quantum fourier analysis - harvard university2019/05/09 · classical fourier duality in the early...

Quantum Fourier Analysis

Zhengwei Liu

Harvard University → Tsinghua University

May 9, 2019, NCGOA, Vanderbilt University

Z. Liu (Harvard) Quantum Fourier Analysis May 9, 2019 1 / 37

Quantum Fourier Analysis

We propose a program of Quantum Fourier Analysis, to investigate analyticaspects of quantum symmetries and their Fourier dualities.


Arthur Jaffe Chunlan Jiang

Yunxiang Ren Jinsong Wu


Fourier Analysis on Groups


Classical Fourier Duality

In the early 1800’s, Joseph Fourier introduced his transformation to solvedifferential equations describing heat.The Fourier transform F on measurable functions f on R is

F(f )(x) =

∫ ∞−∞

f (t)e−2πitxdt .

Convolution for such functions is:

(f1 ∗ f2)(s) =

∫ ∞−∞

f1(t)f2(s − t)dt ,

yielding the Fourier duality

F(f1 ∗ f2) = F(f1)F(f2) . (1)

Pontryagin studied Fourier duality on locally compact abelian groupsthrough their characters. Tannaka and Krein introduced Fourier duality forcompact groups, leading to a categorical understanding of Fourier duality.


Inequalities on RTake

‖f ‖p =

(∫ ∞−∞|f (t)|pdt

)1/p

, 0 < p <∞.

For p > 1, ‖ · ‖p is the p-norm of measurable functions and ‖f ‖∞ is theessential maximum of f .Plancherel formula (1910):

‖F f ‖2 = ‖f ‖2.

Interpolating with the elementary inequality ‖F(f )‖∞ 6 ‖f ‖1, one obtainsthe Hausdorff-Young inequality,

‖F(f )‖q 6 ‖f ‖p , 1 6 p 6 2, 1/p + 1/q = 1 . (2)

Young’s inequality for convolution (1912):

‖f1 ∗ f2‖r 6 ‖f1‖p‖f2‖q . (3)

for p, q, r > 1, 1/p + 1/q − 1/r = 1 .Z. Liu (Harvard) Quantum Fourier Analysis May 9, 2019 6 / 37

Optimal Inequalities

In 1975, Beckner obtained optimal constants. For p, q, r as above andAp = p1/2pq−1/2q, he showed that

‖F(f )‖q 6 Ap‖f ‖p , ‖f1 ∗ f2‖r 6 ApAqA rr−1‖f1‖p‖f2‖q ,

and that Gaussian functions yield equality.


Brascamp-Lieb inequalities

In 1976, Brascamp and Lieb proposed a fundamental inequality:Let Bj : Rn → Rnj , 1 6 j 6 m, be linear maps. Let fj be a non-negative,measurable function on Rnj , and let pj > 0 satisfy

∑mj=1 nj/(pjn) = 1.

Then ∫Rn

m∏j=1

fj ◦ Bj 6 Cm∏j=1

‖fj‖pj , (4)

This includes Young’s inequality, Holder’s inequality, and theLoomis-Whitney inequality as special cases.In 2008, Bennett, Carbery, Christ, and Tao found the optimal constant C ,which is obtained at certain Gaussian functions.


Example: Young’s inequality

Take n = 2, B1(x , y) = x , B2(x , y) = y , B3(x , y) = x + y . Then3∑

j=1

1

pj= 2.

∫Rn

m∏j=1

fj ◦ Bj

=

∫R2

f1(x)f2(y)f3(x + y)dxdy

=

∫R2

f1(x)f2(t − x)f3(t)dxdt

=

∫Rf1 ∗ f2(t)f3(t)dt

6Ap1Ap2Ap3

m∏j=1

‖fj‖pj .


