Quantum Theory: Informational Foundations and Foils by Giulio Chiribella & Robert W. Spekkens

Author:Giulio Chiribella & Robert W. Spekkens
Language: eng
Format: epub
Publisher: Springer Netherlands, Dordrecht

In the limit of large J (or large N) the spin coherent states acquire the properties of “classical” states. The probability of obtaining outcome m of is given by the binomial distribution . In the limit it reduces to the normal distribution:

(4)

where is the width of distribution and is the mean value. The overlap between two spin-coherent states

(5)

becomes exponentially small in the limit of large N.

The uncertainty of measuring is given by the standard deviation . Under the restriction of coarse-grained measurements where the outcomes are merged into “slots” of size much larger than the standard deviation, the Gaussian cannot be distinguished anymore from the delta function [41] and the spin-coherent states become effectively “classical vectors” in three-dimensional space.

There are two independent ways in which large spin-coherent states can be said to induce the properties of the physical space. Firstly, they can be used to define the “reference direction” in a three-dimensional space, though one lacks this notion in the abstract Hilbert space formulation of quantum theory to start with. With no external reference frame only rotationally invariant observables can be measured, such as the total spin length. Consider a “large” spin of length J in a spin-coherent state and a “small” spin of length 1/2. It can be shown that the probability distribution for the outcomes (“aligned”) and (“anti-aligned”) of the total spin length approaches the probability distribution for the outcomes of spin projection of the spin-1/2 along the direction in the classical limit () [42, 45]. In that way, the spin-coherent states define the complete set of measurements for the elementary spin. The set has the same dimensionality and the symmetry as the three-dimensional Euclidian physical space. We call these static properties of the space.

Secondly, spin-coherent states can generate non-trivial dynamics in three-dimensional space. A macroscopic spin in a coherent state can serve as an “external magnetic field” around which another spin can precess, i.e. it serves as a transformation device for the elementary spins [45]. Since there is no preferred direction beside the one defined by the large spin one requires the interaction between the elementary spin and the large spin to be rotationally invariant. To illustrate it consider the situation as given in Fig. 2 (left). A single spin-1/2 particle interacts with N spins prepared in a coherent state along direction . Total interaction Hamiltonian is the sum of all pairwise interactions where labels the interaction between the single spin and nth spin of the macroscopic system. There is only one rotationally invariant Hamiltonian, that is the Heisenberg spin-spin interaction , where is the coupling constant. It can be shown that in the macroscopic limit the elementary spin only negligibly affects the state of a large spin and the dynamics of the elementary spin becomes unitary [45]:

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