# Quantum smooth uncertainty principles for von Neumann bi-algebras

In this article, we prove various smooth uncertainty principles on von Neumann bi-algebras, which unify numbers of uncertainty principles on quantum symmetries, such as subfactors, and fusion bi-algebras etc, studied in quantum Fourier analysis. We also obtain Widgerson-Wigderson type uncertainty principles for von Neumann bi-algebras. Moreover, we give a complete answer to a conjecture proposed by A. Wigderson and Y. Wigderson.

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## 1. Introduction

Uncertainty principles have been investigated for more than hundred years in mathematics and physics inspired by the famous Heisenberg uncertainty principle [6, 14, 21] with significant applications in information theory [2, 3].

Recently quantum uncertainty principles on subfactors, an important type of quantum symmetires [11, 5], have been established for support and for von Neumann entropy in [9] and for Rényi entropy in [18]. These quantum uncertainty principles have been generalized on other types of quantum symmetries, such as Kac algebras [17], locally compact quantum groups [10] and fusion bialgebras [16] etc, in the unified framework of quantum Fourier analysis [8]. Such quantum inequalities were applied in the classification of subfactors [15] and as analytic obstructions of unitary categorifications of fusion rings in [16].

In 2021, A. Wigderson and Y. Wigderson [22] introduced

-Hadamard matrices, as an analogue of discrete Fourier transforms, and they proved various uncertainty principles such as primary uncertainty principles, support uncertainty principles etc. Their work unifies numbers of proofs of uncertainty principles in classical settings.

In this paper, we unify several quantum entropic uncertainty principles on quantum symmetries and we further generalize the results to various smooth entropies. Inspired by the notion of -Hadamard matrices, we introduce -transforms between a pair of finite von Neumann algebras, and we call their combination a von Neumann -bi-algebra. We introduce various smooth entropies and prove the corresponding uncertainty principles for von Neumann -bi-algebras. On one hand, our results generalized numbers of uncertainty principles for quantum symmetries in [9, 16]. On the other hand, these results are slightly stronger than uncertainty principles for -Hadamard matrices in [22]. See Theorems 3.9, 3.13, 3.22 and 3.28.

The primary uncertainty principle for -Hadamard matrices plays a key role in [22] and we call this type of uncertainty principle the Wigderson-Wigderson uncertainty principle. We prove the Wigderson-Wigderson uncertainty principle for von Neumann -bi-algebras in Theorems 2.8 and for subfactors in Theorem 3.19. In [22], A. Wigderson and Y. Wigderson proposed a conjecture on the Wigderson-Wigderson uncertainty principle for the real line . We give a complete answer to the conjecture, see Theorem 4.3 for details.

The paper is organized as follows. In Section 2, we introduce -transforms and von Neumann -bi-algebras with examples from quantum Fourier analysis. We prove some basic uncertainty principles for von Neumann -bi-algebras. In Section 3, we prove uncertainty principles on von Neumann bi-algebras for smooth support and von Neumann entropy perturbed by -norms. We prove Wigderson-Wigderson uncertainty principles on von Neumann bi-algebras, with a better constant in the case of subfactors. In Section 4, we provide a bound for Wigderson-Wigderson uncertainty principle on the real line and this answers a conjecture proposed by A. Wigderson and Y. Wigderson in [22].

###### Acknowledgement.

Zhengwei Liu was supported by NKPs (Grant no. 2020YFA0713000) and by Tsinghua University (Grant no. 100301004). Jinsong Wu was supported by NSFC (Grant no. 11771413 and 12031004).

## 2. von Neumann bi-algebras and k-transforms

In this section, we recall some basic definitions and results about von Neumann algebras. We introduce von Neumann bi-algebras with interesting examples and we prove some basic properties and uncertainty principles.

A von Neumann algebra is said to be finite if it has a faithful normal tracial positive linear functional , see e.g. [13]. We will call this linear functional as trace in the rest of the paper. We denote , for . When , is called the -norm. Moreover, , the operator norm of . It is clear that for .

The following inequalities will be used frequently in the rest of the paper.

###### Proposition  2.1 (Hölder’s inequalities).

For any , we have

1. , where , ;

2. , where , ;

3. , where , .

###### Proof.

See e.g. Theorems 5.2.2 and 5.2.4 in [23]. ∎

###### Notation  2.2.

Suppose and are two finite von Neumann algebras with traces and respectively. Let be a linear map. For any , define

 ∥F∥q→p:=sup{∥F(x)∥p:x∈A, ∥x∥q=1}.
###### Definition  2.3.

