Localization of joint quantum measurements on $\mathbb{C}^d \otimes \mathbb{C}^d$ by entangled resources with Schmidt number at most $d$

Jisho Miyazaki; Seiseki Akibue

arxiv: 2601.02660 · v2 · pith:DXA5PGA4new · submitted 2026-01-06 · 🪐 quant-ph

Localization of joint quantum measurements on mathbb{C}^d otimes mathbb{C}^d by entangled resources with Schmidt number at most d

Seiseki Akibue , Jisho Miyazaki This is my paper

Pith reviewed 2026-05-16 17:46 UTC · model grok-4.3

classification 🪐 quant-ph

keywords localizable measurementsprojection-valued measuresSchmidt numberunitary error basismaximally entangled statesnon-adaptive operationsquditsentanglement

0 comments

The pith

A rank-1 joint measurement on two d-dimensional systems can be localized with entanglement of Schmidt number at most d if and only if its elements form a maximally entangled basis from a nice unitary error basis.

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

The paper gives an if-and-only-if characterization of which rank-1 projection-valued measures on the tensor product of two d-dimensional spaces can be realized by non-adaptive local operations plus shared entanglement whose Schmidt number is bounded by d. It proves that such a measurement is localizable precisely when it coincides with a basis of maximally entangled states generated by a nice unitary error basis. This result imposes concrete algebraic restrictions that are absent in the adaptive setting, where every joint measurement is possible. The characterization resolves an earlier conjecture for two qubits and extends prior work on two-qudit ideal measurements.

Core claim

A rank-1 PVM on C^d ⊗ C^d that contains at least one element of maximal Schmidt rank is localizable by entanglement of Schmidt number at most d if and only if the PVM forms a maximally entangled basis corresponding to a nice unitary error basis.

What carries the argument

The nice unitary error basis, whose associated maximally entangled states satisfy the algebraic commutation and orthogonality conditions required for non-adaptive local implementation.

If this is right

Every localizable rank-1 PVM of this type must consist entirely of maximally entangled states.
The set of states must arise from a unitary error basis that satisfies the additional niceness condition.
All two-qubit rank-1 PVMs localizable with two-qubit entanglement are completely classified by this correspondence.
The same algebraic obstruction applies to ideal measurements on two qudits of any dimension.

Where Pith is reading between the lines

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

Non-adaptive locality therefore forbids many joint measurements that adaptive protocols can still realize.
Explicit constructions of nice unitary error bases would immediately yield families of new localizable measurements.
The same obstruction may appear in other resource theories where the number of local parties or the adaptivity is restricted.

Load-bearing premise

The measurement must be a rank-1 projection-valued measure that includes at least one projector of maximal Schmidt rank.

What would settle it

A concrete rank-1 PVM on C^d ⊗ C^d that contains a maximal-Schmidt-rank element, is localizable with Schmidt number at most d, yet whose states do not arise from any nice unitary error basis.

Figures

Figures reproduced from arXiv: 2601.02660 by Jisho Miyazaki, Seiseki Akibue.

**Figure 2.** Figure 2: FIG. 2. The localization scheme for the ideal measurement [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

read the original abstract

Localizable measurements are joint quantum measurements that can be implemented using only non-adaptive local operations and shared entanglement. We provide a protocol-independent characterization of localizable projection-valued measures (PVMs) by exploiting algebraic structures that any such measurement must satisfy. We first show that a rank-1 PVM on $\mathbb{C}^d\otimes\mathbb{C}^d$ containing an element with the maximal Schmidt rank can be localized using entanglement of a Schmidt number at most $d$ if and only if it forms a maximally entangled basis corresponding to a nice unitary error basis. This reveals strong limitations imposed by non-adaptive local operations, in contrast to the adaptive setting where any joint measurement is implementable. We then completely characterize two-qubit rank-1 PVMs that can be localized with two-qubit entanglement, resolving a conjecture of Gisin and Del Santo, and finally extend our characterization to ideal two-qudit measurements, strengthening earlier results.

