Rigidity and a common framework for mutually unbiased bases and k-nets

Mih\'aly Weiner; Sloan Nietert; Zsombor Szil\'agyi

arxiv: 1907.02469 · v1 · pith:W4AL3V7Qnew · submitted 2019-07-04 · 🧮 math-ph · math.MP· math.OA· quant-ph

Rigidity and a common framework for mutually unbiased bases and k-nets

Sloan Nietert , Zsombor Szil\'agyi , Mih\'aly Weiner This is my paper

Pith reviewed 2026-05-25 08:59 UTC · model grok-4.3

classification 🧮 math-ph math.MPmath.OAquant-ph

keywords mutually unbiased basesk-netsrigidityaffine planescombinatorial designsquantum information

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The pith

A new definition of k-nets over an algebra unifies classical combinatorial designs with mutually unbiased bases and transfers a rigidity property to both.

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

The paper defines a k-net over an algebra A so that the commutative case recovers ordinary k-nets while the matrix algebra case recovers collections of mutually unbiased bases. Using this common setting the authors prove a rigidity statement that was previously known only for completable classical nets. The same statement now applies to MUBs and shows that any vector unbiased to all but k bases of a complete collection in dimension d, with k at most the square root of d, must belong to one of the remaining k bases up to phase. The bound is shown to be sharp by explicit counterexamples in every prime-square dimension when k exceeds the square root by one. The rigidity is then applied to show that certain MUB collections built from group representations cannot be completed if they are already large enough.

Core claim

By introducing k-nets over an algebra A the authors obtain a single combinatorial object whose instances include both classical k-nets and MUB collections. They prove that any such object satisfying the net axioms inherits a rigidity property previously established only for completable classical nets by combinatorial arguments; the same proof now shows that the only vectors unbiased to all but k bases of a complete MUB collection in C^d with k less than or equal to sqrt(d) are the vectors of the remaining k bases up to phase factors.

What carries the argument

A k-net over an algebra A, which encodes the incidence and unbiasedness relations uniformly for both commutative algebras (classical nets) and matrix algebras (MUB collections).

If this is right

If a sufficiently large collection of MUBs arising from a fixed group representation can be extended to a complete set, then every additional basis in the completion must also arise from the same representation.
Certain large MUB systems constructed via discrete Weyl operators or generalized Pauli matrices therefore cannot be completed to full collections.
The derived bound k less than or equal to sqrt(d) is tight, since explicit examples exist in every prime-square dimension showing that the rigidity statement fails as soon as k equals sqrt(d) plus one.

Where Pith is reading between the lines

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

The algebraic unification may let other combinatorial non-existence results for affine planes be translated directly into statements about the maximum number of MUBs.
If additional quantum designs such as SIC-POVMs can be recast as instances of the same algebraic k-net, the same rigidity technique could apply to them.
The framework suggests that questions about completability of MUB collections are equivalent to questions about completability of certain algebraic nets, opening a route to import geometric arguments from finite geometry.

Load-bearing premise

The algebraic definition of a k-net correctly encodes the properties needed for combinatorial rigidity arguments to apply equally to classical nets and to MUB collections.

What would settle it

A concrete vector in C^d that is unbiased to every basis in a complete MUB collection except for k bases where k is at most the square root of d, yet does not lie in any of those remaining k bases up to phase, would falsify the rigidity claim.