Suppose and are two finite von Neumann algebras with traces and respectively. For , a -transform from into is a linear map such that and for any . We call the quintuple a von Neumann -bi-algebra.

###### Example  2.4.

The definition of -transform is inspired by the definition of -Hadamard matrix of A. Wigderson and Y. Wigderson (Definition 2.2 in [22]). In particular, a -Hadamard matrix can be extended to a von Neumann -bi-algebra , such that and are finite-dimensional abelian von Neumann algebras, and are counting measures.

###### Example  2.5.

Let the quintuple be a fusion bialgebra (See Definition 2.12 in [16]), where and are finite-dimensional von Neumann algebras with traces and respectively, and is commutative, and is unitary with respect to -norms. By the quantum Hausdorff-Young inequality , (Theorem 4.5 in [16]), we have that is a von Neumann -bi-algebra.

###### Example  2.6.

Suppose is an irreducible subfactor planar algebra with finite Jones index (See Definition on page 4 in [11]) , . Let be the unnormalized Markov trace of , for , and be the string Fourier transform, which is unitary. Then by the quantum Hausdorff-Young inequality, (Theorem 4.8 and Theorem 7.3 in [9]), we have that for any , and ,

 ∥F∥q→p=(1δ)1−2p.

Therefore, and the quintuple is a von Neumann -bi-algebra.

###### Remark  2.7.

The quantum Hausdorff-Young inequality, Theorem 7.3 in [9], also applies to reducible subfactor planar algebras, and in that case is replaced by certain constant . Then is a von Neumann -bi-algebra.

In [22], Wigderson and Wigderson proved the primary uncertainty principles (See Theorem 2.3 in [22]) for any -Hadamard matrix ,

 (1) ∥v∥1∥Av∥1≥k∥v∥∞∥Av∥∞,v∈Cn,

which is the fundamental result of that paper. We call the inequality as Wigderson-Wigderson uncertainty principle. In this paper, we prove the following quantum version of Wigderson-Wigderson uncertainty principle for von Neumann -bi-algebras. When a von Neumann -bi-algebra is obtained from Example 2.4, then our theorem implies Theorem 2.3 in [22].

###### Theorem  2.8 (The quantum Wigderson-Wigderson uncertainty principle).

Let be a von Neumann -bi-algebra. For any , we have

 ∥x∥1∥F(x)∥1≥k∥x∥∞∥F(x)∥∞.
###### Proof.

When and , we have that , because

 ∥F∥p→q= sup{∥F(x)∥q:x∈A,∥x∥p=1} = sup{|τ(F(x)y∗)|:x∈A,y∈B,∥x∥p=1,∥y∥p=1} = = sup{∥F∗(y)∥q:y∈B,∥y∥p=1} = ∥F∗∥p→q.

This implies that . Then for any , we have

 ∥F(x)∥∞≤∥x∥1,k∥x∥∞≤∥F∗F(x)∥∞≤∥F(x)∥1.

Multiplying the above two inequalities, we obtain

 ∥x∥1∥F(x)∥1≥k∥x∥∞∥F(x)∥∞.

This completes the proof of the theorem. ∎

Using the primary uncertainty principle, A. Wigderson and Y. Wigderson further prove the Donoho-Stark uncertainty principle for arbitrary -Hadamard matrices (See Theorem 3.2 in [22]). In this paper, we prove the Donoho-Stark uncertainty principle for von Neumann -bi-algebras using the quantum Wigderson-Wigderson uncertainty principle. Firstly, let’s recall the notion of the support in a finite von Neumann algebra.

###### Definition  2.9.

Let be a finite von Neumann algebra with a trace . For any , let be the range projection of . The support of is defined as .

The support has been used in the quantum Donoho-Stark uncertainty principles on quantum symmetries such as subfactors and fusion rings, see Theorem 5.2 in [9] and Theorem 4.8 in [16] respectively. We generalize the Donoho-Stark uncertainty principles from these quantum symmetries to von Neumann -bi-algebras.

###### Theorem  2.10 (Quantum Donoho-Stark uncertainty principle).

Let be a von Neuman -bi-algebra. Then for any non-zero operator , we have

 S(x)S(F(x))≥k.
###### Proof.

