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arxiv: 2605.16151 · v1 · pith:GZLP3MY4new · submitted 2026-05-15 · 🪐 quant-ph

Generalized measurement incompatibility

Edwin Peter Lobo , Maria Balanz\'o-Juand\'o , Stefano Pironio This is my paper

Pith reviewed 2026-05-20 18:04 UTC · model grok-4.3

classification 🪐 quant-ph
keywords measurement incompatibilitypartial joint measurabilitydevice-independent protocolsuntrusted measurement devicessemidefinite programmingdetection efficiencyquantum cryptography
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The pith

An adversary with classical side information can perfectly guess all outcomes of quantum measurements if and only if the measurements are partially jointly measurable.

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

This paper generalizes partial joint-measurability so that only chosen subsets of outcomes from each measurement need to be jointly determined by classical variables. It gives two equivalent mathematical definitions of the property and shows both can be checked by solving one semidefinite program. The main result is an if-and-only-if theorem: when a measurement device is untrusted, an adversary limited to classical side information can guess every outcome perfectly precisely when the measurements satisfy this partial joint measurability. The equivalence immediately supplies analytical bounds on how low detector efficiency can fall before generic measurements become partially jointly measurable, which in turn bounds the security of device-independent and semi-device-independent quantum protocols.

Core claim

In the untrusted-device scenario an adversary Eve who holds only classical side information can perfectly guess every outcome of a set of measurements if and only if the measurements are partially jointly measurable, that is, if and only if there exist classical random variables that jointly reproduce the specified subsets of outcomes of every measurement.

What carries the argument

Partial joint-measurability: the property that only a designated subset of outcomes of each measurement can be reproduced by classical post-processing of a single joint measurement.

If this is right

  • Analytical thresholds exist below which the detection efficiency of generic measurements renders them partially jointly measurable.
  • These thresholds directly limit how much detection inefficiency device-independent quantum cryptographic protocols can tolerate while remaining secure.
  • The same thresholds apply to semi-device-independent protocols.
  • Security proofs must treat postselection explicitly to avoid incorrect claims of security.

Where Pith is reading between the lines

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

  • The equivalence may allow tighter security analyses in quantum key distribution by replacing full incompatibility requirements with partial ones when detectors are inefficient.
  • The framework could be used to quantify how much incompatibility is actually needed for other quantum tasks such as state discrimination under loss.
  • An extension to adversaries with quantum side information would be a natural next question left open by the classical-side-information setting.

Load-bearing premise

The adversary has access only to classical side information and possesses no quantum side information about the measurement device.

What would settle it

An explicit set of measurements together with a classical guessing strategy for Eve that succeeds with probability one, yet for which the semidefinite program certifying partial joint-measurability returns a negative answer, would falsify the claimed equivalence.

Figures

Figures reproduced from arXiv: 2605.16151 by Edwin Peter Lobo, Maria Balanz\'o-Juand\'o, Stefano Pironio.

Figure 1
Figure 1. Figure 1: FIG. 1: Left: The overall simulation of the measurements [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Hence θ is close to zero when all measurements are close to a common axis, and it is large when no single axis is well aligned with all of them. In the special case of two measurements, θ = arccos|⃗r1 · ⃗r2| ∈ [0, π/2] (22) is the angle between the two measurement axes and ⃗m is the bisector of the two axes. For θ = π/2, i.e., mutually anticommuting measurements, this yields the bound η ≤ 2 − √ 2 ≈ 0.586. … view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Illustration of the angle [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: The maximum detection efficiency [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
read the original abstract

Quantum measurements can be incompatible, i.e., they can fail to be jointly measurable. Recently, a weaker notion of joint-measurability, called partial joint-measurability, was proposed by Masini et al. in [Quantum 8, 1574 (2024)]. In this work, we further generalize this notion to the setting where only a subset of the outcomes of each measurement is required to be jointly determined by classical variables. We provide two mathematical formulations of partial joint-measurability and show that, like full joint-measurability, it can be decided by solving a single semidefinite program. We prove that in the case of an untrusted measurement device, an adversary Eve, limited to classical side information, can perfectly guess the outcomes of the measurement device if and only if the set of measurements is partially jointly measurable. We derive analytical thresholds on the detection efficiency below which generic measurements become partially jointly measurable. Such bounds directly yield limits on the robustness of device-independent and semi-device-independent quantum cryptographic protocols against detection inefficiency. In particular, our results highlight the importance of a careful treatment of postselection in security analyses.

