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  • Open Access

Optimal Fidelity Estimation from Binary Measurements for Discrete and Continuous Variable Systems

Omar Fawzi1, Aadil Oufkir2, and Robert Salzmann1,*

  • *Contact author: robert.als.salzmann@gmail.com
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PRX Quantum 7, 010309 – Published 15 January, 2026

DOI: https://doi.org/10.1103/qd1c-1fk9

Abstract

Estimating the fidelity between a desired target quantum state and an actual prepared state is essential for assessing the success of experiments. For pure target states, we use functional representations that can be measured directly and determine the number of copies of the prepared state needed for fidelity estimation. In continuous variable (CV) systems, we use the Wigner function, which can be measured via displaced parity measurements. We provide upper and lower bounds on the sample complexity required for fidelity estimation, considering the worst-case scenario across all possible prepared states. For target states of particular interest, such as Fock and Gaussian states, we find that this sample complexity is characterized by the L1-norm of the Wigner function, a measure of Wigner negativity widely studied in the literature, in particular in resource theories of quantum computation. For discrete variable systems consisting of n qubits, we explore fidelity estimation protocols using Pauli string measurements. Similarly as for the CV approach, the sample complexity is shown to be characterized by the L1-norm of the characteristic function of the target state for both Haar random states and stabilizer states. Furthermore, in a general black box model, we prove that, for any target state, the optimal sample complexity for fidelity estimation is characterized by the smoothed L1-norm of the target state. To the best of our knowledge, this is the first time the L1-norm of the Wigner function provides a lower bound on the cost of some information processing task.

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Popular Summary

As quantum technologies advance, so does the need to verify that we are preparing the quantum states we intend to. Fidelity estimation—a way to measure how close a prepared state is to a desired one—is central for this verification. But directly measuring fidelity often requires complex operations that are not feasible in practice. This work investigates how hard fidelity estimation really is when realistic measurement techniques are used and uncovers a fundamental connection: the difficulty of the task is governed by how “nonclassical” the target quantum state is.

We focus on two common experimental settings: one used for systems such as photons, and the other used for systems made of qubits. In each case, we assume we know the target state and ask how hard it is to estimate the fidelity using only basic, binary-outcome measurements. Our main finding is that the number of measurements required depends on how “non classical” the target state is—a property that has long been studied as a marker of quantum advantage. States that are more exotic in this sense are harder to verify, while more classical states can be certified with relatively few measurements.

Beyond fidelity estimation, our work introduces a general framework for analyzing the measurement cost of verifying quantum states and highlights how this cost is tied to the state’s quantum features. In particular, the techniques we develop to prove hardness offer new tools for understanding the complexity of other quantum tasks, potentially shedding light on the fundamental limits of quantum learning and computation more broadly.

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PRX Quantum 7, 010309– Published 15 January, 2026
  • Received 7 April 2025
  • Accepted 22 September 2025
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