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M L e 2 i L π p →L F p H →L ie nuncer- an gives F a edefined be can /2 q q d . 6 eateto ahmtc,Tigu nvriy ejn 004 hn;and China; 100084, Beijing University, Tsinghua Mathematics, of Department q = 5.Further (5). c,d,1,2 stransfor- as where 1, p | /(p − uxagRen Yunxiang , 1). H x ring a fusion 24 as A ref. free references. See is and this groups. results (23); beyond further Lusztig symmetry by for quantum introduced interesting as an rings, fusion is with start us Let Rings Fusion on QFA different 2. many in advances other to are lead We fields. will information. uncertainty QFA quantum that entanglement, in certain problems quantum also other to and QFA relations, approach objects. power- mathematical an a provides distinguish subfactors. provides property” to and product obstruction “Schur categories, ful the fusion how show rings, We fusion between future tions the for goals general some questions. open state some we QFA. and 9, of applications section cite in we paper, Finally, Brascamp–Lieb the Throughout the CFA. in way ers, the H inequality, to universal Young’s similar unites a inequality is propose This inequalities. uncer- We tum an entropy. Eq. yields namely relative inequality, that quantum for pictures principle between tainty inequality algebraic “relative” their a while CFA, in those on to differ. QFA properties in similar extremizers are the quantum of behav- subfactors the transformation whose so Fourier (22); projection projection under a ior of a multiple a is is transform Fourier biprojection A biprojections. hsoe cesatcei distributed under is article access open This Diego. interest.y San competing at no California declare of authors University The J.W.H., Systems and and Sciences; Mathematics of of Academy Academy Chinese and Science, Hampshire paper.y New the of wrote University J.W. Y.R., L.G., and Z.L., Y.R., Reviewers: C.J., Z.L., A.J., A.J., research; and designed research; J.W. performed and J.W. Y.R., and Z.L., C.J., A.J., contributions: Author 2 1 BY-NC-ND) (CC .y 4.0 NoDerivatives License owo orsodnemyb drse.Eal [email protected] or [email protected] Email: addressed. be may correspondence [email protected] whom work.y this to To equally contributed J.W. and Y.R., Z.L., C.J., A.J., 1 nisifny ilhv infiatftr impact. future significant have will infancy, its now quantum analysis, in of Fourier quantum analysis that the believe quantum and We entanglement. entropy to on applications theo- bounds some some with give physics solve we can and it problems, how retical this show introduce we we paper, subject, princi- this mathematical In uncertainty symmetries. quantum with yields for associated It ples communication. inequalities perspective. of this and theory extends insights informa- the analysis and underlies Fourier engineering, it Quantum chemistry, and science, into tion insights theory. gives quantum of It clas- formulation from the ago, range to physics electromagnetism years sical in of applications field 200 Its every mathematics. almost pure over understanding in discovered cornerstone a analysis, remains Fourier Classical Significance F eel nih n nrni tutr,a ela rela- as well as structure, intrinsic and insight reveals QFA establish We QFA. of view unified a give we paper, this In 1 = n uhthat such and , a,1 mdl,wt basis a with Z-module, n isn Wu Jinsong and , raieCmosAttribution-NonCommercial- Commons Creative e,1 htuie ayohrquan- other many unifies that 9, {x 1 , le nqaiy n oth- and inequality, older ¨ x NSLts Articles Latest PNAS 2 , . . . , x c m a Mathematical Yau e A nttt for Institute }, sarn that ring a is m ∈ with N, | f6 of 1 y

