Implementation of quantum measurements using classical resources and only a single ancillary qubit

The work addresses how to implement general quantum measurements (POVMs) on finite-dimensional systems with minimal quantum resources. While many platforms realize projective measurements natively, numerous applications require nonprojective POVMs. Standard constructions (e.g., Naimark’s dilation) typically need auxiliary systems scaling with the system size. The research question is whether arbitrary POVMs on a d-dimensional system can be implemented using only a single ancillary qubit with a constant overhead in sampling complexity. The study develops a probabilistic simulation of general POVMs via d-outcome measurements and postselection, investigates the achievable success probability, and explores implications for resource costs and performance in near-term, noisy quantum devices.

Naimark’s dilation theorem and its generalizations establish that general measurements on N qubits may require N auxiliary qubits when projective measurements are allowed on the combined system. Prior work showed that POVMs can be simulated by convex combinations of projective measurements with classical post-processing and, with postselection, arbitrary POVMs can be implemented using only projective measurements, albeit probabilistically. The resource-theoretic view of measurement simulability characterizes how far a given POVM is from subsets of measurements realizable with bounded outcomes and relates to notions such as critical visibility and robustness. These frameworks inform the success probability attainable via limited-outcome simulators and bound the advantage general POVMs can provide in tasks like state discrimination. The present work builds on these foundations, extending simulation from dichotomic to m-outcome measurements, connecting success probability to operator norms of effects, and analyzing typical (Haar-random) rank-one POVMs.

- General simulation protocol: For an n-outcome POVM M=(M1,...,Mn) on C^d, choose m≤d and partition the outcome set [n] into disjoint subsets Xi with |Xi|≤m−1. For each subset Xi construct an auxiliary (|Xi|+1)-outcome POVM Ni whose effects reproduce λi Mj for j∈Xi and aggregate the rest into a trash outcome n+1 with effect I − λi Σ_{j∈Xi} Mj, where λi is chosen so Ni is a valid POVM. Mix the Ni according to suitable probabilities to obtain an overall POVM L that satisfies Lj = q_succ Mj for j∈[n], thus simulating M with postselection probability q_succ. - Success probability: Theorem 1 shows the scheme implements M with success probability q_succ = (Σ_i ||Mi||)^−1, where ||·|| denotes operator norm. When rank(Mi)≤1 and m≤d, each Ni can be implemented by projective measurements in dimension at most 2d, i.e., requiring only a single auxiliary qubit relative to the d-dimensional system. - Universality: Any POVM on C^d can be decomposed into a convex combination of rank-one POVMs with at most d^2 outcomes followed by post-processing. Applying the scheme to each rank-one component yields a universal implementation using a single ancillary qubit, at the cost of probabilistic success with rate q_succ determined by the chosen partition. - Conjecture: For arbitrary extremal rank-one POVMs M on C^d, there exists a partition with |Xi|≤d−1 such that q_succ is bounded below by a positive constant independent of d. - Analytical results for Haar-random POVMs: Define Haar-random rank-one POVMs via Naimark-inspired construction using a Haar-random unitary on an ancillary-extended Hilbert space and computational-basis measurement. Theorem 2 establishes scaling bounds for q_succ under the standard partition: when simulating with m-outcome measurements, q_succ ≳ O(m/d) for large d, and any protocol satisfies q^{(m)}(M) ≲ O(log d / m). For m=d, with overwhelming probability q_succ exceeds a constant (≥6.74%). Proof techniques employ concentration of measure on U(n), ε-net discretizations, and robustness bounds via state discrimination. - Numerical study: Compute q_succ from Eq. (1) for SIC-POVMs, a family of IC-POVMs, and Haar-random d^2-outcome POVMs up to d=1299, optimizing over up to 24 random partitions per instance. - Noise analysis: Adopt a global depolarizing visibility model η for random circuits. Compare Naimark’s dilation versus the proposed scheme using the total variation distance between ideal and noisy measurement statistics. Proposition 1 provides a lower bound on the average worst-case distance: max_ρ d_TV(p(M^U), p(M^{U,η})) ≥ (1−η) c_Haar with c_Haar ≈ 1/e. Gate-count scaling estimates assume dominant two-qubit errors: Naimark requires circuits on 2N qubits (d=2^N) with g2 = O(16^N), while the proposed scheme uses N+1 qubits with g2 = O(4^N), implying substantially higher visibility and lower statistical deviation for the proposed approach. A bound shows postselection does not significantly worsen the distribution quality for typical Haar-random POVMs.

