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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
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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.

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Maurice A. de Gosson. 2012-05-23. The Unique Pure Gaussian State Determined by the Partial Saturation of the Uncertainty Relations of a Mixed Gaussian State. https://arxiv.org/abs/1205.5222

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