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arxiv: 2605.18291 · v1 · pith:TN2CVVNFnew · submitted 2026-05-18 · 🪐 quant-ph

Quantum randomness beyond projective measurements

Fionnuala Curran This is my paper

Pith reviewed 2026-05-20 10:08 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum randomnessPOVMSIC measurementextremal measurementsdevice-independent randomnessguessing probabilityrandomness certification
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The pith

Unbiased extremal rank-one measurements generate up to the maximum possible quantum randomness in dimensions with SICs.

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

This paper characterizes the amount of intrinsic randomness extractable from any unbiased extremal rank-one POVM acting on any quantum state. It gives a complete explicit solution for the qubit case and shows that the tetrahedral SIC measurement produces the smallest randomness within the class. The authors introduce the skewed SIC family and prove that the theoretical maximum of 2 log d bits can be achieved using device-dependent or source-device-independent protocols in any dimension that admits a SIC measurement.

Core claim

We characterise the randomness generated by any unbiased extremal rank-one measurement acting on any state, solving the problem explicitly in dimension two. Four-outcome qubit measurements of this type are tomographic, so these results hold for fully source-device-dependent randomness too. The tetrahedral symmetric informationally complete (SIC) measurement has the least intrinsic randomness within this class. We also present the skewed SIC family of measurements, and use them to partially solve an open problem: we prove that 2 log d bits of randomness, the maximal amount, can be generated device-dependently (or source-device-independently) in any dimension in which there exists a SIC.

What carries the argument

Guessing probability optimization for unbiased extremal rank-one POVMs, which bounds an eavesdropper's best guess of the outcomes.

If this is right

  • Four-outcome unbiased extremal rank-one measurements on qubits are tomographic and therefore certify randomness even in fully source-device-dependent scenarios.
  • The tetrahedral SIC achieves the minimum randomness among all such four-outcome qubit measurements.
  • The skewed SIC family enables construction of protocols that reach the absolute maximum of 2 log d bits whenever a SIC exists.
  • Maximal randomness extraction holds in both device-dependent and source-device-independent settings for these measurements.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The explicit qubit formulas could be used to select measurements that maximize certified randomness output in practical random number generators.
  • Analogous characterizations for biased or non-rank-one POVMs would extend the current bounds to a wider class of measurements.
  • If SICs exist in every dimension, the result would imply that maximal randomness is always attainable with this type of measurement.

Load-bearing premise

Extremal measurements do not allow information to leak to an eavesdropper in the adversarial scenario.

What would settle it

An explicit calculation or experiment that finds a higher optimal guessing probability for any unbiased extremal rank-one POVM on a qubit than the value derived in the characterization.

Figures

Figures reproduced from arXiv: 2605.18291 by Fionnuala Curran.

Figure 1
Figure 1. Figure 1: Skewed SIC POVMs in dimension two. where Γ = s d − 1 d (1 − γ) , 0 < γ < 1 . (9) If we performed this measurement on the state ρ = Π1, we would obtain the following probability distribution, pj = tr(ρ Nj ) =    γ , j = 1 , 1−γ d 2−1 , j ∈ {2, . . . , d2} . (10) Choosing γ = 1 d 2 gives a uniform distribution, so we can achieve the bound (6). Note, too, that the probability of the obtaining outcome 1 ten… view at source ↗
Figure 2
Figure 2. Figure 2: Unbiased extremal measurements in dimension two. [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Regions of the Bloch ball for an unbiased qubit MIC. [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Two scissors measurements in dimension two. [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The space of average post-measurement states [PITH_FULL_IMAGE:figures/full_fig_p036_5.png] view at source ↗
read the original abstract

