Introduction In many situations, learning from the results of measurements can be regarded as updating one's probability distributions over certain variables. According to Bayesians, this updating should be carried out according to the rule of conditionalization. In the theory of quantum mechanics, there is a rule that tells us how to update the state of a system, given observation of a measurement result. The state of a quantum system is closely related to probability distributions over potential measurements. Therefore we might expect there to be some relation between Bayesian conditionalization and the quantum state-update rule. There have been several suggestions that the state change just is Bayesian conditionalization, appropriately understood, or that it is closely analogous. Bub was the first to make the connection between quantum measurement and Bayesian conditionalization in a 1977 paper, using an approach based on quantum logic. The connection is renewed in discussions by Fuchs and also Jacobs in 2002, where again the analogy between the quantum state update and Bayesian conditionalization is pointed out. At the same time, Fuchs draws attention to a disanalogy – namely that there is an “extra unitary” transformation as part of the measurement in the quantum case. In this chapter, I will first review the proposals of Bub, Jacobs, and Fuchs. I will then show that the presence of the extra unitaries in quantum measurement leads to a difference between classical and quantum measurement in terms of information gain, drawing on results by Nielsen and Fuchs and Jacobs.
Introduction In many situations, learning from the results of measurements can be regarded as updating one's probability distributions over certain variables. According to Bayesians, this updating should be carried out according to the rule of conditionalization. In the theory of quantum mechanics, there is a rule that tells us how to update the state of a system, given observation of a measurement result. The state of a quantum system is closely related to probability distributions over potential measurements. Therefore we might expect there to be some relation between Bayesian conditionalization and the quantum state-update rule. There have been several suggestions that the state change just is Bayesian conditionalization, appropriately understood, or that it is closely analogous. Bub was the first to make the connection between quantum measurement and Bayesian conditionalization in a 1977 paper, using an approach based on quantum logic. The connection is renewed in discussions by Fuchs and also Jacobs in 2002, where again the analogy between the quantum state update and Bayesian conditionalization is pointed out. At the same time, Fuchs draws attention to a disanalogy – namely that there is an “extra unitary” transformation as part of the measurement in the quantum case. In this chapter, I will first review the proposals of Bub, Jacobs, and Fuchs. I will then show that the presence of the extra unitaries in quantum measurement leads to a difference between classical and quantum measurement in terms of information gain, drawing on results by Nielsen and Fuchs and Jacobs.
Quantum theory can be viewed as a generalization of classical probability theory, but the analogy as it has been developed so far is not complete. Whereas the manner in which inferences are made in classical probability theory is independent of the causal relation that holds between the conditioned variable and the conditioning variable, in the conventional quantum formalism, there is a significant difference between how one treats experiments involving two systems at a single time and those involving a single system at two times. In this article, we develop the formalism of quantum conditional states, which provides a unified description of these two sorts of experiment. In addition, concepts that are distinct in the conventional formalism become unified: Channels, sets of states, and positive operator valued measures are all seen to be instances of conditional states; the action of a channel on a state, ensemble averaging, the Born rule, the composition of channels, and nonselective state-update rules are all seen to be instances of belief propagation. Using a quantum generalization of Bayes' theorem and the associated notion of Bayesian conditioning, we also show that the remote steering of quantum states can be described within our formalism as a mere updating of beliefs about one system given new information about another, and retrodictive inferences can be expressed using the same belief propagation rule as is used for predictive inferences. Finally, we show that previous arguments for interpreting the projection postulate as a quantum generalization of Bayesian conditioning are based on a misleading analogy and that it is best understood as a combination of belief propagation (corresponding to the nonselective state-update map) and conditioning on the measurement outcome.
The dynamics of quantum measurements defines a precise relation between the information gained about one physical property of a system and the observable changes in another physical property of the same system. Here, we express this relation in terms of the Hilbert space superpositions of the corresponding eigenstates and show how the probability of an observable physical change can be obtained from the Bayesian update of the probabilities associated with the information obtained in the measurement. Our analysis demonstrates that the superposition principle provides the tightest possible expression of the trade-off between information and back action in a quantum measurement.
