Quantum Bayesianism

Last updated
Each point in the Bloch ball is a possible quantum state for a qubit. In QBism, all quantum states are representations of personal probabilities. Bloch Sphere.svg
Each point in the Bloch ball is a possible quantum state for a qubit. In QBism, all quantum states are representations of personal probabilities.

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]


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.

History and development

British philosopher, mathematician, and economist Frank Ramsey, whose interpretation of probability theory closely matches the one adopted by QBism. 30. Frank Ramsey.jpg
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." [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]

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. [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.

Reception and criticism

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

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]

Relation to other interpretations

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

Other epistemic interpretations

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 BruknerZeilinger interpretation treats some probabilities as objectively fixed. In the BruknerZeilinger 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]

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. [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]

Relational quantum mechanics

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 Ramseyde Finetti school of personalist Bayesianism. [6] [17] Moreover, RQM does not insist that a measurement outcome is necessarily an agent's experience. [17]

Other uses of Bayesian probability in quantum physics

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]

Technical developments and reconstructing quantum theory

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 QBistsFuchs, [26] in particularhave 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]

See also

Related Research Articles

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.

Many-worlds interpretation Interpretation of quantum mechanics which denies the collapse of the wavefunction

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.

EPR paradox Early and influential critique leveled against quantum mechanics

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 Correlation between measurements of quantum subsystems, even when spatially separated

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 Loss of quantum coherence

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.

N. David Mermin American physicist

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.