Heisenberg Uncertainty Principles

In 1927, Heisenberg showed in quantum theory that position andmomentum cannot simultaneously be precisely measured (the Heisenberg’suncertainty principle). This has been reformulated by Kennard and by Weyl.It says that the standard deviation of positiion σx and of momentum σpmust satisfy

σxσp >~2,

where ~ is Planck’s constant.Mathematically, this is a general phenomenon for a pair of noncommutativeoperators x and p = F(−id/dx)F−1, which satisfy

‖xf ‖2 ‖pF f ‖2 > (4π)−1 , ‖f ‖2 = 1 .


Entropic Uncertainty Principles on R

In 1957, Hirschman studied the Shannon entropyH(|f |2) = −

∫∞−∞ |f (x)|2 log |f (x)|2dx of |f |2, proving

H(|f |2) + H(|F(f )|2) > 0 , ‖f ‖2 = 1 .

Everett conjectured the lower bound is log e2 , which is proved by Beckner in

1975. Another quick proof is given by Bialynicki and Birula in 1975. Theextremizers of these inequalities are Gaussians. The Hirschman-Beckneruncertainty principle ensures Heisenberg’s uncertainty principle.


Uncertainty Principles on Finite Abelian Groups

In 1989, Donoho and Stark established an uncertainty principle for functionsf on cyclic groups G in terms of the cardinality of their support:

|S(f )| |S(F(f ))| > |G | ,

here f is a function on G , S(f ) = {x : f (x) 6= 0}, and |A| is the cardinalityof the set A.This type of uncertainty principle has been applied in compressed sensing byCandes-Romberg-Tao and by Donoho in compressed sensing in 2006.


Quantum Fourier Analysis (QFA) on Subfactors


Subfactors

Suppose N ⊂M is a subfactor (of type II1). Jones index theorem (1983):

{[M : N ] := dimN (L2(M))} = {4 cos2 π

n, n = 3, 4, · · · } ∪ [4,∞].

We assume that N ⊂M has finite index δ2.

M is an N -N bimodule.

The N -N bimodule maps homN−N (M) form a C ∗-algebra A.

The multiplication map m :M⊗N M→M is an N −N bimodulemap.

The convolution * on homN−N (M) is x ∗ y = m(x ⊗ y)m∗. It makeshomN−N (M) into another C ∗-algebra B.


Examples: Group Subfactors

When M = N o G , for an outer action of a finite group G , we have[M : N ] = |G |.

M = ⊕g∈GNg as an N -N bimodule

homN−N (M) = A ∼= L∞(G ), with Haar measure

The convolution * is the usual convolution on L1(G ).

B ∼= L(G ), the left regular representation with trace

The identity map on G induces a map F : A → B, which can be consideredas the Fourier transform.


Measurements and Plancherel Formula

For a finite-index subfactor, the C ∗-algebra A has a Haar measure inducedfrom the unique trace of the factors.The C ∗-algebra B has a Dirac measure at the Jones projectione : L2(M)→ L2(N ).We can define Lp spaces for A and B using these measures.

Plancherel Formula: For any x ∈ A,

‖F(x)‖2 = ‖x‖2 .


The Norm of the Fourier Transform

Theorem (L-Wu 19)

Given an irreducible subfactor with index µ, let F be the Fourier transformfrom A to B. For any x ∈ A, ‖x‖2 = 1, and any p, q > 0, we have

‖F(x)‖q 6 K (1/p, 1/q)‖x‖p.

Hausdorff-Young inequality: 1/p + 1/q = 1, 1/2 6 1/p 6 1.

