The Grothendieck formalism applied to physical examples in a single quantum system

Abstract The Grothendieck formalism considers a classical quadratic form C ( θ ), and a quantum quadratic form Q ( θ ). It shows that if C ( θ ) ≤ 1, the Q ( θ ) might take values greater than 1, up to the complex Grothendieck constant k G . The region (1 , k G ) is ‘ultra-quantum’ in the sense that Q can take values in it, but C cannot take values in it. The mathematical formalism is applied to various physical examples. It is shown that many physically interesting density matrices give Q ( θ ) ≤ 1. In some examples Q is the Wigner and Weyl functions, and in that case also Q ( θ ) ≤ 1. In many examples Fourier dual density matrices both give Q ( θ ) ≤ 1, and in this sense Fourier duality does not play an important role within the Grothendieck formalism. Therefore quantumness according to Grothendieck ( Q ε (1 , k G )) is different from quantumness according to other criteria.