  1. 1 2 Timpson, Christopher Gordon (2008). "Quantum Bayesianism: A study" (postscript). Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics. 39 (3): 579–609. arXiv: 0804.2047 . Bibcode:2008SHPMP..39..579T. doi:10.1016/j.shpsb.2008.03.006. S2CID   16775153.
  2. 1 2 Mermin, N. David (2012-07-01). "Commentary: Quantum mechanics: Fixing the shifty split". Physics Today. 65 (7): 8–10. Bibcode:2012PhT....65g...8M. doi: 10.1063/PT.3.1618 . ISSN   0031-9228.
  3. Bub, Jeffrey (2016). Bananaworld: Quantum Mechanics for Primates. Oxford: Oxford University Press. p. 232. ISBN   978-0198718536.
  4. Ladyman, James; Ross, Don; Spurrett, David; Collier, John (2007). Every Thing Must Go: Metaphysics Naturalized . Oxford: Oxford University Press. pp.  184. ISBN   9780199573097.
  5. For "participatory realism," see, e.g.,
    Fuchs, Christopher A. (2017). "On Participatory Realism". In Durham, Ian T.; Rickles, Dean (eds.). Information and Interaction: Eddington, Wheeler, and the Limits of Knowledge. arXiv: 1601.04360 . Bibcode:2016arXiv160104360F. ISBN   9783319437606. OCLC   967844832.
    Fuchs, Christopher A.; Timpson, Christopher G. "Does Participatory Realism Make Sense? The Role of Observership in Quantum Theory". FQXi: Foundational Questions Institute. Retrieved 2017-04-18.
  6. 1 2 3 4 Cabello, Adán (2017). "Interpretations of quantum theory: A map of madness". In Lombardi, Olimpia; Fortin, Sebastian; Holik, Federico; López, Cristian (eds.). What is Quantum Information?. Cambridge University Press. pp. 138–143. arXiv: 1509.04711 . Bibcode:2015arXiv150904711C. doi:10.1017/9781316494233.009. ISBN   9781107142114. S2CID   118419619.
  7. 1 2 Mermin, N. David (2014-03-27). "Physics: QBism puts the scientist back into science". Nature. 507 (7493): 421–423. doi: 10.1038/507421a . PMID   24678539.
  8. Tammaro, Elliott (2014-08-09). "Why Current Interpretations of Quantum Mechanics are Deficient". arXiv: 1408.2093 [quant-ph].
  9. 1 2 Schlosshauer, Maximilian; Kofler, Johannes; Zeilinger, Anton (2013-08-01). "A snapshot of foundational attitudes toward quantum mechanics". Studies in History and Philosophy of Science Part B. 44 (3): 222–230. arXiv: 1301.1069 . Bibcode:2013SHPMP..44..222S. doi:10.1016/j.shpsb.2013.04.004. S2CID   55537196.
  10. 1 2 3 Mermin, N. David (2017-01-01). "Why QBism Is Not the Copenhagen Interpretation and What John Bell Might Have Thought of It". In Bertlmann, Reinhold; Zeilinger, Anton (eds.). Quantum [Un]Speakables II. The Frontiers Collection. Springer International Publishing. pp. 83–93. arXiv: 1409.2454 . doi:10.1007/978-3-319-38987-5_4. ISBN   9783319389851. S2CID   118458259.
  11. Hänsch, Theodor. "Changing Concepts of Light and Matter". The Pontifical Academy of Sciences. Retrieved 2017-04-18.
  12. 1 2 Jaeger, Gregg (2009). "3.7. The radical Bayesian interpretation". Entanglement, information, and the interpretation of quantum mechanics (Online-Ausg. ed.). Berlin: Springer. pp.  170–179. ISBN   978-3-540-92127-1.
  13. 1 2 3 4 5 6 von Baeyer, Hans Christian (2016). QBism: The Future of Quantum Physics. Cambridge, MA: Harvard University Press. ISBN   978-0674504646.
  14. 1 2 3 Stacey, Blake C. (2016-05-28). "Von Neumann Was Not a Quantum Bayesian". Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. 374 (2068): 20150235. arXiv: 1412.2409 . Bibcode:2016RSPTA.37450235S. doi:10.1098/rsta.2015.0235. ISSN   1364-503X. PMID   27091166. S2CID   16829387.
