Entropy in Thermodynamics and Information Theory - Quantum Theory

Quantum Theory

See also: Holographic principle#Energy, matter, and information equivalence See also: Quantum relative entropy

Hirschman showed, cf. Hirschman uncertainty, that Heisenberg's uncertainty principle can be expressed as a particular lower bound on the sum of the classical distribution entropies of the quantum observable probability distributions of a quantum mechanical state, the square of the wave-function, in coordinate, and also momentum space, when expressed in Planck units. The resulting inequalities provide a tighter bound on the uncertainty relations of Heisenberg.

One could speak of the "joint entropy" of the position and momentum distributions in this quantity by considering them independent, but since they are not jointly observable, they cannot be considered as a joint distribution. Note that this entropy is not the accepted entropy of a quantum system, the Von Neumann entropy, −Tr ρ lnρ = −⟨lnρ⟩. In phase-space, the Von Neumann entropy can nevertheless be represented equivalently to Hilbert space, even though positions and momenta are quantum conjugate variables; and thus leads to a properly bounded entropy distinctly different (more detailed) than Hirschman's; this one accounts for the full information content of a mixture of quantum states.

(Dissatisfaction with the Von Neumann entropy from quantum information points of view has been expressed by Stotland, Pomeransky, Bachmat and Cohen, who have introduced a yet different definition of entropy that reflects the inherent uncertainty of quantum mechanical states. This definition allows distinction between the minimum uncertainty entropy of pure states, and the excess statistical entropy of mixtures.)