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 clean algebraic if-and-only-if characterization for which rank-1 PVMs on two qudits are localizable with Schmidt number at most d, and it closes the Gisin-Del Santo conjecture for qubits.

read the letter

This paper pins down exactly when a rank-1 PVM on C^d ⊗ C^d that includes a maximally entangled element can be realized with non-adaptive LOCC using entanglement of Schmidt number ≤ d. The answer is an if-and-only-if statement: the PVM must form a maximally entangled basis tied to a nice unitary error basis. That is the central new piece, and it is stated without protocol dependence, which is a real advantage over earlier constructive approaches. For the qubit case the result is complete, settling the conjecture outright. The qudit extension strengthens what was already known by giving the same algebraic criterion. The work rests on properties of PVMs and unitary error bases rather than explicit constructions, so the limitations of non-adaptive operations come out sharply against the adaptive setting where any joint measurement works. The restriction to rank-1 PVMs with at least one max-Schmidt element is declared up front, so the claim stays within its stated regime and does not overreach. No circular definitions or fitted parameters appear in the abstract description. The algebraic route looks promising for checking, though the full derivations would need a close read to confirm there are no hidden restrictions in the steps. For anyone working on LOCC, measurement theory, or the power of entanglement resources, the paper supplies a usable classification tool. It is worth referee time; the core characterization is sharp enough to justify sending it out even if some technical details need polishing in review.

Referee Report

0 major / 2 minor

Summary. The manuscript claims to provide a protocol-independent characterization of localizable projection-valued measures (PVMs) on C^d ⊗ C^d. It shows that a rank-1 PVM containing an element with the maximal Schmidt rank can be localized using entanglement of Schmidt number at most d if and only if it forms a maximally entangled basis corresponding to a nice unitary error basis. Additionally, it completely characterizes two-qubit rank-1 PVMs localizable with two-qubit entanglement, resolving a conjecture by Gisin and Del Santo, and extends the characterization to ideal two-qudit measurements.

Significance. If the algebraic derivations hold, this work is significant for revealing strong limitations of non-adaptive local operations on joint quantum measurements, in contrast to adaptive settings. The protocol-independent iff characterization via nice unitary error bases is a clear strength, as is the resolution of the Gisin-Del Santo conjecture for two qubits and the strengthening of earlier two-qudit results. These provide concrete, falsifiable conditions on when limited entanglement suffices for localization.

minor comments (2)

[Abstract] Abstract: the term 'nice unitary error basis' appears without a brief definition or forward reference; adding one would aid readers new to the concept.
The restriction to rank-1 PVMs with at least one maximal-Schmidt-rank element is explicitly stated but could be reiterated in the statement of the main theorem to prevent misreading as a fully general result.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of the main results, and recommendation for minor revision. The referee correctly identifies the protocol-independent characterization via nice unitary error bases, the resolution of the Gisin-Del Santo conjecture for two qubits, and the strengthening of prior two-qudit results as key contributions.

Circularity Check

0 steps flagged

No significant circularity detected in the algebraic characterization

full rationale

The paper derives an if-and-only-if characterization of localizable rank-1 PVMs on C^d ⊗ C^d (those containing a maximal-Schmidt-rank element) directly from algebraic properties that any such measurement must satisfy, specifically that it must form a maximally entangled basis corresponding to a nice unitary error basis. This is a self-contained mathematical argument resting on the definitions of PVMs, Schmidt rank, and unitary error bases; no equation reduces to a fitted parameter, no prediction is constructed from its own inputs, and no load-bearing step relies on a self-citation chain whose validity is presupposed by the present work. The two-qubit and two-qudit extensions are likewise presented as independent complete characterizations resolving external conjectures, with all restrictions (rank-1, presence of maximal-Schmidt element) explicitly declared rather than smuggled in. The derivation therefore remains non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard definitions and algebraic properties of projection-valued measures, Schmidt rank, and nice unitary error bases drawn from existing quantum information literature; no new free parameters or invented entities are introduced.

axioms (2)

standard math Standard properties of rank-1 projection-valued measures on tensor-product spaces
Invoked throughout the characterization of localizable PVMs.
domain assumption Existence and algebraic closure properties of nice unitary error bases
Used as the target class in the iff statement; drawn from prior literature on unitary error bases.

pith-pipeline@v0.9.0 · 5475 in / 1210 out tokens · 45662 ms · 2026-05-16T17:46:25.758594+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/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

a rank-1 PVM on C^d ⊗ C^d containing an element with the maximal Schmidt rank can be localized using entanglement of a Schmidt number at most d if and only if it forms a maximally entangled basis corresponding to a nice unitary error basis
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat recovery / Peano structure unclear

?

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

Theorem 1: {U†j Ui} is a nice error basis for any j (Latin-square property)

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

38 extracted references · 38 canonical work pages

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ideal measurement

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which subspace

corresponds to the case d = 2 and where all W s are identity isomorphisms. The localization protocol for the basis (22) is divided into three steps. Refer to Fig. 2 for clariﬁcation. The reference system has d2 × d2 dimension, and the resource state is the same as Eq. (21). First, Alice and Bob per- form the “which subspace” ideal measurement on their loc...