  15. Jaynes, E. T. (1990). "Probability in Quantum Theory". In Zurek, W. H. (ed.). Complexity, Entropy, and the Physics of Information. Redwood City, CA: Addison-Wesley. p. 381.
  16. 1 2 Gefter, Amanda. "A Private View of Quantum Reality". Quanta. Retrieved 2017-04-24.
  17. 1 2 3 4 Fuchs, Christopher A.; Schlosshauer, Maximilian; Stacey, Blake C. (2014-05-10). "My Struggles with the Block Universe". arXiv: 1405.2390 [quant-ph].
  18. Keynes, John Maynard (2012-01-01). "F. P. Ramsey". Essays in biography. Martino Fine Books. ISBN   978-1614273264. OCLC   922625832.
  19. 1 2 3 4 5 Fuchs, Christopher A.; Schack, Rüdiger (2013-01-01). "Quantum-Bayesian coherence". Reviews of Modern Physics. 85 (4): 1693–1715. arXiv: 1301.3274 . Bibcode:2013RvMP...85.1693F. doi:10.1103/RevModPhys.85.1693. S2CID   18256163.
  20. Fine, Arthur (2016-01-01). "The Einstein–Podolsky–Rosen Argument in Quantum Theory". In Zalta, Edward N. (ed.). The Stanford Encyclopedia of Philosophy (Fall 2016 ed.). Metaphysics Research Lab, Stanford University.
  21. 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.
  22. 1 2 3 4 Fuchs, Christopher A.; Stacey, Blake C. (2016-12-21). "QBism: Quantum Theory as a Hero's Handbook". arXiv: 1612.07308 [quant-ph].
  23. 1 2 3 Fuchs, Christopher A.; Mermin, N. David; Schack, Ruediger (2014-07-22). "An introduction to QBism with an application to the locality of quantum mechanics". American Journal of Physics. 82 (8): 749–754. arXiv: 1311.5253 . Bibcode:2014AmJPh..82..749F. doi:10.1119/1.4874855. ISSN   0002-9505. S2CID   56387090.
  24. Fuchs, Christopher A. (2010-03-26). "QBism, the Perimeter of Quantum Bayesianism". arXiv: 1003.5209 [quant-ph].
  25. Caves, Carlton M.; Fuchs, Christopher A.; Schack, Ruediger (2002-01-01). "Quantum probabilities as Bayesian probabilities". Physical Review A. 65 (2): 022305. arXiv: quant-ph/0106133 . Bibcode:2002PhRvA..65b2305C. doi:10.1103/PhysRevA.65.022305. S2CID   119515728.
  26. 1 2 C. A. Fuchs, "Quantum Mechanics as Quantum Information (and only a little more),'' in Quantum Theory: Reconsideration of Foundations, edited by A. Khrennikov (Växjö University Press, Växjö, Sweden, 2002), pp. 463543. arXiv:quant-ph/0205039.
  27. "International School of Physics "Enrico Fermi"". Italian Physical Society. Retrieved 2017-04-18.
  28. 1 2 3 Mermin, N. David (2013-01-28). "Annotated Interview with a QBist in the Making". arXiv: 1301.6551 [quant-ph].
  29. 1 2 von Rauchhaupt, Ulf (9 February 2014). "Philosophische Quantenphysik : Ganz im Auge des Betrachters". Frankfurter Allgemeine Sonntagszeitung (in German). 6. p. 62. Retrieved 2017-04-18.
  30. "Q3: Quantum Metaphysics Panel". Vimeo. 13 February 2016. Retrieved 2017-04-18.
  31. 1 2 Fuchs, Christopher A. (2017). "Notwithstanding Bohr, the Reasons for QBism". Mind and Matter. 15: 245–300. arXiv: 1705.03483 . Bibcode:2017arXiv170503483F.
  32. 1 2 Nauenberg, Michael (2015-03-01). "Comment on QBism and locality in quantum mechanics". American Journal of Physics. 83 (3): 197–198. arXiv: 1502.00123 . Bibcode:2015AmJPh..83..197N. doi:10.1119/1.4907264. ISSN   0002-9505. S2CID   117823345.
