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The British Journal for the Philosophy of Science

An accuracy-based approach to quantum conditionalization

Alexander Meehan, Jeremy Steeger

Published Nov 6, 2023• 0 citations
Publisher

Abstract

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:

DOI
10.1086/728049

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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:
Can an agent's predictions reveal whether its reasoning is based on a classical or a quantum model? Using known results on matrix factorizations, here we show that quantum Bayesian retrodiction can produce one-step-ahead forecasts incompatible with every classical Bayesian explanation based on learning about a fixed but unknown sampling law from conditionally independent observations. The separation has a sharp threshold in the number of outcomes: for any measurement with at most four outcomes, the quantum predictions admit a classical realization, irrespective of the prior state and the finite Hilbert-space dimension, whereas a five-outcome measurement on a single qubit already violates an explicit classical bound. This bound holds for arbitrary classical latent states, prior probabilities, and sampling laws, and involves only the agent's initial and updated predictions. Thus, in this setting, quantum Bayesian retrodiction is strictly more expressive than classical Bayesian learning, and this difference can be witnessed without observing or specifying the agent's internal description.
At the intersection of the foundations of statistics and the foundations of quantum theory, one comes across a hotly-debated question: what are the probabilities in quantum theories? Do they all reflect the subjective opinions of various agents, or do some of them reflect objective features of the world? One answer—Lewisian dualism, a view that takes some quantum probabilities to be subjective and others to be objective—has gained near-consensus among philosophers of science. However, our most sophisticated defense of this view (due to Deutsch, Wallace, Saunders, Myrvold, Greaves, and others) has two significant problems. First, it requires a very particular approach to the measurement problem (namely, the Everett interpretation). Second, it relies on the pragmatic goals of scientists (e.g., their aim to avoid losses in bets) rather than their epistemic goals (e.g., their aim to accurately describe the world). I give a new defense of dualism that rectifies both of these problems. I generalize the Deutsch-Wallace argument to defend the objective probabilities in a host of realist approaches to measurement, and I modify the accuracy-dominance argument for probabilism to defend the subjective probabilities in these approaches. I also critically examine several previous attempts to interpret all quantum probabilities subjectively.
The Petz recovery map is considered one of the key candidates for the quantum analogue of Bayesian inference and Jeffrey's conditionalization. Since, there seems to be a natural connection between Bayesian inference and the notion of state tomography, it is natural to ask if the Petz recovery can be used for this latter task. In this paper, we discuss such recent results on iterated Petz recovery and relate them to Bayesian approaches to quantum state tomography. We highlight the limitations of direct Petz iteration and show how an extended Petz construction, by lifting the inference problem to a classical distribution over candidate quantum states, recovers the structure of Bayesian and maximum likelihood tomography. This provides perspective on the modifications or nuances required for a Petz approach to quantum retrodiction to perform quantum state tomography.
QBism is currently one of the most widely discussed 'subjective' interpretations of quantum mechanics. Its key move is to say that quantum probabilities are personalist Bayesian probabilities and that the quantum state represents subjective degrees of belief. Even probability-one predictions are considered subjective assignments expressing the agent's highest possible degree of certainty about what they will experience next. For most philosophers and physicists this means that QBism is simply too subjective. Even those who agree with QBism that the wave function should not be reified and that we should look for alternatives to standard Psi-ontic interpretations often argue that QBism must be abandoned because it detaches science from objectivity. The problem is that from the QBist perspective it is hard to see how objectivity could enter science. In this paper, I introduce and motivate an interpretation of quantum mechanics that takes QBism as a starting point, is consistent with all its virtues, but allows objectivity to enter from the get-go. This is the view that quantum probabilities should be understood as objective degrees of epistemic justification.
Estimating the state of an open quantum system monitored over time requires incorporating information from past measurements (filtering) and, for improved accuracy, also from future measurements (smoothing). While classical smoothing is well understood within a Bayesian framework, its quantum generalization has been challenging, leading to distinct and seemingly incompatible approaches. In this work, we demonstrate that quantum state smoothing hinges on a uniquely quantum feature: the fundamental dependence of retrodiction on prior correlations. We introduce auxiliary systems into the prior to represent specified correlations formed during preparation and evolution, and formulate quantum state smoothing as extended Bayesian retrodiction. This framework identifies previous approaches as different choices of the extended prior, and naturally extends it to other choices that have not been considered before. We also give an information-theoretic characterization of the choices of prior, in terms of the average entropy of the smoothed states. Our results establish quantum state smoothing as a fundamentally retrodictive process just like classical smoothing, with proper quantum features clearly identified.
A mixed quantum state can be taken as capturing an unspecified form of ignorance; or as describing the lack of knowledge about the true pure state of the system ("proper mixture"); or as arising from entanglement with another system that has been disregarded ("improper mixture"). These different views yield identical density matrices and therefore identical predictions for future measurements. But when used as prior beliefs for inferring the past state from later observations ("retrodiction"), they lead to different updated beliefs. This is a purely quantum feature of Bayesian agency. Based on this observation, we establish a framework for retrodicting on any quantum belief and we prove a necessary and sufficient condition for the equivalence of beliefs. We also illustrate how these differences have operational consequences in quantum state recovery.