MATHEMATICS Pm s s [B1] xj xk = s=1 Nj ,k xs , with Nj ,k ∈ N, and Any w ∈ P1,+ satisfies the spherical condition, and any x ∈ [B2] there exists an involution ∗ on {1, 2, ... , m} such that Pn,+ satisfies the “reflection-positivity” condition, 1 ∗ Nj ,k = δj ,k∗ , inducing an antiisomorphism of A, given by xk := ∗ ∗ ∗ xk∗ and xk xj = (xj xk ) . From a fusion ring, one can construct two C ∗-algebras with [1] faithful tracial states and a unitary Fourier transform between them. We recall this construction and the QFA on fusion rings ∗ ∗ The action of planar tangles turns Pn,± into C -algebras, and as studied in ref. 25. Define a unital -algebra over C with basis the trace gives a Hilbert-space representation by the GNS con- {xj }, with multiplication given by property [B1], and with adjoint ∗ struction. We set A = P2,+ and B = P2,−. For elements in A, given by in [B2]. Define a linear functional τ given by the Dirac one has pictorial representations for x, multiplication xy, the measure on the basis, τ(xj ) = δj ,1. From [B2] we infer that τ is Fourier transform F(x), and the trace tr(x) as follows: a strictly positive trace, and therefore it defines an inner prod- uct. This gives the C ∗-algebra B by the Gelfand–Naimark–Segal (GNS) construction. One can define a second (in this case abelian) C ∗-algebra A from A with multiplication , adjoint #, given by another strictly positive linear functional d on A. Let −1 [A1] xj  xk = δj ,k d(xj ) xj , ∗ A # Define the convolution product on by [A2] xj = xj . Here, d(·) is defined to be linear, by sending a basis element xj [2] to d(xj ), the operator norm of the fusion matrix Mj , with entries t (Mj )s,t = Nj ,s . This is called the Perron–Frobenius dimension of ∗ ∗ xj . The trace d (resp. τ) on the finite-dimensional C -algebra The C -algebra B has a similar pictorial representation. These p q A (resp. B) defines L norms on A (resp. L norms on B) by pictures not only make QFA transparent; they also provide a kak = d(|a|p )1/p and kbk = τ(|b|q )1/q . Then A and B are precise framework for proofs. Ap Bq two C ∗-algebras with the same basis. We use the classical nota- Theorem 3.1 (Schur product theorem; theorem 4.1 in ref. 20). For any 0 x, y ∈ A (or 0 x, y ∈ B), one has 0 x ∗ y. tion F : x 7→ xb for the Fourier transform as the linear map from 6 6 6 A B F(x ) = x Proof. Since tr is faithful, it suffices to show that tr ((x ∗ y)z)> to defined by j j . The Fourier transform can be ∗ extended to a map from Lp (A, d) to Lq (B, τ) for 1/p + 1/q = 1. 0 for any 0 6 z ∈ A. Since A is a C -algebra, there exist elements 1 1 1 2 2 2 Plancherel’s formula follows as: kF(x)kB2 = kxkA2 . Similarly, x , y , z ∈ A, such that one has τ((F(x))∗F(y)) = d(x # ◦ y). We summarize several results in ref. 25 about QFA on fusion rings, including the quantum Schur product theo- rem (QSP), the quantum Hausdorff–Young inequality (QHY) with 1/p + 1/q = 1, the quantum Young inequality (QY) [3] with 1/p + 1/q = 1 + 1/r, and the basic quantum uncertainty principles (QUP)—defined in terms of the von Neumann 2 # # 2 entropy, HA(|x| ) = −d((x  x)  log(x  x)), and HB(|xb| ) = ∗ ∗ [4] −τ((xb xb) log(xb xb)), and in terms of the support SA(x) = d(R(x)), SB(xb) = τ(R(xb)), where R(·) is the range projection. Theorem 2.1. For nonzero x, y ∈ A, Here, we infer positivity from the reflection positivity in Eq. 1. −1 [QSP]: F (xbyb) >A 0, whenever both x, y >A 0. This proof works on B by switching the shading, since the dual of [QHY]: kxbkBq 6 kxkAp , for 1 6 p 6 2. a subfactor is also a subfactor. −1 The Schur product property holds on both A and B for subfac- [QY]: kF (xbyb)kAr 6 kxkAp kykAq , for 1 6 p, q, r 6 ∞. 2 2 2 2 tors. On fusion rings, it holds on A but not necessarily on B. We [QUP−1]: HA(|x| ) + HB(|x| ) + 2kxkA2 log kxkA2 > 0 . b discuss this essential difference revealed by QFA in more detail [QUP-2] : SA(x)SB(x) > 1. b in section 4.

