Causality and Complementarity

Henry J. Folse, Niels Henrik David Bohr, Jan Faye

Mohd Asad Siddiqui, Md Qutubuddin, Tabish Qureshi

Wave--particle duality is a cornerstone of quantum mechanics, traditionally formulated under definite causal order. We investigate how complementarity is modified when the temporal order of operations is coherently superposed, as in the quantum switch. We show that no universal, state-independent, linear additive complementarity relation exists that combines spatial coherence, path distinguishability, and causal coherence within a common unit-normalized bound. This result follows from explicit extremal constructions in the quantum-switch family: although the three quantities are derived from the same global switch state, the standard spatial duality relation can be saturated while the reduced control coherence is simultaneously maximal. Tracing out the order qubit recovers the standard duality relation at the level of the reduced quanton--detector state, while coherence associated with alternative causal orders remains inaccessible in this reduced description. To characterize this contribution, we introduce \emph{causal coherence}, defined as the coherence of the reduced order qubit in the causal-order basis. We further show that complementarity admits a state-dependent entropic formulation based on incompatible measurements on the causal degree of freedom, which arises from a universal uncertainty principle and provides a canonical operationally meaningful description. These results show that operational formulations of complementarity can depend on causal structure and cannot, in general, be fully characterized at the level of reduced quantum states alone.

Irving Kirsch, Michael E. Hyland

John P. Losee

Philosophers have discussed the relationship of cause and effect from ancient times through our own. Prior to the work of Niels Bohr, these discussions presupposed that successful causal attribution implies explanation. The success of quantum theory challenged this presupposition. Bohr introduced a principle of complementarity that provides a new way of looking at causality and explanation.In this succinct review of the history of these discussions, John Losee presents the philosophical background of debates over the cause-effect relation. He reviews the positions of Aristotle, Rene Descartes, Isaac Newton, David Hume, Immanuel Kant, and John Stuart Mill. He shows how nineteenth-century theories in physics and chemistry were informed by a dominant theory of causality and how specific developments in physics provided the background for the emergence of quantum theory.Problems created for the causality implies explanation thesis by the emergence of quantum theory are reviewed in detail. Losee evaluates Bohr's proposals to apply a principle of complementarity within physics, biology, and psychology. He also discusses the feasibility of using complementarity as a principle of interpretation within Christian theology. This volume, which includes an in-depth index, is an essential addition to the libraries of advanced undergraduate and graduate students, philosophers, and those interested in causality and explanation.

Yuuki Sugiyama, Akira Matsumura, Kazuhiro Yamamoto

It has long been debated whether gravity should be quantized or not. Recently, the authors in [Sci. Rep. 6, 22777 (2016); Proc. Natl. Acad. Sci. U.S.A. 106, 3035 (2009)] discussed the inconsistency between causality and complementarity in a Gedankenexperiment involving the quantum superposition of massive/charged bodies, and Belenchia et al. [Phys. Rev. D 98, 126009 (2018); Int. J. Mod. Phys. D 28, 1943001 (2019)] resolved the inconsistency by requiring the quantum radiation and vacuum fluctuations of gravitational/electromagnetic field. Stimulated by their works, we reanalyze the consistency between the two physical properties, causality and complementarity, according to the quantum field theory. In this analysis, we consider a Gedankenexperiment inspired by [Sci. Rep. 6, 22777 (2016); Proc. Natl. Acad. Sci. U.S.A. 106, 3035 (2009); Phys. Rev. D 98, 126009 (2018); Int. J. Mod. Phys. D 28, 1943001 (2019)], in which two charged particles coupled with a photon field are in a superposition of two trajectories. First, we observe that causality is satisfied by the retarded propagation of the photon field. Next, by introducing an inequality between visibility and which-path information, we show that the quantum radiation and vacuum fluctuations of the photon field ensure complementarity. We further find that the Robertson inequality associated with the photon field leads to the consistency between causality and complementarity in our Gedankenexperiment. Finally, we mention that a similar feature appears in the quantum field of gravity.