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[1] [1]

and partially entangled bases, such as { |00⟩, |11⟩, |01⟩ + |10⟩ √ 2 , |01⟩ − |10⟩√ 2 } (named “pBSM” in [14]). Among iso-entangled bases, the higher-dimensional generalization [27] of the elegant joint measurement [28] has the maximal Schmidt rank but is not maximally entangled for any dimension. Merely being a maximally entangled basis is not enough to ...

work page

[2] [2]

ideal measurement

as a conjecture. Pauwels et al. [14] showed that the theorem holds when one relies on the blind-teleportation protocol, where the BB84 is called π 2 -twisted basis mea- surement. Consequently, the ‘if’ direction can be shown using their protocol. In contrast, the ‘only if’ direction requires a separate proof, since the theorem permits ar- bitrary localiza...

work page

[3] [3]

which subspace

corresponds to the case d = 2 and where all W s are identity isomorphisms. The localization protocol for the basis (22) is divided into three steps. Refer to Fig. 2 for clariﬁcation. The reference system has d2 × d2 dimension, and the resource state is the same as Eq. (21). First, Alice and Bob per- form the “which subspace” ideal measurement on their loc...

work page

[4] [4]

Gisin and F

N. Gisin and F. Del Santo, Towards a measurement the- ory in QFT: ”Impossible” quantum measurements are possible but not ideal, Quantum 8, 1267 (2024)

work page 2024

[5] [5]

Aharonov and D

Y. Aharonov and D. Z. Albert, States and ob- servables in relativistic quantum ﬁeld theories, Phys. Rev. D 21, 3316 (1980)

work page 1980

[6] [6]

Aharonov and D

Y. Aharonov and D. Z. Albert, Can we make sense out of the measurement process in relativistic quantum me- chanics?, Phys. Rev. D 24, 359 (1981)

work page 1981

[7] [7]

Aharonov, D

Y. Aharonov, D. Z. Albert, and L. Vaidman, Measurement process in relativistic quantum theory, Phys. Rev. D 34, 1805 (1986)

work page 1986

[8] [8]

Popescu and L

S. Popescu and L. Vaidman, Causality con- straints on nonlocal quantum measurements, Phys. Rev. A 49, 4331 (1994)

work page 1994

[9] [9]

Beckman, D

D. Beckman, D. Gottesman, A. Kitaev, and J. Preskill, Measurability of wilson loop operators, Phys. Rev. D 65, 065022 (2002)

work page 2002

[10] [10]

Sorkin, Impossible measurements on quantum ﬁelds, in Directions in General Relativity: Proceedings of the 1993 International Symposium, Maryland , Vol

R. Sorkin, Impossible measurements on quantum ﬁelds, in Directions in General Relativity: Proceedings of the 1993 International Symposium, Maryland , Vol. 2, edited by B. L. Hu and T. A. Jacobson (Cambridge University Press, 1993) pp. 293–305

work page 1993

[11] [11]

Fraser and M

D. Fraser and M. Papageorgiou, Note on episodes in the history of modeling measurements in local spacetime re- gions using qft, Eur. Phys. J. H 48, 14 (2023)

work page 2023

[12] [12]

Beckman, D

D. Beckman, D. Gottesman, M. A. Nielsen, and J. Preskill, Causal and localizable quantum operations, Phys. Rev. A 64, 052309 (2001)

work page 2001

[13] [13]

Vaidman, Instantaneous measurement of nonlocal variables, Phys

L. Vaidman, Instantaneous measurement of nonlocal variables, Phys. Rev. Lett. 90, 010402 (2003)

work page 2003

[14] [14]

Groisman, B

B. Groisman, B. Reznik, and L. Vaidman, In- stantaneous measurements of nonlocal variables, J. Mod. Opt. 50, 943 (2003)

work page 2003

[15] [15]

S. R. Clark, A. J. Connor, D. Jaksch, and S. Popescu, En- tanglement consumption of instantaneous nonlocal quan- tum measurements, New J. Phys. 12, 083034 (2010)

work page 2010

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Beigi and R

S. Beigi and R. K¨ onig, Simpliﬁed instantaneous non-local quantum computation with applications to position- based cryptography, New J. Phys. 13, 093036 (2011)

work page 2011

[17] [17]

Pauwels, A

J. Pauwels, A. Pozas-Kerstjens, F. Del Santo, and N. Gisin, Classiﬁcation of joint quantum measure- ments based on entanglement cost of localization, Phys. Rev. X 15, 021013 (2025)

work page 2025

[18] [18]

Pauwels, A

J. Pauwels, A. Pozas-Kerstjens, F. Del Santo, and N. Gisin, Classiﬁcation of joint quantum measurements based on entan glement cost of localization (2024), arXiv preprint, arXiv:2408.00831 [quant-ph]

work page arXiv 2024

[19] [19]

