15 research outputs found
Quantum analog of the original Bell inequality for two-qudit states with perfect correlations/anticorrelations
For an even qudit dimension we introduce a class of two-qudit
states exhibiting perfect correlations/anticorrelations and prove via the
generalized Gell-Mann representation that, for each two-qudit state from this
class, the maximal violation of the original Bell inequality is bounded from
above by the value - the upper bound attained on some two-qubit states.
We show that the two-qudit Greenberger-Horne-Zeilinger (GHZ) state with an
arbitrary even exhibits perfect correlations/anticorrelations and
belongs to the introduced two-qudit state class. These new results are
important steps towards proving in general the upper bound on
quantum violation of the original Bell inequality. The latter would imply that
similarly as the Tsirelson upper bound specifies the quantum analog
of the CHSH inequality for all bipartite quantum states, the upper bound
specifies the quantum analog of the original Bell inequality for
all bipartite quantum states with perfect correlations/ anticorrelations.
Possible consequences for the experimental tests on violation of the original
Bell inequality are briefly discussed.Comment: 16 page
Evaluating the Maximal Violation of the Original Bell Inequality by Two-Qudit States Exhibiting Perfect Correlations/Anticorrelations
We introduce the general class of symmetric two-qubit states guaranteeing the
perfect correlation or anticorrelation of Alice and Bob outcomes whenever some
spin observable is measured at both sites. We prove that, for all states from
this class, the maximal violation of the original Bell inequality is upper
bounded by 3/2 and specify the two-qubit states where this quantum upper bound
is attained. The case of two-qutrit states is more complicated. Here, for all
two-qutrit states, we obtain the same upper bound 3/2 for violation of the
original Bell inequality under Alice and Bob spin measurements, but we have not
yet been able to show that this quantum upper bound is the least one. We
discuss experimental consequences of our mathematical study.Comment: 12 p
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Quantum probability in decision making from quantum information representation of neuronal states
The recent wave of interest to modeling the process of decision making with the aid of the quantum formalism gives rise to the following question: ‘How can neurons generate quantum-like statistical data?’ (There is a plenty of such data in cognitive psychology and social science.) Our model is based on quantum-like representation of uncertainty in generation of action potentials. This uncertainty is a consequence of complexity of electrochemical processes in the brain; in particular, uncertainty of triggering an action potential by the membrane potential. Quantum information state spaces can be considered as extensions of classical information spaces corresponding to neural codes; e.g., 0/1, quiescent/firing neural code. The key point is that processing of information by the brain involves superpositions of such states. Another key point is that a neuronal group performing some psychological function F is an open quantum system. It interacts with the surrounding electrochemical environment. The process of decision making is described as decoherence in the basis of eigenstates of F. A decision state is a steady state. This is a linear representation of complex nonlinear dynamics of electrochemical states. Linearity guarantees exponentially fast convergence to the decision state
Mental States Follow Quantum Mechanics During Perception and Cognition of Ambiguous Figures
Ubiquitous Quantum Structure: From Psychology to Finance
Quantum-like structure is present practically everywhere. Quantum-like (QL) models, i.e. models based on the mathematical formalism of quantum mechanics and its generalizations can be successfully applied to cognitive science, psychology, genetics, economics, finances, and game theory. This book is not about quantum mechanics as a physical theory. The short review of quantum postulates is therefore mainly of historical value: quantum mechanics is just the first example of the successful application of non-Kolmogorov probabilities, the first step towards a contextual probabilistic description of natural, biological, psychological, social, economical or financial phenomena. A general contextual probabilistic model (Växjö model) is presented. It can be used for describing probabilities in both quantum and classical (statistical) mechanics as well as in the above mentioned phenomena. This model can be represented in a quantum-like way, namely, in complex and more general Hilbert spaces. In this way quantum probability is totally demystified: Born's representation of quantum probabilities by complex probability amplitudes, wave functions, is simply a special representation of this type