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Pretty-good simulation of all quantum measurements by projective measurements

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Pith's one-line read The paper proves that adding a fixed amount of depolarizing noise, with visibility $c = 0.02$, makes every generalized quantum measurement (POVM) simulable by randomized projective measurements in any finite dimension, with no ancillary…

desk verdict Constant-factor simulation of arbitrary POVMs by projective measurements, resolving a conjecture with q=1/8 and c=0.02; the main caveat is the non-constructive Kadison-Singer dependence. read the letter →

arxiv 2501.09339 v1 pith:7XJJ3DCH submitted 2025-01-16 quant-ph math-phmath.MP

Micha{l} Kotowski , Micha{l} Oszmaniec This is my paper · ORCID
classification quant-phmath-phmath.MP MSC 81P1581P45 PACS 03.65.Ta03.67.-a
keywords generalizedquantummeasurementsprojectivesimulationdepolarizingnoiseKadison-SingerproblemNaimarkextensionpostselectionmeasurementincompatibilitycircuitknitting
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper asks how much the most general quantum measurements (POVMs) can outperform the simple projective measurements of textbook quantum mechanics. It answers that the advantage is bounded by a constant factor: after applying depolarizing noise with a fixed visibility $c = 0.02$, every POVM on a $d$-dimensional system becomes a convex combination of projective measurements, with no ancilla needed. It also shows that every POVM can be simulated with constant postselection probability $q = 1/8$ by measurements that use just one auxiliary qubit. Because these constants do not grow with $d$, general measurements cannot offer a dimension-dependent asymptotic edge in state discrimination, shadow tomography, or other linear tasks; the paper also derives new bounds on local hidden variable models and measurement incompatibility.

What carries the argument

The argument runs through three pieces. (1) A fine-graining lemma converts any POVM into one with rank-one effects of nearly equal magnitude. (2) The solution of the Kadison-Singer problem is invoked as a partition theorem: for any such nearly-flat POVM, the outcome set can be partitioned into blocks of size at most $d/2$ so that the simulation protocol of Theorem 2 succeeds with probability $q \ge 0.068/(1+\delta)$; this is the step that makes $q$ independent of $d$. (3) The dimension-deficient Naimark theorem shows that a nearly projective measurement with at most $d/2$ rank-one effects can be twirled, by random phases and unitaries on the orthogonal complement of the span of its effects, into a genuine convex combination of projective measurements, and an auxiliary lemma quantifies the amount of depolarizing noise tolerated by such measurements. Pasting the post-processing through these steps yields the constant $c = 0.02$.

What would settle it

Compute $t_{\mathrm{SP}}(M) = \max\{t : \Phi_t(M) \in \mathrm{SP}(\mathbb{C}^d)\}$ for a fixed sequence of POVMs, for example the symmetric informationally complete POVMs in increasing dimension; the claim predicts a uniform lower bound of $0.02$ for every $d$, so finding any $M_d$ with $t_{\mathrm{SP}}(M_d) < 0.02$, or a numerical violation of $q = 1/8$ for the single-qubit simulation, would refute Result 2.

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Extended reading notes

Core claim

The paper establishes two dimension-independent simulation theorems. First, every $d$-dimensional POVM can be realized, with postselection probability $q = 1/8$, as a convex combination of measurements that each require only a single ancillary qubit plus classical post-processing. Second, the depolarized measurement $\Phi_c(M)$ with $c = 0.02$ belongs to $\mathrm{SP}(\mathbb{C}^d)$ for every POVM $M$: it is exactly a convex combination of projective measurements on $\mathbb{C}^d$ itself, needing no ancilla. The constants $q$ and $c$ do not depend on $d$, so the asymptotic power of general measurements over projective ones is bounded by a constant factor in every linear information-processing task considered, and the paper derives consequences for state discrimination, shadow tomography, circuit knitting, local hidden variable models, and joint measurability of noisy POVMs.

Load-bearing premise

The argument stands on the existence, asserted by the Kadison-Singer solution, of a partition of the outcomes of any nearly-flat rank-one measurement into blocks of size at most $d/2$ whose simulation success probability stays constant as the dimension grows, and nothing in the paper provides an efficient way to find that partition.

