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Certifying entanglement dimensionality by random Pauli sampling

Changhao Yi

TL;DR

This work presents a basis-independent protocol to certify the Schmidt number $χ$ of bipartite pure states by random Pauli measurements and two local random unitaries. The core idea is that the full correlation matrix $T$ has rank $χ^2$, so a suitably chosen projected matrix $T_S$ should retain rank $χ^2$; observing a higher rank certifies entanglement dimensionality beyond $χ$. A key result bounds the required Pauli samples as $K = O(μ_0 \log(χ/η))$ with $μ_0 = d \max_{P \in \mathcal{P}_0} \sum_{i,j<χ} |<l_i|P|l_j>|^2$, yielding worst-case $O(d χ)$ but typical/ Haar-delocalized scenarios give $K = \widetilde{O}(χ^2)$. The approach is robust to depolarizing noise and supported by numerical experiments on toy and many-body states, indicating scalable, practical entanglement certification for high-dimensional systems and opening routes to pseudorandom-unitary-based randomized measurement protocols.

Abstract

We introduce a Pauli-measurement-based algorithm to certify the Schmidt number of $n$-qubit pure states. Our protocol achieves an average-case sample complexity of $\caO(\mathrm{poly}(n)χ^2)$, a substantial improvement over the $\caO(2^n χ)$ worst-case bound. By utilizing local pseudorandom unitaries, we ensure the worst case can be transformed into the average-case with high probability. This work establishes a scalable approach to high-dimensional entanglement certification and introduces a proof framework for random Pauli sampling.

Certifying entanglement dimensionality by random Pauli sampling

TL;DR

This work presents a basis-independent protocol to certify the Schmidt number of bipartite pure states by random Pauli measurements and two local random unitaries. The core idea is that the full correlation matrix has rank , so a suitably chosen projected matrix should retain rank ; observing a higher rank certifies entanglement dimensionality beyond . A key result bounds the required Pauli samples as with , yielding worst-case but typical/ Haar-delocalized scenarios give . The approach is robust to depolarizing noise and supported by numerical experiments on toy and many-body states, indicating scalable, practical entanglement certification for high-dimensional systems and opening routes to pseudorandom-unitary-based randomized measurement protocols.

Abstract

We introduce a Pauli-measurement-based algorithm to certify the Schmidt number of -qubit pure states. Our protocol achieves an average-case sample complexity of , a substantial improvement over the worst-case bound. By utilizing local pseudorandom unitaries, we ensure the worst case can be transformed into the average-case with high probability. This work establishes a scalable approach to high-dimensional entanglement certification and introduces a proof framework for random Pauli sampling.
Paper Structure (14 sections, 11 theorems, 58 equations, 3 figures)

This paper contains 14 sections, 11 theorems, 58 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Schmidt number certification
  3. Proof idea
  4. Robustness to noises
  5. Numerical results
  6. Conclusion
  7. Background on random matrix theory
  8. Background on Pauli groups
  9. Proof of Lemma \ref{['lem: Tdecomposition']}
  10. Proof of the main theorem
  11. Proof of Theorem \ref{['theorem:main']}
  12. Proof of Corollary \ref{['coro:main']}
  13. Proof of Lemma \ref{['lem:muU']}
  14. Computational basis scenario

Key Result

Lemma 1

If $|\psi\rangle$ has Schmidt rank $\chi$, then $\rank(T) = \chi^2$.

Figures (3)

  • Figure 1: Schmidt number certification by random Pauli sampling. Here $\rho$ is a bipartite quantum state on $\mathcal{H}_{AB}$; $U_A, U_B$ are two samples of local Haar random unitaries; $P,Q$ are two Pauli operators from the random set $\mathcal{S}$. The figure shows how to compute one entry of the projected CM $T_\mathcal{S}$. Note that we use the same random unitary $U_A \otimes U_B$ to compute all entries of $T_{\mathcal{S}}$.
  • Figure 2: Performance of random projection for the trial state $|\phi\rangle = \sum_{i=0}^3 |ii\rangle/2$. The vertical axis represents the size of the top $16$ rescaled singular values. As revealed by the figure, under the original standard basis, only a few singular values of $T$ are retained. Under the randomly rotated bases, $K = 64$ number of samples suffices to recover the 16 singular values robustly.
  • Figure 3: Performance of random projection for the ground state of free fermion model and strongly correlated model, respectively. The vertical axis represents the size of the top $256$ rescaled singular values. The trends of this model are consistent with that in Fig. \ref{['fig:trialstate']}. However, the effect of the random unitary layer is not as obvious.

Theorems & Definitions (16)

  • Lemma 1
  • Theorem 1: Informal
  • Theorem 2
  • Corollary 1
  • Proposition 1
  • Theorem 3
  • Corollary 2
  • proof
  • proof
  • Lemma 2
  • ...and 6 more