- Constructive scheme: A general n-outcome POVM M can be simulated via convex combinations of m-outcome measurements and postselection with success probability q_succ = (Σ_i ||Mi||)^−1. For rank-one effects and m≤d, each simulator is implementable with projective measurements in dimension ≤2d, i.e., only one ancillary qubit beyond the d-dimensional system. - Universality with single ancilla: Since any POVM decomposes into rank-one POVMs with ≤d^2 outcomes, the scheme enables universal POVM implementation on C^d with a single auxiliary qubit and constant overhead if q_succ is lower bounded by a dimension-independent constant. - Conjecture support: Analytical results for rank-one Haar-random POVMs show that for m=d the success probability is bounded below by a constant; specifically, with overwhelming probability q_succ ≥ 6.74%, and numerically ≈25%. More generally, q_succ scales like O(m/d) for large d under the standard partition, while any protocol satisfies q^{(m)}(M) ≤ O(log d / m). - Numerical evidence: For d^2-outcome POVMs up to d=1299, q_succ approaches ≈25% for SIC-POVMs and Haar-random POVMs, and remains ≥≈20% for the considered IC-POVM family, optimizing over up to 24 random partitions. - Resource-theoretic implications: Bounds relate q^{(m)}(M) to critical visibility and robustness against m-outcome simulation. A constant lower bound on q^{(d)} implies no asymptotically increasing advantage of general POVMs over d-outcome simulable measurements for general quantum state discrimination tasks. - Noise robustness: Under visibility η≈exp(−r2 g2), the proposed scheme uses O(4^N) two-qubit gates versus O(16^N) for Naimark’s implementation (d=2^N). Proposition 1 yields a worst-case TV distance lower bound proportional to (1−η)/e, and analysis shows postselection minimally impacts distribution quality. Consequently, the proposed scheme achieves exponentially higher fidelity than Naimark’s for typical random POVMs.

The results demonstrate that general POVMs on C^d can be simulated using only d-outcome measurements, classical randomness, and postselection with a success probability that, for broad classes of POVMs, is bounded below by a constant independent of d. This addresses the central question of reducing quantum resources: a single auxiliary qubit suffices to implement arbitrary nonadaptive measurement protocols with only constant sampling overhead. Analytical bounds for Haar-random POVMs and extensive numerics for SIC and IC ensembles support the conjecture’s validity and suggest that typical nonprojective measurements do not provide an asymptotically growing advantage over d-outcome simulable measurements in tasks like state discrimination. Practically, the scheme offers better noise performance than Naimark-based implementations due to the smaller Hilbert-space extension, yielding shorter circuits and significantly improved visibility. From a resource-theory perspective, the success probability bounds translate into limits on robustness and critical visibility, constraining the operational advantage of general POVMs under outcome constraints. The approach is also relevant to nonlocality and randomness generation, where it tightens tolerable noise thresholds and may inform local models for general POVM measurements.

This work introduces a practical and universal scheme to implement arbitrary POVMs using classical randomness, post-processing, and postselection from m-outcome measurements, with m=d realizable using only a single ancillary qubit. The scheme’s success probability has a simple operator-norm expression and is conjectured to be bounded below by a dimension-independent constant for extremal rank-one POVMs, a claim proven for typical Haar-random POVMs and supported numerically for SIC and IC families up to dimension 1299. Noise analysis indicates a substantial advantage over Naimark’s dilation due to dramatically reduced circuit size. Future directions include: (1) proving the conjecture in full generality; (2) devising efficient algorithms to find optimal partitions and compile circuits from given POVM descriptions; (3) quantifying real-time costs of randomization and postprocessing and optimizing implementations accordingly; and (4) investigating relations between success probability and other POVM properties (e.g., entanglement cost).

- The central conjecture (dimension-independent constant lower bound on q_succ for all extremal rank-one POVMs) is not proven generally; rigorous guarantees are established only for typical Haar-random POVMs. - Optimal partition selection to maximize q_succ is a difficult combinatorial problem; the numerical study uses optimization over a limited number of random partitions. - Numerical evidence, while extensive, covers specific POVM families and random instances up to finite dimensions. - The noise analysis employs simplified models (global depolarizing visibility and gate-count scaling); comprehensive assessments under realistic hardware-specific noise remain for future work. - The scheme is probabilistic and requires postselection, discarding a fraction of outcomes, which may be costly for tasks sensitive to sampling efficiency.