We already have, from Theorem 2.8, that for any nonzero ,

 ∥x∥1∥F(x)∥1≥k∥x∥∞∥F(x)∥∞.

Thus, all we need is to bound the 1-norm by the support of , which can be implemented through Hölder’s inequality, for any ,

 ∥x∥1=∥R(x)x∥1≤∥x∥∞∥R(x)∥1=∥x∥∞S(x).

Applying this bound to both and , we obtain the result.

###### Remark  2.11.

Our theorem is a generalization of the Donoho-Stark uncertainty principle in [4] and some variations,

1. In Example 2.4, Theorem 2.10 implies Theorem 3.2 in [22];

2. In Example 2.5, Theorem 2.10 implies Theorem 4.8 in [16];

3. In Example 2.6, Theorem 2.10 implies Theorem 5.2 in [9].

## 3. Quantum smooth uncertainty principles

In this section, we prove a series of smooth uncertainty principles for von Neumann bi-algebras. We firstly prove the quantum smooth support uncertainty principles in §3.1. Then we proceed to prove quantum Wigderson-Wigderson uncertainty principles for general -norms, , and give an example concerning the quantum Fourier transform on subfactor planar algebras in §3.2. Finally, we also prove quantum smooth Hirschman-Becker uncertainty principles in §3.3.

### 3.1. Quantum smooth support uncertainty principles

We firstly introduce a new smooth support which is slightly different from the classical smooth support.

###### Definition  3.1.

Let be a finite von Neumann algebera with a trace . Let and . For any element , we define the smooth support to be

 Spϵ(x)=inf{τM(HR(x)):H∈M, 0≤H≤I, ∥(I−H)x∥p≤ϵ∥x∥p},

where is the range projection of .

###### Remark  3.2.

Since the set

 S(ϵ,p,x):={H∈M:  0≤H≤I, ∥(I−H)x∥p≤ϵ∥x∥p}

is compact in the weak operator topology and the trace is normal, there exits an such that .

###### Remark  3.3.

Take , then and this implies . In this case, .

Besides Definition 3.1, there are three kinds of notions of the smooth support.

###### Definition  3.4.

Let be a finite von Neumann algebra with a trace . Let and . For any element , define

 f1(ϵ,p,x) :=inf{τM(R(y)):y∈M, ∥x−y∥p≤ϵ∥x∥p}, f2(ϵ,p,x) :=inf{τM(R(Hx)):H∈M, 0≤H≤I, ∥(I−H)x∥p≤ϵ∥x∥p}, f3(ϵ,p,x) :=inf{τM(R(Qx)):Q∈M, Q=Q∗=Q2, ∥(I−Q)x∥p≤ϵ∥x∥p}.
###### Proposition  3.5.

For any , we have

 Spϵ(x)≤f1(ϵ,p,x)=f2(ϵ,p,x)=f3(ϵ,p,x).
###### Proof.

It is clear that .

For any , we claim that

 τM(R(R(y)x))≤τM(R(y)),∥(I−R(y))x∥p≤∥x−y∥p.

If the claim holds, then . Since , the first inequality holds.

Next, we prove the second inequality in the claim. It is enough to prove that . Since is an operator-monotone function, it reduces to prove . For any normal state on , by the Cauchy-Schwartz inequality, we have

 2|ρ(y∗x)|= 2|⟨x,y⟩ρ| = 2|⟨R(y)x,y⟩ρ| ≤ 2⟨R(y)x,R(y)x⟩12ρ⟨y,y⟩12ρ ≤ ρ(x∗R(y)x)+ρ(y∗y).

Therefore,

 ρ(x∗y)+ρ(y∗x)≤ρ(x∗R(y)x)+ρ(y∗y).

Rearranging the above inequality, we obtain

 ρ((x−R(y)x)∗(x−R(y)x))≤ρ((x−y)∗(x−y)).

Thus,

 (x−R(y)x)∗(x−R(y)x)≤(x−y)∗(x−y).

The claim holds and we have .

For any , , we have

 τM(R(x)H)≤τM(|R(x)H|)≤τM(∥R(x)H∥R(Hx))≤τM(R(Hx)).

The first inequality is true by Hölder’s inequality. The second one uses the fact that , . The last inequality is due to . So we have

 Spϵ(x)≤f2(ϵ,p,x).

In summary, the statement holds. ∎

In [22], A. Wigderson and Y. Wigderson introduced the following smooth support for the finite-dimensional and abelian case.

