On the probabilistic postulate of quantum mechanics

Leon W. Cohen

Roger A. Hegstrom, Alexandra J. MacDermott

Several recent studies have suggested that incompatible variables, which play an essential role in quantum mechanics (QM), are, somewhat surprisingly, not necessarily unique to QM. To investigate this possibility and obtain a better understanding of two central postulates of QM, namely the commutator postulate and the Born postulate, we introduce a classical probabilistic theoretical framework which is more general than QM and contains QM as a special case. We call this framework the General Incompatible Variables (GIV) theory, and we show that not only QM systems but also any probabilistic systems (classical or quantal) that possess incompatible variables will exhibit the quantal properties of uncertainty and interference, and we illustrate this with a roulette-like classical system (which we call the Arrow) that shows precisely these properties. We show that QM emerges naturally from the GIV framework when the fundamental variables are taken to be symmetries, so that the incompatibility of the QM variables is actually just the incompatibility of the corresponding symmetries (or their generators). Specifically, when the variables are taken to be the elements of the Poincare group (the symmetry of spacetime) we find that the commutator postulate and the Born postulate follow automatically, and are therefore no longer postulates. QM thus emerges as a special case of GIV theories for which the variables are symmetries.

Terrence L. Fine

Giacomo Mauro D’Ariano

Quantum Mechanics (QM) is a very special probabilistic theory, yet we don't know which operational principles make it so. All axiomatization attempts suffer at least one postulate of a mathematical nature. Here I will analyze the possibility of deriving QM as the mathematical representation of a "fair operational framework", i.e. a set of rules which allows the experimenter to make predictions on future "events" on the basis of suitable "tests", e.g. without interference from uncontrollable sources. Two postulates need to be satisfied by any fair operational framework: NSF: "no-signaling from the future"--for the possibility of making predictions on the basis of past tests; PFAITH: "existence of a preparationally faithful state"--for the possibility of preparing any state and calibrating any test. I will show that all theories satisfying NSF admit a C*-algebra representation of events as linear transformations of effects. Based on a very general notion of dynamical independence, it is easy to see that all such probabilistic theories are "non-signaling without interaction" ("non-signaling" for short)--another requirement for a fair operational framework. Postulate PFAITH then implies the "local observability principle", the tensor-product structure for the linear spaces of states and effects, the impossibility of bit commitment and additional features, such an operational definition of transpose, a scalar product for effects, weak-selfduality of the theory, and more. Dual to Postulate PFAITH an analogous postulate for effects would give additional quantum features, such as teleportation. However, all possible consequences of these postulates still need to be investigated, and it is not clear yet if we can derive QM from the present postulates only. [CONTINUES on manuscript]

Zhaoqi Wu, Chuanxi Zhu, Xiaozhi Zhang

Abstract The incompatibility of dynamics postulate (unitary evolution) and the measurement postulate (wave-packet collapse) of quantum mechanics has recently been solved by Zurek from an information transfer perspective. Luo gave his derivation by relaxing the repeatability postulate. In this paper, we reconsider Luo’s derivation in the setting of general probabilistic theory (GPT). We also introduce the concept of sub- and super-fidelity in GPT and discuss their properties.

Terrence L. Fine