  33. Bacciagaluppi, Guido (2014-01-01). "A Critic Looks at QBism". In Galavotti, Maria Carla; Dieks, Dennis; Gonzalez, Wenceslao J.; Hartmann, Stephan; Uebel, Thomas; Weber, Marcel (eds.). New Directions in the Philosophy of Science. The Philosophy of Science in a European Perspective. Springer International Publishing. pp. 403–416. doi:10.1007/978-3-319-04382-1_27. ISBN   9783319043814.
  34. Norsen, Travis (2014). "Quantum Solipsism and Non-Locality" (PDF). Int. J. Quant. Found. John Bell Workshop.
  35. Wallace, David (2007-12-03). "The Quantum Measurement Problem: State of Play". arXiv: 0712.0149 [quant-ph].
  36. DeBrota, John B.; Fuchs, Christopher A. (2017-05-17). "Negativity Bounds for Weyl-Heisenberg Quasiprobability Representations". Foundations of Physics. 47 (8): 1009–1030. arXiv: 1703.08272 . Bibcode:2017FoPh...47.1009D. doi:10.1007/s10701-017-0098-z. S2CID   119428587.
  37. Fuchs, Christopher A.; Mermin, N. David; Schack, Ruediger (2015-02-10). "Reading QBism: A Reply to Nauenberg". American Journal of Physics. 83 (3): 198. arXiv: 1502.02841 . Bibcode:2015AmJPh..83..198F. doi:10.1119/1.4907361.
  38. Stairs, Allen (2011). "A loose and separate certainty: Caves, Fuchs and Schack on quantum probability one" (PDF). Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics. 42 (3): 158–166. Bibcode:2011SHPMP..42..158S. doi:10.1016/j.shpsb.2011.02.001.
  39. Fuchs, Christopher A.; Schack, Rüdiger (2015-01-01). "QBism and the Greeks: why a quantum state does not represent an element of physical reality". Physica Scripta. 90 (1): 015104. arXiv: 1412.4211 . Bibcode:2015PhyS...90a5104F. doi:10.1088/0031-8949/90/1/015104. ISSN   1402-4896. S2CID   14553716.
  40. Mermin, N. David (2012-11-30). "Measured responses to quantum Bayesianism". Physics Today. 65 (12): 12–15. Bibcode:2012PhT....65l..12M. doi: 10.1063/PT.3.1803 . ISSN   0031-9228.
  41. Mermin, N. David (2013-06-28). "Impressionism, Realism, and the aging of Ashcroft and Mermin". Physics Today. 66 (7): 8. Bibcode:2013PhT....66R...8M. doi:10.1063/PT.3.2024. ISSN   0031-9228.
  42. 1 2 Healey, Richard (2016). "Quantum-Bayesian and Pragmatist Views of Quantum Theory". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University.
  43. Mohrhoff, Ulrich (2014-09-10). "QBism: A Critical Appraisal". arXiv: 1409.3312 [quant-ph].
  44. Marchildon, Louis (2015-07-01). "Why I am not a QBist". Foundations of Physics. 45 (7): 754–761. arXiv: 1403.1146 . Bibcode:2015FoPh...45..754M. doi:10.1007/s10701-015-9875-8. ISSN   0015-9018. S2CID   119196825.
    Leifer, Matthew. "Interview with an anti-Quantum zealot". Elliptic Composability. Retrieved 10 March 2017.
  45. Marchildon, Louis (2015). "Multiplicity in Everett's interpretation of quantum mechanics". Studies in History and Philosophy of Modern Physics. 52 (B): 274–284. arXiv: 1504.04835 . Bibcode:2015SHPMP..52..274M. doi:10.1016/j.shpsb.2015.08.010. S2CID   118398374.
  46. Schlosshauer, Maximilian; Claringbold, Tangereen V. B. (2015). "Entanglement, scaling, and the meaning of the wave function in protective measurement". Protective Measurement and Quantum Reality: Towards a New Understanding of Quantum Mechanics. Cambridge University Press. pp. 180–194. arXiv: 1402.1217 . doi:10.1017/cbo9781107706927.014. ISBN   9781107706927. S2CID   118003617.
  47. Barnum, Howard N. (2010-03-23). "Quantum Knowledge, Quantum Belief, Quantum Reality: Notes of a QBist Fellow Traveler". arXiv: 1003.4555 [quant-ph].
  48. Appleby, D. M. (2007-01-01). "Concerning Dice and Divinity". AIP Conference Proceedings. 889: 30–39. arXiv: quant-ph/0611261 . Bibcode:2007AIPC..889...30A. doi:10.1063/1.2713444.
