Certifying unknown-basis pure-state entanglement dimensionality with few measurement settings

Technical Summary: Certifying Unknown-Basis Pure-State Entanglement Dimensionality with Few Measurement Settings

Problem Statement
Quantum entanglement is a central resource for quantum information processing, and its characterization beyond simple separability is a longstanding objective. Specifically, the Schmidt number (or entanglement dimensionality) quantifies the number of local degrees of freedom participating in entanglement. While high-dimensional entanglement offers advantages in communication, channel discrimination, and key distribution, experimentally certifying the Schmidt number of prepared states is challenging. Full state tomography is infeasible due to exponential cost. Existing tomography-free protocols face a fundamental trade-off:

• Basis-dependent methods (e.g., fidelity witnesses) use a constant number of measurement settings but require prior knowledge of the Schmidt basis. They lose certification power under local basis misalignment, a common issue in strongly correlated many-body systems or photonic platforms where Schmidt modes depend on experimental drift.
• Basis-independent methods (e.g., correlation matrix protocols) avoid this assumption but typically require a number of measurement settings that grows at least linearly with the local dimension $d$, often necessitating high-order unitary designs that are experimentally demanding.

The paper addresses the need for a protocol that is simultaneously basis-independent, measurement-setting-efficient, and experimentally accessible for high-dimensional bipartite pure states.

Methodology
The author proposes a protocol that certifies the Schmidt number $\chi$ of a $2n$-qubit bipartite pure state $|\psi\rangle$ (where local dimension $d=2^n$) using single-copy measurements. The core of the method relies on the property that the rank of the full correlation matrix (CM) of a pure state equals the square of its Schmidt number ($\text{rank}(G_\psi) = \chi^2$).

1. Projected Correlation Matrix (CM): Instead of measuring the full $d^2 \times d^2$ CM, the protocol constructs a "projected" CM, $G_{\psi, A}$, by sampling a small set of Pauli strings $A = \{a_0, a_1, \dots, a_S\}$. The rank of this projected matrix is bounded by $\chi^2$. If $\text{rank}(G_{\psi, A}) > r^2$, it certifies that the Schmidt number is at least $r+1$.
2. Randomized Preprocessing: A critical innovation is the application of local random unitaries $U_A \otimes U_B$ (sampled from the Haar random unitary ensemble or an approximate unitary design) to the target state before measurement.
   • Without preprocessing, if the Schmidt basis aligns poorly with the measurement basis (e.g., computational basis), the CM is sparse, and preserving its rank requires $\Omega(d)$ Pauli strings.
   • With preprocessing, the incoherence parameter $\mu(U)$, which measures the overlap between Pauli operators and the Schmidt basis, drops from $O(d\chi)$ to $O(\chi^2 \log d)$. This "scrambling" spreads the matrix entries, making the projected CM dense and allowing the rank to be preserved with only $\tilde{O}(\chi^2)$ Pauli strings with high probability.
3. Algorithm: The protocol involves:
   • Sampling a random sequence of Pauli strings.
   • Applying local random unitaries to the state.
   • Measuring the state in the eigenbases of the sampled Pauli strings (and their tensor products) to estimate the entries of the projected CM.
   • Calculating the singular values of the estimated CM and comparing them against a noise-induced threshold to count the number of significant singular values.

Key Contributions and Results

• Sample Complexity: The protocol achieves a total sample complexity of $\tilde{O}(d^2 \chi^2 \lambda_{lb}^{-2})$, where $\lambda_{lb}$ is the lower bound on non-zero Schmidt coefficients. Crucially, the number of distinct measurement settings is reduced to $\tilde{O}(\chi^4)$, which is nearly independent of the local dimension $d$ for constant $\chi$. This represents an exponential reduction in settings compared to low-rank tomography or worst-case basis-independent methods.
• Theoretical Guarantees:
  • Theorem 1: Proves that with randomized preprocessing, $S = \tilde{O}(\chi^2 \log(d\chi^2\eta^{-1}) \log(\chi^2\eta^{-1}))$ Pauli strings suffice to preserve the rank of the projected CM with high probability.
  • Theorem 2: Demonstrates that without randomized preprocessing, $\Omega(d)$ settings are necessary in the worst case (e.g., when Schmidt vectors are computational basis states).
  • Theorem 3: Establishes that high-order approximate unitary designs (specifically 2q-designs with $q = \Theta(\log(d\chi\eta^{-1}))$) can replace Haar random unitaries, requiring only polylogarithmic circuit depth.
• Robustness: The protocol is proven sound against uncorrelated local Pauli noise, as such noise acts diagonally in the Pauli basis and does not increase the rank of the correlation matrix. However, the author notes that globally correlated noise (e.g., depolarizing noise) can inflate the observed rank by one, requiring a conservative adjustment in certification thresholds.
• Numerical Validation: Simulations on random Matrix Product States (MPS) with $2n=28$ qubits and $\chi=4$ confirm that the randomized preprocessing makes the rank of the projected CM visible and robust against local Pauli noise, whereas the protocol fails without it.

Significance and Claims
The paper claims to establish a measurement-setting-efficient approach to high-dimensional entanglement certification. Its primary significance lies in resolving the trade-off between basis independence and experimental feasibility. By leveraging random local unitaries, the method avoids the exponential scaling of measurement settings associated with full tomography or worst-case basis-independent methods, while remaining robust to local basis misalignment.

The author positions this work as complementary to fidelity-based methods. While fidelity estimation is efficient when the target state is known, it fails under unknown local rotations. In contrast, this protocol requires no prior knowledge of the Schmidt basis and remains effective for states that are difficult to predict or represent classically. The author also suggests that the projected CM viewpoint and the incoherence parameter may have broader applications in testing structural properties like nonstabilizerness and operator spreading, drawing conceptual parallels to out-of-time-ordered correlators (OTOCs).