Editorial extensions

If this is right

  • In minimal-error state discrimination, the best POVM can beat the best projective measurement by at most the factor $1/c = 50$ in success probability, for any ensemble in any dimension.
  • In classical shadow tomography with traceless observables, a general POVM can reduce the worst-case variance bound by at most the constant factor $1/c^2 = 2500$ compared with a projectively simulable measurement.
  • Noisy isotropic states of two qudits admit local hidden variable models for all POVMs up to visibility $c\log(d)/d$, and general noisy pure states up to $c\log(d)/d^2$, improving over previous results.
  • All POVMs on $\mathbb{C}^d$ become jointly measurable after depolarizing with visibility $t \le c\log(d)/d$, a bound that matches the known projective-measurement scaling up to the constant $c$.
  • Sampling from the output of any $2N$-qubit unitary can be emulated by randomization over $N+1$-qubit subcircuits with success probability $q = 1/8$, giving a circuit-knitting scheme with constant probabilistic overhead.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the result is correct, any task whose figure of merit is a linear functional of measurement statistics can separate POVMs from projective measurements by at most a constant factor in any dimension, ruling out exponential POVM advantages in principle.
  • The partition guaranteed by the Kadison-Singer solution is not known to be efficiently findable; the paper shows random partitions achieve $q \approx \Theta(1/\log d)$, so determining whether constant-$q$ partitions can be found in polynomial time would decide whether the circuit-knitting scheme becomes a practical algorithm.
  • The dimension-deficient technique may extend from measurements to quantum channels, instruments, and combs, suggesting that constant-factor simulation of general processes by low-ancilla devices is a general structural phenomenon rather than a special property of measurements.
  • The joint-measurability bound is tight in $d$ up to the constant $c$, making it natural to test numerically whether $c$ can be pushed to 1 for specific families; if it can, POVMs would share exactly the same compatibility noise threshold as projective measurements.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 5 minor

Summary. The paper proves two structural results about generalized quantum measurements. Result 1 states that every POVM on C^d can be simulated, up to a constant postselection probability q = 1/8, by a convex combination of POVMs each implementable with a single auxiliary qubit and classical post-processing. Result 2 states that for every POVM M, the depolarized measurement Φ_{0.02}(M) belongs to SP(d), i.e., it is a convex combination of ancilla-free projective measurements with classical post-processing. The proof proceeds by fine-graining an arbitrary POVM into a nearly flat rank-one POVM (Lemma 1), using the Marcus–Spielman–Srivastava solution of Kadison–Singer to partition the outcomes into small groups with constant success probability (Theorem 4), and then using a new dimension-deficient Naimark theorem to implement the resulting nearly projective POVMs by projective measurements under depolarizing noise (Lemma 3 / Theorem 6). The paper derives applications to state discrimination, shadow tomography, circuit knitting, hidden-variable models for noisy two-qudit states, and joint measurability of noisy POVMs.

Significance. The significance is high: the paper settles in the affirmative a conjecture from the authors' earlier work [28] and establishes that, asymptotically in dimension, general POVMs offer only a constant-factor advantage over projective measurements (with or without a single-qubit ancilla) for the tasks considered. The proof is detailed and the main logical chain is internally consistent; the constants c = 0.02 and q = 1/8 are derived from the construction rather than fitted. The reliance on the non-constructive Kadison–Singer theorem is explicitly disclosed, and Appendix C provides an efficiently generatable random partition achieving q = Θ(1/log d), so the algorithmic limitation is clearly separated from the existential claims. The applications (improved POVM-locality ranges, tight incompatibility robustness up to constant factors, and the circuit-knitting scheme) are nontrivial and correctly derived from the main results.

minor comments (5)
  1. [Section IV, Step 3] The displayed bound t_NP ≥ 0.3/(1+δ) is not correct for all δ ∈ (0,1]. Using A_i ≥ 0.47/(1+δ) and the worst case |W| = |W⊥| = d/2 in Lemma 3 gives t_NP ≥ 0.47/(2(1+δ)−0.47), which is below 0.3/(1+δ) for δ ≳ 0.085. Since the final constant c = 0.02 only requires sufficiently small δ, the main result is unaffected, but the bound should be restated with its validity range or replaced by the exact expression.
  2. [Appendix A.1, Lemma 4] When x_i is rational and k_i is chosen so that k_i x_i is an integer, the remainder α_i in Eq. (A2) equals 0, contradicting the stated range 1−Δ ≤ α_i ≤ 1. The intended construction still works if the zero-size remainder is discarded (or if one instead uses a common-denominator construction of equal parts of size 1/k), but the proof as written is internally inconsistent.
  3. [Section IV, Lemma 3] The statement says Φ_t(N) ∈ SP(d) for t equal to the displayed minimum; it should say 'for all t ≤ ...' to match Lemma 7 and to justify the later use of mixing with Φ_0(N).
  4. [Section V, proof of Lemma 2] The passage treating C as a free parameter via 'C can be effectively regarded as an unconstrained real parameter' is hand-wavy; a rigorous argument should fix C first and then choose ε sufficiently small and an integer r with rε close to C, using continuity of the bounds in C.
  5. [Section III.D, Proposition 3] There are several typos, including a missing period after 'Let ρ be a N-qubit state' and duplicated words ('for for', 'by by'); also 'transforming classical states on N+1 bits into states on 2N bits' should read 'transforming probability distributions on N+1 bits into probability distributions on 2N bits'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: central constants are derived, not fitted, and the external Kadison–Singer input is independent of the claimed conclusion.