  49. See Chalmers, Matthew (2014-05-07). "QBism: Is quantum uncertainty all in the mind?". New Scientist. Retrieved 2017-04-09. Mermin criticized some aspects of this coverage; see Mermin, N. David (2014-06-05). "QBism in the New Scientist". arXiv: 1406.1573 [quant-ph].
    See also Webb, Richard (2016-11-30). "Physics may be a small but crucial fraction of our reality". New Scientist. Retrieved 2017-04-22.
    See also Ball, Philip (2017-11-08). "Consciously quantum". New Scientist. Retrieved 2017-12-06.
  50. von Baeyer, Hans Christian (2013). "Quantum Weirdness? It's All in Your Mind". Scientific American. 308 (6): 46–51. Bibcode:2013SciAm.308f..46V. doi:10.1038/scientificamerican0613-46. PMID   23729070.
  51. 1 2 Ball, Philip (2013-09-12). "Physics: Quantum quest". Nature. 501 (7466): 154–156. Bibcode:2013Natur.501..154B. doi: 10.1038/501154a . PMID   24025823.
  52. Siegfried, Tom (2014-01-30). "'QBists' tackle quantum problems by adding a subjective aspect to science". Science News. Retrieved 2017-04-20.
  53. Waldrop, M. Mitchell. "Painting a QBist Picture of Reality". fqxi.org. Retrieved 2017-04-20.
  54. Frank, Adam (2017-03-13). Powell, Corey S. (ed.). "Materialism alone cannot explain the riddle of consciousness". Aeon. Retrieved 2017-04-22.
  55. Folger, Tim (May 2017). "The War Over Reality". Discover Magazine. Retrieved 2017-05-10.
  56. Ball, Philip (2018). Beyond Weird: Why Everything You Thought You Knew About Quantum Physics is Different. London: Penguin Random House. ISBN   9781847924575. OCLC   1031304139.
  57. Ananthaswamy, Anil (2018). Through Two Doors at Once: The Elegant Experiment That Captures the Enigma of Our Quantum Reality. New York: Penguin Random House. ISBN   9781101986097. OCLC   1089112651.
  58. Rickles, Dean (2019). "Johntology: Participatory Realism and its Problems". Mind and Matter. 17 (2): 205–211.
  59. Bitbol, Michel (2020). "A Phenomenological Ontology for Physics: Merleau-Ponty and QBism". In Wiltsche, Harald; Berghofer, Philipp (eds.). Phenomenological Approaches to Physics. Synthese Library (Studies in Epistemology, Logic, Methodology, and Philosophy of Science). 429. Springer. pp. 227–242. doi:10.1007/978-3-030-46973-3_11. ISBN   978-3-030-46972-6. OCLC   1193285104.
  60. de La Tremblaye, Laura (2020). "QBism from a Phenomenological Point of View: Husserl and QBism". In Wiltsche, Harald; Berghofer, Philipp (eds.). Phenomenological Approaches to Physics. Synthese Library (Studies in Epistemology, Logic, Methodology, and Philosophy of Science). 429. Springer. pp. 243–260. doi:10.1007/978-3-030-46973-3_12. ISBN   978-3-030-46972-6. OCLC   1193285104.
  61. Peres, Asher (2002-03-01). "Karl Popper and the Copenhagen interpretation". Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics. 33 (1): 23–34. arXiv: quant-ph/9910078 . Bibcode:2002SHPMP..33...23P. doi:10.1016/S1355-2198(01)00034-X.
    Żukowski, Marek (2017-01-01). "Bell's Theorem Tells Us Not What Quantum Mechanics Is, but What Quantum Mechanics Is Not". In Bertlmann, Reinhold; Zeilinger, Anton (eds.). Quantum [Un]Speakables II. The Frontiers Collection. Springer International Publishing. pp. 175–185. arXiv: 1501.05640 . doi:10.1007/978-3-319-38987-5_10. ISBN   9783319389851. S2CID   119214547.