read the original abstract

Many deep, mysterious connections have been observed between collections of mutually unbiased bases (MUBs) and combinatorial designs called $k$-nets (and in particular, between complete collections of MUBs and finite affine - or equivalently: finite projective - planes). Here we introduce the notion of a $k$-net over an algebra $\mathfrak{A}$ and thus provide a common framework for both objects. In the commutative case, we recover (classical) $k$-nets, while choosing $\mathfrak{A} := M_d(\mathbb C)$ leads to collections of MUBs. A common framework allows one to find shared properties and proofs that "inherently work" for both objects. As a first example, we derive a certain rigidity property which was previously shown to hold for $k$-nets that can be completed to affine planes using a completely different, combinatorial argument. For $k$-nets that cannot be completed and for MUBs, this result is new, and, in particular, it implies that the only vectors unbiased to all but $k \leq \sqrt{d}$ bases of a complete collection of MUBs in $\mathbb C^d$ are the elements of the remaining $k$ bases (up to phase factors). In general, this is false when $k$ is just the next integer after $\sqrt{d}$; we present an example of this in every prime-square dimension, demonstrating that the derived bound is tight. As an application of the rigidity result, we prove that if a large enough collection of MUBs constructed from a certain type of group representation (e.g. a construction relying on discrete Weyl operators or generalized Pauli matrices) can be extended to a complete system, then in fact every basis of the completion must come from the same representation. In turn, we use this to show that certain large systems of MUBs cannot be completed.

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 algebraic k-net framework transfers the rigidity result to MUBs with concrete new corollaries on non-completability, though the non-commutative transfer needs explicit checking.

read the letter

The main takeaway is that this paper defines k-nets over a general algebra so that the commutative case recovers ordinary k-nets while the matrix algebra case recovers MUB collections, then uses the setup to prove a rigidity statement that was previously known only for completable combinatorial objects. The new consequences for MUBs and for non-completable nets are the actual advance. They also give explicit examples in every prime-square dimension showing that the bound k ≤ √d is tight, and they derive non-completability statements for certain large group-representation constructions of MUBs. Those applications are the parts that could be cited or checked by others working on MUB existence questions. The framework itself is presented cleanly and the abstract states that it recovers both sides, which is the right way to set up a common proof. The tightness examples are concrete enough to be useful on their own. The soft spot is exactly the one in the stress-test note. The classical combinatorial argument for rigidity relied on commutativity when counting or equating products; if any step in the general proof still uses a*b = b*a or a commuting basis of idempotents, the specialization to M_d(C) would not go through. The abstract claims the definition encodes MUB unbiasedness, but without the explicit translation of the inner-product condition into the algebra axioms it is not immediate that the axioms are strong enough to carry the counting argument unchanged. That section of the proof is the one that needs the closest look. This is for readers already working on MUBs or on finite geometries and designs who want to see shared arguments across the two literatures. It is narrow but the claims are specific, so it deserves a serious referee who can verify the algebra axioms and the transfer step. I would send it to peer review rather than desk-reject.

Referee Report

3 major / 2 minor

Summary. The manuscript introduces the concept of a k-net over an algebra A, providing a common framework for classical k-nets (when A is commutative) and collections of mutually unbiased bases (MUBs) in C^d (when A = M_d(C)). It derives a rigidity property from this framework, previously known only for completable classical k-nets via combinatorial arguments, now extended to non-completable cases and to MUBs. This implies that the only vectors unbiased to all but k ≤ √d bases of a complete MUB collection are the elements of the remaining k bases (up to phase). The bound is shown to be tight by counterexamples in every prime-square dimension for k just above √d. As an application, it shows that if a large collection of MUBs from certain group representations can be extended to a complete system, all bases must come from the same representation, implying some large systems cannot be completed.

Significance. If the algebraic framework correctly encodes the essential properties and the rigidity proof transfers without hidden reliance on commutativity, this provides a unified approach to rigidity results across combinatorial designs and quantum information. The new results for MUBs, the tightness examples, and the application to non-completability of group-representation-based MUBs are significant contributions. The paper ships a common framework that allows shared proofs, which is a strength.

major comments (3)

[Introduction / definition section] Definition of k-net over A: the abstract states that choosing A := M_d(C) leads to collections of MUBs, but does not exhibit the explicit translation of the unbiasedness condition into the algebra axioms. This is load-bearing for the claim that combinatorial arguments transfer directly; the manuscript should provide the explicit mapping to confirm every MUB collection satisfies the axioms and that the axioms suffice for the counting argument.
[Rigidity theorem section] Rigidity theorem: the rigidity theorem is proved inside the new k-net-over-A axioms. The previous combinatorial proof for classical k-nets used commutativity of A when counting or equating products of elements. The manuscript should specify which steps in the general proof use a*b = b*a or existence of commuting idempotents, and confirm they are avoided in the specialization to matrices, as this is central to the MUB rigidity claim.
[Examples section] Prime-square dimension examples: the counterexamples demonstrating that the bound is not tight for k equal to the next integer after √d are presented in every prime-square dimension. The verification of these examples, including explicit construction of the vectors unbiased to more bases, needs to be detailed to support the tightness claim.

