schrödinger equation. The theory is named after Louis de Broglie (1892–1987) and David Bohm (1917–1992).

An **interpretation of quantum mechanics** is an attempt to explain how the mathematical theory of quantum mechanics "corresponds" to reality. Although quantum mechanics has held up to rigorous and extremely precise tests in an extraordinarily broad range of experiments, there exist a number of contending schools of thought over their interpretation. These views on interpretation differ on such fundamental questions as whether quantum mechanics is deterministic or stochastic, which elements of quantum mechanics can be considered real, and what is the nature of measurement, among other matters.

**Bell's theorem** proves that quantum physics is incompatible with local hidden-variable theories. It was introduced by physicist John Stewart Bell in a 1964 paper titled "On the Einstein Podolsky Rosen Paradox", referring to a 1935 thought experiment that Albert Einstein, Boris Podolsky and Nathan Rosen used to argue that quantum physics is an "incomplete" theory. By 1935, it was already recognized that the predictions of quantum physics are probabilistic. Einstein, Podolsky and Rosen presented a scenario that, in their view, indicated that quantum particles, like electrons and photons, must carry physical properties or attributes not included in quantum theory, and the uncertainties in quantum theory's predictions were due to ignorance of these properties, later termed "hidden variables". Their scenario involves a pair of widely separated physical objects, prepared in such a way that the quantum state of the pair is entangled.

In quantum mechanics, **wave function collapse** occurs when a wave function—initially in a superposition of several eigenstates—reduces to a single eigenstate due to interaction with the external world. This interaction is called an "observation". It is the essence of a measurement in quantum mechanics which connects the wave function with classical observables like position and momentum. Collapse is one of two processes by which quantum systems evolve in time; the other is the continuous evolution via the Schrödinger equation. Collapse is a black box for a thermodynamically irreversible interaction with a classical environment. Calculations of quantum decoherence show that when a quantum system interacts with the environment, the superpositions *apparently* reduce to mixtures of classical alternatives. Significantly, the combined wave function of the system and environment continue to obey the Schrödinger equation. More importantly, this is not enough to explain wave function collapse, as decoherence does not reduce it to a single eigenstate.

**Quantum decoherence** is the loss of quantum coherence. In quantum mechanics, particles such as electrons are described by a wave function, a mathematical representation of the quantum state of a system; a probabilistic interpretation of the wave function is used to explain various quantum effects. As long as there exists a definite phase relation between different states, the system is said to be coherent. A definite phase relationship is necessary to perform quantum computing on quantum information encoded in quantum states. Coherence is preserved under the laws of quantum physics.

In quantum physics, a **measurement** is the testing or manipulation of a physical system in order to yield a numerical result. The predictions that quantum physics makes are in general probabilistic. The mathematical tools for making predictions about what measurement outcomes may occur were developed during the 20th century and make use of linear algebra and functional analysis.

The **quantum Zeno effect** is a feature of quantum-mechanical systems allowing a particle's time evolution to be arrested by measuring it frequently enough with respect to some chosen measurement setting.

In quantum mechanics, the **measurement problem** considers how, or whether, wave function collapse occurs. The inability to observe such a collapse directly has given rise to different interpretations of quantum mechanics and poses a key set of questions that each interpretation must answer.

In physics, **complementarity** is a conceptual aspect of quantum mechanics that Niels Bohr regarded as an essential feature of the theory. The complementarity principle holds that objects have certain pairs of complementary properties which cannot all be observed or measured simultaneously. An example of such a pair is position and momentum. Bohr considered one of the foundational truths of quantum mechanics to be the fact that setting up an experiment to measure one quantity of a pair, for instance the position of an electron, excludes the possibility of measuring the other, yet understanding both experiments is necessary to characterize the object under study. In Bohr's view, the behavior of atomic and subatomic objects cannot be separated from the measuring instruments that create the context in which the measured objects behave. Consequently, there is no "single picture" that unifies the results obtained in these different experimental contexts, and only the "totality of the phenomena" together can provide a completely informative description.

