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arXiv:1709.07256 · Physica A Statistical Mechanics and its Applications

New entropic inequalities for qubit and unimodal Gaussian states

Julio A. López-Saldívar, Octavio Castaños, Margarita A. Man’ko, Vladimir I. Man’ko

Published Sep 23, 2017• 10 citations
arXivHTMLPDF
DOI
10.1016/j.physa.2017.09.027

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We introduce the average confidence width Δax=∫01Δcx(θ) dθΔ_{a} x=\int_0^1Δ_{c} x (θ)\,\mathrm{d}θΔa​x=∫01​Δc​x(θ)dθ: the confidence width Δcx(θ)Δ_{c} x(θ)Δc​x(θ)---the measure of the smallest region of position space carrying probability θθθ---averaged over all confidence levels. It is the first moment of the decreasing rearrangement of the position density, an L1L^1L1 (mean-absolute-deviation) measure of localization, and Δax ΔapΔ_{a} x\,Δ_{a} pΔa​xΔa​p is dilation invariant. We study the uncertainty relation Δax Δap≥c∗ℏΔ_{a} x\,Δ_{a} p\ge c^\ast\hbarΔa​xΔa​p≥c∗ℏ for pure and mixed states. A mean--entropy argument with the Bialynicki-Birula--Mycielski relation gives c∗≥π/ec^\ast\geπ/ec∗≥π/e, while the ground state ψ0ψ_0ψ0​ of the Fourier-invariant operator ∣x∣+∣p∣|x|+|p|∣x∣+∣p∣, with E0=1.1040744E_0=1.1040744E0​=1.1040744, gives c∗≤E02≈1.2190<4/πc^\ast\le E_0^2\approx1.2190<4/πc∗≤E02​≈1.2190<4/π: Gaussian states, which saturate the Heisenberg--Kennard and entropic relations, are not optimal here. We prove that ψ0ψ_0ψ0​ is nondegenerate, nodeless, even, self-dual, and symmetric decreasing, with algebraic tails, and that the sharp hybrid relations Δax⋅2⟨∣p−p0∣⟩≥E02ℏΔ_{a} x \cdot 2 \langle |p-p_0| \rangle \ge E_0^2\hbarΔa​x⋅2⟨∣p−p0​∣⟩≥E02​ℏ and 2⟨∣x−x0∣⟩⋅Δap≥E02ℏ2\langle|x-x_0| \rangle \cdotΔ_{a} p\ge E_0^2\hbar2⟨∣x−x0​∣⟩⋅Δa​p≥E02​ℏ hold for all states, so Δax Δap≥E02ℏΔ_{a} x\,Δ_{a} p\ge E_0^2\hbarΔa​xΔa​p≥E02​ℏ whenever either density is symmetric unimodal; the second variation of Δax ΔapΔ_{a} x\,Δ_{a} pΔa​xΔa​p at ψ0ψ_0ψ0​ is positive definite apart from the symmetry directions. Extensive numerical minimization, including coherent states, never finds a product below E02ℏE_0^2\hbarE02​ℏ and consistently returns ψ0ψ_0ψ0​, supporting the conjecture c∗=E02c^\ast=E_0^2c∗=E02​. In two dimensions the relation is sharp and saturated by Gaussians, identifying the dimension as the origin of the non-Gaussian optimizer in one dimension. Finally, we discuss other complementary pairs and the relation of the width to interferometric predictability and visibility.