The superposition principle is a founding principle of quantum mechanics; we have already made use of it in interpreting the Young's slit experiment. Quantum mechanics is a linear theory, and so it is natural that vector spaces play an important role in it. We shall see that a physical state is represented mathematically by a vector in a space whose characteristics we shall define; this is called the space of states . A second founding principle, which can also be deduced from the Young's slit experiment, is the existence of probability amplitudes. These probability amplitudes will be represented mathematically by scalar products defined on the space of states. In the theory of waves, the use of complex numbers is just a convenience, but in quantum mechanics the probability amplitudes are fundamentally complex numbers – the scalar product will a priori be a complex number. Physical properties like momentum, position, energy, and so on will be represented by operators acting in the space of states. In this chapter we shall introduce the essential properties of Hilbert spaces, that is, vector spaces on which a positive-definite scalar product is defined, and we shall limit ourselves to the case of finite dimension. This restriction will be lifted later on, because the space of states is in general of infinite dimension. The mathematical theory of Hilbert spaces of infinite dimension is much more complicated than that of spaces of finite dimension, and we shall put off studying them until Chapter 7. The reader familiar with vector spaces of finite dimension and operators in such spaces can proceed directly to Chapter 3 after reviewing the notation.
We study the soliton modes carrying fractional quantum numbers in one-dimensional fermionic systems. In particular, we consider the solitons in the one-dimensional fermionic superfluids. For the s -wave order parameter with phase twisted by an angle φ , the complex Z 2 soliton may occur carrying a localized fractional quantum number φ /(2 π ). If the system is finite with length L , we show the existence of a uniform background − φ /(2 πL ) which, though vanishing in the thermodynamic limit, is essential to maintain the conservation of the total integer quantum number. This analysis is also applicable to other systems with fractional quantum numbers, thus providing a mechanism to understand the compatibility of the emergence of fractional quantum number in the thermodynamic limit of a finite system with only integer quantum numbers. For the p -wave pairing case, the Majorana zero mode may occur associated with a real Z 2 soliton, and the fractionalized quantum number is the dimension of the single particle Hilbert space, which turns out to be 1/2. Again for a finite system with length L , there is an accompanying uniform dimension density −1/(2 L ) to maintain the dimension of the Hilbert space invariant. We conjecture a connection of the dimension density of one-dimensional solitons with the quantum dimension of topological excitations.
A key goal in the foundations of quantum mechanics is to identify operational constraints characterizing physical theories. Bell inequalities do so for classical probability theory, while Tsirelson's bound provides a first step for quantum mechanics. Here, we address the dual question: Can one certify that all correlations predicted by quantum theory are actually realizable? In this work, we construct a finite number of two-body correlations such that the only probabilistic theory that (1) realizes them, and (2) does so in a way that is stable under iterated teleportation, is quantum theory. For
n quantum systems of dimension
d each, Condition (1) can be verified using
poly(d,n) measurement settings. Condition (2) may be understood as a hierarchy of tests, one for each number
N of teleportation steps. There is thus a sense in which finite-dimensional quantum theory can be self-tested. In particular, one can certify the existence of Bell inequality violations larger than any that have been directly observed.