[Paper Review] Extending Hudson's theorem to mixed quantum states

Aikaterini Mandilara, Evgueni Karpov+1 more|Nazarbayev University Repository (Nazarbayev University)|Aug 18, 2008

Quantum Information and Cryptography1 references3 citations

TL;DR

This paper extends Hudson's theorem—originally stating that only Gaussian states have positive Wigner functions in pure states—to mixed quantum states by deriving analytical upper and lower bounds on non-Gaussianity. Using Lagrange multipliers and the Cauchy-Schwarz inequality, the authors bound the non-Gaussianity of mixed states with positive Wigner functions in terms of their purity and the purity of the corresponding Gaussian state, revealing that non-Gaussianity increases with decreasing purity, though the upper bound is unphysical due to non-physical extremal solutions.

ABSTRACT

According to Hudson's theorem, any pure quantum state with a positive Wigner function is necessarily a Gaussian state. Here, we make a step towards the extension of this theorem to mixed quantum states by finding upper and lower bounds on the degree of non-Gaussianity of states with positive Wigner functions. The bounds are expressed in the form of parametric functions relating the degree of non-Gaussianity of a state, its purity, and the purity of the Gaussian state characterized by the same covariance matrix. Although our bounds are not tight, they permit us to visualize the set of states with positive Wigner functions.

Motivation & Objective

• To extend Hudson’s theorem—valid for pure states—to mixed quantum states with positive Wigner functions.
• To characterize the set of mixed states with positive Wigner functions by quantifying their non-Gaussianity relative to a reference Gaussian state with the same covariance matrix.
• To derive analytically computable bounds on non-Gaussianity that depend only on the purity of the state and its Gaussian counterpart.
• To explore the physicality and tightness of these bounds, particularly the upper bound derived via extremization.
• To provide a framework for understanding de-Gaussification in continuous-variable quantum information, especially for non-Gaussian state engineering.

Proposed method

• Formulate the trace overlap Tr(ρρG) using the Wigner function integral representation: Tr(ρρ′) = 2π∫∫ Wρ(x,p)Wρ′(x,p) dx dp.
• Apply symplectic transformations to reduce the Gaussian state to a thermal state, preserving purity, Wigner function positivity, and trace overlap.
• Use the method of Lagrange multipliers to extremize Tr(ρρG) under the constraint of fixed second moments (covariance matrix) and positivity of Wρ.
• Derive an upper bound on non-Gaussianity δ[ρ,ρG] by solving the extremization problem, yielding a parametric upper bound δ^u.ult(μ[ρG]).
• Apply the Cauchy-Schwarz inequality to the Wigner representation to obtain a lower bound on δ[ρ,ρG], yielding δ^l.ult via Tr(ρρG) ≤ √(μ[ρG]μ[ρ]).
• Assess physicality of the upper bound by testing overlap with Fock states; conclude that the extremal solution Wρ^ex is unphysical for finite μ[ρG] > 0.

Experimental results

Research questions

• RQ1 Can Hudson’s theorem on positive Wigner functions for pure states be extended to mixed quantum states?
• RQ2 What are the bounds on non-Gaussianity for mixed states with positive Wigner functions, given their purity and that of the corresponding Gaussian state?
• RQ3 Is the upper bound on non-Gaussianity derived via extremization physically realizable?
• RQ4 How do the purity of the state and its Gaussian reference relate to the maximum allowable non-Gaussianity?
• RQ5 What is the relationship between the positivity of Wigner functions and the Heisenberg uncertainty principle in mixed states?

Key findings

• The ultimate upper bound on non-Gaussianity δ^u.ult depends only on the purity of the Gaussian reference state μ[ρG], making it experimentally accessible.
• The upper bound δ^u.ult is unphysical, as the extremal Wigner function Wρ^ex fails to have positive overlap with all Fock states for finite μ[ρG] > 0.
• A lower bound δ^l.ult on non-Gaussianity is derived using the Cauchy-Schwarz inequality, valid for all states with positive or negative Wigner functions.
• The region of possible non-Gaussianity is confined between δ^u.ult and δ^l.ult in a 3D space of μ[ρ], μ[ρG], and δ[ρ,ρG].
• The bound μ[ρ] = (8/9)μ[ρG] is derived as a lower limit on the purity of a mixed state given its Gaussian reference, consistent with known uncertainty relations.
• The results confirm a deep link between Hudson’s theorem and the Heisenberg uncertainty principle, as the bound μ[ρ] = (8/9)μ[ρG] is asymptotically tight and realized by positive Wigner functions.