The mathematical expression of generalized complementarity

Yunlong Xiao, Kun Fang, Gilad Gour

The uncertainty principle bounds the uncertainties about incompatible measurements, clearly setting quantum theory apart from the classical world. Its mathematical formulation via uncertainty relations, plays an irreplaceable role in quantum technologies. However, neither the uncertainty principle nor uncertainty relations can fully describe the complementarity between quantum measurements. As an attempt to advance the efforts of complementarity in quantum theories, we formally propose a complementary information principle, significantly extending the one introduced by Heisenberg. First, we build a framework of black box testing consisting of pre- and post-testing with two incompatible measurements, introducing a rigorous mathematical expression of complementarity with definite information causality. Second, we provide majorization lower and upper bounds for the complementary information by utilizing the tool of semidefinite programming. In particular, we prove that our bounds are optimal under majorization due to the completeness of the majorization lattice. Finally, as applications to our framework, we present a general method to outer-approximating all uncertainty regions and also establish fundamental limits for all qualified joint uncertainties.

Eric Jakobsson

Harald Atmanspacher A

Two key concepts of quantum theory, complementarity and entanglement, are considered with respect to their significance in and beyond physics. An axiomatically formalized, weak version of quantum theory, more general than the ordinary quantum theory of physical systems, is described. Its mathematical structure generalizes the algebraic approach to ordinary quantum theory. The crucial formal feature leading to complementarity and entanglement is the non‐commutativity of observables.The ordinary Hilbert space quantum mechanics can be recovered by stepwise adding the necessary features. This provides a hierarchy of formal frameworks of decreasing generality and increasing specificity. Two concrete applications, more specific than weak quantum theory and more general than ordinary quantum theory, are discussed: (i) complementarity and entanglement in classical dynamical systems, and (ii) complementarity and entanglement in the bistable perception of ambiguous stimuli.

Lucien Morren

Vladimir V. Kuz'min

The author examines the reality of an object in a situation. Such a given is due to the corresponding actual (intentional) property that forms the situation with the studied object in the act of attention. In a situation, an object has its own way of being – a mode. Modes are conditionally divided into strong and weak. In epistemological terms, the reality of an object in a situation has two states relative to each other, depending on the position of the cognizing subject in relation to this object. In accordance with this, the object is known either intensionally or extensionally. A phenomenological approach is used. The difference between internal and external observers in the constituted situation is shown. Such a distinction does not occur at all, but depends on the actual property in the situation. The act of cognition introduces asymmetry into the reality of its relational states: one of them will be relevant (with a strong mode), the other – only possible (with a weak mode) in the same situation. This asymmetry, in particular, manifests itself in wave-particle dualism and is expressed through the principles of complementarity and uncertainty. In particular, it is shown that the complementarity principle indicates the duality of the given object, expressed through its relational states, and the uncertainty principle indicates the difference of modes (strong and weak) of these states in the same situation. In support of what has been said, some other examples from physics, as well as from mathematics and everyday life are given.

Kalervo V. Laurikainen