arxiv: 2512.15199 · v2 · submitted 2025-12-17 · 🪐 quant-ph

Recognition: 2 theorem links

· Lean Theorem

Sharing quantum indistinguishability with multiple parties

Lemieux Wang , Hanwool Lee , Joonwoo Bae , Kieran Flatt

Authors on Pith no claims yet

Pith reviewed 2026-05-16 22:14 UTC · model grok-4.3

classification 🪐 quant-ph

keywords quantum state discriminationweak measurementsmax relative entropysequential protocolsquantum uncertaintyindistinguishabilitymulti-party quantum information

0 comments

The pith

A sequential scheme lets multiple parties share the max relative entropy uncertainty from one party's state discrimination using weak measurements.

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

The paper presents a protocol in which a single quantum system undergoes weak measurements so that several parties can each perform maximum-confidence discrimination in sequence. Each party extracts some of the indistinguishability resource, quantified as max relative entropy uncertainty, without fully collapsing the state for the next user. A reader would care because this distributes a quantum resource across parties while preserving the possibility of further extraction, directly relevant to multi-user cryptography and randomness generation.

Core claim

The central claim is that a sequential state-discrimination scheme based on maximum-confidence measurements and weak measurements enables multiple parties to share the quantum uncertainty, measured by max relative entropy, that originates from a single party's discrimination task on one physical system. The scheme is illustrated for ensembles both with and without symmetries, showing that each successive party can still obtain a useful discrimination outcome while the shared uncertainty remains quantifiable.

What carries the argument

Sequential state-discrimination protocol that interleaves weak measurements with maximum-confidence discrimination so that max relative entropy uncertainty generated by one party is shared across multiple parties on the same system.

If this is right

Multiple parties obtain quantifiable shares of the same quantum uncertainty resource without requiring separate systems.
The protocol applies equally to symmetric and asymmetric state ensembles.
Sequential information extraction is bounded by the persistence of useful discrimination after each weak measurement.
The approach supplies a concrete method for sharing indistinguishability in multi-party quantum protocols.

Where Pith is reading between the lines

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

The scheme could be combined with existing quantum key distribution protocols to let several users share uncertainty bounds from one source.
Optimizing the strength of each weak measurement might allow a larger number of parties while keeping each discrimination above a minimum confidence threshold.
The same construction might extend to other uncertainty measures beyond max relative entropy if the discrimination step can be adjusted accordingly.

Load-bearing premise

Weak measurements can be realized such that each successive party can still perform a useful maximum-confidence discrimination on the same physical system while the shared uncertainty remains quantifiable in max relative entropy.

What would settle it

An experiment implementing the first party's weak measurement followed by the second party's maximum-confidence discrimination, where the measured max relative entropy for the second party deviates significantly from the value predicted by the sharing scheme.

Figures

Figures reproduced from arXiv: 2512.15199 by Hanwool Lee, Joonwoo Bae, Kieran Flatt, Lemieux Wang.

**Figure 2.** Figure 2: FIG. 2: Sequential maximum confidence measurements are applied to an ensemble of two mixed [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3: Sequential MCM is implemented on an ensemble of symmetric states. After each [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4: Sequential MCM is implemented on an ensemble of geometrically uniform states. Party [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5: Sequential maximum confidence measurements are applied to three lifted geometrically [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6: The direction of the mirror-symmetric ensemble evolves depending on the initial angle. [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7: Numerical results of sequential MCM of the mirror symmetric ensemble. State 1 is [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗

read the original abstract

Quantum indistinguishability of non-orthogonal quantum states is a valuable resource in quantum information applications such as cryptography and randomness generation. In this article, we present a sequential state-discrimination scheme that enables multiple parties to share quantum uncertainty, in terms of the max relative entropy, generated by a single party. Our scheme is based upon maximum-confidence measurements and takes advantages of weak measurements to allow a number of parties to perform state discrimination on a single quantum system. We review known sequential state discrimination and show how our scheme would work through a number of examples where ensembles may or may not contain symmetries. Our results will have a role to play in understanding the ultimate limits of sequential information extraction and guide the development of quantum resource sharing in sequential settings.

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.

Desk Editor's Note private letter to a colleague

The paper gives a concrete sequential scheme using weak max-confidence measurements to let multiple parties share max-relative-entropy uncertainty from one system, with examples for symmetric and asymmetric ensembles, but it needs explicit bounds on how much uncertainty survives after several parties.

