[Paper Review] Quantum Information Theory and Free Semialgebraic Geometry: One Wonderland Through Two Looking Glasses

Gemma De las Cuevas, Tim Netzer|arXiv (Cornell University)|Feb 8, 2021

Quantum Mechanics and Applications49 references4 citations

TL;DR

This paper establishes a deep connection between quantum information theory and free semialgebraic geometry by showing how both fields study noncommutative positivity and convexity in tensor product spaces from complementary perspectives. It demonstrates that insights from one field—such as quantum correlations, separability, and tensor network positivity—can be translated into geometric characterizations in free semialgebraic geometry, and vice versa, yielding new computational and conceptual tools, including undecidability results for infinite-size translationally invariant systems.

ABSTRACT

We illustrate how quantum information theory and free (i.e. noncommutative) semialgebraic geometry often study similar objects from different perspectives. We give examples in the context of positivity and separability, quantum magic squares, quantum correlations in non-local games, and positivity in tensor networks, and we show the benefits of combining the two perspectives. This paper is an invitation to consider the intersection of the two fields, and should be accessible for researchers from either field.

Motivation & Objective

• To bridge quantum information theory and free semialgebraic geometry by identifying shared mathematical structures in noncommutative positivity and convexity.
• To demonstrate that quantum information problems—such as separability, quantum correlations, and tensor network positivity—can be reframed as geometric questions in free semialgebraic sets.
• To show that geometric tools from free semialgebraic geometry can yield new insights into quantum information problems, including undecidability results for infinite systems.
• To promote cross-disciplinary collaboration by making the connections accessible to researchers in both fields.

Proposed method

• Using the framework of free (noncommutative) semialgebraic geometry, the paper studies positivity and convexity in tensor product spaces as geometric objects.
• It compares quantum information concepts—like positive semidefinite matrices, POVMs, and quantum magic squares—with free semialgebraic constructs such as free convex hulls and operator systems.
• The paper applies moment-based approximations via sums of squares to bound the distance of a matrix to the positive semidefinite cone using only low-order traces.
• It reduces problems in quantum tensor networks to noncommutative positivity questions, using the matrix mortality problem to prove undecidability of positivity for translationally invariant systems.
• It formulates quantum correlations in non-local games as membership in the free convex hull of a free independence model.
• It uses dilation theorems and invariant decompositions to analyze structure in tensor product spaces under group actions.

Experimental results

Research questions

• RQ1 How do quantum information theory and free semialgebraic geometry approach the same mathematical objects—such as positive semidefinite matrices in tensor products—differently?
• RQ2 Can geometric characterizations from free semialgebraic geometry improve the analysis of quantum correlations in non-local games?
• RQ3 What is the role of noncommutativity in determining the decidability of positivity in tensor networks, especially in translationally invariant settings?
• RQ4 How can moment-based approximations using low-degree polynomials be used to estimate the distance of a matrix to the positive semidefinite cone?
• RQ5 To what extent can the free convex hull of PVMs recover the set of POVMs, and what does this imply for quantum measurement theory?

Key findings

• The problem of determining whether the translationally invariant tensor network state τₙ(ρ) is positive semidefinite for all n ∈ ℕ is undecidable when d, r ≥ 7.
• The set of quantum correlations realizable in non-local games corresponds to membership in the free convex hull of the free independence model, providing a geometric characterization.
• For fixed n, the distance of a tensor network state ρ to the positive semidefinite cone can be bounded using low-order moments tr(ρᵏ) and polynomial approximations of the positive part function f(x) = max(x,0).
• Sums of squares approximations of f(x) enable efficient computation of bounds via semidefinite programming, even when full diagonalization is infeasible.
• The dilation of any POVM to a PVM corresponds to the free convex hull of the set of PVMs being equal to the set of all POVMs, a result that holds in the noncommutative setting.
• The framework of invariant decompositions in tensor product spaces applies beyond quantum systems, including to multivariate symmetric polynomials with positivity constraints.