Quantum Fourier transform and tomographic Rényi entropic inequalities

Abstract

We show that the Rényi entropy associated with spin tomograms of quantum states satisfies new inequalities that depend on the quantum Fourier transform. We obtain the limiting inequality for the von Neumann entropy of quantum spin states and a new kind of entropy associated with the quantum Fourier transform. We consider possible connections with subadditivity and strong subadditivity conditions for tomographic entropies and von Neumann entropies.

Probability representation of spin states and inequalities for unitary matrices

“…The problem of the entanglement properties of a two-spin quantum state is thus mapped to the problem of the properties of elements of the unitary matrix uu 0 and related to the orthostochastic matrix |uu 0 | 2 . The separability of quantum state (12) reduces to finding joint probability distribution (13), which can be represented in the form of a convex sum of uncorrelated probability distributions for each subsystem [22],…”

“…The entropic inequalities for tomograms of spin states were considered in [13]. Here, we obtain some inequalities for unitary matrices associated with these entropic inequalities.…”

“…The development of the probability representation of quantum states in this aspect was presented in [12], [13].…”

“…Some well-known inequalities for Shannon and Rényi entropies in probability theory can be used in the tomographic approach to study new properties of quantum states [13], [22]. On the other hand, in the theory of entanglement, there are aspects that need a better elucidation, for example, an explication of the role of eigenvectors and eigenvalues of the density matrix of a composite system, which might elucidate the difference between the separable and the entangled states.…”

Probability Description and Entropy of Classical and Quantum Systems

“…Using expressions of tomograms in terms of unitary matrices, some inequalities for unitary matrix can be obtained [39].…”