read the letter

This paper presents a sequential state-discrimination scheme that lets several parties each pull some max-relative-entropy uncertainty from a single quantum system. They combine maximum-confidence measurements with weak measurements so the state survives for the next user. The abstract and examples show the idea working on both symmetric and asymmetric ensembles, and they review the existing sequential-discrimination literature before laying out their construction. That combination and the explicit examples are the main new pieces. The examples appear to demonstrate that the uncertainty can be distributed without one party taking everything. The soft spot is the scaling with more parties. Successive weak measurements will disturb the state, and without a general bound or calculations showing that the remaining max relative entropy stays useful after three or four parties, the claim for arbitrary multiple parties rests on the specific cases shown. If the disturbance grows faster than the sharing, later parties get vanishing returns. This is aimed at people working on multi-party quantum information tasks such as cryptography or randomness generation who already know the basic tools of weak measurements and uncertainty relations. A reader looking for new constructions in sequential settings would get value from the examples even if the general bound is missing. I would send it for peer review. The construction is specific enough and the examples give something concrete to check, so referees can evaluate whether the sharing holds up beyond the cases presented.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes a sequential state-discrimination scheme that employs weak measurements combined with maximum-confidence discrimination to enable multiple parties to share the quantum uncertainty—quantified via max relative entropy—generated by a single party's preparation of non-orthogonal states. It reviews prior sequential discrimination protocols and illustrates the approach with examples of symmetric and asymmetric ensembles.

Significance. If the central claim holds with explicit disturbance bounds, the result would clarify the ultimate limits of sequential information extraction from a single quantum system and provide a concrete mechanism for multi-party sharing of indistinguishability resources, with potential relevance to sequential quantum cryptography and randomness generation protocols.

major comments (2)

[Examples (symmetric and asymmetric ensembles)] The examples section does not supply a general bound (or even explicit numerical values for k>2) demonstrating that the extracted max-relative-entropy remains non-negligible after successive weak measurements; without this, the multi-party sharing claim rests on unverified extrapolation from the two-party case.
[Scheme description] The weak-measurement implementation is described only at the level of the abstract and review; no explicit Kraus operators or post-measurement state expressions are given that would allow verification that each successive maximum-confidence discrimination still extracts a positive, quantifiable fraction of the original distinguishability.

minor comments (1)

[Introduction] Notation for the max-relative-entropy quantity is introduced without an explicit equation reference in the main text; a numbered definition would improve traceability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We address each major comment below and indicate the revisions we will make to strengthen the presentation and support the claims.

read point-by-point responses

Referee: [Examples (symmetric and asymmetric ensembles)] The examples section does not supply a general bound (or even explicit numerical values for k>2) demonstrating that the extracted max-relative-entropy remains non-negligible after successive weak measurements; without this, the multi-party sharing claim rests on unverified extrapolation from the two-party case.

Authors: We acknowledge that the current examples primarily illustrate the two-party case and do not include explicit numerical values or a general bound for k>2. In the revised manuscript we will add explicit calculations for k=3 and k=4 using specific weak-measurement strengths, demonstrating that the extracted max-relative-entropy remains positive and non-negligible. We will also include a general argument showing how the extracted quantity scales with the number of parties and the measurement strength parameter, thereby removing the need for extrapolation. revision: yes
Referee: [Scheme description] The weak-measurement implementation is described only at the level of the abstract and review; no explicit Kraus operators or post-measurement state expressions are given that would allow verification that each successive maximum-confidence discrimination still extracts a positive, quantifiable fraction of the original distinguishability.

Authors: We agree that explicit operators are required for verification. In the revised manuscript we will supply the Kraus operators for the weak measurements performed by each successive party and derive the corresponding post-measurement states. These expressions will explicitly show that each maximum-confidence discrimination extracts a positive, quantifiable fraction of the original distinguishability, controlled by the tunable weak-measurement strength. revision: yes

Circularity Check

0 steps flagged

No circularity: scheme builds on external established concepts without self-referential reduction

full rationale

The paper's derivation reviews known sequential state discrimination, maximum-confidence measurements, and weak measurements as independent inputs, then illustrates the multi-party sharing scheme via explicit examples on symmetric and asymmetric ensembles. No equations or steps reduce the claimed max-relative-entropy sharing to a fitted parameter, self-defined quantity, or self-citation chain by construction. The central construction is presented as an application of prior results rather than a tautological renaming or re-derivation of its own distinguishability metric.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The scheme rests on standard quantum measurement postulates and the definition of max relative entropy; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)

standard math Standard postulates of quantum mechanics for state preparation, evolution, and measurement.
The protocol presupposes the usual Hilbert-space formalism and the existence of weak and maximum-confidence measurements.

pith-pipeline@v0.9.0 · 5417 in / 1107 out tokens · 37959 ms · 2026-05-16T22:14:30.443131+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Cx = qx 2^Dmax(ρx||ρ) ... sequential MCM with Kraus operators Kx = √α |φx⟩⟨ϕx| minimizing trace distance D(S(j),S(j+1))
IndisputableMonolith/Foundation/DimensionForcing.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

equally high confidence iff POVM elements linearly independent; weak measurements for linearly dependent ensembles

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages

[2]

P. J. Brown and R. Colbeck, Arbitrarily many independent observers can share the nonlocality of a single maximally entangled qubit pair, Phys. Rev. Lett.125, 090401 (2020)

work page 2020
[3]

S. M. Barnett and S. Croke, Quantum state discrimination, Advances in Optics and Photonics1, 238 (2009)

work page 2009
[4]

