David W. Cohen

Reduction of Theoretical Uncertainty in Quantum Computing

A theoretical framework is developed to evaluatethe amount of intrinsic uncertainty, as distinguishedfrom operational uncertainty (noise), inherent inquantum computation. The temporal evolution of states in quantum computing is analyzeddiagramatically, providing a visual tool for therefining of quantum algorithms to help achieve minimaluncertainty and maximal efficiency, as well as forbetter understanding of the quantum entanglements crucial to quantumcomputing.

Minimal supports in quantum logics and Hilbert space

It is shown that if a fully atomic, complete orthomodular lattice satisfies a “minimal support condition” (m.s.c.), then it satisfies Pirons axioms, and is thereby shown to be the projection lattice of a generalized Hilbert space. It is shown, conversely, that m.s.c. holds in Hilbert space subspace lattices. The physical justification for m.s.c. is provided in the context of a property lattice ℒ(A, ∑) for a realistic entity (A, ∑) in the sense of Foulis-Piron-Randall. This context provides a clear focus on key issues in the debate over the appropriateness of requiring quantum logics to be represented over Hilbert spaces.

Ultimate Stochastic Entities

We take the view that everything that is known about a physical system can be described by a “stochastic entity” (Å, Δ), which consists of a “manual” Å of experiments that can be performed on the system, and a set Δ of possible stochastic states (probability measures) on the logic of the manual. We next consider what happens when new information about the system is learned and describe precisely how one then obtains a new stochastic entity more elaborate than the first. Finally, we show that as information about the system continues to grow, the increasingly elaborate stochastic entities describing the system necessarily acquire mathematical properties often assumed for mathematical convenience in papers on quantum mechanics.

Experiments, Measure and Integration

We learn about our physical universe by doing experiments. That is, first we do something such as flip a coin, or touch a hot stove, or measure how long it takes a marble to drop from a certain height. Then we record what happens after we do it—the coin comes up heads, we get burned, the marble takes 6 seconds to drop. What we record is called an outcome of the experiment. We identify an experiment by its outcome set, so we can write C = (heads, tails) to denote the coin flip experiment.

State Space and Gleason’s Theorem

We begin this chapter by defining the state space of a general logic. We examine the geometric structure of the state space and use it to define the notions of pure states, mixtures of states, and physical properties. Next we define an observable on a logic, allowing us to consider physical experiments whose outcome sets are more general than the finite subsets of ℜ we saw in Chapter 1. We shall be guided by Lemma 1B.10, however, when we define the expected value of an observable as the integral of the identity function on ℜ with respect to a measure determined by a state on the logic.

Hilbert Space Basics

We assume that you know the definition of a vector space V = (V, +, ·) over the field ℭ of complex numbers. Perhaps you recall defining an “inner product” between vectors in a vector space and using the inner product to consider angles between vectors. A ‘Hilbert space’ is a vector space with an inner product. We begin by defining an inner product space.

The Hilbert Space Model for Quantum Mechanics and the EPR Dilemma

Throughout this book we have used the phrase “quantum physics” to mean the collection of all the philosophical viewpoints and all the alternative ways of dealing with a physical universe behaving in contradiction to classical Newtonian physics, whereas “quantum mechanics” means a specific set of working hypotheses about the physical universe. Up to now we have been exploring quantum physics in general. Part A of this chapter is a brief history of quantum mechanics. In Part B we outline a set of working hypotheses for quantum mechanics. In Part C we discuss a philosophical dilemma arising from these hypotheses and articulated in a famous paper written in 1935 by Einstein, Podolsky, and Rosen.

The Logic of Nonclassical Physics

In this chapter we introduce a mathematical formulation for the foundations of quantum physics.

Subspaces in Hilbert Space

In Chapter 3 we used two-dimensional Hilbert space to model a manual for measuring electron spin, because every experiment had only two outcomes: spin-up and spin-down. Of course, many physical experiments have outcome sets that are not finite. A measurement of energy, position, or momentum of a moving particle usually means obtaining an outcome from an infinite number of possibilities. In this chapter we shall learn about infinite dimensional Hilbert spaces. In particular, we shall consider the subspace structure of infinite dimensional Hilbert spaces. These provide the logics for orthodox quantum physics.

Maps on Hilbert Spaces

As with all mathematical structures, it is important to know what kinds of functions from one Hilbert space to another preserve the important properties of the structure. Besides mathematical interest, the functions on Hilbert space that we study in this chapter provide a crucial link to the notion of an “observable physical quantity” in the Hilbert space formulation of quantum mechanics.