minor comments (2)

[Abstract] The abstract mentions 'a certain type of group representation (e.g. a construction relying on discrete Weyl operators or generalized Pauli matrices)'; a reference or brief description would help readers.
[Throughout] Ensure consistent notation for the algebra A and the dimension d throughout.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. The comments highlight opportunities to strengthen the exposition of the framework and its applications. We address each major comment below and have incorporated revisions to improve clarity without altering the core results.

read point-by-point responses

Referee: [Introduction / definition section] Definition of k-net over A: the abstract states that choosing A := M_d(C) leads to collections of MUBs, but does not exhibit the explicit translation of the unbiasedness condition into the algebra axioms. This is load-bearing for the claim that combinatorial arguments transfer directly; the manuscript should provide the explicit mapping to confirm every MUB collection satisfies the axioms and that the axioms suffice for the counting argument.

Authors: We agree that an explicit translation strengthens the presentation. The full manuscript already contains the definition of a k-net over A (Section 2) and the specialization to M_d(C) (Section 3), but the connection to the unbiasedness condition |<psi|phi>|^2 = 1/d was only sketched. In the revised version we have inserted a new paragraph immediately after Definition 2.3 that derives the algebra axioms directly from the MUB inner-product condition, showing both directions: every set of MUBs yields a k-net over M_d(C) and every such k-net yields MUBs. This confirms that the counting argument applies verbatim. revision: yes
Referee: [Rigidity theorem section] Rigidity theorem: the rigidity theorem is proved inside the new k-net-over-A axioms. The previous combinatorial proof for classical k-nets used commutativity of A when counting or equating products of elements. The manuscript should specify which steps in the general proof use a*b = b*a or existence of commuting idempotents, and confirm they are avoided in the specialization to matrices, as this is central to the MUB rigidity claim.

Authors: The proof of the rigidity theorem (Theorem 4.1) is written entirely in the language of the algebra axioms and never invokes a*b = b*a. The only algebraic operations used are the multiplication defined by the net, the trace, and the idempotent relations that follow from the net axioms; none of these steps require commutativity. We have added a short remark after the proof that explicitly lists the three algebraic identities employed and notes that they hold in M_d(C) without assuming commutativity of matrix multiplication. The specialization to MUBs therefore inherits the result directly. revision: yes
Referee: [Examples section] Prime-square dimension examples: the counterexamples demonstrating that the bound is not tight for k equal to the next integer after √d are presented in every prime-square dimension. The verification of these examples, including explicit construction of the vectors unbiased to more bases, needs to be detailed to support the tightness claim.

Authors: We have expanded the examples section (now Section 5.2) with explicit matrix representations of the additional vectors for the smallest prime-square dimensions (d=9,25,49) and a uniform construction for general p^2 that uses the finite-field affine plane and a fixed set of Weyl operators. Each construction is accompanied by a direct computation verifying that the extra vectors are unbiased to exactly k+1 bases while satisfying the net axioms. The remaining dimensions follow by the same pattern, which is now stated as a proposition with a reference to the explicit formulas. revision: yes

Circularity Check

0 steps flagged

No significant circularity; new algebraic framework supplies independent derivation

full rationale

The paper introduces a novel definition of k-net over an algebra A that recovers both classical k-nets (commutative case) and MUB collections (A = M_d(C)). The rigidity property is then derived inside this framework; the prior combinatorial proof for classical k-nets is cited as using a completely different argument, while the MUB case is explicitly new. No equation or claim reduces by construction to a fitted parameter, self-citation chain, or renamed input. The derivation chain is therefore self-contained and does not meet any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no free parameters, background axioms, or invented physical entities are identifiable. The central new object is the definition of k-net over an algebra, which functions as a unifying mathematical construction rather than an ad-hoc postulate.

pith-pipeline@v0.9.0 · 5895 in / 1301 out tokens · 34463 ms · 2026-05-25T08:59:54.060569+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.

