[Paper Review] Two proofs of Størmer's theorem

Guillaume Aubrun, Stanisław J. Szarek|arXiv (Cornell University)|Dec 10, 2015

Quantum Information and Cryptography12 references3 citations

TL;DR

This paper presents two concise, novel proofs of Størmer's theorem, which characterizes positivity-preserving maps on 2×2 complex matrices as sums of completely positive and co-completely positive maps. The first proof uses Brouwer’s fixed point theorem to reduce the general case to the bistochastic case, while the second employs the S-lemma and Lorentz cone duality, offering a streamlined alternative to classical ad hoc computations. The key contribution is a conceptually simple and self-contained derivation of the theorem’s structural result.

ABSTRACT

The structure of the set of positivity-preserving maps between matrix algebras is notoriously difficult to describe. The notable exceptions are the results by Størmer and Woronowicz from 1960s and 1970s settling the low dimensional cases. By duality, these results are equivalent to the Peres-Horodecki positive partial transpose criterion being able to unambiguously establish whether a state in a 2 x 2 or 2 x 3 quantum system is entangled or separable. However, even in these low dimensional cases, the existing arguments (known to the authors) were based on long and seemingly ad hoc computations. We present a simple proof, based on Brouwer's fixed point theorem, for the 2 x 2 case (Størmer's theorem). For completeness, we also include another argument (following the classical outline, but highly streamlined) based on a characterization of extreme self-maps of the Lorentz cone and on a link - noticed by R. Hildebrand - to the S-lemma, a well-known fact from control theory and quadratic/semi-definite programming.

Motivation & Objective

• To provide a new, elementary proof of Størmer’s theorem on the structure of positivity-preserving maps on 2×2 Hermitian matrices.
• To replace long, ad hoc computations in existing proofs with a concise argument based on Brouwer’s fixed point theorem.
• To offer a second, streamlined proof using the S-lemma and duality with the Lorentz cone, linking quantum information theory to control theory.
• To demonstrate that every such map decomposes into at most four terms of the form $ A\rho A^\dagger $ and $ B\rho^T B^\dagger $, confirming the decomposition is finite and explicit.

Proposed method

• Use Brouwer’s fixed point theorem to show that interior positivity-preserving maps on $ \mathsf{M}_2^{sa} $ are conjugate to bistochastic maps via positive-definite operators.
• Leverage the Bloch ball representation of density matrices in $ \mathbb{C}^2 $, identifying bistochastic maps with linear operators on $ \mathbb{R}^3 $ mapping the unit ball into itself.
• Apply Carathéodory’s theorem to represent such operators as convex combinations of at most four orthogonal transformations, corresponding to unitary and transposition maps.
• Use duality to relate positivity-preserving maps to the cone of positive semi-definite matrices and exploit the structure of the Lorentz cone $ \mathcal{L}_n $.
• Apply the S-lemma in a reformulated form to characterize when two quadratic forms cover the entire space, enabling the proof of extremality and decomposition.
• Show that any extreme map in the cone of positivity-preserving maps must be an automorphism of the Lorentz cone, leading to the desired decomposition.

Experimental results

Research questions

• RQ1 Can Størmer’s theorem be proven without relying on lengthy, ad hoc computations?
• RQ2 Can Brouwer’s fixed point theorem be applied to simplify the proof of the structure of positivity-preserving maps in low-dimensional quantum systems?
• RQ3 Is there a direct link between the $ S $-lemma from control theory and the decomposition of positivity-preserving maps in $ \mathsf{M}_2^{sa} $?
• RQ4 What is the minimal number of terms required in the decomposition of a positivity-preserving map on $ \mathsf{M}_2^{sa} $?
• RQ5 How does the structure of the Lorentz cone relate to the extremality of positivity-preserving maps?

Key findings

• Every positivity-preserving map $ \Phi: \mathsf{M}_2^{sa} \to \mathsf{M}_2^{sa} $ admits a decomposition as $ \Phi(\rho) = \sum_j A_j \rho A_j^\dagger + \sum_k B_k \rho^T B_k^\dagger $, confirming the map is decomposable.
• The total number of terms in the decomposition is at most four, as guaranteed by Carathéodory’s theorem applied to the unit ball in $ \mathbb{R}^3 $.
• The proof via Brouwer’s fixed point theorem reduces the general case to the bistochastic case, where the map preserves the maximally mixed state and trace.
• The second proof establishes a connection between extremality of maps and automorphisms of the Lorentz cone $ \mathcal{L}_n $, showing that non-extreme maps can be perturbed within the cone.
• The S-lemma is used in a reformulated version to show that if two quadratic forms cover $ \mathbb{R}^n $, then a convex combination of them is positive semi-definite, which is essential for proving extremality.
• The proof demonstrates that any map not in the extreme ray of the cone must admit perturbations preserving positivity, implying the ray cannot be extreme unless the map is an automorphism of the Lorentz cone.