    Camilleri, Kristian (2009-02-01). "Constructing the Myth of the Copenhagen Interpretation". Perspectives on Science. 17 (1): 26–57. doi:10.1162/posc.2009.17.1.26. ISSN   1530-9274. S2CID   57559199.
  62. Peres, Asher (1984-07-01). "What is a state vector?". American Journal of Physics. 52 (7): 644–650. Bibcode:1984AmJPh..52..644P. doi:10.1119/1.13586. ISSN   0002-9505.
    Caves, Carlton M.; Fuchs, Christopher A.; Schack, Rüdiger (2007-06-01). "Subjective probability and quantum certainty". Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics. Probabilities in quantum mechanics. 38 (2): 255–274. arXiv: quant-ph/0608190 . Bibcode:2007SHPMP..38..255C. doi:10.1016/j.shpsb.2006.10.007. S2CID   119549678.
  63. Harrigan, Nicholas; Spekkens, Robert W. (2010-02-01). "Einstein, Incompleteness, and the Epistemic View of Quantum States". Foundations of Physics. 40 (2): 125–157. arXiv: 0706.2661 . Bibcode:2010FoPh...40..125H. doi:10.1007/s10701-009-9347-0. ISSN   0015-9018. S2CID   32755624.
  64. Spekkens, Robert W. (2007-01-01). "Evidence for the epistemic view of quantum states: A toy theory". Physical Review A. 75 (3): 032110. arXiv: quant-ph/0401052 . Bibcode:2007PhRvA..75c2110S. doi:10.1103/PhysRevA.75.032110. S2CID   117284016.
  65. 1 2 Leifer, Matthew S.; Spekkens, Robert W. (2013). "Towards a Formulation of Quantum Theory as a Causally Neutral Theory of Bayesian Inference". Phys. Rev. A. 88 (5): 052130. arXiv: 1107.5849 . Bibcode:2013PhRvA..88e2130L. doi:10.1103/PhysRevA.88.052130. S2CID   43563970.
  66. Bub, Jeffrey; Pitowsky, Itamar (2010-01-01). "Two dogmas about quantum mechanics". In Saunders, Simon; Barrett, Jonathan; Kent, Adrian; Wallace, David (eds.). Many Worlds?: Everett, Quantum Theory & Reality. Oxford University Press. pp. 433–459. arXiv: 0712.4258 . Bibcode:2007arXiv0712.4258B.
  67. Duwell, Armond (2011). "Uncomfortable bedfellows: Objective quantum Bayesianism and the von Neumann–Lüders projection postulate". Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics. 42 (3): 167–175. Bibcode:2011SHPMP..42..167D. doi:10.1016/j.shpsb.2011.04.003.
  68. Brukner, Časlav; Zeilinger, Anton (2001). "Conceptual inadequacy of the Shannon information in quantum measurements". Physical Review A. 63 (2): 022113. arXiv: quant-ph/0006087 . Bibcode:2001PhRvA..63b2113B. doi:10.1103/PhysRevA.63.022113. S2CID   119381924.
    Brukner, Časlav; Zeilinger, Anton (2009). "Information Invariance and Quantum Probabilities". Foundations of Physics. 39 (7): 677–689. arXiv: 0905.0653 . Bibcode:2009FoPh...39..677B. doi:10.1007/s10701-009-9316-7. S2CID   73599204.
  69. Khrennikov, Andrei (2016). "Reflections on ZeilingerBrukner information interpretation of quantum mechanics". Foundations of Physics. 46 (7): 836–844. arXiv: 1512.07976 . Bibcode:2016FoPh...46..836K. doi:10.1007/s10701-016-0005-z. S2CID   119267791.
  70. 1 2 3 Baez, John (2003-09-12). "Bayesian Probability Theory and Quantum Mechanics" . Retrieved 2017-04-18.
  71. Youssef, Saul (1991). "A Reformulation of Quantum Mechanics" (PDF). Modern Physics Letters A . 6 (3): 225–236. doi:10.1142/S0217732391000191.
    Youssef, Saul (1994). "Quantum Mechanics as Bayesian Complex Probability Theory". Modern Physics Letters A . 9 (28): 2571–2586. arXiv: hep-th/9307019 . doi:10.1142/S0217732394002422. S2CID   18506337.
  72. Streater, R. F. (2007). Lost Causes in and beyond Physics . Springer. p.  70. ISBN   978-3-540-36581-5.
