Quantum Bayesianism - Alchetron, The Free Social Encyclopedia

Quantum Bayesianism most often refers to an interpretation of quantum theory, also called QBism, which gives a "subjective Bayesian account of quantum probability." Rooted in the prior work of Carlton Caves, Christopher Fuchs, and Ruediger Schack during the early 2000s, QBism itself is primarily associated with Fuchs and Schack and has more recently been advocated by David Mermin. QBism draws from the fields of quantum information and Bayesian probability, claiming to correct, clarify, and extend the viewpoint on quantum mechanics that is commonly presented in textbooks. The QBist interpretation is related to, but distinct from, the views of the various physicists that are often grouped together as the Copenhagen interpretation.

The term "Quantum Bayesianism" may sometimes refer more generically to the use of a Bayesian or personalist (aka "subjective") treatment of the probabilities that appear in quantum theory. QBism, in particular, has been referred to as "the radical Bayesian interpretation". It attempts, on a philosophical level, to provide an understanding of quantum theory, and on a more technical level, to derive as much of quantum theory from informational considerations as possible. The remainder of this article focuses primarily on QBism.

QBism deals with common questions in the interpretation of quantum theory about the nature of wavefunction superposition, non-locality, and entanglement. According to QBism, many, but not all, aspects of the quantum formalism are subjective in nature. For example, in this interpretation, a quantum state is not an element of reality—instead it represents the degrees of belief an agent has in the outcomes of measurements. As the interpretation of quantum mechanics is important to philosophers of science, some compare the idea of degree of belief and its application in QBism with the idea of anti-realism. The originators of the interpretation strongly disagree with this characterization, proposing instead that the theory more properly aligns with a kind of realism they call "participatory realism".

In addition to presenting an interpretation of the existing mathematical structure of quantum theory, QBists have advocated a research program of reconstructing quantum theory from basic physical principles whose QBist character is manifest. These reconstruction efforts are still incomplete. The principles which comprise the QBism interpretation are enumerated in the Core Positions section. The most developed QBist reformulation of quantum theory is described in the Urgleichung section. Alternative reconstruction efforts are in the beginning stages.

QBist foundational research stimulated interest in symmetric, informationally-complete, positive operator-valued measures (SIC-POVMs), which now have applications in quantum theory outside of foundational studies. Likewise, a quantum version of the de Finetti theorem, introduced by Caves, Fuchs, and Schack (independently reproving a result found using different means by Störmer) to provide a Bayesian understanding of the idea of an "unknown quantum state", has found application elsewhere, in topics like quantum key distribution and entanglement detection.

History and Development

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." The present form of QBism developed out of efforts to separate these parts using the tools of quantum information theory and personalist Bayesian probability theory.

There are many interpretations of probability theory. Broadly speaking, these interpretations fall into one of two categories: those which assert that a probability is an objective property of reality and those which assert that a probability is a subjective, mental construct which an agent may use to quantify their ignorance or degree of belief in a proposition. QBism begins by asserting that all probabilities, even those appearing in quantum theory, are most properly viewed as members of the latter category. Specifically, QBism adopts the personalist Bayesian position of Italian mathematician Bruno de Finetti.

The advantages of adopting this view of probability are twofold in QBism. First, it suggests that the Einstein Podolsky Rosen (EPR) criterion of reality should be rejected because it identifies "probability one" assignments with elements of reality preexisting the quantum measurement outcomes. A personalist Bayesian considers all probabilities, even those equal to unity, to be degrees of belief. Therefore, a QBist does not conclude, as many interpretations of quantum theory do, that quantum mechanics is a nonlocal theory. Second, if probabilities are degrees of belief, and one can calculate probabilities from a wavefunction, then wavefunctions must be degrees of belief themselves. With respect to a minimal, informationally complete POVM, this point is made especially clear: A quantum state is then conceptually no more than a single probability distribution over the possible outcomes.

Fuchs introduced the term QBism and outlined the interpretation in more or less its present form in 2010, carrying further and demanding consistency of the ideas first broached in reference with Caves and Schack and in reference. Several subsequent papers, notably references, and, have expanded and elaborated these foundations.

Core Positions

According to QBism, quantum theory is a tool which an agent may use to help manage his or her expectations, more like probability theory than a conventional physical theory. Quantum theory, QBism claims, is not a theory of reality, but rather a guide for decision making which has been shaped by some as yet unidentified aspects of reality. The principle tenets of QBism are as follows:

All probabilities, including probability-1 assignments, are valuations that an agent ascribes to his or her degrees of belief in possible outcomes. As they define and update probabilities, quantum states (density operators), channels (completely positive trace-preserving maps), and measurements (positive operator-valued measures) are also the personal judgements of an agent.
The Born rule rule is normative, not descriptive or prescriptive. It is a relation to which an agent should strive to adhere in his or her probability and quantum state assignments.
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.

The Urgleichung

The most extensively explored QBist reformulation of quantum theory involves the use of SIC-POVMs to rewrite quantum states (either pure or mixed) as a set of probabilities defined over the outcomes of a "Bureau of Standards" measurement. That is, if one expresses a density matrix as a probability distribution over the outcomes of a SIC-POVM experiment, one can reproduce all the statistical predictions implied by the density matrix from the SIC-POVM probabilities instead. The Born rule then takes the role of relating one valid probability distribution to another, rather than of deriving probabilities from something apparently more fundamental. In this alternative form, the developers have taken to calling the Born rule the "urgleichung", German for "primal equation", because of the large portion of quantum theory which follows from this single postulate.