J. A. Bergou, Discrimination of quantum states, Journal of Modern Optics57, 160 (2010)

work page 2010
[5]

Bae and L.-C

J. Bae and L.-C. Kwek, Quantum state discrimination and its applications, Journal of Physics A: Mathematical and Theoretical48, 083001 (2015)

work page 2015
[6]

Croke, E

S. Croke, E. Andersson, S. M. Barnett, C. R. Gilson, and J. Jeffers, Maximum confidence quantum measurements, Physical review letters96, 070401 (2006). 20

work page 2006
[8]

H. Lee, K. Flatt, C. Roch i Carceller, J. B. Brask, and J. Bae, Maximum-confidence measurement for qubit states, Physical Review A106, 032422 (2022)

work page 2022
[9]

Silva, N

R. Silva, N. Gisin, Y. Guryanova, and S. Popescu, Multiple observers can share the nonlocality of half of an entangled pair by using optimal weak measurements, Phys. Rev. Lett.114, 250401 (2015)

work page 2015
[10]

C. W. Helstrom, Detection theory and quantum mechanics, Information and Control10, 254 (1967)

work page 1967
[11]

I. D. Ivanovic, How to differentiate between non-orthogonal states, Physics Letters A123, 257 (1987)

work page 1987
[12]

Dieks, Overlap and distinguishability of quantum states, Physics Letters A126, 303 (1988)

D. Dieks, Overlap and distinguishability of quantum states, Physics Letters A126, 303 (1988)

work page 1988
[13]

Peres, How to differentiate between non-orthogonal states, Physics Letters A128, 19 (1988)

A. Peres, How to differentiate between non-orthogonal states, Physics Letters A128, 19 (1988)

work page 1988
[14]

P. J. Mosley, S. Croke, I. A. Walmsley, and S. M. Barnett, Experimental realization of maximum confidence quantum state discrimination for the extraction of quantum information, Phys. Rev. Lett. 97, 193601 (2006)

work page 2006
[15]

Datta, Min-and max-relative entropies and a new entanglement monotone, IEEE Transactions on Information Theory55, 2816 (2009)

N. Datta, Min-and max-relative entropies and a new entanglement monotone, IEEE Transactions on Information Theory55, 2816 (2009)

work page 2009
[16]

M¨ uller-Lennert, F

M. M¨ uller-Lennert, F. Dupuis, O. Szehr, S. Fehr, and M. Tomamichel, On quantum r´ enyi entropies: A new generalization and some properties, Journal of Mathematical Physics54(2013)

work page 2013
[17]

Van Erven and P

T. Van Erven and P. Harremos, R´ enyi divergence and kullback-leibler divergence, IEEE Transactions on Information Theory60, 3797 (2014)

work page 2014
[18]

Konig, R

R. Konig, R. Renner, and C. Schaffner, The operational meaning of min-and max-entropy, IEEE Transactions on Information theory55, 4337 (2009)

work page 2009
[19]

Wilde,Quantum Information Theory, Quantum Information Theory (Cambridge University Press, 2013)

M. Wilde,Quantum Information Theory, Quantum Information Theory (Cambridge University Press, 2013)

work page 2013
[20]

Bergou, E

J. Bergou, E. Feldman, and M. Hillery, Extracting information from a qubit by multiple observers: Toward a theory of sequential state discrimination, Physical review letters111, 100501 (2013)

work page 2013
[21]

H. Lee, K. Flatt, and J. Bae, Sequential quantum maximum-confidence discrimination, Phys. Rev. A 112, 052206 (2025)

work page 2025
[22]

U. Herzog, Optimized maximum-confidence discrimination of n mixed quantum states and application to symmetric states, Physical Review A—Atomic, Molecular, and Optical Physics85, 032312 (2012)

work page 2012
[23]

Andersson, S

E. Andersson, S. M. Barnett, C. R. Gilson, and K. Hunter, Minimum-error discrimination between three mirror-symmetric states, Physical Review A65, 052308 (2002)

work page 2002
[24]

J. B. Brask, A. Martin, W. Esposito, R. Houlmann, J. Bowles, H. Zbinden, and N. Brunner, Megahertz- rate semi-device-independent quantum random number generators based on unambiguous state dis- crimination, Phys. Rev. Appl.7, 054018 (2017)

work page 2017
[25]

Roch i Carceller, K

C. Roch i Carceller, K. Flatt, H. Lee, J. Bae, and J. B. Brask, Quantum vs noncontextual semi-device- independent randomness certification, Phys. Rev. Lett.129, 050501 (2022)

work page 2022
[26]

C. R. I. Carceller, H. Lee, J. B. Brask, K. Flatt, and J. Bae, Sequential semi-device-independent quantum randomness certification (2025), arXiv:2510.19445 [quant-ph]

work page arXiv 2025
[27]

Takagi and B

R. Takagi and B. Regula, General resource theories in quantum mechanics and beyond: Operational characterization via discrimination tasks, Phys. Rev. X9, 031053 (2019)

work page 2019