Definition 2: collection of orthogonal projections N satisfying equivalence relation on 'P=Q or PQ=0', τ(PQ)=1/dim(A) for distinct classes, and class sums to I.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

Theorem 1: if N is k-net of order d over A with k≤√d then any P in Span(N) with τ(P)=1/d lies in N.

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

21 extracted references · 21 canonical work pages · 1 internal anchor

[1]

The limitati ons of nice mutually unbiased bases

M. Aschbacher, A. M. Childs, and P. Wocjan. “The limitati ons of nice mutually unbiased bases”. In: J. Algebraic Combin. 25.2 (2007), pp. 111–123

work page 2007
[2]

A new proof for the existence of m utu- ally unbiased bases

S. Bandyopadhyay et al. “A new proof for the existence of m utu- ally unbiased bases”. In: Algorithmica 34.4 (2002), pp. 512–528

work page 2002
[3]

On discrete structures in finite Hilbert spaces

I. Bengtsson and K. Zyczkowski. “On discrete structures in ﬁnite Hilbert spaces”. In: arXiv preprint (2017), arXiv:1701.07902

work page internal anchor Pith review Pith/arXiv arXiv 2017
[4]

Finite nets II: uniqueness and imbedding

R. H. Bruck. “Finite nets II: uniqueness and imbedding”. In: Canad. J. Math. 13.2 (1963), pp. 421–457

work page 1963
[5]

Davidson

K. Davidson. C*-algebras by Example . Fields Institute for Re- search in Mathematical Sciences Toronto: Fields Institute mono- graphs. American Mathematical Society, 1996. REFERENCES 25

work page 1996
[6]

Self-testing mutually unbi ased bases in the prepare-and-measure scenario

M. Farkas and J. Kaniewski. “Self-testing mutually unbi ased bases in the prepare-and-measure scenario”. In: Phys.Rev.A99.3 (2019)

work page 2019
[7]

Geometrical description of quantal sta te determi- nation

I. D. Ivanović. “Geometrical description of quantal sta te determi- nation”. In: J. Phys. A 14.12 (1981)

work page 1981
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On the monomiality of n ice error bases

A. Klappenecker and M. Rötteler. “On the monomiality of n ice error bases”. In: IEEE T. Inform. Theory 51.3 (2005), pp. 1084– 1089

work page 2005
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Unitary Error Bases: C on- structions, Equivalence, and Applications

A. Klappenecker and M. Rötteler. “Unitary Error Bases: C on- structions, Equivalence, and Applications”. In: Proceedings of the 15th International Conference on Applied Algebra, Algebra ic Al- gorithms and Error-correcting Codes. AAECC’03. Toulouse, Fran- ce: Springer-Verlag, 2003, pp. 139–149

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Nice Complete Sets of Pairwise Quasi-Or thogonal Masas

S. Krusekamp. “Nice Complete Sets of Pairwise Quasi-Or thogonal Masas”. PhD thesis. Universitäts-und Landesbibliothek Mü nster, 2014

work page 2014
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Unextendible Mutually Unbiased Bas es from Pauli Classes

P. Mandayam et al. “Unextendible Mutually Unbiased Bas es from Pauli Classes”. In: Quantum Inf. Comput. 14.9&10 (2014), pp. 823– 844

work page 2014
[12]

Complementarity in quantum systems

D. Petz. “Complementarity in quantum systems”. In: Rep. Math. Phys. 59 (2 2007), pp. 209–224

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Complementarity and the algebraic structure of 4-level quantum systems

D. Petz, A. Szántó, and M. Weiner. “Complementarity and the algebraic structure of 4-level quantum systems”. In: J. Inﬁn. Dim. Anal. Quantum Probability and Related Topics 12.1 (2009), pp. 99– 116

work page 2009
[14]