K(1/p,1/q):=

0

1q

12

•1•

1p

12•

1•µ1q− 1

2

RTF

RF 1

µ1p

+ 1q−1

RT


Extremizers

Bisch 94: intermediate subfactors ↔ biprojectionsJiang-L-Wu 16: modulation and translation → bishiftsbishifts of biprojections ∼ Gaussian functions

Analysis Algebra

Regions Extremizers

1/p + 1/q > 1, 1/p > 1/2 trace-one projections

1/p + 1/q = 1, 1/2 < 1/p < 1 bishifts of biprojections

1/p = 1, 1/q = 0 extremal elements

1/p = 1/2, 1/q = 1/2 A1/p + 1/q < 1, 0 < 1/q < 1/2 Fourier transform of trace-one projections

1/q = 0, 0 6 1/p < 1 extremal unitary elements

1/q = 1/2, 0 6 1/p < 1/2 unitary elements

1/q > 1/2, 1/p = 1/2 Fourier transform of unitary elements

1/q > 1/2, 1/p < 1/2 biunitary elements if exist

Table: Table for extremizersZ. Liu (Harvard) Quantum Fourier Analysis May 9, 2019 18 / 37

Renyi entropic uncertainty principles

For p ∈ (0, 1) ∪ (1,∞), the Renyi entropy of order p of x in A is defined as

hp(x) =p

1− plog ‖x‖p.

h1(x) = H(x) = tr2(−‖x‖ log ‖x‖).

Theorem (L-Wu 19)

Let x ∈ A be such that ‖x‖2 = 1. Then for any p, q > 0,

(1/p − 1/2)hp/2(|x |2) + (1/2− 1/q)hq/2(|F(x)|2) > − logK (1/p, 1/q).

The weights 1/p − 1/2 and 1/2− 1/q can be modified as 1.


Uncertainty Principles

When 1/p, 1/q → 1/2, we recover the Hirschman-Beckner uncertaintyprinciple:

Theorem (Jiang-L-Wu 16)

For any x ∈ A, we have

H(|x |2) + H(|F(x)|2) > −2‖x‖22 log ‖x‖2

2.

Moreover, “ = ” holds if and only if x is a bi-shift of a biprojection.

When 1/p, 1/q →∞, we recover the Donoho-Stark uncertainty principle:


For any x ∈ A, we have

S(F(x))S(x) > δ2.

Moreover, “ = ” holds if and only if x is a bi-shift of a biprojection.


Main Results on Subfactors

Schur product theorem (May Not hold on the dual of fusion rings)

Hausdorff-Young inequality

Young’s inequality (Hold on Temperley-Lieb-Jones iff δ2=Jones Index)

Hirschman-Beckner uncertainty principle

Donoho-Stark uncertainty principle

Sum set estimate (Suggested by Terrence Tao in 2014)

The characterization of operators which attain the equality of theabove inequalities

Hardy uncertainty principle

Renyi entropic uncertainty principle

Block maps (“Renormalization maps”, non-linear, even new for Z2)(Central limit theorem for finite-index subfactors)


Bishifts of Biprojections


For any non-zero x ∈ A, the following are equivalent:

(1) x is a bi-shift of a biprojection;

(2) x is an extremal bi-partial isometry;

(3) S(x)S(F(x)) = µ;

(4) H(|x |2) + H(|F(x)|2) = −2‖x‖22 log ‖x‖2

2;

(5) ‖x ∗ x∗‖r = ‖x‖t‖x‖s for some 1 < r , t, s <∞ such that1r + 1 = 1

t + 1s ;

(6) ‖x ∗ x∗‖r = ‖x‖1‖x‖r for some 1 < r <∞;

(7) ‖x ∗ x∗‖r = ‖x‖t‖x‖s for any 1 6 r , t, s 6∞ such that 1r + 1 = 1

t + 1s ;

(8) ‖F(x)‖ tt−1

= ‖x‖t for some 1 < t < 2;

(9) ‖F(x)‖ tt−1

= ‖x‖t for any 1 6 t 6 2;


Fourier Analysis on Groups

2004

1977

1974

1989-90

H.Y.

N.F. Y.

H.B.

D.S.

S.S.

R.

Min

I.S.

E.P.

E.F.

E.N.

