In the Brussels’ operational-realistic approach to cognition, concepts are treated as entities that can be in different states and be subjected to measurements performed by human minds. The Brussels group also considered two other examples of bipartite systems, the famous Bohm-EPR state of two spin-1/2 particles forming a singlet state (microscopic level) and a model of breakable elastic bands (macroscopic level) to show that they violate Bell-CHSH inequalities. Sandeau et al. employed entangled states of classical Maxwell fields coupling the polarization and spatial parity degrees of freedom in order to implement the analogue of non-unitary transformations, leading to maximal correlations beyond Tsirelson’s bound. Their approach relies on the realization at the level of classical Maxwell fields of a so-called positive operator valued measure. POVM stands for ‘positive operator valued measure’. They are the most general kinds of measurement. In standard quantum mechanics one uses ideal measurements which are one-dimensional projection operators.
In the Brussels’ operational-realistic approach to cognition, concepts are treated as entities that can be in different states and be subjected to measurements performed by human minds. The Brussels group also considered two other examples of bipartite systems, the famous Bohm-EPR state of two spin-1/2 particles forming a singlet state (microscopic level) and a model of breakable elastic bands (macroscopic level) to show that they violate Bell-CHSH inequalities. Sandeau et al. employed entangled states of classical Maxwell fields coupling the polarization and spatial parity degrees of freedom in order to implement the analogue of non-unitary transformations, leading to maximal correlations beyond Tsirelson’s bound. Their approach relies on the realization at the level of classical Maxwell fields of a so-called positive operator valued measure. POVM stands for ‘positive operator valued measure’. They are the most general kinds of measurement. In standard quantum mechanics one uses ideal measurements which are one-dimensional projection operators.
We have recently presented a general scheme enabling quantum modeling of different types of situations that violate Bell's inequalities [1]. In this paper, we specify this scheme for a combination of two concepts. We work out a quantum Hilbert space model where 'entangled measurements' occur in addition to the expected 'entanglement between the component concepts', or 'state entanglement'. We extend this result to a macroscopic physical entity, the 'connected vessels of water', which maximally violates Bell's inequalities. We enlighten the structural and conceptual analogies between the cognitive and physical situations which are both examples of a nonlocal non-marginal box modeling in our classification.
The scientific community is becoming more and more interested in the research that applies the mathematical formalism of quantum theory to model human decision-making. In this paper, we provide the theoretical foundations of the quantum approach to cognition that we developed in Brussels. These foundations rest on the results of two decade studies on the axiomatic and operational-realistic approaches to the foundations of quantum physics. The deep analogies between the foundations of physics and cognition lead us to investigate the validity of quantum theory as a general and unitary framework for cognitive processes, and the empirical success of the Hilbert space models derived by such investigation provides a strong theoretical confirmation of this validity. However, two situations in the cognitive realm, `question order effects' and `response replicability', indicate that even the Hilbert space framework could be insufficient to reproduce the collected data. This does not mean that the mentioned operational-realistic approach would be incorrect, but simply that a larger class of measurements would be in force in human cognition, so that an extended quantum formalism may be needed to deal with all of them. As we will explain, the recently derived `extended Bloch representation' of quantum theory (and the associated `general tension-reduction' model) precisely provides such extended formalism, while remaining within the same unitary interpretative framework.
The physical meaning of the basic quantum mechanical concepts (such as the wave function, reduction, state preparation and measurement, the projection postulate, and the uncertainty principle) is clarified using realistic experimental procedures and employing classical analogies whenever possible. Photon polarization measurement and particle coordinate and momentum measurement are considered as examples, as also are Einstein–Podolsky–Rosen correlations, Aharonov–Bohm effects, quantum teleportation, etc. Various nonclassicality criteria of quantum models, including photon antibunching and the violation of the Bell inequality, are discussed.
A bipartite Bell inequality is derived which is maximally violated on the two-qubit state space if measurements describable by positive operator valued measure (POVM) elements are allowed, rather than restricting the possible measurements to projective ones. In particular, the presented Bell inequality requires POVMs in order to be maximally violated by a maximally entangled two-qubit state. This answers a question raised by N. Gisin [in Quantum Reality, Relativistic Causality, and Closing the Epistemic Circle: Essays in Honour of Abner Shimony, edited by W. C. Myrvold and J. Christian (Springer, The Netherlands, 2009), pp. 125--138].
Abstract We provide a general description of the phenomenon of entanglement in bipartite systems, as it manifests in micro and macro physical systems, as well as in human cognitive processes. We do so by observing that when genuine coincidence measurements are considered, the violation of the ‘marginal laws’, in addition to the Bell-CHSH inequality, is also to be expected. The situation can be described in the quantum formalism by considering the presence of entanglement not only at the level of the states, but also at the level of the measurements. However, at the “local” level of a specific joint measurement, a description where entanglement is only incorporated in the state remains always possible, by adopting a fine-tuned tensor product representation. But contextual tensor product representations should only be considered when there are good reasons to describe the outcome-states as (non-entangled) product states. This will not in general be true, hence, the entanglement resource will have to generally be allocated both in the states and in the measurements. In view of the numerous violations of the marginal laws observed in physics’ laboratories, it remains unclear to date if entanglement in micro-physical systems is to be understood only as an ‘entanglement of the states’, or also as an ‘entanglement of the measurements’. But even if measurements would also be entangled, the corresponding violation of the marginal laws (also called ‘no-signaling conditions’) would not for this imply that a superluminal communication would be possible.
Positive operator valued measurements (POVMs) are the most general measurements that can be performed on quantum systems.A POVM is described by set of positive operators instead of orthogonal projections, and it can always be traced back to joint von Neumann measurements on the system and on an ancilla attached to it (Busch et. al, 1991;Peres, 1993).* Note on nomenclature: subscript letters denote the appropriate state spaces for vectors and density operators throughout the document.The same symbol with different subscripts intends to mean that the same quantum information is represented on different systems.