Knill, Non-binary unitary error bases and quantum codes (1996)

E. Knill, Non-binary unitary error bases and quantum codes (1996)

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[20] [20]

Klappenecker and M

A. Klappenecker and M. R¨ otteler, Unitary error bases: Constructions, equivalence, and applications, in Ap- plied Algebra, Algebraic Algorithms and Error-Correcting Codes, edited by M. Fossorier, T. Høholdt, and A. Poli (Springer Berlin Heidelberg, Berlin, Heidelberg, 2003) pp. 139–149

work page 2003

[21] [21]

Klappenecker and M

A. Klappenecker and M. Rotteler, On the monomiality of nice error bases, IEEE Trans. Inf. Theory 51, 1084 (2005)

work page 2005

[22] [22]

D¨ ur, G

W. D¨ ur, G. Vidal, and J. I. Cirac, Three qubits can be entangled in two inequivalent ways, Phys. Rev. A 62, 062314 (2000)

work page 2000

[23] [23]

B. M. Terhal and P. Horodecki, Schmidt number for den- sity matrices, Phys. Rev. A 61, 040301 (2000)

work page 2000

[24] [24]

Schmid, T

D. Schmid, T. C. Fraser, R. Kunjwal, A. B. Sainz, E. Wolfe, and R. W. Spekkens, Understanding the in- terplay of entanglement and nonlocality: motivating and developing a new branch of entanglement theory, Quantum 7, 1194 (2023)

work page 2023

[25] [25]

Groisman and L

B. Groisman and L. Vaidman, Nonlo- cal variables with product-state eigenstates, J. Phys. A: Math. Gen. 34, 6881 (2001)

work page 2001

[26] [26]

Chiribella, G

G. Chiribella, G. M. D’Ariano, and P. Perinotti, Theoretical framework for quantum networks, Phys. Rev. A 80, 022339 (2009)

work page 2009

[27] [27]

Chitambar, D

E. Chitambar, D. W. Leung, L. Manˇ cinska, M. Ozols, and A. Winter, Everything You Always Wanted to Know About LOCC (But Were Afraid to Ask), Communications in Mathematical Physics 328, 303 (2014)

work page 2014

[28] [28]

Musto and J

B. Musto and J. Vicary, Quantum latin squares and uni- tary error bases, Quantum Inf. Comput. 16, 1318 (2016)

work page 2016

[29] [29]

Del Santo, J

F. Del Santo, J. Czartowski, K. ˙Zyczkowski, and N. Gisin, Iso-entangled bases and joint measurements, Phys. Rev. Res. 6, 023085 (2024)

work page 2024

[30] [30]

Czartowski and K

J. Czartowski and K. ˙Zyczkowski, Bipartite quantum measurements with optimal single-sided distinguishabil- ity, Quantum 5, 442 (2021)

work page 2021

[31] [31]

Gisin, Entanglement 25 years after quantum tele- portation: Testing joint measurements in quantum net- works, Entropy 21, 10.3390/e21030325 (2019)

N. Gisin, Entanglement 25 years after quantum tele- portation: Testing joint measurements in quantum net- works, Entropy 21, 10.3390/e21030325 (2019)

work page doi:10.3390/e21030325 2019

[32] [32]

While a diﬀerent deﬁnition of localizability is used in [9], their example persists to be localizable in our criterion as well

work page

[33] [33]

J. S. Bell, On the einstein podolsky rosen paradox, Physics Physique Fizika 1, 195 (1964)

work page 1964

[34] [34]

Brunner, D

N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, Bell nonlocality, Rev. Mod. Phys. 86, 419 (2014)

work page 2014

[35] [35]

Mayers and A

D. Mayers and A. C.-C. Yao, Self testing quantum appa- ratus, Quantum Inf. Comput. 4, 273 (2004)

work page 2004

[36] [36]

ˇSupi´ c and J

I. ˇSupi´ c and J. Bowles, Self-testing of quantum systems: a review, Quantum 4, 337 (2020)

work page 2020

[37] [37]

Buscemi, All entangled quantum states are nonlocal, Phys

F. Buscemi, All entangled quantum states are nonlocal, Phys. Rev. Lett. 108, 200401 (2012)

work page 2012

[38] [38]

Zjawin, D

B. Zjawin, D. Schmid, M. J. Hoban, and A. B. Sainz, Quantifying EPR: the resource theory of nonclassicality of common-cause assemblages, Quantum 7, 926 (2023). Appendix A: Proof of Lemma 1 In this appendix, we do not employ the double–ket notation as we also deal with degenerate operators Mc, Aa, Bb and ψ R. We prove Lemma 1 by explicitly constructing the...

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