S. S. : Sum set estimateN. F. : Norm of Fourier transformH. Y. : Hausdorff-Young inequalityY. : Young’s inequalityH. B. : Hirschman-Beckner uncertainty principleD. S. : Donoho-Stark uncertainty principle

R. : Renyi entropic uncertainty principle

I. S. : Exact inverse sum set theoremE. F. : Extremal functions of Hausdorff-Young in-equalityE. N. : Extremizers of the norm of the Fourier trans-formE. P. : Extremal Pairs of Young’s inequality

Min : Minimizers of the uncertainty principles


Quantum Fourier Analysis on Subfactors

S.P.H.Y.

N.F. Y.

H.B.

D.S.

S.S.

R.

Min

I.S.

E.P.

E.F.

E.N.

Inequalities

Bi-shifts ofbiprojections

S. P. : Schur Product theoremS. S. : Sum set estimateN. F. : Norm of Fourier transformH. Y. : Hausdorff-Young inequalityY. : Young’s inequalityH. B. : Hirschman-Beckner uncertainty principleD. S. : Donoho-Stark uncertainty principle

R. : Renyi entropic uncertainty principle

I. S. : Exact inverse sum set theoremE. F. : Extremal functions of Hausdorff-YounginequalityE. N. : Extremizers of the norm of the FouriertransformE. P. : Extremal Pairs of Young’s inequality

Min : Minimizers of the uncertainty principles

Figure: It shows the logic net of our proofs about the quantum Fourier analysis onsubfactors.


Topological Brascamp-Lieb inequality

Quanhua Xu: Brascamp-Lieb inequality on subfactors?

Brascamp-Lieb inequality:∫Rn

m∏j=1

fj ◦ Bj 6 Cm∏j=1

‖fj‖pj , (5)

Bj : Rn → Rnj , fj : Rnj → R+, pj > 0,∑m

j=1 nj/(pjn) = 1.

Answer: topological Brascamp-Lieb inequality!


Topological Quantum Field Theory

A 1+1 Topological Quantum Field Theory is a monoidal functor F fromthe category of (oriented) 1+1 cobordisms Cob to the category of finitedimensional vector spaces Vec. (TQFT was studied by Witten, Atiyah,Jones, Reshetikhin, Turaev, Viro and many others).

Oriented Circle → Vector space V (or its dual)Oriented Surface with boundary → Multi-linear transformation on V

Disjoint Union → ⊗Gluing Boundary → 〈·, ·〉

→ hom(V 2,V 2)


Surface Algebras

A surface algebra with reflection positivity is a monoidal *-functor F fromthe category of 1+1 cobordisms with non-intersecting strings CobS to Vec.It extends Jones’ subfactor planar algebras from the plane to surfaces.(Liu 2019 CMP)

Oriented Circle with n points → Hilbert space Hn

Reflection → Riesz representation

→ hom(H2n,H2

n)


Pictorial Fourier Duality

In surface algebras, we can represent the string Fourier transform Fs , themultiplication and the convolution on H4 as the action of the followingsurface tangles respectively:

, , .

3D Pictorial Fourier duality: the 90◦ rotation around the z−axis.(For shaded tangles, Fs : H4,+ → H4,−)In general, a surface tangle (with 4 boundary points on each input/outputdisc) is a multi-linear map on H4.


Topological Brascamp-Lieb inequality

Brascamp-Lieb inequality:

‖m∏j=1

fj ◦ Bj‖1 6 Cm∏j=1

‖fj‖pj , (6)

Bj : Rn → Rnj , fj : Rnj → R+, pj > 0,∑m

j=1 nj/(pjn) = 1.Topological Brascamp-Lieb inequality:∣∣∣∣∣∣tr

m∏j=1

Tj(xj)

∣∣∣∣∣∣ 6 Cm∏j=1

‖xj‖pj , (7)

where C is the best constant.(1) B∗j → a surface tangle Tj with kj input discs and n out put discs

(2) R→ H4, xj → vectors in Hnj4


Topological identity for pj ’s.

