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arXiv:1712.09475 · Journal of Fourier Analysis and Applications

A Refinement of the Robertson–Schrödinger Uncertainty Principle and a Hirschman–Shannon Inequality for Wigner Distributions

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

Published Feb 22, 2018• 22 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1007/s00041-018-9602-x

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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 ρ\rho ρ . We prove sharp estimates which critically depend on the purity of ρ\rho ρ and which are saturated in the Gaussian case.
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.
Let ρ the density matrix of a mixed Gaussian state. Assuming that one of the Robertson–Schrödinger uncertainty inequalities is saturated by ρ, e.g. , we show that there exists a unique pure Gaussian state whose Wigner distribution is dominated by that of ρ and having the same variances and covariance Δ ρ X 1 , Δ ρ P 1 and Δ ρ ( X 1 , P 1 ) as ρ. This property can be viewed as an analytic version of Gromov's non-squeezing theorem in the linear case, which implies that the intersection of a symplectic ball by a single plane of conjugate coordinates determines the radius of this ball. We conclude by giving a short geometric proof of the fact that pure Gaussian states are the only quantum states saturating the Robertson–Schrödinger uncertainty inequalities.
Let ρ the density matrix of a mixed Gaussian state. Assuming that one of the Robertson--Schrödinger uncertainty inequalities is saturated by ρ, e.g. (Δ^{ρ}X_1)^2(Δ^{ρ}P_1)^2=Δ^{ρ}(X_1,P_1)^2+(1/4)\hbar^2, we show that there exists a unique pure Gaussian state whose Wigner distribution is dominated by that of ρ and having the same variances and covariance Δ^{ρ}X_1,Δ^{ρ}P_1, and Δ^{ρ}(X_1,P_1) as ρ. This property can be viewed as an analytic version of Gromov's non-squeezing theorem in the linear case, which implies that the intersection of a symplectic ball by a single plane of conjugate coordinates determines the radius of this ball.
Letthe density matrix of a mixed Gaussian state. Assuming that one of the Robertson-Schrodinger uncertainty inequalities is sat- urated by �, e.g. (� � X1) 2 (� � P1) 2 = � � (X1,P1) 2 + 1 ~ 2 , we show that there exists a unique pure Gaussian state whose Wigner distribution is dominated by that ofand having the same variances and covariance � � X1,� � P1, and � � (X1,P1) as �. This property can be viewed as an analytic version of Gromov's non-squeezing theorem in the linear case, which implies that the intersection of a symplectic ball by a single plane of conjugate coordinates determines the radius of this ball.
We report a refinement of Robertson-Schroedinger uncertainty relation via Wigner-Yanase skew information. Besides the well known quantum uncertainty arising from the noncommutativity of observables, there is classical uncertainty arising from the mixedness of the states that is quantified by the difference between the variance and the skew information. Our refined uncertainty relation for canonical observables is saturated by all the Gaussian states, pure or mixed, and thus provides an alternative measure for the non-Gaussianity of quantum states. Generalizations to the case of metric adjusted skew information are presented, unifying and refining most of previous results.
We extend an uncertainty principle due to Beurling into a characterization of Hermite functions. More precisely, all functions f on \mathbb{R}^d which may be written as P(x)\exp (-\langle Ax, x\rangle) , with A a real symmetric definite positive matrix, are characterized by integrability conditions on the product f(x) \widehat{f}(y) . We then obtain similar results for the windowed Fourier transform (also known, up to elementary changes of functions, as the radar ambiguity function or the Wigner transform). We complete the paper with a sharp version of Heisenberg's inequality for this transform.