Probability and Observation Theory.

Sergio M. Focardi, Frank J. Fabozzi, Davide Mazza

In this article, the authors discuss the use of quantum probability, that is, the probability theory of quantum mechanics, for option pricing and for finance in general. The authors discuss the motivations for applying quantum probability to finance. The critical issues are replacing random variables with operators, self-reflexivity of markets, and the existence of incompatible observations. The authors outline quantum probability theory, quantum stochastic processes, and the pricing of options in a quantum context. TOPICS:Options, portfolio theory, portfolio construction Key Findings • Quantum probability theory is a probabilistic theory of observations. Observations can change the system and be incompatible. • Quantum probability offers a more empirically faithful handling of large events and of uncertainty. • A better theory of valuation is offered by quantum probability theory than classical probability theory.

Koji Nagata, Tadao Nakamura, Han J.F. Geurdes +3 more

We consider the two quantum measurement theories for measuring a single Pauli observable. We assume also the existence of a classical probability space for the two measurement theories. We cannot avoid the Kochen-Specker (KS) contradiction when we measure the Pauli observable by using the projective measurement theory if we introduce a classical probability space. The results of measurement are either $+1$ or $-1$ (in $\hbar/2$ unit) when we consider a spin-1/2 system. The projective measurement theory does not accept a classical probability space when we measure the Pauli observable. We propose a new measurement theory based on the truth values, i.e., the truth T (1) for true and the falsity F (0) for false. The results of measurement are either $+1$ or 0 (in $\hbar/2$ unit). We avoid the KS contradiction when we measure the Pauli observable by using the new measurement theory if we introduce a classical probability space. The new measurement theory accepts a classical probability space when we measure the Pauli observable.

Brian P. Anderson

Ask Ellingsen

In quantum mechanics, not everything that can be observed can be observed simultaneously. Observational data exhibits \emph{contextuality} -- a generalisation of nonlocality -- if the result of an observation is necessarily dependent on which combination of observables was measured. This article gives a mathematical introduction to contextuality, emphasising its nature as a general feature of probability theory and logic, rather than of any particular quantum theory.

ARMEN E. ALLAHVERDYAN, Arshag Danageozian

Quantum theory does not provide a unique definition for the joint probability of two noncommuting observables, which is the next important question after the Born's probability for a single observable. Instead, various definitions were suggested, e.g., via quasiprobabilities or via hidden-variable theories. After reviewing open issues of the joint probability, we relate it to quantum imprecise probabilities, which are noncontextual and are consistent with all constraints expected from a quantum probability. We study two noncommuting observables in a two-dimensional Hilbert space and show that there is no precise joint probability that applies for any quantum state and is consistent with imprecise probabilities. This contrasts with theorems by Bell and Kochen-Specker that exclude joint probabilities for more than two noncommuting observables, in Hilbert space with dimension larger than two. If measurement contexts are included into the definition, joint probabilities are not excluded anymore, but they are still constrained by imprecise probabilities.