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A Closed Form for Moment-Based Entanglement Tests Associated to the PPT Criterion

Zachary P. Bradshaw, Margarite L. LaBorde

TL;DR

This work provides a closed-form description of moment-based entanglement tests associated with the PPT criterion by leveraging Newton's identities, Descartes' rule, and generating-function techniques. It links elementary symmetric polynomials to moments of the partial transpose via Bell and cycle-index polynomials, yielding a concrete inequality framework that is equivalent to PPT and can be interpreted through graph zeta functions. The authors also outline a quantum-circuit approach to estimate the necessary moments efficiently and demonstrate the necessity of checking the full moment hierarchy up to the rank. The results deepen the mathematical structure underlying PPT-based entanglement tests and offer a graph-theoretic perspective that could guide future witnesses and implementations on quantum hardware.

Abstract

Neven et al. have explored an unexpected alliance between the mathematical insights of Sir Isaac Newton and René Descartes which culminates in the reduction of the Positive Partial Transpose (PPT) criterion to an equivalent hierarchy of entanglement tests based on the moments of the partial transpose. By repurposing these classical results in the context of modern quantum theory, they illuminate new pathways for entanglement verification. Here, we expand on this work by providing a closed form for the inequalities defining these entanglement tests and producing an equivalent set of graph theoretic conditions on the weighted graph induced by the partial transpose.

A Closed Form for Moment-Based Entanglement Tests Associated to the PPT Criterion

TL;DR

This work provides a closed-form description of moment-based entanglement tests associated with the PPT criterion by leveraging Newton's identities, Descartes' rule, and generating-function techniques. It links elementary symmetric polynomials to moments of the partial transpose via Bell and cycle-index polynomials, yielding a concrete inequality framework that is equivalent to PPT and can be interpreted through graph zeta functions. The authors also outline a quantum-circuit approach to estimate the necessary moments efficiently and demonstrate the necessity of checking the full moment hierarchy up to the rank. The results deepen the mathematical structure underlying PPT-based entanglement tests and offer a graph-theoretic perspective that could guide future witnesses and implementations on quantum hardware.

Abstract

Neven et al. have explored an unexpected alliance between the mathematical insights of Sir Isaac Newton and René Descartes which culminates in the reduction of the Positive Partial Transpose (PPT) criterion to an equivalent hierarchy of entanglement tests based on the moments of the partial transpose. By repurposing these classical results in the context of modern quantum theory, they illuminate new pathways for entanglement verification. Here, we expand on this work by providing a closed form for the inequalities defining these entanglement tests and producing an equivalent set of graph theoretic conditions on the weighted graph induced by the partial transpose.

Paper Structure

This paper contains 7 sections, 8 theorems, 59 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Newton's Identities and Descartes' Rule of Sign
  3. Entanglement Criteria
  4. Closed Form Derivation
  5. Implementation on Quantum Computers
  6. Graph Theoretic Equivalent Conditions
  7. Conclusion

Key Result

Lemma 1

Let $n\ge k\ge1$. Then Moreover, for $k>n\ge1$, we have

Figures (4)

  • Figure 1: Circuit for estimating $\mathop{\mathrm{Tr}}\nolimits[(\rho^\Gamma)^k]$.
  • Figure 2: A calculation of Eq. \ref{['eq:closedform']} for a three qubit GHZ state (a) and a two qubit Werner state (b) with $p =0.75$.
  • Figure 3: A calculation of Eq. \ref{['eq:closedform']} for a "butterfly metrology state" as given in kobrin2024universal obtained from evolution under the Rydberg XY Hamiltonian.
  • Figure 4: Graph induced by the partial transpose of the Bell state $\frac{1}{\sqrt{2}}(\ket{00}+\ket{11})$.

Theorems & Definitions (14)

  • Lemma 1: Newton's Identities
  • proof
  • Lemma 2: Descartes' Rule of Sign
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Proposition 1
  • proof
  • ...and 4 more