We investigate the quantum interaction between two quantum sys- tems in terms of
C∗ -algebras.We introduce the
co rrelation CP-map between the bipartite systems and study the operational and statistical structure of the quantum interaction.We then generalize the informational quantities into this quantum setting, and introduce a new entropy of CP-maps, which vanishes at extreme CP-maps in CP-convexity.Introduction.We prepare two quantum states described by density operators
ρ and
σ on Hilbert spaces
H1 and
K1 respectively, and so initially we have the compound state
ω0=ρ⊗σ on the tensor product Hilbert space
H1⊗K1 .Suppose that there exists an interaction between the two subsystems for some time interval, and that the systems are in equilibrium with the exterior for this time period, then the compound state is changed to a state
ω described by
ω=U(ρ⊗σ)U∗ where
U is a unitary operator representing the quantum evolution of the systems.After the interaction, the states of the subsystems on
H2 and
K2 are described by
TrK2w and
RH2ω respectively.Then the channels
φ∗ from
T(H1) to
T(H2) , and
ϕ∗ from
T(K1) to
T(K2) , are defined by
φ∗(ρ)=hK2ω=rbK2U(ρ⊗σ)U∗ and
ϕ∗(σ)=bH2ω=hH2U(ρ⊗σ)U∗ ,The detailed version of this paper will be submitted for publication elsewhere.Typaeet by
AA#?fflQuantum permutations, or magic unitaries, have in recent years been explored in the context of identifying `genuinely quantum' isometries of graphs. Here, we import this tool in physics, showing that quantum permutations yield a generalisation of quantum reference frames in a discrete setting that is reminiscent of the passage from special to general relativity. We show that the typical quantum reference frames framework corresponds to quantum permutations classified as `classical' in the mathematical literature, and which we demonstrate are quantum controlled transformations (superpositions of classical coordinate maps). Genuinely quantum permutations (i) allow to construct \emph{non--commuting} quantum reference frames (ii) correspond to \emph{local}, as opposed to global, superpositions of transformations. Strikingly, we find that the non-commutativity of quantum fields, when used as reference systems, is exactly what implies that the change of frame is achieved through a genuine quantum permutation. We illustrate the above with several examples in both first and second quantization formalism, which demonstrate (a) simultaneous control on non--commuting variables, (b) the existence of bipartite states that can be localized with a genuine quantum permutation and cannot be localized with the usual quantum reference frame transformations (without introducing additional degrees of freedom), (c) extension of the Ising model symmetries to genuinely quantum permutations, and (d) extension of the symmetries of a scalar field action on curved spacetime to genuinely quantum permutations. While we have in mind applications in quantum gravity, we expect our formalism to be of interest in a wide range of topics in quantum information.