Uncertainty principles for the Whittaker Wigner transform

Lai Tien Minh

Amit Samanta, Santanu Sarkar

We prove an uncertainty type principle for finite linear combinations of Fourier-Wigner transforms.This is equivalent to proving a finite rank theorem for the Weyl transform.

Jean-Pierre Antoine, Romain Murenzi, Pierre Vandergheynst +1 more

M. Casas, H. Krivine, Joan Martorell

Nuno. C.Dias, M. de Grosson, João. N.Prata

We propose a refinement of the Robertson–Schrödinger uncertainty principle (RSUP) using Wigner distributions. This new principle is stronger than the RSUP. In particular, and unlike the RSUP, which can be saturated by many phase space functions, the refined RSUP can be saturated by pure Gaussian Wigner functions only. Moreover, the new principle is technically as simple as the standard RSUP. In addition, it makes a direct connection with modern harmonic analysis, since it involves the Wigner transform and its symplectic Fourier transform, which is the radar ambiguity function. As a by-product of the refined RSUP, we derive inequalities involving the entropy and the covariance matrix of Wigner distributions. These inequalities refine the Shanon and the Hirschman inequalities for the Wigner distribution of a mixed quantum state . We prove sharp estimates which critically depend on the purity of and which are saturated in the Gaussian case.

Nuno Costa Dias, Maurice A. de Gosson, João Nuno Prata

We propose a refinement of the Robertson–Schrödinger uncertainty principle (RSUP) using Wigner distributions. This new principle is stronger than the RSUP. In particular, and unlike the RSUP, which can be saturated by many phase space functions, the refined RSUP can be saturated by pure Gaussian Wigner functions only. Moreover, the new principle is technically as simple as the standard RSUP. In addition, it makes a direct connection with modern harmonic analysis, since it involves the Wigner transform and its symplectic Fourier transform, which is the radar ambiguity function. As a by-product of the refined RSUP, we derive inequalities involving the entropy and the covariance matrix of Wigner distributions. These inequalities refine the Shanon and the Hirschman inequalities for the Wigner distribution of a mixed quantum state $$\rho $$ . We prove sharp estimates which critically depend on the purity of $$\rho $$ and which are saturated in the Gaussian case.