3. QFA on Subfactors Applications. An important application of the Schur product the- Modern subfactor theory was initiated by Jones in 1983 by his orem comes in the classification of abelian subfactor planar remarkable index theorem (26). A subfactor of type II1 is an algebras (20). One can regard this as the fundamental theorem inclusion of II1 factors N ⊂ M, and its index describes the rel- for abelian subfactors, extending the fundamental theorem for ative size of these two factors. Jones’ index theorem asserts that finite abelian groups. This was the first classification that requires 2 2 π the index δ of a subfactor belongs to the set {4 cos ( n ): n = a bound neither on the Jones index nor on the dimension. 3, 4, · · · } ∪ [4, ∞], and every possible value can be realized as the Theorem 3.2 (theorem 6.7 in ref. 20). An irreducible subfactor index of a subfactor. Subfactor theory turns out to be a natural planar algebra is abelian if and only if it is a free product of the framework to study quantum symmetry in statistical physics and simplest (Temperley–Lieb–Jones) planar algebras and finite abelian quantum field theory (e.g., ref. 27). Subfactor planar algebras groups. (28) provide a pictorial tool to study subfactor theory. A pla- Also one obtains a geometric proof of the (mathematical- nar algebra P• = {Pn,± : n > 0} is a family of finite-dimensional physics version of) reflection-positivity condition. vector spaces with an action of the operad of planar tangles, sim- Theorem 3.3. Reflection positivity holds for Hamiltonians on ilar to topological quantum field theory (29). One represents an planar para algebras (theorem 7.1 in ref. 30) and on Levin–Wen element in Pn,± (called an n-box) by a labeled rectangle with models (theorem 3.2 in ref. 31). 2n strings attached to its boundary. Each vector space Pn,± is equipped with an involution ∗, which is compatible with the invo- Quantum Hausdorff–Young Inequalities. Many estimates for the ∗ lution of planar tangles. This involution is called the adjoint and norm Kp,q = kFkLp →Lq have been established in the quantum given pictorially by vertical reflection. case, including when p, q are not dual (21, 32). Instead of

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C , L any for let space and derivatives, spatial Connes’ the of commutant the on weight ful of commutant the Let unitary, on (35). multiplicative transform case and special Fourier group a quantum the is which algebra of Kac groups, quantum compact Kac the on infinite-dimensional relies noncommutative This The on of groups. theory results symmetry. quantum compact are quantum locally and there infinite algebras “yes”; for is established whether answer ask be might as can One such subfactors. QFA symmetry, finite-index quantum and finite rings on fusion focus results previous Groups The Quantum Compact Locally on QFA 5. ring fusion the Therefore, (where matrices these of table eigenvalue The rmtodmninl(D otredmninl(D space, (3D) three-dimensional algebras planar to of (2D) extension surface the two-dimensional introduced formalizing from 41, Z.L. ref. framework. in pictorial algebras a in axiomatized Algebras. 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MATHEMATICS outlined in ref. 42. Surface algebras are an extensive framework P2,−). Let Dϕ (resp. Dψ) be the density operator of ϕ (resp. to capture additional pictorial features of Fourier analysis. ψ), namely, ϕ(·) = tr(Dϕ ·). Now, we define a Fourier transform p q For any subfactor planar algebra, the actions of planar tangles Fp,ϕ,ψ : L (P2,+, tr) → L (P2,−, tr) for 1 6 p 6 2, q = p/(p − can be further extended to the actions of surface tangles. (The 1) as arrow in planar diagrams corresponds to the $ sign in planar 1/p 1/2 1/q−1/2 Fp,ϕ,ψ(xDϕ ) = F(xDϕ )Dψ . [10] algebras. The clockwise/anticlockwise orientation of the arrow indicates the input/output disc in surface algebras.) One can rep- This Fourier transform is represented pictorially as follows, resent Fourier transform, multiplication, and convolution as the action of the following surface tangles in the 3D space: [11]

[8] From Plancherel’s theorem for F, we infer Plancherel’s theorem Using 3D pictures, one can consider the Fourier duality for sur- for Fp,ϕ,ψ, face tangles with multiple inputs and outputs. The 3D formalism 1/2 1/2 1/2 has provided deep insights into quantum information, algebraic kF2,ϕ,ψ(xDϕ )k2 = kF(xDϕ )k2 = kxDϕ k2. [12] identities, and various other connections with physics. One can consider a finite-dimensional Kac algebra K as A Theorem 7.1 (Relative, quantum Hausdorff–Young inequal- ity). Let • be an irreducible subfactor planar algebra and let ϕ, ψ and its dual K as B, with a Fourier transform from K to K P b b be faithful states on . Then for any x ∈ , 1 p 2, and defined analogously to section 5. The pair of Kac algebras K and P2,± P2,+ 6 6 dual 2 6 q = p/(p − 1), we have Kb can be understood as A and B for the surface algebra. The comultiplication is given by the picture: 1/p 1/p kFp,ϕ,ψ(xDϕ )kq 6 Kp,ϕ,ψkxDϕ kp . [13]

−2/p 1/q−1/2 1/2−1/p Here, Kp,ϕ,ψ = δ kDψ k∞kF(Dϕ )k1. Pictorially,

The Hopf-axiom that the comultiplication is an algebraic homo- morphism reduces to the string–genus relation of surface tangles given in equation 17 of ref. 42.