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 / 3 minor

Summary. The manuscript generalizes partial joint-measurability of quantum measurements to the case where only specified subsets of outcomes from each measurement must be jointly determined by a classical variable. It supplies two equivalent mathematical characterizations (one SDP-based), proves that both are decidable via a single semidefinite program, establishes an if-and-only-if equivalence between this generalized partial joint-measurability and the ability of an adversary limited to classical side information to perfectly guess the relevant outcome subsets on an untrusted device, and derives analytical detection-efficiency thresholds below which generic measurements become partially jointly measurable, with direct implications for the robustness of device-independent and semi-device-independent cryptographic protocols.

Significance. If the central equivalence and SDP formulations hold, the work supplies a practical, computable criterion for a weakened form of incompatibility that is directly tied to guessing attacks with classical side information. The analytical efficiency bounds and the explicit link to post-selection in security proofs constitute a concrete advance for the analysis of DI/SDI protocols under realistic detection losses.

major comments (1)
  1. §4, Theorem 1: the proof of the guessing equivalence relies on the existence of a classical random variable that jointly reproduces the specified outcome subsets; it would be useful to see an explicit construction of the joint POVM from the SDP solution (or vice versa) to confirm that the reduction is tight and does not introduce additional assumptions on the dimension of the classical variable.
minor comments (3)
  1. Abstract and §1: the citation to Masini et al. (Quantum 8, 1574, 2024) should include the arXiv identifier or DOI for completeness.
  2. Figure 1 caption: the labeling of the outcome subsets in the diagram is slightly ambiguous; adding an explicit legend or equation reference would improve clarity.
  3. §5.2: the derivation of the analytical detection-efficiency threshold assumes a specific form for the measurements; a brief remark on how the bound generalizes to arbitrary POVMs would be helpful.

Simulated Author's Rebuttal

1 responses · 0 unresolved

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

read point-by-point responses

  1. Referee: §4, Theorem 1: the proof of the guessing equivalence relies on the existence of a classical random variable that jointly reproduces the specified outcome subsets; it would be useful to see an explicit construction of the joint POVM from the SDP solution (or vice versa) to confirm that the reduction is tight and does not introduce additional assumptions on the dimension of the classical variable.

    Authors: We thank the referee for this helpful suggestion to strengthen the exposition. The proof of Theorem 1 establishes the equivalence by showing that a joint POVM for the relevant outcome subsets exists if and only if there exists a classical random variable whose marginals reproduce the specified outcome probabilities. The SDP in the manuscript is formulated directly over the joint POVM elements (with the appropriate marginal constraints for the subsets), so any feasible SDP solution yields an explicit joint POVM. From this joint POVM one constructs the classical random variable by labeling its outcomes with the tuples of the relevant measurement outcomes and setting the probabilities via the Born rule with respect to the input state. Conversely, given a classical random variable that reproduces the subsets, the joint POVM is recovered by grouping the classical outcomes according to the specified subsets and defining the corresponding POVM elements as the sum of the projectors for each group. The alphabet size of the classical variable is at most the product of the cardinalities of the relevant outcome subsets, which is finite and introduces no additional dimensional assumptions beyond those already present in the finite-outcome setting. We will add a dedicated paragraph with this explicit bidirectional construction immediately following the statement of Theorem 1 in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper introduces a generalization of partial joint-measurability to subsets of outcomes and supplies two independent mathematical characterizations (one SDP-based). It then proves their equivalence and derives the guessing equivalence for an adversary with only classical side information directly from the existence of a joint classical variable that reproduces the specified outcome subsets. The central if-and-only-if result follows from this construction without reducing to a fitted parameter, self-citation chain, or redefinition of the target quantity. The cited prior work (Masini et al.) is external and concerns only the original notion; the present argument is self-contained against the stated classical-side-information model and contains no load-bearing self-reference or ansatz smuggling.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, invented entities, or ad-hoc axioms are stated. The work rests on standard quantum mechanics and the computational tractability of semidefinite programs for joint-measurability problems.

axioms (1)
  • domain assumption Quantum measurements are described by positive operator-valued measures on a Hilbert space
    Standard framework invoked for all joint-measurability questions in quantum information.

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Reference graph

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