Unitary operator bases

J. Schwinger. “Unitary operator bases”. In: Proc. Natl. Acad. Sci. USA 46.4 (1960), pp. 570–579

work page 1960
[15]

Complementary decompositions and unexten dible mu- tually unbiased bases

A. Szántó. “Complementary decompositions and unexten dible mu- tually unbiased bases”. In: Linear Alg. Appl. 496 (2016), pp. 392– 406

work page 2016
[16]

Unextendible mutually unbiased bases (after Mandayam, Bandyopadhyay, Grassl and Wootters)

K. Thas. “Unextendible mutually unbiased bases (after Mandayam, Bandyopadhyay, Grassl and Wootters)”. In: Entropy 18.11 (2016), p. 395

work page 2016
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A gap for the maximum number of mutually unbi ased bases

M. Weiner. “A gap for the maximum number of mutually unbi ased bases”. In: Proc. Amer. Math. Soc. 141 (2013), pp. 1963–1969

work page 2013
[18]

On orthogonal systems of matrix algebras

M. Weiner. “On orthogonal systems of matrix algebras”. In: Lin- ear Alg. Appl. 433 (2010), pp. 520–533

work page 2010
[19]

New Construction of Mutually Unb iased Bases in Square Dimensions

P. Wocjan and T. Beth. “New Construction of Mutually Unb iased Bases in Square Dimensions”. In: Quantum Info. Comput. 5.2 (2005), pp. 93–101

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Quantum measurements and ﬁnite geomet ry

W. K. Wootters. “Quantum measurements and ﬁnite geomet ry”. In: Found. Phys. 36.1 (2006), pp. 112–126

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[21]

Optimal State-Determi nation by Mutually Unbiased Measurements

W. K. Wootters and B. D. Fields. “Optimal State-Determi nation by Mutually Unbiased Measurements”. In: Ann. Phys. 191.2 (1989), pp. 363–381. 26 REFERENCES Sloan Nietert, Dept. of Computer Science, Cornell Universit y, Ithaca, NY 14853, USA E-mail address : sbn45@cornell.edu Zsombor Szilágyi, MTA-BME Lendület Quantum Information The ory Research Group, Dep...

work page 1989

[1] [1]

The limitati ons of nice mutually unbiased bases

M. Aschbacher, A. M. Childs, and P. Wocjan. “The limitati ons of nice mutually unbiased bases”. In: J. Algebraic Combin. 25.2 (2007), pp. 111–123

work page 2007

[2] [2]

A new proof for the existence of m utu- ally unbiased bases

S. Bandyopadhyay et al. “A new proof for the existence of m utu- ally unbiased bases”. In: Algorithmica 34.4 (2002), pp. 512–528

work page 2002

[3] [3]

On discrete structures in finite Hilbert spaces

I. Bengtsson and K. Zyczkowski. “On discrete structures in ﬁnite Hilbert spaces”. In: arXiv preprint (2017), arXiv:1701.07902

work page internal anchor Pith review Pith/arXiv arXiv 2017

[4] [4]

Finite nets II: uniqueness and imbedding

R. H. Bruck. “Finite nets II: uniqueness and imbedding”. In: Canad. J. Math. 13.2 (1963), pp. 421–457

work page 1963

[5] [5]

Davidson

K. Davidson. C*-algebras by Example . Fields Institute for Re- search in Mathematical Sciences Toronto: Fields Institute mono- graphs. American Mathematical Society, 1996. REFERENCES 25

work page 1996

[6] [6]

Self-testing mutually unbi ased bases in the prepare-and-measure scenario

M. Farkas and J. Kaniewski. “Self-testing mutually unbi ased bases in the prepare-and-measure scenario”. In: Phys.Rev.A99.3 (2019)

work page 2019

[7] [7]

Geometrical description of quantal sta te determi- nation

I. D. Ivanović. “Geometrical description of quantal sta te determi- nation”. In: J. Phys. A 14.12 (1981)

work page 1981

[8] [8]