  73. Brukner, Časlav (2017-01-01). "On the Quantum Measurement Problem". In Bertlmann, Reinhold; Zeilinger, Anton (eds.). Quantum [Un]Speakables II. The Frontiers Collection. Springer International Publishing. pp. 95–117. arXiv: 1507.05255 . doi:10.1007/978-3-319-38987-5_5. ISBN   9783319389851. S2CID   116892322.
    Marlow, Thomas (2006-03-07). "Relationalism vs. Bayesianism". arXiv: gr-qc/0603015 .
    Pusey, Matthew F. (2018-09-18). "An inconsistent friend". Nature Physics . 14 (10): 977–978. doi:10.1038/s41567-018-0293-7. S2CID   126294105.
  74. Cabello, Adán; Gu, Mile; Gühne, Otfried; Larsson, Jan-Åke; Wiesner, Karoline (2016-01-01). "Thermodynamical cost of some interpretations of quantum theory". Physical Review A. 94 (5): 052127. arXiv: 1509.03641 . Bibcode:2016PhRvA..94e2127C. doi:10.1103/PhysRevA.94.052127. S2CID   601271.
  75. Smerlak, Matteo; Rovelli, Carlo (2007-02-26). "Relational EPR". Foundations of Physics. 37 (3): 427–445. arXiv: quant-ph/0604064 . Bibcode:2007FoPh...37..427S. doi:10.1007/s10701-007-9105-0. ISSN   0015-9018. S2CID   11816650.
  76. Rovelli, Carlo (1996-08-01). "Relational quantum mechanics". International Journal of Theoretical Physics. 35 (8): 1637–1678. arXiv: quant-ph/9609002 . Bibcode:1996IJTP...35.1637R. doi:10.1007/BF02302261. ISSN   0020-7748. S2CID   16325959.
  77. Tucci, Robert R. (1995-01-30). "Quantum bayesian nets". International Journal of Modern Physics B. 09 (3): 295–337. arXiv: quant-ph/9706039 . Bibcode:1995IJMPB...9..295T. doi:10.1142/S0217979295000148. ISSN   0217-9792. S2CID   18217167.
  78. Moreira, Catarina; Wichert, Andreas (2016). "Quantum-Like Bayesian Networks for Modeling Decision Making". Frontiers in Psychology. 7: 11. doi:10.3389/fpsyg.2016.00011. PMC   4726808 . PMID   26858669.
  79. Jones, K. R. W. (1991). "Principles of quantum inference". Annals of Physics. 207 (1): 140–170. Bibcode:1991AnPhy.207..140J. doi:10.1016/0003-4916(91)90182-8.
  80. Bužek, V.; Derka, R.; Adam, G.; Knight, P. L. (1998). "Reconstruction of Quantum States of Spin Systems: From Quantum Bayesian Inference to Quantum Tomography". Annals of Physics. 266 (2): 454–496. Bibcode:1998AnPhy.266..454B. doi:10.1006/aphy.1998.5802.
  81. Granade, Christopher; Combes, Joshua; Cory, D. G. (2016-01-01). "Practical Bayesian tomography". New Journal of Physics. 18 (3): 033024. arXiv: 1509.03770 . Bibcode:2016NJPh...18c3024G. doi:10.1088/1367-2630/18/3/033024. ISSN   1367-2630. S2CID   88521187.
  82. Størmer, E. (1969). "Symmetric states of infinite tensor products of C*-algebras". J. Funct. Anal. 3: 48–68. doi:10.1016/0022-1236(69)90050-0. hdl: 10852/45014 .
  83. Caves, Carlton M.; Fuchs, Christopher A.; Schack, Ruediger (2002-08-20). "Unknown quantum states: The quantum de Finetti representation". Journal of Mathematical Physics. 43 (9): 4537–4559. arXiv: quant-ph/0104088 . Bibcode:2002JMP....43.4537C. doi:10.1063/1.1494475. ISSN   0022-2488. S2CID   17416262.
  84. J. Baez (2007). "This Week's Finds in Mathematical Physics (Week 251)" . Retrieved 2017-04-18.
  85. Renner, Renato (2005-12-30). "Security of Quantum Key Distribution". arXiv: quant-ph/0512258 .
  86. Doherty, Andrew C.; Parrilo, Pablo A.; Spedalieri, Federico M. (2005-01-01). "Detecting multipartite entanglement" (PDF). Physical Review A. 71 (3): 032333. arXiv: quant-ph/0407143 . Bibcode:2005PhRvA..71c2333D. doi:10.1103/PhysRevA.71.032333. S2CID   44241800.
