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Physical Review A

Quantum Bayes' rule affirming consistency in measurement inferences in quantum mechanics

Mohit Lal Bera, Manabendra Nath Bera

Published Jul 31, 2023• 0 citations
Publisher

Abstract

The traditional Bayes' rule lays the foundation for causal reasoning and finding relations between cause (input) and effect (output). This causal reasoning is universally applied to all physical processes to establish causal relations. Here we show that it does not establish correct causal correspondence between quantum causes and effects in general. In fact, there are instances within the framework of quantum mechanics where the use of the traditional Bayes' rule leads to inconsistencies in quantum measurement inferences. We consider two such cases, inspired by Frauchiger-Renner's and Hardy's setups, where the traditional Bayes' rule results in paradoxical situations even after assuming quantum mechanics as a nonlocal theory. As a remedy, we introduce an input-output causal relation using the reasoning based on a quantum Bayes' rule. It applies to general quantum processes even when a cause (or effect) is in coherent superposition with other causes (or effects), involves nonlocal correlations as allowed by quantum mechanics, and applies in the cases where causes belonging to one system induce effects in some other system as happens in quantum measurement processes. This enables us to propose a resolution to the contradictions that appear in the context of Frauchiger-Renner's and Hardy's setups. Our results thereby affirm that quantum mechanics, equipped with a quantum Bayes' rule, can indeed consistently describe the use of itself.

DOI
10.1103/physreva.108.012224

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The traditional Bayes' rule lays the foundation for causal reasoning and finding relations between cause (input) and effect (output). This causal reasoning is universally applied to all physical processes to establish causal relations. Here we show that it does not establish correct causal correspondence between quantum causes and effects in general. In fact, there are instances within the framework of quantum mechanics where the use of the traditional Bayes' rule leads to inconsistencies in quantum measurement inferences. We consider two such cases, inspired by Frauchiger-Renner's and Hardy's setups, where the traditional Bayes' rule results in paradoxical situations even after assuming quantum mechanics as a nonlocal theory. As a remedy, we introduce an input-output causal relation using the reasoning based on a quantum Bayes' rule. It applies to general quantum processes even when a cause (or effect) is in coherent superposition with other causes (or effects), involves nonlocal correlations as allowed by quantum mechanics, and applies in the cases where causes belonging to one system induce effects in some other system as happens in quantum measurement processes. This enables us to propose a resolution to the contradictions that appear in the context of Frauchiger-Renner's and Hardy's setups. Our results thereby affirm that quantum mechanics, equipped with a quantum Bayes' rule, can indeed consistently describe the use of itself.
Classical Bayes' rule lays the foundation for the classical causal relation between cause (input) and effect (output). This causal relation is believed to be universally true for all physical processes. Here we show, on the contrary, that it is inadequate to establish correct correspondence between cause and effect in quantum mechanics. In fact, there are instances within the framework of quantum mechanics where the use of classical Bayes' rule leads to inconsistencies in quantum measurement inferences, such as Frauchiger-Renner's paradox. Similar inconsistency also appears in the context of Hardy's setup even after assuming quantum mechanics as a non-local theory. As a remedy, we introduce an input-output causal relation based on quantum Bayes' rule. It applies to general quantum processes even when a cause (or effect) is in coherent superposition with other causes (or effects), involves nonlocal correlations as allowed by quantum mechanics, and in the cases where causes belonging to one system induce effects in some other system as it happens in quantum measurement processes. This enables us to propose a resolution to the contradictions that appear in the context of Frauchiger-Renner's and Hardy's setups. Our results thereby affirm that quantum mechanics, equipped with quantum Bayes' rule, can indeed consistently explain the use of itself.
Quantum measurement is widely regarded as paradoxical due to the apparent role of observers, wave function collapse, and mutually inconsistent descriptions of reality. We argue that these paradoxes are not intrinsic to quantum mechanics, but arise from a category error: treating observer-dependent inferences as observer-independent physical facts.We introduce a causality-first framework in which quantum states represent open causal structures, and measurement corresponds to a scale-dependent transition to causal closure. No privileged observer, stochastic collapse mechanism, or many-worlds ontology is required. Within this framework, celebrated paradoxes such as Wigner’s Friend and the Frauchiger–Renner no-go theorem dissolve, as their contradictions rely on an implicit violation of causal consistency.Our proposal does not modify quantum dynamics nor introduce new postulates. Instead, it clarifies when quantum processes may enter causal chains. The result is a realist, observer-independent account of measurement that preserves standard quantum theory while eliminating its most persistent conceptual pathologies.
An analysis using classical stochastic processes is used to construct a consistent system of quantum counterfactual reasoning. When applied to a counterfactual version of Hardy's paradox, it shows that the probabilistic character of quantum reasoning together with the ``one framework'' rule prevents a logical contradiction, and there is no evidence within this approach for any nonlocal influences. Counterfactual reasoning can support a realistic interpretation of standard quantum theory (measurements reveal what is actually there) under appropriate circumstances.
Recent extended formulations of the Wigner's friend thought experiment throw the measurement problem of quantum mechanics into sharper relief. Here I respond to an invitation by Renner to provide a consistent and concrete set of rules for quantum mechanics which can avoid the apparent paradox formulated by Frauchiger and Renner [Nat. Comm. 9, 3711 (2018)]. I propose a slight addition to standard textbook quantum mechanics, in the form of two rules, which avoids the paradox. The first specifies when a given quantum dynamics can be interpreted as a measurement. Potentially any dynamics can, but doing so depends on the context of other performed operations. The second requires that a joint context be used to determine whether several different dynamical evolutions can all be interpreted as measurement. The paradox is then avoided because not every participant in the Frauchiger-Renner thought experiment regards the stated dynamical descriptions of the experiment as measurements.
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.