Rescaling transformations and the Grothendieck bound formalism in a single quantum system

The Grothedieck bound formalism is studied using `rescaling transformations', in the context of a single quantum system. The rescaling transformations enlarge the set of unitary transformations (which apply to isolated systems), with transformations that change not only the phase but also the absolute value of the wavefunction, and can be linked to irreversible phenomena (e.g., quantum tunnelling, damping and amplification, etc). A special case of rescaling transformations are the dequantisation transformations, which map a Hilbert space formalism into a formalism of scalars. The Grothendieck formalism considers a `classical' quadratic form ${\cal C}(\theta)$ which takes values less than $1$, and the corresponding `quantum' quadratic form ${\cal Q}(\theta)$ which takes values greater than $1$, up to the complex Grothendieck constant $k_G$. It is shown that ${\cal Q}(\theta)$ can be expressed as the trace of the product of $\theta$ with two rescaling matrices, and ${\cal C}(\theta)$ can be expressed as the trace of the product of $\theta$ with two dequantisation matrices. Values of ${\cal Q}(\theta)$ in the `ultra-quantum' region $(1,k_G)$ are very important, because this region is classically forbidden (${\cal C}(\theta)$ cannot take values in it). An example with ${\cal Q}(\theta)\in (1,k_G)$ is given, which is related to phenomena where classically isolated by high potentials regions of space, communicate through quantum tunnelling. Other examples show that `ultra-quantumness' according to the Grothendieck formalism (${\cal Q}(\theta)\in (1,k_G)$), is different from quantumness according to other criteria (like quantum interference or the uncertainty principle).