[Paper Review] Strongly symmetric spectral convex bodies are Jordan algebra state spaces

Howard Barnum, Joachim Hilgert|arXiv (Cornell University)|Apr 7, 2019

Quantum Computing Algorithms and Architecture49 references10 citations

TL;DR

This paper establishes that finite-dimensional convex compact sets satisfying spectrality and strong symmetry are precisely the normalized state spaces of finite-dimensional simple Euclidean Jordan algebras and simplices. The result shows that the absence of higher-order interference—previously assumed in characterizations—is redundant when spectrality and strong symmetry are present, offering a purely convex-geometric characterization of quantum-like systems and narrowing the class of viable general probabilistic theories to Jordan-algebraic structures.

ABSTRACT

We show that the strongly symmetric spectral convex compact sets are precisely the normalized state spaces of finite-dimensional simple Euclidean Jordan algebras and the simplices. Spectrality is the property that every state has a convex decomposition into perfectly distinguishable pure states; strong symmetry is transitivity, for each integer N, of the affine automorphism group of the state space on lists of N perfectly distinguishable pure states. Additional assumptions combine with this theorem to give simple characterizations of finite-dimensional complex quantum state space. Important aspects of quantum and classical thermodynamics and of query complexity have been generalized to classes of general probabilistic theories (GPTs) satisfying natural postulates including or implying spectrality and strong symmetry; our result shows that these apply to a narrower class of theories than might have been hoped. Sorkin's notion of irreducibly k-th order interference has been studied in the GPT framework and looked for in experiments. Our result shows that the assumption of no higher-order (k > 2) interference, used along with spectrality and strong symmetry to characterize the same class of Jordan-algebraic convex sets by Barnum, Mueller, and Ududec in arXiv:1403.4147, was superfluous. It also implies that Lee and Selby's extension, on the assumption that interference has fixed maximal degree k, of the important order square root of N lower bound on the quantum black-box query complexity of searching N possibilities for one having a desired property (which is achieved by Grover's quantum algorithm), to a class of theories satisfying certain postulates allowing the formulation of a generalized notion of query algorithm, actually applies in the Jordan-algebraic setting where higher-order interference is not possible.

Motivation & Objective

• To characterize the convex compact sets that satisfy spectrality and strong symmetry in finite dimensions.
• To show that these properties alone, without assuming the absence of higher-order interference, fully characterize the state spaces of finite-dimensional simple Euclidean Jordan algebras and simplices.
• To clarify the role of higher-order interference in general probabilistic theories by demonstrating its redundancy under spectrality and strong symmetry.
• To provide a convex-geometric foundation for understanding quantum and classical systems in the framework of general probabilistic theories.

Proposed method

• The authors analyze the symmetry group actions on convex sets, focusing on the transitive action on lists of perfectly distinguishable pure states.
• They use the polar representation of symmetric spaces and the structure of the Lie algebra decomposition g = k ⊕ p to study the action of the automorphism group on the state space.
• They identify the fundamental weights and Weyl group orbits in the context of Euclidean Jordan algebras, particularly for the An−1 series.
• They show that the convex hull of the Weyl group orbit of a traceless projection (e.g., e₁₁ − I/n) generates a simplex in the traceless matrices, which corresponds to the state space under affine isomorphism.
• They establish that the normalized state space is affinely isomorphic to the convex hull of the K-orbit of e₁₁ in the unit-trace matrices, via translation by I/n.
• They leverage the Madden-Robertson construction to link fundamental weights to orbitopes and verify that the orbitopes of the first and last nodes of the Coxeter diagram yield isomorphic simplices.

Experimental results

Research questions

• RQ1 Which finite-dimensional convex compact sets satisfy both spectrality and strong symmetry?
• RQ2 Is the assumption of no higher-order interference necessary to characterize Jordan-algebraic state spaces when spectrality and strong symmetry are already assumed?
• RQ3 Can the state spaces of finite-dimensional simple Euclidean Jordan algebras be fully characterized using only convex-geometric properties like spectrality and strong symmetry?
• RQ4 How do the geometric structures of orbitopes and Weyl group actions relate to the physical properties of general probabilistic theories?
• RQ5 What is the role of the traceless and unit-trace matrix subspaces in constructing the normalized state space from the symmetric space structure?

Key findings

• Finite-dimensional convex compact sets that are spectral and strongly symmetric are exactly the normalized state spaces of finite-dimensional simple Euclidean Jordan algebras and simplices.
• The absence of higher-order interference is not required as an independent postulate, since it follows from spectrality and strong symmetry.
• The state space of a simple Euclidean Jordan algebra is affinely isomorphic to the convex hull of the K-orbit of a rank-one projection in the unit-trace matrices.
• The Weyl group action on the traceless diagonal matrices (e.g., e₁₁ − I/n) generates a simplex whose convex hull corresponds to the state space under affine isomorphism.
• The fundamental weights λ₁ and λₙ₋₁ in the An−1 case correspond to orbitopes that are affinely isomorphic to simplices, confirming the duality and symmetry of the construction.
• The polar body of the simplex generated by the Weyl orbit of λ₁ is again a simplex, consistent with the self-duality of the Coxeter diagram for An−1.