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arXiv:1205.5222

The Unique Pure Gaussian State Determined by the Partial Saturation of the Uncertainty Relations of a Mixed Gaussian State

Maurice A. de Gosson

Published May 23, 2012• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.1205.5222

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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.
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.
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.
In order that a bivariate normal probability distribution in phase space with variances ${\mathrm{\ensuremath{\sigma}}}_{\mathit{q}}$,${\mathrm{\ensuremath{\sigma}}}_{\mathit{p}}$ and covariance ${\mathrm{\ensuremath{\sigma}}}_{\mathit{q},}$p may correspond to a Wigner distribution of a pure or a mixed state, it is necessary and sufficient that Heisenberg's uncertainty relation in Schr\"odinger form ${\mathrm{\ensuremath{\sigma}}}_{\mathit{q}}$${\mathrm{\ensuremath{\sigma}}}_{\mathit{p}}$-${\mathrm{\ensuremath{\sigma}}}_{\mathit{q},}$p2{\mathit{p}}^{2}p2\ensuremath{\ge}${\mathrm{\ensuremath{\Elzxh}}}^{2}$/4 should be satisfied. The diagonalization of the corresponding density matrix entails a correspondence between the statistical and the physical properties of temperature-dependent oscillator states; the expansion of the density matrix into coherent states allows a physical interpretation in phase space.
Abstract We show that a proper expression of the uncertainty relation for a pair of canonically-conjugate continuous variables relies on entropy power, a standard notion in Shannon information theory for real-valued signals. The resulting entropy-power uncertainty relation is equivalent to the entropic formulation of the uncertainty relation due to Bialynicki-Birula and Mycielski, but can be further extended to rotated variables. Hence, based on a reasonable assumption, we give a partial proof of a tighter form of the entropy-power uncertainty relation taking correlations into account and provide extensive numerical evidence of its validity. Interestingly, it implies the generalized (rotation-invariant) Schrödinger–Robertson uncertainty relation exactly as the original entropy-power uncertainty relation implies Heisenberg relation. It is saturated for all Gaussian pure states, in contrast with hitherto known entropic formulations of the uncertainty principle.