Causal isomorphism and complementarity: Setting the record straight

David Favrholdt

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

János A. Bergou, E. Bagán, John Calsamiglia +1 more

To set the record straight, we present a brief history of interferometric complementarity relations, including those recently rediscovered by other groups, and results for multi-path interferometers and finite groups employing l1 and entropic coherence measures.

Rolando P. Hong Enriquez

Two structural facts anchor the theory of indefinite causal order, and this paper proves both as explicit theorems and checks them line by line in a machine proof assistant. The first is a record fact: a process without a definite order must keep, on its control memory, a record of the orders it superposes, and erasing that record by blurring the control returns a plain mixture of definite orders. The second is a classification fact: on the canonical slice of processes, a single straight-line certificate decides exactly which processes are ordered and which are not. Working over an abstract algebra of order operators, so that each result holds in any faithful concrete model, the paper proves a short, connected set of statements. Blurring the order record always lands the process in the definite-order set, which is the mathematical heart of record-erasure. A process sits in that definite-order set exactly when it is unchanged by blurring, so that any surviving control coherence is an irreducible record. The number of orders a process populates is at most the size of its control memory, which is the record form of the memory cost studied elsewhere in the program. On the canonical slice the certificate takes a specific value that varies in a straight-line way across the slice, so that indefiniteness holds precisely on one side of a line, with an explicit decomposition into definite orders on the other side. And a tempting finer diagnostic built from the process's intrinsic flow is shown to over-count, because it tracks a product of the two dials rather than the true boundary. Every theorem is stated over an abstract ring so that it transfers to any faithful model, and every statement in the paper has been checked in a machine proof assistant with no assumptions beyond the three standard foundations of mathematics; the correspondence to the formal development is tabulated in an appendix. The contribution is a small, dependable, reusable core that turns the program's paper-level results about indefinite causal order into a durable, independently checkable formal library, and by forcing complete rigour it guards against hidden gaps of the kind that informal arguments can conceal. The paper is careful about scope, noting that the definite-order set reached by full blurring is contained in, but does not exhaust, the whole ordered set, and that the finer flow-based diagnostic is only a necessary sign of indefiniteness, not a sufficient one. What it secures is a machine-checked foundation on which the geometric, thermodynamic, and resource-theoretic parts of the program can safely rest.

Douglas Michael Snyder

Today it is important to see James’s original complementarity in 1890 independently of Bohr’s discussions on complementarity in order that James’s conception can be properly evaluated as the model on its own merits for quantum measurement. On the basis of new evidence and a deeper analysis of previous evidence presented here, it is very reasonable to conclude based on Bohr’s own words that Bohr read James’s complementarity between approximately 1908 and 1912. An Appendix provides the relevant text from both Bohr’s Nobel biography and his interview with the American Institute of Physics to support this conclusion. Bohr’s description of significant events in his life are matched with dates in his Nobel biography, adding support to the conclusion. How Bohr specifically adapted James’s complementarity to physics is presented. Bohr’s discussion of “mental activity” is compared here to James’s definition of complementarity focusing on consciousness and is found to be extremely similar to James’s definition. In addition, a side by side comparison of James’s definition of complementarity and Bohr’s most important and well-known definition of complementarity for physics is presented. This comparison shows just how Bohr adapted James’s complementarity to Bohr’s complementarity for physics. Examining the chapters “The Stream of Thought”” (chapter 9 in James’s “The Principles of Psychology”) and “The Relation of Minds to Other Things” (chapter 8 in “The Principles of Psychology”) shows that the very evidence that James used to develop complementarity in “The Relation of Minds to Other Things” is discussed in “The Stream of Thought,” a chapter that Bohr acknowledged reading and concerning which he had a very high opinion. It is also made clear in the paper that Bohr was aware that he had a choice to adopt complementarity for physics along the lines of James’s complementarity, which he had read, and that he rejected this possibility. Instead, in his complementarity for physics, Bohr adopted a very different approach fundamentally based on the necessity of a physical interaction between a physical system and a physical measuring apparatus with an uncontrolled aspect to obtain a specific measurement result. Bohr attempted to apply the uncertainty principle to this uncontrolled feature of the physical interaction. Bohr’s approach has been the widely adopted framework for understanding quantum measurement since 1927. The historical description of complementarity in Bohr’s Nobel biography is discussed in light of the evidence presented and is shown to be incorrect. It is important to set the record straight regarding the origin of complementarity so that James’s complementarity is not seen as an appendage of Bohr’s version of complementarity for physics. Bohr’s attempt to extrapolate what his Nobel biography incorrectly states was complementarity originally developed by Bohr, which was for physics, to other fields confuses and detracts from the importance of James’s complementarity which was in fact the original. Bohr’s detour with his proposed necessity (and its proposed fundamental significance) of a physical interaction in the measuring process with an uncontrolled aspect in quantum mechanics has distracted attention away from the possible significance of James’s original concept in psychology to quantum measurement, including its evaluation on its own merits as to whether it provides the correct conceptual framework for this measurement.