The **Born rule** is a key postulate of quantum mechanics which gives the probability that a measurement of a quantum system will yield a given result. In its simplest form, it states that the probability density of finding a particle at a given point, when measured, is proportional to the square of the magnitude of the particle's wavefunction at that point. It was formulated by German physicist Max Born in 1926.

In mathematical physics, **Gleason's theorem** shows that the rule one uses to calculate probabilities in quantum physics, the Born rule, can be derived from the usual mathematical representation of measurements in quantum physics together with the assumption of non-contextuality. Andrew M. Gleason first proved the theorem in 1957, answering a question posed by George W. Mackey, an accomplishment that was historically significant for the role it played in showing that wide classes of hidden-variable theories are inconsistent with quantum physics. Multiple variations have been proven in the years since. Gleason's theorem is of particular importance for the field of quantum logic and its attempt to find a minimal set of mathematical axioms for quantum theory.

**Nathaniel David Mermin** is a solid-state physicist at Cornell University best known for the eponymous Mermin–Wagner theorem, his application of the term "boojum" to superfluidity, his textbook with Neil Ashcroft on solid-state physics, and for contributions to the foundations of quantum mechanics and quantum information science.

The **Koopman–von Neumann mechanics** is a description of classical mechanics in terms of Hilbert space, introduced by Bernard Koopman and John von Neumann in 1931 and 1932, respectively.

In quantum mechanics, **weak measurements** are a type of quantum measurement that results in an observer obtaining very little information about the system on average, but also disturbs the state very little. From Busch's theorem the system is necessarily disturbed by the measurement. In the literature weak measurements are also known as unsharp, fuzzy, dull, noisy, approximate, and gentle measurements. Additionally weak measurements are often confused with the distinct but related concept of the weak value.

**Quantum foundations** is a discipline of science that seeks to understand the most counter-intuitive aspects of quantum theory, reformulate it and even propose new generalizations thereof. Contrary to other physical theories, such as general relativity, the defining axioms of quantum theory are quite ad hoc, with no obvious physical intuition. While they lead to the right experimental predictions, they do not come with a mental picture of the world where they fit.

* Quantum Theory: Concepts and Methods* is a 1993 quantum physics textbook by Israeli physicist Asher Peres.

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*Bananaworld: Quantum Mechanics for Primates*. Oxford: Oxford University Press. p. 232. ISBN 978-0198718536. - ↑ Ladyman, James; Ross, Don; Spurrett, David; Collier, John (2007).
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See also Webb, Richard (2016-11-30). "Physics may be a small but crucial fraction of our reality".*New Scientist*. Retrieved 2017-04-22.

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*Quantum*.**4**: 338. arXiv: 1911.12456 . Bibcode:2019arXiv191112456S. doi:10.22331/q-2020-09-30-338. ISSN 2521-327X. - ↑ Busch, Paul; Lahti, Pekka (2009-01-01). "Lüders Rule". In Greenberger, Daniel; Hentschel, Klaus; Weinert, Friedel (eds.).
*Compendium of Quantum Physics*. Springer Berlin Heidelberg. pp. 356–358. doi:10.1007/978-3-540-70626-7_110. ISBN 9783540706229. - ↑ van de Wetering, John (2018). "Quantum theory is a quasi-stochastic process theory".
*Electronic Proceedings in Theoretical Computer Science*.**266**(2018): 179–196. arXiv: 1704.08525 . doi:10.4204/EPTCS.266.12. S2CID 53635011. - ↑ Fuchs, Christopher A.; Stacey, Blake C. (2016-01-01). "Some Negative Remarks on Operational Approaches to Quantum Theory". In Chiribella, Giulio; Spekkens, Robert W. (eds.).
*Quantum Theory: Informational Foundations and Foils*. Fundamental Theories of Physics. Springer Netherlands. pp. 283–305. arXiv: 1401.7254 . doi:10.1007/978-94-017-7303-4_9. ISBN 9789401773027. S2CID 116428784. - ↑ Chiribella, Giulio; Cabello, Adán; Kleinmann, Matthias. "The Observer Observed: a Bayesian Route to the Reconstruction of Quantum Theory".
*FQXi: Foundational Questions Institute*. Retrieved 2017-04-18.

- 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].

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