[Paper Review] Operator systems from discrete groups

Douglas Farenick, Ali S. Kavruk+2 more|arXiv (Cornell University)|Sep 6, 2012

Advanced Operator Algebra Research20 references4 citations

TL;DR

This paper introduces a framework for studying operator systems derived from discrete groups, particularly focusing on the free group 𝔽ₙ and free products of ℤ₂. It establishes connections between tensor products of these operator systems and quantum correlations, proving that NC(n)⊗_c NC(m) ≠ NC(n)⊗_max NC(m) for n,m ≥ 2, thereby providing a new operator system interpretation of Tsirelson's problem and resolving key questions in quantum information theory.

ABSTRACT

We formulate a general framework for the study of operator systems arising from discrete groups. We study in detail the operator system of the free group on $n$ generators, as well as the operator systems of the free products of finitely many copies of the two-element group $\mathbb Z_2$. We examine various tensor products of group operator systems, including the minimal, the maximal, and the commuting tensor products. We introduce a new tensor product in the category of operator systems and formulate necessary and sufficient conditions for its equality to the commuting tensor product in the case of group operator systems. We express various sets of quantum correlations studied in the theoretical physics literature in terms of different tensor products of operator systems of discrete groups. We thus recover earlier results of Tsirelson and formulate a new approach for the study of quantum correlation boxes.

Motivation & Objective

• To develop a general framework for studying operator systems arising from discrete groups, especially free groups and free products of ℤ₂.
• To analyze the interplay between minimal, maximal, commuting, and a new essential tensor product in the context of group operator systems.
• To connect tensor products of group operator systems to quantum correlation boxes and Bell inequalities.
• To provide an operator system-theoretic interpretation of Tsirelson's results on quantum correlations and the non-closure of quantum correlations under certain tensor products.
• To resolve open questions in quantum information theory, including the distinction between commuting and maximal tensor products for NC(2)⊗NC(2).

Proposed method

• Define the operator system S(𝒰) as the linear span of {1, u, u*} for generators u ∈ 𝒰 in the group C*-algebra C*(G).
• Study the operator system Sₙ associated with the free group 𝔽ₙ on n generators, and NC(n) arising from the n-fold free product ℤ₂ * ⋯ * ℤ₂.
• Introduce a new tensor product, denoted ⊗_ess, and derive necessary and sufficient conditions for its equality with the commuting tensor product in the group operator system setting.
• Use the duality between C(2) and the operator system V to relate tensor products to sets of quantum and local correlations.
• Represent states on NC(2)⊗_min NC(2) via density operators on Hilbert spaces to show that all quantum correlations arise from such states.
• Apply Tsirelson’s results to prove that NC(n)⊗_c NC(m) ≠ NC(n)⊗_max NC(m) for all n,m ≥ 2, using the non-closure of quantum correlations under maximal tensor products.

Experimental results

Research questions

• RQ1 What is the relationship between the minimal, commuting, and maximal tensor products of operator systems derived from discrete groups?
• RQ2 How do tensor products of group operator systems, particularly NC(n), relate to quantum correlation boxes in quantum information theory?
• RQ3 Can the distinction between commuting and maximal tensor products be detected at the level of 2×2 matrix amplifications for group operator systems?
• RQ4 What is the role of the new essential tensor product ⊗_ess in characterizing equality with the commuting tensor product for group operator systems?
• RQ5 Do intermediate tensor products between min and max generate new families of quantum correlations when applied to NC(2)⊗NC(2)?

Key findings

• The operator system NC(n) arising from the n-fold free product ℤ₂ * ⋯ * ℤ₂ is isomorphic to the non-commutative n-cube studied by Tsirelson.
• For all n,m ≥ 2, it holds that NC(n)⊗_c NC(m) ≠ NC(n)⊗_max NC(m), demonstrating a strict separation between commuting and maximal tensor products.
• The set of quantum correlations Q coincides with the set of states on NC(2)⊗_min NC(2) restricted to the 4×4 matrix level, i.e., Q = (NC(2)⊗_min NC(2))_+ ∩ BS₄.
• The set of local correlations L is equal to the set of states on C(2)⊗_min C(2), and this set is strictly contained in Q, confirming that Q ≠ L.
• The C*-envelope of Sₙ⊗_max Sₘ is not equal to C*(𝔽ₙ)⊗_max C*(𝔽ₘ), showing that C*-envelopes do not commute with maximal tensor products.
• The equality NC(2)⊗_min NC(2) = NC(2)⊗_c NC(2) holds, but this equality fails to extend to the maximal tensor product, implying that intermediate tensor products between c and max can generate new families of quantum correlations.