Take the (shaded) tangle T , such that

T (⊗jxj) = tr

m∏j=1

Tj(xj)

.

We define the genus of the topological Brascamp-Lieb inequality as thegenus of T .Suppose T has r+ unshaded regions, r− shaded regions, m+ unshadedinputs with parameters pj ,+, 1 6 j 6 m+, and m+ shaded inputs withparameters pk,−, 1 6 k 6 m−.(3) Genus-zero identity: (two are equivalent)

r+ −m− −m+∑j=1

p−1j ,+ +

m−∑j=1

p−1k,− − 1 = 0 ;

r− −m+ −m−∑j=1

p−1k,− +

m+∑j=1

p−1j ,+ − 1 = 0 .


Optimal Inequalities

The genus-0 topological Brascamp-Lieb inequality includes Hausdorff-Younginequality, Holder inequality, and Young’s inequality.We gave the best constant of the first three inequalities which are achievedat bishifts of biprojections.


Questions on Topological Brascamp-Lieb Inequalities:

There are three central problems for the Brascamp-Lieb inequality on Rn.1. The finiteness of the best constant.2. Whether the best constant can be achieved as Gaussian functions?3. Whether all extremizers are Gaussian?

Since H4 is finite dimensional, the best constant C of the topologicalBrascam-Lieb inequality is finite and the extremizer exists by thecompactness. We ask the following questions:1. In which case, the best constant is achieved by biprojections.2. Whether the extremizers are all bishifts of biprojections.The topological Brascamp-Lieb inequality also suggests an inequality on Rn

generalizing both the Brascamp-Lieb inequality and the Hausdorff-Younginequality!


Depth two Subfactors and Kac algebras

Symanski 94 (claimed by Ocneanu)finite-index irreducible depth-2 subfactors ↔ finite dimensional Kacalgebras.

The co-multiplication of the Kac algebra H4 is given by the followingsurface tangle:

.

The Hopf-axiom reduces to the string-genus relation of surface tangles(Jaffe-L18).

Enock-Nest 96infinite-index irreducible depth-2 subfactors ↔ discrete (or compact)Kac algebras, (based on certain regular conditions.)

Inspired by this connection, we also investigated the QFA on infinitequantum symmetries, such as infinite dimensional Kac algebras and locallycompact quantum groups. Various inequalities have been established.


Quantum Fourier Analysis (QFA) on Various Quantum Symmetries

Infinite type Kac AlgebrasLocally Compact Quantum Groups

Topological Type Planar algebrasSurface algebras

TQFTs

Categorical Type Unitary Fusion CategoriesUnitary Modular Tensor Categories

Multibody system Multiple qubitsMultiple quonsLattice modes

Tensor networks


Thank you!

A paper on Quantum Fourier Analysis will be posted soon!


A Question on Biprojections

Given an irreducible, finite-index subfactor, we obtain two C ∗-algebras Aand B as above. The two C ∗-algebras A and B are P2,+ and P2,− interms of subfactor planar algebras.

Conjecture

For any ε > 0, there is a ε′ (depending on ε and the index), such that ifx ∈ B, ‖x − P‖2 < ε′ and ‖Fs(x)− λQ‖2 < ε′, for some projectionsP ∈ B,Q ∈ A and some positive scalar λ, then there is a biprojection B,such that ‖x − B‖ < ε.


A Question on entropic Uncertainty principles

• William Helton’s observation.


Take h(t) = −t log t. For any x ∈ A, φ = φ = tr , we have

φ ◦ h(|x |2) + φ ◦ h(|F(x)|2) > h ◦ φ(|x |2) + h ◦ φ(|F(x)|2).

Question: For which φ and h, the above inequality holds?


quantum fourier analysis - harvard university2019/05/09 · classical fourier duality in the early...

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