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arXiv:1902.03607 · IEEE Transactions on Information Theory

Quantum Measurement as Marginalization and Nested Quantum Systems

Hans‐Andrea Loeliger, Pascal O. Vontobel

Published Dec 24, 2019• 3 citations
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Abstract

In prior work, we have shown how the basic concepts and terms of quantum mechanics relate to factorizations and marginals of complex-valued quantum mass functions, which are generalizations of joint probability mass functions. In this paper, using quantum mass functions, we discuss the realization of measurements in terms of unitary interactions and marginalizations. It follows that classical measurement results strictly belong to local models, i.e., marginals of more detailed models. Classical variables that are created by marginalization do not exist in the unmarginalized model, and different marginalizations may yield incompatible classical variables. These observations are illustrated by the Frauchiger-Renner paradox, which is analyzed (and resolved) in terms of quantum mass functions. Throughout, the paper uses factor graphs to represent quantum systems/models with multiple measurements at different points in time.

DOI
10.1109/tit.2019.2961377

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In prior work, we have shown how the basic concepts and terms of quantum mechanics relate to factorizations and marginals of complex-valued quantum mass functions, which are generalizations of joint probability mass functions. In this paper, using quantum mass functions, we discuss the realization of measurements in terms of unitary interactions and marginalizations. It follows that classical measurement results strictly belong to local models, i.e., marginals of more detailed models. Classical variables that are created by marginalization do not exist in the unmarginalized model, and different marginalizations may yield incompatible classical variables. These observations are illustrated by the Frauchiger-Renner paradox, which is analyzed (and resolved) in terms of quantum mass functions. Throughout, the paper uses factor graphs to represent quantum systems/models with multiple measurements at different points in time.
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.
MEASUREMENT INTERACTIONS In the present interpretation terms such as ‘measurement,’ ‘observation,’ ‘find,’ ‘classical,’ and ‘apparatus’ do not appear in the formulation of quantum mechanics. Yet it must of course be true that quantum mechanics can be used to predict features of the results of measurements – in particular their statistics. What is required, then, is an account in quantum mechanical terms of those physical interactions capable of being used to perform measurements of quantum dynamical variables, from which it follows that the predicted probabilities for specified behavior of certain quantum systems in these interactions coincide with what, on a more conventional interpretation, are simply postulated as the quantum mechanical probabilities for various measurement results on a system with a given quantum state. It is necessary to see how this consequence may be derived, not only in order to explicate the Born rules, but also to explain the role of quantum states within the present interpretation. The first step toward the required account is to analyze the nature of an interaction capable of being used to perform a measurement of the value of a quantum dynamical variable in a quantum system. Any such interaction will be called an M -type interaction, whether or not it is in fact used to perform a measurement. The basic characteristic of an M -type interaction is that it establishes a correlation between the initial dynamical state of one quantum system σ (the object system) and the final dynamical state of another quantum system α (the apparatus system).
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.
It is known that the classical framework of causal models is not general enough to allow for causal reasoning about quantum systems. While the framework has been generalized in a variety of different ways to the quantum case, much of this work leaves open whether causal concepts are fundamental to quantum theory, or only find application at an emergent level of classical devices and measurement outcomes. Here, we present a framework of quantum causal models, with causal relations defined in terms intrinsic to quantum theory, and the central object of study being the quantum process itself. Following Allen et al., Phys. Rev. X 7, 031021 (2017), the approach defines quantum causal relations in terms of unitary evolution, in a way analogous to an approach to classical causal models that assumes underlying determinism and situates causal relations in functional dependences between variables. We show that any unitary quantum circuit has a causal structure corresponding to a directed acyclic graph, and that when marginalising over local noise sources, the resulting quantum process satisfies a Markov condition with respect to the graph. We also prove a converse to this statement. We introduce an intrinsically quantum notion that plays a role analogous to the conditional independence of classical variables, and (generalizing a central theorem of the classical framework) show that d-separation is sound and complete for it in the quantum case. We present generalizations of the three rules of the classical do-calculus, in each case relating a property of the causal structure to a formal property of the quantum process, and to an operational statement concerning the outcomes of interventions. In addition, we introduce and derive similar results for classical split-node causal models, which are more closely analogous to quantum causal models than the classical causal models that are usually studied.
Quantum mass functions (QMFs), which are tightly related to decoherence functionals, were introduced by Loeliger and Vontobel [IEEE Trans. Inf. Theory, 2017, 2020] as a generalization of probability mass functions toward modeling quantum information processing setups in terms of factor graphs. Simple quantum mass functions (SQMFs) are a special class of QMFs that do not explicitly model classical random variables. Nevertheless, classical random variables appear implicitly in an SQMF if some marginals of the SQMF satisfy some conditions; variables of the SQMF corresponding to these “emerging” random variables are called classicable variables. Of particular interest are jointly classicable variables. In this paper we initiate the characterization of the set of marginals given by the collection of jointly classicable variables of a graphical model and compare them with other concepts associated with graphical models like the sets of realizable marginals and the local marginal polytope. In order to further characterize this set of marginals given by the collection of jointly classicable variables, we generalize the CHSH inequality based on the Pearson correlation coefficients, and thereby prove a conjecture proposed by Pozsgay et al. A crucial feature of this inequality is its nonlinearity, which poses difficulties in the proof.