On the monomiality of n ice error bases

A. Klappenecker and M. Rötteler. “On the monomiality of n ice error bases”. In: IEEE T. Inform. Theory 51.3 (2005), pp. 1084– 1089

work page 2005

[9] [9]

Unitary Error Bases: C on- structions, Equivalence, and Applications

A. Klappenecker and M. Rötteler. “Unitary Error Bases: C on- structions, Equivalence, and Applications”. In: Proceedings of the 15th International Conference on Applied Algebra, Algebra ic Al- gorithms and Error-correcting Codes. AAECC’03. Toulouse, Fran- ce: Springer-Verlag, 2003, pp. 139–149

work page 2003

[10] [10]

Nice Complete Sets of Pairwise Quasi-Or thogonal Masas

S. Krusekamp. “Nice Complete Sets of Pairwise Quasi-Or thogonal Masas”. PhD thesis. Universitäts-und Landesbibliothek Mü nster, 2014

work page 2014

[11] [11]

Unextendible Mutually Unbiased Bas es from Pauli Classes

P. Mandayam et al. “Unextendible Mutually Unbiased Bas es from Pauli Classes”. In: Quantum Inf. Comput. 14.9&10 (2014), pp. 823– 844

work page 2014

[12] [12]

Complementarity in quantum systems

D. Petz. “Complementarity in quantum systems”. In: Rep. Math. Phys. 59 (2 2007), pp. 209–224

work page 2007

[13] [13]

Complementarity and the algebraic structure of 4-level quantum systems

D. Petz, A. Szántó, and M. Weiner. “Complementarity and the algebraic structure of 4-level quantum systems”. In: J. Inﬁn. Dim. Anal. Quantum Probability and Related Topics 12.1 (2009), pp. 99– 116

work page 2009

[14] [14]

Unitary operator bases

J. Schwinger. “Unitary operator bases”. In: Proc. Natl. Acad. Sci. USA 46.4 (1960), pp. 570–579

work page 1960

[15] [15]

Complementary decompositions and unexten dible mu- tually unbiased bases

A. Szántó. “Complementary decompositions and unexten dible mu- tually unbiased bases”. In: Linear Alg. Appl. 496 (2016), pp. 392– 406

work page 2016

[16] [16]

Unextendible mutually unbiased bases (after Mandayam, Bandyopadhyay, Grassl and Wootters)

K. Thas. “Unextendible mutually unbiased bases (after Mandayam, Bandyopadhyay, Grassl and Wootters)”. In: Entropy 18.11 (2016), p. 395

work page 2016

[17] [17]

A gap for the maximum number of mutually unbi ased bases

M. Weiner. “A gap for the maximum number of mutually unbi ased bases”. In: Proc. Amer. Math. Soc. 141 (2013), pp. 1963–1969

work page 2013

[18] [18]

On orthogonal systems of matrix algebras

M. Weiner. “On orthogonal systems of matrix algebras”. In: Lin- ear Alg. Appl. 433 (2010), pp. 520–533

work page 2010

[19] [19]

New Construction of Mutually Unb iased Bases in Square Dimensions

P. Wocjan and T. Beth. “New Construction of Mutually Unb iased Bases in Square Dimensions”. In: Quantum Info. Comput. 5.2 (2005), pp. 93–101

work page 2005

[20] [20]

Quantum measurements and ﬁnite geomet ry

W. K. Wootters. “Quantum measurements and ﬁnite geomet ry”. In: Found. Phys. 36.1 (2006), pp. 112–126

work page 2006

[21] [21]

Optimal State-Determi nation by Mutually Unbiased Measurements

W. K. Wootters and B. D. Fields. “Optimal State-Determi nation by Mutually Unbiased Measurements”. In: Ann. Phys. 191.2 (1989), pp. 363–381. 26 REFERENCES Sloan Nietert, Dept. of Computer Science, Cornell Universit y, Ithaca, NY 14853, USA E-mail address : sbn45@cornell.edu Zsombor Szilágyi, MTA-BME Lendület Quantum Information The ory Research Group, Dep...

work page 1989