Efficient Verification of Entangled Measurements with Local States

Abstract: We develop a framework for quantum measurement verification (QMV) using only local state preparations. For locally transitive and irreducible projective measurements, we prove that symmetry reduces locality constrained QMV to quantum state verification of a single basis state, thereby reducing protocol design to the optimization of homogeneous verification operators. We apply the framework to generalized Bell measurements, single-parameter measurements on two qubits, elegant joint measurements, and stabilizer state induced measurements, and derive explicit local protocols together with closed form verification operators, success probabilities, and sample complexities. We further show that homogeneous QMV protocols can estimate measurement fidelity directly from observed passing frequencies.

• The paper establishes a practical quantum measurement verification framework by reducing the problem to a single quantum state verification using local product states.
• It leverages group-theoretic analysis and stabilizer-twirling to design efficient protocols with tight failure probability and sample complexity bounds.
• The approach enables direct measurement fidelity estimation with robust performance, crucial for applications in quantum communication and device certification.

Efficient Verification of Entangled Measurements with Local States

Introduction and Motivation

The verification of quantum measurements is critical in contemporary quantum information processing, where practical hardware is vulnerable to imperfect state preparation and noisy measurement operations. Quantum measurement verification (QMV) addresses the need for certifying that a measurement device implements a specified measurement protocol, especially for nonlocal, entangled measurements, using physically feasible input states. Complete measurement tomography is resource-intensive; the paper establishes a practical QMV framework constructed solely from local product states, ensuring efficiency under the relevant experimental constraints (2606.21355).

Framework for Local Quantum Measurement Verification

The paper formulates QMV as a statistical hypothesis test distinguishing an ideal measurement from a faulty measurement channel, characterized by a positive-operator-valued measure (POVM) $\{M_\theta\}_{\theta \in \Theta}$. The verification protocol uses only local product state preparations and accepts outcomes according to the support induced by the ideal measurement. Crucially, the protocol leverages symmetry properties—local transitivity and irreducibility of the von Neumann measurement—to reduce the locality-constrained QMV optimization to a single quantum state verification (QSV) problem. Under these symmetries, protocol design simplifies to minimizing the second largest eigenvalue of a homogeneous verification operator. This reduction is formally established via group-theoretic analysis and stabilizer-twirling arguments, offering analytic sample complexity bounds.

Explicit Local Verification Protocols for Representative Measurements

The framework is instantiated for four important classes of entangled measurements:

• Generalized Bell Measurement (GBM): For qudits of prime local dimension, a protocol based solely on mutually unbiased bases (MUBs) is shown to achieve failure probability scaling as $P(\varepsilon) = 1 - \frac{d}{d+1}\varepsilon$, with sample complexity $$N \approx \frac{d+1}{d} \frac{1}{\varepsilon}\log(1/\delta)$$. The locality constraint incurs only a constant factor penalty compared to the unconstrained optimal scaling.
• Single-parameter measurements on two qubits: The constructed local protocol maintains the same asymptotic scaling as the Bell measurement except at product endpoints, where unconstrained strategies recover the optimal scaling.
• Elegant Joint Measurement (EJM): For partial entanglement, the homogeneous protocol exhibits a parameter-dependent penalty, $$P(\varepsilon) = 1 - \frac{2}{3(2-\sin\kappa)}\varepsilon$$. At maximal entanglement, the protocol recovers Bell measurement efficiency.
• Stabilizer State-Induced Measurements: On $n$ qudits, the protocol based on stabilizer generators yields $$P(\varepsilon) = 1 - \frac{d^n - d^{n-1}}{d^n-1}\varepsilon$$ and optimal scaling relative to stabilizer state verification.

Each construction provides explicit local test states, verification operators, acceptance criteria, and analytic expressions for sample complexity. The symmetry reduction is proven to apply broadly under local transitivity and irreducibility, allowing optimality arguments to use stabilizer and MUB identities.

Direct Estimation of Measurement Fidelity

Homogeneous QMV protocols permit direct estimation of measurement fidelity via observed passing frequencies. The expected passing rate, under arbitrary measurement, is affine in the fidelity: $p_{\mathrm{pass}} = \beta + (1-\beta)F$, where $\beta$ is determined by the protocol. The estimator is unbiased, with standard deviation bounded as $$\Delta(\widehat{F}) \leq \frac{1}{2(1-\beta)\sqrt{N}}$$; thus sample complexity for estimation is $N = O(1/\varepsilon_{\mathrm{est}}^2)$, which is quadratically worse than verification, but offers robustness and operational simplicity for practical deployment.

Implications and Outlook

The symmetry-driven reduction of QMV enables efficient certification of entangled measurements using local inputs, under realistic constraints, with analytic guarantees on failure probability and sample complexity. The results imply feasibility for device-level certification of measurements crucial for protocols such as teleportation, entanglement swapping, and quantum communication. Optimally efficient verification, previously available for pure state and process certification, is now extended to measurement certification under locality, closing the theoretical loop for verification theory.

Future directions include extending the reduction to measurements lacking local transitivity or irreducibility, incorporating imperfect test state preparation, non-i.i.d. device behavior, and rigorous finite-sample confidence bounds. These developments are essential for scaling QMV protocols to large-scale and heterogeneous quantum architectures.

Conclusion

The paper synthesizes symmetry-driven verification theory to yield efficient local protocols for entangled measurement certification, providing explicit constructions, tight failure probability bounds, and sample complexities. The symmetry reduction unifies QMV under the broader verification framework, with practical implications for robust quantum computing and communication platforms (2606.21355).