The unpredictability of quantum physics gives rise to intrinsic randomness. In an adversarial scenario, any additional degrees of freedom must be attributed to an eavesdropper with correlations to the measurement set-up. The true randomness is then quantified by the probability that she correctly guesses the measurement outcomes, optimised over all possible strategies. Extremal measurements are appealing here, since they do not allow information to leak to such an eavesdropper. Beyond projective measurements, however, a simple question remains open: how much intrinsic randomness can be generated by a given extremal measurement? In a step towards solving it, we characterise the randomness generated by any unbiased extremal rank-one measurement acting on any state, solving the problem explicitly in dimension two. Four-outcome qubit measurements of this type are tomographic, so these results hold for fully source-device-dependent randomness too. The tetrahedral symmetric informationally complete (SIC) measurement, we find, has the least intrinsic randomness within this class. We also present the skewed SIC family of measurements, and use them to partially solve an open problem: we prove that $2 \log d$ bits of randomness, the maximal amount, can be generated device-dependently (or source-device-independently) in any dimension in which there exists a SIC measurement.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript characterizes the intrinsic randomness (via guessing probability in an adversarial setting) generated by unbiased extremal rank-one POVMs acting on arbitrary states. It provides an explicit solution to this problem in dimension two. It introduces the skewed SIC family of measurements and proves that SIC measurements achieve the maximal 2 log d bits of randomness (device-dependently or source-device-independently) in any dimension admitting a SIC.

Significance. If the central derivations hold, the work constitutes a concrete step toward quantifying randomness from non-projective extremal measurements. The d=2 solution and the SIC-based construction for maximal randomness supply explicit, usable results for randomness extraction protocols in quantum information. The approach relies on standard quantum information tools (guessing-probability optimization over extremal POVMs) with no free parameters or invented entities.

major comments (1)
  1. [d=2 characterization section] The explicit solution for d=2 (stated in the abstract and presumably derived in the main text) is load-bearing for the claim that four-outcome qubit measurements are tomographic; the derivation should explicitly verify that the unbiased rank-one extremal condition suffices to close the optimization without additional assumptions on the state.
minor comments (2)
  1. [Introduction] The abstract and introduction would benefit from a brief statement of the open problem being partially solved, with a forward reference to the relevant theorem or proposition.
  2. [Skewed SIC section] Notation for the skewed SIC family should be introduced with a clear definition (e.g., via a parameter or equation) before its use in the maximal-randomness construction.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript, positive evaluation of its significance, and recommendation for minor revision. We address the single major comment below.

read point-by-point responses

  1. Referee: [d=2 characterization section] The explicit solution for d=2 (stated in the abstract and presumably derived in the main text) is load-bearing for the claim that four-outcome qubit measurements are tomographic; the derivation should explicitly verify that the unbiased rank-one extremal condition suffices to close the optimization without additional assumptions on the state.

    Authors: We appreciate the referee highlighting the need for explicit verification here. In the d=2 section, the guessing probability is optimized by parameterizing all unbiased extremal rank-one POVMs (which in dimension 2 are necessarily four-outcome) and then minimizing the maximum eigenvalue of the associated operator over all input states. The unbiasedness condition fixes the trace of each element to 1/4, while extremality and rank-one together constrain the possible directions such that the convex optimization over states closes without requiring any further restrictions (e.g., no purity or eigenstate assumptions). In the revised manuscript we have inserted a short clarifying paragraph immediately after the main derivation that makes this closure explicit, thereby confirming that the tomographic property holds for arbitrary states and that the randomness quantification is fully source-device-dependent. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central results follow from an explicit analytic characterization of the guessing probability for unbiased extremal rank-one POVMs (solved in d=2) together with a direct construction showing that SIC measurements saturate the 2 log d bound in dimensions where they exist. These steps rest on the standard SDP formulation of the adversarial guessing probability, the stated premises of unbiasedness and rank-one extremality, and known properties of SIC POVMs; none of the load-bearing claims reduce by construction to fitted parameters, self-definitions, or unverified self-citations. The derivation is therefore self-contained against external quantum-information benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the work relies on standard quantum mechanics without introducing new free parameters, axioms beyond domain assumptions, or invented entities.

axioms (1)
  • domain assumption Quantum mechanics framework for states, POVMs, and adversarial randomness quantification via guessing probability
    The paper builds directly on established quantum information concepts for extremal measurements and eavesdropper models.

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Reference graph

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    The final vectors are then ⃗ r3 = (cosδ,−sinδsinα,sinδcosα) ⃗ r4 = (cosδ,sinδsinα,−sinδcosα)

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