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Multipartite parity bounds and total correlation

James Tian

Abstract

This paper studies multipartite observables formed from sums of local self-adjoint contractions on tensor product Hilbert spaces. The square of such a sum has a parity structure: after decomposing each local product into commutator and anticommutator parts, the odd parity terms cancel and only even parity contributions remain. This yields a norm bound in terms of a family of pairwise defect weights built from local commutator and anticommutator norms. These defect weights also control an information theoretic estimate. The excess of the observable expectation above the product state threshold is shown to necessarily carry a definite amount of total correlation. Under a natural $\ell^{2}$-type bound on each local family, this product state threshold becomes explicit, which leads to a fully explicit lower bound on total correlation. A simple depolarizing example illustrates the resulting decay mechanism under local noise.

Multipartite parity bounds and total correlation

Abstract

This paper studies multipartite observables formed from sums of local self-adjoint contractions on tensor product Hilbert spaces. The square of such a sum has a parity structure: after decomposing each local product into commutator and anticommutator parts, the odd parity terms cancel and only even parity contributions remain. This yields a norm bound in terms of a family of pairwise defect weights built from local commutator and anticommutator norms. These defect weights also control an information theoretic estimate. The excess of the observable expectation above the product state threshold is shown to necessarily carry a definite amount of total correlation. Under a natural -type bound on each local family, this product state threshold becomes explicit, which leads to a fully explicit lower bound on total correlation. A simple depolarizing example illustrates the resulting decay mechanism under local noise.
Paper Structure (5 sections, 5 theorems, 155 equations)

This paper contains 5 sections, 5 theorems, 155 equations.

Table of Contents

  1. Introduction
  2. Multipartite Parity Bounds
  3. Static Correlation Bounds
  4. Correlation Decay Under Local Noise
  5. Concluding Remarks

Key Result

Proposition 2.1

For each $r\in\left\{ 1,\dots,n\right\}$, let $H_{r}$ be a complex Hilbert space. Let be self-adjoint contractions. Set and define For $1\leq i<j\leq m$, define Then

Theorems & Definitions (16)

  • Proposition 2.1
  • proof
  • Example 2.2
  • Theorem 3.1
  • proof
  • Example 3.2: CHSH total correlation bound
  • Remark 3.3
  • Theorem 3.4
  • proof
  • Corollary 3.5
  • ...and 6 more