A Universal Inequality. In a subfactor planar/surface algebra P, where the Fourier transform, the multiplication, and the convolution can be realized by planar/surface tangles. In general, a surface [14] L tangle is a multilinear map on Pn,±. Now, we give a pic- n∈N torial inequality in the quantum case, motivated by the classical Brascamp–Lieb inequality. We replace the dual of the linear Proof. We give the elementary and insightful picture proof: n kj map Bj : R → R by a surface tangle Tj with kj input discs and n output discs; moreover, the n-output discs are identical for different j :

m m Y Y T (x ) C kx k , [9] j j 6 j pj j =1 1 j =1 and C is the best constant. This topological inequality includes the quantum Hausdorff– Young inequality, quantum Holder¨ inequality, and quantum version of Young’s inequality. The best constants of these three inequalities are achieved at biprojections. For those familiar with the Quon language (42), we can consider the pictorial inequalities whose Tj s are surface tan- gles with braided charged strings. In particular, if all of the inputs and outputs are 2-boxes, corresponding to qudits, then T j can be any Clifford transformation on qudits. These Clifford The first inequality is a consequence of the quantum Holder¨ transformations can be considered as a quantum analog of the inequality. The second inequality is a consequence of the quan- n kj dual of a linear transformation Bj :(Zd ) → (Zd ) . Consider- tum Hausdorff–Young inequality (theorem 4.8 in ref. 21). Here, ing the action on density matrices, the n-qudit Clifford gates on the constant is Ke = sup Ke (y), with Ke (y) equal to the Pauli matrices are symplectic transformations on 2n-dimensional kykp =1 symplectic spaces over Zd . following picture: 7. Relative Inequalities, Entropy, and Uncertainty Here, we present a relative, quantum, Hausdorff–Young inequality. This leads to a relative, quantum, entropic uncertainty principle. Let P• be an irreducible subfactor planar algebra with the Markov trace tr. Let ϕ (resp. ψ) be a faithful state on P2,+ (resp.

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MATHEMATICS ∞ n 1 n 2 n projections P, Q and constant λ, then there is a biprojection B, Conjecture 9.6. For any f ∈ L (R ) ∩ L (R ) ∩ L (R ), f such that kx − Bk < ε. converges either to 0 or to a Gaussian function, under the action n Question 9.5. Can one characterize the extremizers for the of the iteration of the block map 2 Bλ. ∞ n 1 n 2 n uncertainty principles on n-boxes, for n > 3? Conjecture 9.7. For any f ∈ L (R ) ∩ L (R ) ∩ L (R ), the Hirschman–Beckner entropy decreases under the action of the block n Block Renormalization Map and Quantum Central Limit Theorem. map 2 Bλ. The same question remains for finite cyclic groups. The block map Bλ is a composition of convolution and multiplication, ACKNOWLEDGMENTS. We are grateful for informative discussions with a number of colleagues and especially thank Dietmar Bisch, Kaifeng Bu, Pavel Etingof, Xun Gao, Leonard Gross, J. William Helton, Vaughan Jones, Christopher King, Robert Lin, Sebastian Palcoux, Terence Tao, Zhenghan Wang, Feng Xu, and Quanhua Xu. Z.L. and J.W. thank Harvard University for hospitality during early work on this paper. This research was supported in part by Grant TRT 0159 on mathematical picture language from the Templeton Religion Trust. A.J. was partially supported by Army Research Office Grant W911NF1910302. C.J. was partially supported by National Nat- The limit points of the iteration of the block map are all bipro- ural Science Foundation of China (NSFC) Grant 11831006. Z.L. was partially jections for finite-index, irreducible subfactors (47). We regard supported by Grant 100301004 from Tsinghua University. J.W. was partially this result as a quantum 2D central limit theorem. supported by NSFC Grant 11771413.

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