Modal Quantum Theory

Abstract

The remarkable features of quantum theory are best appreciated by comparing the theory to other possible theories-what Spekkens calls "foil" theories [1]. The most celebrated example of this approach was Bell's analysis [2], which showed that entangled quantum systems have statistical properties unlike any hypothetical local hidden variable model. More recently, there have been several efforts to give quantum theory an operational axiomatic foundation [3,4,5]. In these efforts, a general abstract framework is posited to describe system preparations, choices of measurement, observed results of measurement, and probabilities. Many possible theories can be expressed in the framework. The axioms embody fundamental aspects of quantum theory that uniquely identify it among them. A striking lesson of this work is that familiar quantum theory can be characterized by axioms that seem to have little to do with the traditional quantum machinery of states and observables in Hilbert space. The Hilbert space structure is "derived" from the operational axioms.These approaches are based on two distinct concepts of generalization. First, we generalize within quantum theory to give the theory its most general form. For example, we generalize state vectors to density operators as a

Thermodynamical cost of some interpretations of quantum theory

“…Toy theories with this property have been shown to retain fundamental features of quantum theory [34,35]. This solution would imply that quantum theory is only a continuous idealization of a deeper discrete theory.…”

Spin and rotations in Galois field quantum mechanics

“…The key variation introduced in [5] is the replacement of the Hilbert space of an N -level quantum system, H C = C N , with a discrete vector space over a finite field: [7,10]. Here, Z q denotes the Galois field GF (q), where q = p n with p prime and n ∈ N. For the n = 1 case, GF (p) is simply Z p = Z/pZ.…”

“…The six basis transformations in S 3 can be mapped onto the six rotations of the dihedral group D 3 which keep the equilateral triangle abc in figure 1 invariant. ‡ Schumacher and Westmoreland call it an effect in [7] Thus, our 'spins' transform in an analogous way to canonical spin under 'rotations.' However, a significant difference also exists.…”

Absolute Quantum Theory (after Chang, Lewis, Minic and Takeuchi), and a Road to Quantum Deletion

“…In [9], the authors have shown that similarly, one cannot clone an unknown state in Modal Quantum Theory over prime fields. In [14], we obtained the general result that one cannot clone an unknown state in a General Quantum Theory over any division ring.…”

“…In many papers, alternative Quantum Theories have been proposed for classical Quantum Theory (in complex Hilbert spaces, following the København interpretation). For instance, there are a number of papers on "Modal Quantum Theories" (MQTs), which consider similar theories over finite fields (see e. g. [9,7]). Whether the motivation is that these simply serve as toy models for the classical theory, or that they maybe come closer to physical reality, is arguable.…”