Consider a d -dimensional quantum system. If a set of d 2 rank-1 projectors Π ^ i satisfying

H ^ i = 1 d Π ^ i ρ ^ P ( H i ) = tr ρ ^ H ^ i H i ρ ^ { D ^ j } Q ( D j ) = tr ρ ^ D ^ j P ( D j | H i ) ≡ tr Π ^ i D ^ j D j Π ^ i P ( D j | H i ) { D j } ρ ^ ′ H i ρ ^ ′ = Π ^ i D j H i P ( D j | H i )

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

It is important to recognize that the urgleichung does not replace the law of total probability as quantum theory does not, according to QBism, generalize or invalidate probability theory in any way. Rather, the urgleichung and the law of total probability apply in different scenarios because P ( D j ) and Q ( D j ) refer to different situations. P ( D j ) is the probability that an agent assigns for obtaining outcome D j on her second of two planned measurements, that is, for obtaining outcome D j after first making the SIC measurement and obtaining one of the H i outcomes. Q ( D j ) , on the other hand, is the probability an agent assigns for obtaining outcome D j when she doesn't 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.

A fully personalist account of quantum theory must account for quantum dynamics in Bayesian terms. It may be tempting to view state evolution via the Schrödinger equation as it is presented in the conventional quantum formalism as a physical process. What is evolving in time if not an element of reality? It turns out that the SIC basis again provides an answer. Consider a quantum state ρ ^ with SIC representation P ( H i ) . After some time t , quantum theory instructs us to use the unitarily rotated state U ^ ρ ^ U ^ † with SIC representation P t ( H i ) = tr [ ( U ^ ρ ^ U ^ † ) H ^ i ] = tr [ ρ ^ ( U ^ † H ^ i U ^ ) ] . 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 { D j } = { U ^ † H ^ j U ^ } of the original quantum state ρ ^ . Then the Schrödinger equation is completely captured in the urgleichung for this measurement:

Those QBists who find this approach promising are pursuing a complete reconstruction of quantum theory featuring the urgleichung as the key postulate. Progress towards this goal can be found in reference.

Reception

Reactions to the QBism interpretation have ranged from delight to outrage. Some who have criticized QBism claim that it fails to meet the goal of resolving paradoxes in quantum theory. Mohrhoff, for example, criticized QBism from the standpoint of Kantian philosophy. Others may have misunderstood the claims of QBism. See Nauenberg's article which prompted the reply. Still others find QBism internally self-consistent, but do not subscribe to the interpretation. Several critiques of QBism which arose in response to Mermin's Physics Today article and his replies to these comments may be found in the Physics Today readers' forum. Section 2 of the Stanford Encyclopedia of Philosophy entry on QBism contains a list of objections and replies to the interpretation. A curated list of various other published criticisms of QBism can be found on page 2278 of reference.

Nobel laureate Theodor Hänsch has promoted the QBism interpretation.

Copenhagen Interpretations

The views of many physicists (Bohr, Heisenberg, Rosenfeld, von Weizsäcker, Peres, etc.) are often grouped together as the "Copenhagen interpretation" of quantum mechanics. Several authors have deprecated this terminology, claiming that it is historically misleading and obscures differences between physicists that are as important as their similarities. QBism shares many characteristics in common with the ideas often labaled as "the Copenhagen interpretation", but the differences are important; to conflate them or to regard QBism as a minor modification of the points of view of Bohr or Heisenberg, for instance, would be a substantial misrepresentation. QBism takes probabilities to be personal judgments of the individual agent who is using quantum mechanics while the Copenhagen view holds that probabilities are given by something ontologically prior, namely a wavefunction. QBism considers a measurement to be any action that an agent takes to elicit a response from the world and the outcome of that measurement to be the experience the world's response induces back on that agent. As a consequence, communication between agents is the only means by which different agents can attempt to compare their internal experiences. Most variants of the Copenhagen interpretation, however, hold that the outcomes of experiments are agent-independent pieces of reality for anyone to access. Although not yet claiming to provide an overt underlying ontology, QBism claims that such changes resolve the obscurities that many critics have found in the Copenhagen interpretation by supplanting the role that quantum theory plays. Rather than a mechanics of reality, QBism claims that quantum theory is a normative tool which an agent may use to better navigate reality.

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. An article by Stacey (a collaborator of Fuchs) disagrees, arguing that the views expressed in that book on the nature of quantum states and the interpretation of probability are not compatible with QBism, or indeed, with any position that might be called Quantum Bayesianism.

Relational Quantum Mechanics

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

Other Epistemic Views

Approaches to quantum mechanics, like QBism, which treat quantum states as expressions of information, knowledge, belief, or expectation are called "epistemic" interpretations. These approaches differ from each other in what they consider quantum states to be information or expectations 'about', as well as in the technical features of the mathematics they employ.

Other Uses of Bayesian Probability in Quantum Physics

QBism should be distinguished from other applications of Bayesian probability in quantum physics. For example, quantum computer science uses Bayesian networks, which find applications in "medical diagnosis, monitoring of processes, and genetics". (A Bayesian framework is also used for neural networks.) Bayesian inference has also been applied in quantum theory for updating probability densities over quantum states, and MaxEnt methods have been used in similar ways.

References

Quantum Bayesianism Wikipedia

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