A core tenet of Bayesian epistemology is that rational agents update by conditionalization.Accuracy arguments in favour of this norm are well-known.Meanwhile, scholars working in quantum probability and quantum state estimation have proposed multiple updating rules, all of which look prima facie like analogues of Bayesian conditionalization.The most common are Lüders conditionalization and Bayesian mean estimation (BME).Some authors also endorse a lesser-known alternative that we call retrodiction.We show how one can view Lüders and BME as complementary rules, and we give expected accuracy and accuracy dominance arguments for both.By contrast, we find that retrodiction is accuracy-dominated, at least on many measures of accuracy.Bayesian norms do not immediately carry over to this setting.(Here and in what follows, we assume that there are objective quantum probabilities, and that quantum systems have true, objective quantum states.The idea that there are both subjective and objective probabilities, and we can discuss their relationship, is sometimes known as Lewisian dualism (Wallace [2012]).)While it is unclear how to translate standard Bayesian norms into this quantum operatortheoretic setting, we can draw a natural analogy.Usually, the Bayesian represents both credences and states of the world as real-valued functions over an algebra of propositions; we can view credences as 'estimates' of the truth values assigned by a given state. 1 In quantum theory, one represents states with bounded, linear operators on a Hilbert space.It turns out that one can also use such operators to encode estimates of quantum states.We adopt this paradigm to construct natural analogues of the accuracy arguments, affording novel Bayesian justifications for several quantum updating rules.It is controversial whether such a non-classical approach to probability is necessary once we fix an appropriate solution to the measurement problem (Loewer, Healey [1994, 2020]).However, even if it is not necessary, this alternate setting is immensely popular.Experimenters typically work within the framework of Hilbert-space operators, as this framework houses most approaches to quantum state estimation (that is, the reconstruction of quantum states based on measurement outcomes; see Jones, Blume-Kohout, Granade et al. [1991Granade et al. [ , 2010Granade et al. [ , 2016]]).So, the status of Bayesian norms in this setting seems well worth exploring.Interestingly, authors have proposed multiple updating norms in this framework, all of which look prima facie like analogues of Bayesian conditionalization:
I review the formalism of classical extensions of quantum mechanics introduced by Beltrametti and Bugajski, and compare it to the classical representations discussed e.g. by Busch, Hellwig and Stulpe and recently used by Fuchs in his discussion of quantum mechanics in terms of standard quantum measurements. I treat the problem of finding Bayesian analogues of the state transition associated with measurement in the canonical classical extension as well as in the related 'uniform' classical representation. In the classical extension, the analogy is extremely good.
Introduction and overview Much that is studied in quantum measurement theory, and thus in this book, is very different from that studied in its classical counterpart, Bayesian inference. While quantum measurement theory is in one sense a minimal extension of the latter to quantum states, quantum measurements cause dynamical changes that never need appear in a classical theory. This has a multitude of ramifications, many of which you will find in this book. One such consequence is a limit to the information that can be obtained about the state of a quantum system, and this results in relationships between information gained and dynamics induced that do not exist in classical systems. Because of the limits to information extraction, choosing the right kinds of measurements becomes important in a way that it never was for classical systems. Finding the best measurements for a given purpose, and working out how to realize these measurements, is often nontrivial. The relationship between information and disturbance impacts the control of quantum systems using measurements and feedback, and we discuss this application in Chapter 5. The ability to extract information about quantum systems is also important in using quantum systems to make precise measurements of classical quantities such as time, acceleration, and the strengths of electric and magnetic fields. This subject is referred to as quantum metrology, and we discuss it in Chapter 6.
In the Bayesian approach to probability theory, probability quantifies a degree of belief for a single trial, without any a priori connection to limiting frequencies. Despite being prescribed by a fundamental law, probabilities for individual quantum systems can be understood within the Bayesian approach. We argue that the distinction between classical and quantum probabilities lies not in their definition, but in the nature of the information they encode. In the classical world, maximal information about a physical system is complete in the sense of providing definite predictions for all possible questions that can be asked of the system. In the quantum world, maximal information is not complete and cannot be completed. Using this distinction, we show that any Bayesian probability assignment in quantum mechanics must have the form of the quantum probability rule, that maximal information about a quantum system leads to a unique quantum-state assignment, and that quantum theory provides a stronger connection between probability and measured frequency than can be justified classically. Finally we give a Bayesian formulation of quantum-state tomography. There are excellent reasons for interpreting quantum states as states of knowledge. A classic