full rationale

The derivation chain is self-contained in the relevant sense. Result 2 follows from Lemma 1 (nearly flat fine-graining), Theorem 4 (existence of a good partition via the Marcus–Spielman–Srivastava solution to Kadison–Singer), Lemma 2, and Lemma 3/Theorem 6 (dimension-deficient Naimark theorem). Result 1 follows from Theorem 4 and Theorem 5. The constants c = 0.02 and q = 1/8 are obtained by explicit inequalities such as q(M,S) ≥ (1+√C)^{-2} and t_N = min_i |W^⊥|A_i/(|W|(1−A_i)+|W^⊥|), with parameter choices C = 5, κ = 1/2; they are not fitted to data or extracted from the target result. The only external mathematical input is the Kadison–Singer theorem, which has stated assumptions that do not include the target claims. The prior protocol [28] is cited and used, but the paper's contribution is precisely to prove the existence of a partition with constant success probability that [28] had left as a conjecture; the conclusion is not assumed. The non-constructive nature of the partition is explicitly disclosed in Section III.G and Appendix C, and it affects algorithmic applicability, not the validity of the existence claims. No self-citation chain is load-bearing for the central theorem, and no definition smuggles in the conclusion. Hence no circularity.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central claims rest on standard quantum measurement theory, the proven Kadison-Singer theorem, and the authors' prior simulation protocol. The only hand-chosen numbers are proof parameters C and κ, which set the specific constants 0.02 and 1/8 but are not fitted to data. No new physical entities are introduced.

free parameters (2)
  • Kadison-Singer partition parameter C = 5 for Result 2; 1 for Result 1 (k=2 case)
    Chosen by hand in Lemma 5 to optimize the product c = q · t_NP in the proof of Result 2. The reported constants c = 0.02 and q = 1/8 are derived from these choices; it is a proof parameter, not a fit to experimental data.
  • Subpartition size fraction κ = 1/2 (Result 2 proof); 1 or 1−1/d (Result 1, k=2)
    Controls the maximum subset size |S'_β| ≤ κd. Choosing κ = 1/2 enables the dimension-deficient Naimark theorem (l ≤ d/2); the k=2 case requires κ close to 1. Chosen by hand as a proof parameter.
assumptions (6)
  • standard math Naimark extension theorem (Theorem 1 of the paper)
    Used in the standard realization of POVMs via projective measurements on extended spaces; foundational in quantum measurement theory.
  • standard math Marcus-Spielman-Srivastava solution to Kadison-Singer in POVM form (Theorem 3, from [38])
    Gives the partition norm bound that drives Theorem 4 and hence the dimension-independent constants.
  • domain assumption Simulation protocol of [28] (Theorem 2 of the paper)
    The probabilistic POVM simulation protocol with success probability q(M,S) = (Σ_β ∥Σ_{i∈Sβ} M_i∥)^{-1}; taken as a black-box tool.
  • domain assumption Local hidden variable models for projective measurements from [29,30]
    Used in Proposition 4 to construct POVM-local models for isotropic and general pure states; these models allow arbitrary POVMs on Alice's side.
  • domain assumption Reduction of continuous-outcome POVMs to at most d^2 outcomes [49]
    Used in Remark 1 to extend finite-outcome results to general continuous-outcome measurements.
  • standard math Self-duality of the depolarizing channel and its commutation with coarse-graining
    The central mechanism for moving depolarizing noise from measurements to states (Eq. (5) and surrounding text).

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Cite this review

Pith. "Pith review of Pretty-good simulation of all quantum measurements by projective measurements." pith.science (2026). https://pith.science/paper/7XJJ3DCH

@misc{pith2026250109339,
  author       = {Pith},
  title        = {Pith review of: Pretty-good simulation of all quantum measurements by projective measurements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7XJJ3DCH}},
  note         = {Machine review of arXiv:2501.09339}
}
abstract

In quantum theory general measurements are described by so-called Positive Operator-Valued Measures (POVMs). We show that in $d$-dimensional quantum systems an application of depolarizing noise with constant (independent of $d$) visibility parameter makes any POVM simulable by a randomized implementation of projective measurements that do not require any auxiliary systems to be realized. This result significantly limits the asymptotic advantage that POVMs can offer over projective measurements in various information-processing tasks, including state discrimination, shadow tomography or quantum metrology. We also apply our findings to questions originating from quantum foundations by asymptotically improving the range of visibilities for which noisy pure states of two qudits admit a local model for generalized measurements. As a byproduct, we give asymptotically tight (in terms of dimension) bounds on critical visibility for which all POVMs are jointly measurable. On the technical side we use recent advances in POVM simulation, the solution to the celebrated Kadison-Singer problem, and a method of approximate implementation of nearly projective POVMs by a convex combination of projective measurements, which we call dimension-deficient Naimark theorem. Finally, some of our intermediate results show (on information-theoretic grounds) the existence of circuit-knitting strategies allowing to simulate general $2N$ qubit circuits by randomization of subcircuits operating on $N+1$ qubit systems, with a constant (independent of $N$) probabilistic overhead.

Figures

Figures reproduced from arXiv: 2501.09339 by the authors.

Figure 1
Figure 1. FIG. 1. We tackle the problem of minimizing the size of the ancilla space necessary to implement a general POVM [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. A circuit knitting method originating from the POVM simulation protocol in Result [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Structure of the main results of the paper. Some of the arrows are labelled by auxiliary lemmas used in the proof of the respective result. We [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

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Reference graph

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.