A note on the quantum Wielandt inequality

• URL: http://arxiv.org/abs/2504.21638v1
• Date: Wed, 30 Apr 2025 13:40:53 GMT
• Title: A note on the quantum Wielandt inequality
• Authors: Owen Ekblad,
• Abstract summary: We show how to extend operator algebraic methods introduced by Rahaman to prove that the index of primitivity of any primitive Schwarz map is at most $2(D-1)2$.<n>We briefly discuss of how this relates to a conjecture of Perez-Garcia, Verstraete, Wolf and Cirac concerning properties of parent Hamiltonians of matrix product states.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this note, we show how to extend operator algebraic methods introduced by Rahaman to prove that the index of primitivity of any primitive Schwarz map is at most $2(D-1)^2$, where $D$ is the dimension of the underlying matrix algebra. This inequality was first proved by Rahaman for Schwarz maps which were both unital and trace preserving. We show here that the assumption of unitality is automatic (up to normalization) for primitive Schwarz maps, but, in general, not all primitive unital Schwarz maps are trace preserving. Therefore, the precise purpose of this note is to showcase how to apply the proof of Rahaman to arbitrary primitive Schwarz maps. As a corollary of this theorem, we show that the index of primitivity of any primitive 2-positive map is at most $2(D-1)^2$, so in particular this bound holds for arbitrary primitive completely positive maps. We briefly discuss of how this relates to a conjecture of Perez-Garcia, Verstraete, Wolf and Cirac concerning properties of parent Hamiltonians of matrix product states.

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