I study the class of problems efficiently solvable by a quantum computer, given the ability to “postselect” on the outcomes of measurements. I prove that this class coincides with a classical complexity class called PP, or Probabilistic Polynomial-Time. Using this result, I show that several simple changes to the axioms of quantum mechanics would let us solve PP-complete problems efficiently. The result also implies, as an easy corollary, a celebrated theorem of Beigel, Reingold, and Spielman that PP is closed under intersection, as well as a generalization of that theorem due to Fortnow and Reingold. This illustrates that quantum computing can yield new and simpler proofs of major results about classical computation.

Can NP-complete problems be solved efficiently in the physical universe? I survey proposals including soap bubbles, protein folding, quantum computing, quantum advice, quantum adiabatic algorithms, quantum-mechanical nonlinearities, hidden variables, relativistic time dilation, analog computing, Malament-Hogarth spacetimes, quantum gravity, closed timelike curves, and “anthropic computing. ” The section on soap bubbles even includes some “experimental ” results. While I do not believe that any of the proposals will let us solve NP-complete problems efficiently, I argue that by studying them, we can learn something not only about computation but also about physics. 1

This recreational paper investigates what happens if we change quantum mechanics in several ways. The main results are as follows. First, if we replace the 2-norm by some other p-norm, then there are no nontrivial norm-preserving linear maps. Second, if we relax the demand that norm be preserved, we end up with a

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Quantum information science is a source of task-related axioms whose consequences can be explored in general settings encompassing quantum mechanics, classical theory, and more. Quantum states are compendia of probabilities for the outcomes of possible operations we may perform on a system: ‘‘operational states.’ ’ I discuss general frameworks for ‘‘operational theories’ ’ (sets of possible operational states of a system), in which convexity plays key role. The main technical content of the paper is in a theorem that any such theory naturally gives rise to a ‘‘weak effect algebra’ ’ when outcomes having the same probability in all states are identified and in the introduction of a notion of ‘‘operation algebra’ ’ that also takes account of sequential and conditional operations. Such frameworks are appropriate for investigating what things look like from an ‘‘inside view,’ ’ i.e., for describing perspectival information that one subsystem of the world can have about another. Understandinghow such views can combine, and whether an overall ‘‘geometric’ ’ picture (‘‘outside view’’) coordinating them all can be had, even if this picture is very different in structure from the perspectives within it, is the key to whether we may be able to achieve a unified, ‘‘objective’ ’ physical view in which quantum mechanics is the appropriate description for certain perspectives, or whether quantum mechanics is truly telling us we must go beyond this ‘‘geometric’ ’ conception of physics.

Recent experiments appear to have revealed the possibility of the existence of quantum entanglement between spatially separated human subjects. In addition, a similar condition might exist between basins containing human neurons adhering to printed circuit boards. In both instances, preliminary data indicates what appear to be non-local correlations between brain electrical activities in the case of the human subjects and also non-local correlations between neuronal basin electrical activities, implying entanglement at the macroscopic level. If the ongoing expanded research and the analysis of same continues to support this hypothesis, it may then make it possible to simultaneously address some of the fundamental problems facing us in both physics and biology through the adoption of an interdisciplinary empirical approach based on Bell’s experimental philosophy, with the goal of unifying these two fields.

A detailed list of postulates is formulated in an algebraic setting. These postulates are sufficient to entail the standard time evolution governed by the Schrodinger- or Dirac equation. They are also necessary in a strong sense: Dropping any one of the postulates allows for other types of time-evolution, as is demonstrated with examples. Some philosophical remarks shall hint on possible further investigations. 1 1 Introduction Time has several appearances in physics: A1: In general relativity it is itself an object of the theory. Unified with space it is in mutual interaction with matter. A2: For smaller systems it is an external parameter, used to describe the dynamics, the timeevolution. A3: There is a symmetry of all the physical laws, they are invariant under translation in time and, more generally, under transformations belonging to the Poincar'e or Galilei-group. A4: The basic laws are also invariant under time-reflection, if coupled with changes of charges and of parity. A...

urt, World Scientific, Singapora (2002). advances, that will demand a more straightforward connection between the theory and the type of manipulations and control to be executed in the laboratory. Although operational quantum mechanics is in full development, we must admit that the time has not yet come for it to function as a `better to apply and more easy to use' theory for experimentation. The reason is that the operational quantum structures that have been elaborated, while carefully but boldly aiming at physical clearness and transparency, stumble upon a lot of problems of purely technical mathematical nature. Quantum mechanics is not only conceptually a very di#cult theory, it entails also a very sophisticated mathematical apparatus. It becomes even more and more clear that both dimensions of di#culty, the conceptual one and the mathematical structural one, are linked in a profound way. It has been shown that some of the deep conceptual problems of quantum mechanics -- the so

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