Quantum measurements are fundamental to quantum information protocols, extracting classical information from quantum systems. While projective measurements are readily implemented on many platforms, many applications necessitate more general quantum measurements, known as Positive Operator-Valued Measures (POVMs). Implementing POVMs typically requires additional resources. A generalized Naimark's dilation theorem indicates that a general measurement on *N* qubits demands *N* auxiliary qubits when projective measurements are used randomly on the combined system. This paper focuses on reducing these resources, specifically aiming to minimize the number of auxiliary qubits needed for implementing arbitrary POVMs. A key question revolves around the quantitative advantage of generalized measurements over simpler projective ones. While POVMs are essential for tasks like quantum state discrimination, tomography, metrology, and randomness generation, it's unclear what advantage they provide over projective measurements, particularly given that non-projective measurements can be realized via randomization and post-processing of simpler measurements. This work investigates the relationship between *d*-outcome POVMs and general POVMs on a *d*-dimensional Hilbert space. The authors aim to demonstrate that general quantum measurements don't offer an asymptotically increasing advantage over *d*-outcome POVMs for quantum state discrimination as *d* grows. They generalize a POVM simulation method from previous work, employing randomized implementation of restricted-class POVMs, followed by post-processing and postselection. The central goal is to show that a protocol exists to simulate a broad class of POVMs using *d*-outcome POVMs and postselection, with a success probability that remains above a constant irrespective of the dimension *d*. The method uses *d*-outcome POVMs implemented via projective measurements in a 2*d* dimensional Hilbert space, effectively using a single auxiliary qubit. Existing schemes require von Neumann instruments on a system extended by a single auxiliary qubit. This new scheme is simpler as it only requires a single projective measurement in each round, eliminating the need to consider post-measurement states. The authors conjecture that the success probability remains above a dimension-independent constant for all POVMs and provide evidence through analytical and numerical results.

The paper extensively cites existing literature on quantum measurements and their applications. Key references discuss quantum error-correcting codes, measurement-based quantum computation, quantum communication, quantum state discrimination, quantum metrology, and randomness generation. The authors review previous work on Naimark's dilation theorem and its implications for implementing POVMs. They also discuss the simulation of POVMs via randomized implementations of restricted classes of POVMs and the use of post-processing and postselection techniques. The literature review highlights the open question of the quantitative advantage of POVMs over projective measurements, particularly in the context of state discrimination and other quantum information tasks. The authors contextualize their work within existing research on POVM simulability and resource theories, referencing studies on the simulability of POVMs using measurements with a bounded number of outcomes and the relationship between simulability and quantities like white noise critical visibility and robustness.

The core methodology revolves around a novel POVM simulation protocol. This protocol utilizes probabilistic implementation of POVMs with a limited number of outcomes (at most *m*), coupled with classical post-processing and postselection. The authors present Theorem 1, which establishes a lower bound on the success probability of simulating an *n*-outcome POVM using measurements with at most *m* outcomes and postselection. The success probability, *q*_succ, is defined as (Σⁿᵢ₌₁ ||Mᵢ||)⁻¹, where Mᵢ are the effects of the target POVM and ||Mᵢ|| denotes the operator norm. The proof constructs auxiliary measurements Nᵢ, each with *m* + 1 outcomes, designed to mimic the target measurement for specific subsets of outcomes. A convex combination of these measurements, weighted according to a probability distribution, simulates the target POVM with the calculated success probability. The theorem establishes that if the effects of the target POVM are rank-one, and *m* ≤ *d*, the protocol can be implemented using projective measurements in a 2*d* dimensional space, which can be achieved with a single ancillary qubit. The authors further discuss how arbitrary POVMs can be simulated using this protocol by decomposing them into convex combinations of rank-one POVMs. This leads to their central conjecture: for arbitrary extremal rank-one POVMs, there exists a partition of outcomes such that the success probability *q*_succ is bounded below by a positive constant, independent of the dimension *d*. This conjecture's validity would imply that any nonadaptive measurement protocol can be realized using only a single ancillary qubit with constant sampling overhead. The authors then investigate this conjecture through analytical and numerical methods. They analyze Haar-random POVMs, proving in Theorem 2 that the success probability scales as *m*/d for simulating Haar-random POVMs using *m*-outcome measurements. They also present numerical results for SIC-POVMs and IC-POVMs in dimensions up to 1299, showing consistently high success probabilities. Finally, they analyze the impact of noise on the implementation, comparing their proposed method with the standard Naimark's scheme using a depolarizing noise model. Proposition 1 provides a lower bound for the average worst-case distance between ideal and noisy implementation of Haar-random POVMs. The authors find that their scheme exhibits exponentially higher fidelity than the Naimark's scheme due to the reduced number of ancillary qubits.

The paper's key findings center around the proposed POVM simulation scheme and the supporting evidence for its efficiency and robustness. Theorem 1 provides a general lower bound on the success probability of simulating POVMs using measurements with a bounded number of outcomes and postselection. The success probability is shown to be inversely proportional to the sum of the operator norms of the POVM effects. Theorem 2 analyzes the performance of the scheme for Haar-random POVMs, showing that the success probability scales as m/d, where m is the number of outcomes in the simulating POVM and d is the dimension. This result highlights the scalability of the proposed method. Numerical simulations on SIC-POVMs and IC-POVMs in high dimensions (up to 1299) demonstrate consistently high success probabilities, typically around 20-25%, further supporting the conjecture that the success probability remains above a dimension-independent constant. The noise analysis, comparing the proposed scheme to Naimark's dilation, reveals that the new method achieves exponentially higher fidelity in the presence of noise due to the significantly lower number of required two-qubit gates. This is evidenced by Proposition 1 which provides a lower bound for the average worst-case distance between ideal and noisy implementation of Haar-random POVMs. The observed high success probabilities in the numerical experiments, combined with the analytical results, provide strong evidence in favor of the authors' central conjecture.

The findings suggest that general POVMs do not offer a significant asymptotic advantage over *d*-outcome POVMs for quantum state discrimination, at least in terms of sampling complexity. The high success probability of the proposed simulation scheme implies that the resources required for implementing general POVMs can be significantly reduced, using only a single auxiliary qubit and classical post-processing. This has important implications for experimental implementations of POVMs in near-term quantum devices where noise is a major limiting factor. The reduced number of qubits needed for implementation leads to exponentially lower noise compared to standard methods. The paper's conjecture, if proven true, would represent a significant step forward in our understanding of quantum measurement resource theory and its applications. The results also have implications for other areas, such as nonlocality, quantum randomness generation and the construction of local models for entangled quantum states.

This paper introduces a novel scheme for implementing general quantum measurements using only classical resources and a single ancillary qubit. The method's efficiency is supported by both analytical and numerical results, indicating a consistently high success probability irrespective of the dimension. This finding challenges the notion of an asymptotic advantage of general POVMs over *d*-outcome measurements in sampling-based quantum information tasks. Future work could focus on proving the central conjecture, developing algorithms for automated circuit generation, and further exploring the scheme's applications in various quantum information processing tasks.

The main limitation lies in the unproven central conjecture that the success probability remains above a constant independent of the dimension for all POVMs. While the numerical simulations provide strong evidence, a rigorous mathematical proof remains an open problem. The noise analysis relies on a specific depolarizing noise model; investigating more realistic noise models would enhance the applicability of the results. The numerical simulations were performed on a limited set of partitions; exploring a broader range of partitions might reveal even higher success probabilities. Finally, the real-time cost of randomization and post-processing remains to be fully quantified.