Detecting high-dimensional entanglement by randomized product projections

1. Problem Statement

High-dimensional quantum entanglement is a critical resource for quantum communication, cryptography, and metrology, offering increased information capacity and noise resistance. However, characterizing this entanglement (specifically the Schmidt Number, or SN) faces significant experimental bottlenecks:

• Scalability Issues: Conventional methods, such as the Mutually Unbiased Bases (MUBs) approach or Direct Fidelity Estimation, require a number of measurement settings that scales poorly with dimension $d$. For example, MUBs typically require $O(md)$ settings (where $m$ is the number of bases), and direct fidelity estimation often requires $d(d+1)$ projections.
• Hardware Limitations: Many physical platforms (e.g., photonic time-frequency systems, integrated optics) cannot simultaneously control or measure coherent superpositions of many basis states. They are often limited to measuring single basis states or pairs of states.
• Data Scarcity: Performing a large number of distinct measurements is infeasible, leading to limited experimental data. Existing protocols often require infinite sampling or high-order statistical moments (e.g., 4-designs) to achieve robustness, which is experimentally demanding.

2. Methodology

The authors propose a Randomized Product Projections protocol to estimate the Schmidt Number efficiently. The core methodology involves:

A. Randomized Measurement Strategy

Instead of fixed bases, Alice and Bob apply random local operations to the bipartite state $\rho$ before measuring a fixed product observable.

1. Unitary Randomization: Apply a random unitary $U$ (sampled from the Haar measure or a unitary 2-design) to both parties: $U \otimes U$.
2. Orthogonal Randomization: Apply a random orthogonal matrix $O$ (sampled from the Haar measure or an orthogonal 2-design) to both parties: $O \otimes O$.
3. Measurement: Measure a rank-1 product observable $M^{\otimes 2} = |j\rangle\langle j| \otimes |j\rangle\langle j|$ (a single basis state projection).

B. Theoretical Framework

The protocol utilizes the first-order moments of the measurement outcomes to reconstruct the fidelity $F(\rho)$ with a maximally entangled state $|\phi_d^+\rangle$.

• Moments: The authors define two statistical moments based on the expectation values of the randomized measurements:
  • $R(\rho) = \int dU \, \text{tr}[(U \otimes U)\rho(U^\dagger \otimes U^\dagger) M^{\otimes 2}]$
  • $Q(\rho) = \int dO \, \text{tr}[(O \otimes O)\rho(O^T \otimes O^T) M^{\otimes 2}]$
• Fidelity Estimation: Using Schur-Weyl duality, they derive a linear relationship between these moments and the fidelity:
  $$F(\rho) = (d+2)Q(\rho) - (d+1)R(\rho)$$
• Schmidt Number Criterion: Once $F(\rho)$ is estimated, the Schmidt number $\mu$ is bounded by $\mu \ge \lceil d \cdot F(\rho) \rceil$.

C. Classical Post-Processing Algorithm

Since infinite sampling is impossible, the authors introduce a statistical algorithm to estimate the SN lower bound from a finite dataset ($N$ random operations):

1. Calculate individual fidelity estimates $F^{(l)}_e$ for each random operation pair.
2. Construct a confidence interval $[F_{lb}, F_{ub}]$ for the true fidelity using the sample mean and the Student's t-distribution (robust for small sample sizes, $N < 30$).
3. Use the lower bound $F_{lb}$ to calculate a conservative estimate of the Schmidt number $\mu_{est}$.

3. Key Contributions

1. Resource Efficiency (Single Channel Control):
   The protocol requires measuring only a single basis state (rank-1 projection). This reduces the experimental overhead from controlling $d$ channels (as in MUBs) to just one channel, making it compatible with platforms like integrated optics and time-frequency domain systems.

2. Optimal Scaling of Random Operations:

   • General States: The number of random operations $N$ scales as $O(d^2)$, consistent with direct fidelity estimation.
   • Noisy Maximally Entangled States: For states subject to typical noise (depolarizing, dephasing, or random noise), the required $N$ scales as $O(1)$, independent of the dimension $d$. This is a significant improvement over MUBs ($O(md)$) and direct fidelity estimation.

3. Robustness to Noise:
   The first-order moments are invariant under orthogonal transformations ($O \otimes O$), making the fidelity estimation robust against specific types of measurement noise that plague other methods.

4. Statistical Rigor for Small Data:
   The paper provides a rigorous algorithm using the t-distribution to obtain high-confidence lower bounds on the Schmidt number even with a limited number of experimental runs ($N \approx 30$), addressing the challenge of data scarcity.

4. Results

• Numerical Simulations: The authors simulated the protocol for local dimensions $d = 20, 30, 40, 50$ on isotropic states ($\rho_v^{iso}$) with varying noise levels.
• Performance Comparison:
  • With $N=30$ (totaling 60 projections), the randomized projection method provided more accurate Schmidt number estimations than the 3-MUBs criterion for states with low purity (high noise).
  • For $d=20$, the method outperformed 3-MUBs for noise parameters $v < 0.75$. As $d$ increased, the threshold where the new method outperforms MUBs shifted to higher noise levels, indicating superior scalability.
  • The method successfully detected entanglement in states where the 3-MUBs criterion failed or provided loose bounds.
• Variance Analysis: Theoretical analysis confirmed that for noisy maximally entangled states, the variance of the estimator is independent of $d$, explaining the $O(1)$ scaling of required samples.

5. Significance

• Experimental Feasibility: This work bridges the gap between theoretical entanglement detection and practical implementation. By reducing the requirement to a single measurement channel, it enables the certification of high-dimensional entanglement on current and near-future experimental platforms that lack multi-channel control.
• Scalability: The $O(1)$ scaling for noisy states suggests that as quantum systems grow in dimension, the cost of verifying high-dimensional entanglement does not necessarily explode, provided the states are not perfectly pure.
• General Applicability: The framework is adaptable to various physical systems (photons, cold atoms, trapped ions) and opens avenues for characterizing high-dimensional quantum channels and multipartite entanglement in future research.

In summary, the paper presents a resource-efficient, experimentally friendly, and statistically robust protocol for certifying high-dimensional entanglement, overcoming the scalability and control limitations of existing methods.