Detecting entanglement from few partial transpose moments and their decay via weight enumerators

1. Problem Statement

The rapid advancement of experimental quantum devices has created an urgent need for efficient methods to certify quantum entanglement. The gold standard for entanglement detection is the Positive Partial Transposition (PPT) criterion, which states that a separable state must have a positive semi-definite (PSD) partial transpose ($\rho^\Gamma$). However, full PPT certification requires diagonalizing $\rho^\Gamma$, a task that scales exponentially with the number of qubits and is experimentally infeasible for large systems.

Recent approaches utilize Partial Transpose (PT) moments, defined as $p_k = \text{Tr}[(\rho^\Gamma)^k]$, to relax the PPT criterion. While criteria like the Descartes and Stieltjes conditions use these moments, they typically require access to all moments up to a certain order $m$ ($p_1, \dots, p_m$). Measuring higher-order moments ($k > 3$) is experimentally demanding, often requiring multi-copy measurements (e.g., $k$-cycle tests) that are difficult to implement for large $k$. The paper addresses the challenge of detecting entanglement using fewer PT moments while maintaining high detection power and robustness against noise.

2. Methodology

The authors employ a combination of linear algebra inequalities, spectral theory, and concepts from quantum error correction (specifically weight enumerators).

• Generalization of PPT Relaxations: The paper derives a new criterion based on Hölder's inequality (specifically a generalization of the Cauchy-Schwarz inequality used in the Stieltjes criterion). Instead of requiring all moments up to $m$, the authors show that comparing any three moments of orders $k < l < m$ is sufficient to detect entanglement.
• Quantitative Bounds: The framework is extended to provide lower bounds on entanglement measures (specifically Logarithmic Negativity) using only a subset of PT moments.
• Spectral Completeness: The authors investigate the conditions under which the Stieltjes hierarchy (based on the positivity of Hankel matrices formed by moments) becomes equivalent to the full PPT criterion. They link the number of required moments to the number of distinct eigenvalues in the spectrum of $\rho^\Gamma$.
• Quantum Weight Enumerators (QWEs): To analyze how PT moments decay under local depolarizing noise, the authors introduce a new family of Quantum Weight Enumerators. These generalize the Shor-Laflamme enumerators (used in quantum error correction) to capture the decay of higher-order moments ($k \geq 3$) under noise, utilizing Fourier analysis on finite Abelian groups (the stabilizer group) for efficient computation.

3. Key Contributions

A. The $(k, l, m)$-PPT Criterion

The paper introduces a simple yet powerful relaxation of the PPT criterion (Theorem 1).

• Theorem: For integers $k < l < m$, define $x = (m-l)/(m-k)$. If a state satisfies:
  $$p_l > p_k^x p_m^{1-x}$$
  then the state is NPT entangled (Negative Partial Transpose).
• Significance: This allows experimentalists to detect entanglement by measuring only three specific moments ($p_k, p_l, p_m$) rather than the full sequence up to $m$. For example, measuring $p_3, p_4, p_5$ is sufficient to apply the $(3,4,5)$-PPT criterion, avoiding the need to measure $p_2$ (purity) or higher orders if not needed.

B. Quantitative Entanglement Bounds

The authors generalize previous lower bounds for Logarithmic Negativity ($E_N$).

• Theorem 2: They derive a bound on $E_N$ using only two moments $p_l$ and $p_m$ (and the normalization $p_1=1$), showing that strong violations of the moment inequalities correspond to significant quantitative entanglement.

C. Equivalence of Stieltjes and PPT Criteria

The paper proves that the Stieltjes criterion is as powerful as the full PPT criterion under specific spectral conditions (Theorem 3).

• Result: If $\rho^\Gamma$ has $N$ distinct eigenvalues, the Stieltjes criterion with order $m \geq 2N - 1$ detects all NPT states.
• Application:
  • For globally depolarized stabilizer states, $N=3$, meaning the Stieltjes-5 criterion ($m=5$) is sufficient to reproduce the full PPT threshold.
  • For locally depolarized GHZ states, the required $m$ scales linearly with system size ($m \approx 2n+5$), but for small $n$, low-order moments (e.g., $m=3, 5, 7$) already suffice.

D. Quantum Weight Enumerators (QWEs)

The authors define QWEs to analytically describe the decay of PT moments under local white noise.

• Framework: They express $p_k[(D_\epsilon^\otimes \rho)^\Gamma]$ as a polynomial in $(1-\epsilon)$ weighted by QWEs $c_w^{(k)}$.
• Efficiency: Using Fourier analysis on the stabilizer group, they show that these enumerators can be computed efficiently without exponential scaling in the moment order $k$. This allows for the prediction of noise thresholds for complex states like the Absolutely Maximally Entangled (AME) state.

4. Results and Performance Analysis

The authors benchmarked their criteria against standard methods (Fidelity, Purity, Descartes, Stieltjes) using locally depolarized GHZ states and 6-qubit AME states.

• GHZ States:

  • Scaling: Traditional criteria like Fidelity and Purity suffer from noise thresholds that scale as $O(n^{-1})$, becoming ineffective for large $n$.
  • Stieltjes Performance: The Stieltjes-$m$ criteria (for $m=5, 7$) exhibit much better scaling ($O(n^{-\alpha})$ with $\alpha \approx 0.84$), allowing detection of entanglement in states with hundreds of qubits even at noise levels $\epsilon \approx 10^{-2}$.
  • Three-Moment Advantage: The $(k, l, m)$-PPT criteria (e.g., $(3,4,5)$) achieve performance nearly identical to the full Stieltjes-$m$ criterion but require measuring fewer moments (e.g., skipping $p_2$). For $n=300$ and $\epsilon=10^{-2}$, the $(3,4,5)$-PPT criterion detects entanglement with ~25% less experimental overhead than Stieltjes-5.

• AME States:

  • Using the QWE framework, the authors calculated noise thresholds for 6-qubit AME states across different bipartitions ($1|5, 2|4, 3|3$).
  • The Stieltjes-5 criterion was found to be extremely powerful, reproducing the full PPT threshold for the $3|3$ bipartition.
  • The $(3,4,5)$-PPT criterion offered a sweet spot: it was stronger than the standard $p_3$-PPT (which failed for some bipartitions) but required less data than Stieltjes-5.

5. Significance and Conclusion

This work significantly advances the practicality of entanglement detection in near-term quantum devices:

1. Reduced Experimental Overhead: By proving that comparing just three moments is sufficient, the paper enables entanglement certification in regimes where measuring all moments up to a high order is impossible.
2. Scalability: The identified criteria (Stieltjes-$m$ and $(k,l,m)$-PPT) maintain detection capabilities for large systems ($n \sim 100$) where fidelity-based methods fail.
3. Theoretical Unification: The introduction of Quantum Weight Enumerators provides a unified mathematical framework to predict the robustness of entanglement detection under noise, bridging the gap between quantum information theory, error correction, and experimental constraints.
4. Practical Guidance: The paper provides concrete recommendations for experimentalists (e.g., using the $(3,4,5)$-PPT criterion for GHZ states) to maximize detection probability while minimizing resource consumption.

In summary, Miller and Eisert demonstrate that "less is more" in entanglement detection: carefully selected partial transpose moments can replicate the power of full spectral analysis, making robust entanglement certification feasible for the next generation of large-scale quantum computers.