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The complexity of semidefinite programs for testing $k$-block-positivity

Qian Chen, Benoît Collins

Abstract

We extend \cite{chen2025srkbp} by analyzing the complexity of the $k$-block-positivity testing algorithm. In this paper, we investigate a symmetry reduction scheme based on rectangular shaped Young diagrams. Connecting the complexity to the dimensions of irreducible representations of $\mathrm{U}(d)$, we derive an explicit formula for the complexity, which also clarifies why the semidefinite program hierarchy collapses in the $k=d$ case.

The complexity of semidefinite programs for testing $k$-block-positivity

Abstract

We extend \cite{chen2025srkbp} by analyzing the complexity of the -block-positivity testing algorithm. In this paper, we investigate a symmetry reduction scheme based on rectangular shaped Young diagrams. Connecting the complexity to the dimensions of irreducible representations of , we derive an explicit formula for the complexity, which also clarifies why the semidefinite program hierarchy collapses in the case.
Paper Structure (23 sections, 14 theorems, 62 equations)

This paper contains 23 sections, 14 theorems, 62 equations.

Table of Contents

  1. Preliminary
  2. Notation
  3. Optimization problems for testing $k$-block positivity
  4. Auxiliary spaces with $U^{\otimes k}$-symmetry
  5. Representing feasible pure states
  6. Optimization problem based on $k$-purification
  7. Semidefinite programs for testing $k$-block positivity
  8. Reduced semidefinite programs indexed by Young diagrams
  9. SDP symmetry reduction
  10. Two questions
  11. Relaxation of the trace constraint
  12. An important construction
  13. Weak trace constraint
  14. Hierarchical SDPs on rectangular Young diagrams
  15. Tensor power representation and Schur function
  16. ...and 8 more sections

Key Result

Theorem 1

Setting $N$ by $N+k-1=k n$ with integer $n$. The SDP complexity of the reduced SDP indexed by a rectangular Young diagram $(n^{k})$ is That is to say, the SDP complexity is $O(n^{k(d-k)})$ after symmetry reduction, compared with the unreduced one $O((kd)^{N+1})$.

Theorems & Definitions (30)

  • Theorem 1: The SDP complexity of rectangular scheme
  • Definition 2: Optimization: testing $k$-block-positivity
  • Definition 3: $k$-purification and dualization
  • Proposition 4
  • proof
  • Definition 5: Optimization: testing $k$-block-positivity via $k$-purification
  • Proposition 6
  • Definition 7: $k$-block-positivity testing SDP with $N$-Bose symmetric extension
  • Definition 8: Reduced SDP: ${\mathrm{U}}(k)$-symmetry
  • Proposition 9
  • ...and 20 more