  87. Chiribella, Giulio; Spekkens, Rob W. (2016). "Introduction". Quantum Theory: Informational Foundations and Foils. Fundamental Theories of Physics. 181. Springer. pp. 1–18. arXiv: 1208.4123 . doi:10.1007/978-94-017-7303-4. ISBN   978-94-017-7302-7. S2CID   118699215.
  88. Technical references on SIC-POVMs include the following:
    Scott, A. J. (2006-01-01). "Tight informationally complete quantum measurements". Journal of Physics A: Mathematical and General. 39 (43): 13507–13530. arXiv: quant-ph/0604049 . Bibcode:2006JPhA...3913507S. doi:10.1088/0305-4470/39/43/009. ISSN   0305-4470. S2CID   33144766.
    Wootters, William K.; Sussman, Daniel M. (2007). "Discrete phase space and minimum-uncertainty states". arXiv: 0704.1277 [quant-ph].
    Appleby, D. M.; Bengtsson, Ingemar; Brierley, Stephen; Grassl, Markus; Gross, David; Larsson, Jan-Åke (2012-05-01). "The Monomial Representations of the Clifford Group". Quantum Information & Computation. 12 (5–6): 404–431. arXiv: 1102.1268 . Bibcode:2011arXiv1102.1268A. ISSN   1533-7146.
    Hou, Zhibo; Tang, Jun-Feng; Shang, Jiangwei; Zhu, Huangjun; Li, Jian; Yuan, Yuan; Wu, Kang-Da; Xiang, Guo-Yong; Li, Chuan-Feng (2018-04-12). "Deterministic realization of collective measurements via photonic quantum walks". Nature Communications . 9 (1): 1414. arXiv: 1710.10045 . Bibcode:2018NatCo...9.1414H. doi:10.1038/s41467-018-03849-x. ISSN   2041-1723. PMC   5897416 . PMID   29650977.
  89. Appleby, Marcus; Flammia, Steven; McConnell, Gary; Yard, Jon (2017-04-24). "SICs and Algebraic Number Theory". Foundations of Physics. 47 (8): 1042–1059. arXiv: 1701.05200 . Bibcode:2017FoPh..tmp...34A. doi:10.1007/s10701-017-0090-7. ISSN   0015-9018. S2CID   119334103.
  90. Fuchs, Christopher A.; Schack, Rüdiger (2010-01-08). "A Quantum-Bayesian Route to Quantum-State Space". Foundations of Physics. 41 (3): 345–356. arXiv: 0912.4252 . Bibcode:2011FoPh...41..345F. doi:10.1007/s10701-009-9404-8. ISSN   0015-9018. S2CID   119277535.
  91. Appleby, D. M.; Ericsson, Åsa; Fuchs, Christopher A. (2010-04-27). "Properties of QBist State Spaces". Foundations of Physics. 41 (3): 564–579. arXiv: 0910.2750 . Bibcode:2011FoPh...41..564A. doi:10.1007/s10701-010-9458-7. ISSN   0015-9018. S2CID   119296426.
  92. Rosado, José Ignacio (2011-01-28). "Representation of Quantum States as Points in a Probability Simplex Associated to a SIC-POVM". Foundations of Physics. 41 (7): 1200–1213. arXiv: 1007.0715 . Bibcode:2011FoPh...41.1200R. doi:10.1007/s10701-011-9540-9. ISSN   0015-9018. S2CID   119102347.
  93. 1 2 3 Appleby, Marcus; Fuchs, Christopher A.; Stacey, Blake C.; Zhu, Huangjun (2016-12-09). "Introducing the Qplex: A Novel Arena for Quantum Theory". The European Physical Journal D. 71 (7). arXiv: 1612.03234 . Bibcode:2017EPJD...71..197A. doi:10.1140/epjd/e2017-80024-y. S2CID   119240836.
  94. Słomczyński, Wojciech; Szymusiak, Anna (2020-09-30). "Morphophoric POVMs, generalised qplexes, and 2-designs". Quantum . 4: 338. arXiv: 1911.12456 . Bibcode:2019arXiv191112456S. doi:10.22331/q-2020-09-30-338. ISSN   2521-327X.
  95. 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.
  96. 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.
  97. 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.
  98. 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.