###### Definition  3.6.

(See Definition 3.15 in [22]) Let , , and be the counting measure. Let and . For an operator , the support-size of is defined to be

 |supppϵ(x)|=min{τM(Q): Q∈M, Q=Q∗=Q2, ∥(I−Q)x∥p≤ϵ∥x∥p}.
###### Remark  3.7.

When is finite-dimensional and abelian and is the counting measure, then is equal to . In this case, .

###### Lemma  3.8.

For any , we have is continuous with respect to .

###### Proof.

When , take an such that

 Spϵ(x)=τM(HR(x)),∥(I−H)x∥p≤ϵ∥x∥p,0≤H≤I.

Let , then . Moreover, we have

 τM(H′R(x))=(1−c)τM(R(x))+cSpϵ(x),∥(I−H′)x∥p≤cϵ∥x∥p.

Therefore,

 (2) Spϵ(x)≤Spcϵ(x)≤(1−c)τM(R(x))+cSpϵ(x).

So

 limc→1−Spcϵ(x)=Spϵ(x).

When , replacing by and by in Inequality (2), we have

 Spcϵ(x)≤Spϵ(x)≤(1−1c)τM(R(x))+1cSpcϵ(x).

So

 limc→1+Spcϵ(x)=Spϵ(x).

From the above discussions, is continuous with respect to .

We have the following quantum smooth support uncertainty principle.

###### Theorem  3.9 (The quantum L1 smooth support uncertainty principle).

Let the quintuple be a von Neumann -bi-algebra and be a non-zero operator. For any , we have

 S1ϵ(x)S1η(F(x))≥k(1−ϵ)(1−η).
###### Proof.

Take a positive operator in such that

 S1ϵ(x)=d(R(x)H),∥(I−H)x∥1≤ϵ∥x∥1,0≤H≤I.

By Hölder’s inequality, we have

 d(|x∗|(I−H))≤d(|x∗(I−H)|)=∥(I−H)x∥1≤ϵ∥x∥1.

Thus

 S1ϵ(x) =d(R(x)H) =1∥x∥∞d(∥x∥∞R(x)H) ≥1∥x∥∞d(|x∗|H)=1∥x∥∞d(|x∗|)−1∥x∥∞d(|x∗|(I−H)) ≥∥x∥1∥x∥∞−1∥x∥∞d(|x∗(I−H)|) ≥∥x∥1∥x∥∞(1−ϵ).

Repeating the above process for , we obtain

 S1η(F(x))≥∥F(x)∥1∥F(x)∥∞(1−η).

Multiplying these two inequalities, we have

 S1ϵ(x)S1η(F(x))≥∥x∥1∥x∥∞⋅∥F(x)∥1∥F(x)∥∞(1−ϵ)(1−η)≥k(1−ϵ)(1−η).

The second inequality uses Theorem 2.8, the quantum Wigderson-Wigderson uncertainty principle. ∎

###### Remark  3.10.

We can obtain Theorem 2.10 from Theorem 3.9 by assuming .

Applying Theorem 3.9 to the quantum Fourier transform on subfactor planar algebras, we obtain the following corollary.

###### Corollary  3.11.

Suppose is an irreducible subfactor planar algebra with finite Jones index . Let be the Fourier transform from onto . Then for any non-zero -box , we have

 S1ϵ(x)S1η(F(x))≥δ2(1−ϵ)(1−η).

When in Definition 3.1, we are able to choose a positive contraction in the abelian *-subalgebra generated by such that the support-size is exactly the trace of . More precisely, we have

###### Proposition  3.12.

Suppose is a finite von Neumann algebra with a trace . Let , and let be the abelian von Neumann subalgebra generated by in . For any , we have

 S2ϵ(x)=min{τM(H):H∈N, 0≤H≤R(x), ∥(I−H)x∥2≤ϵ∥x∥2}.
###### Proof.

Let be the trace-preserving conditional expectation from onto . For any , . Take , then

 τM(H′)=τM(Φ(H)R(x))=τM(R(x)H),

and and .

Note that any pure state on is multiplicative, so , for any . Moreover. is a state on , by the Cauchy-Schwartz inequality, . So , and therefore .

Take , then

 ∥(I−H′)x∥22 =∥Φ(I−H)R(x)x∥22 =τM(|Φ(I−H)|2|x∗|2) ≤τM(Φ(|I−H|2)|x∗|2) =τM(|I−H|2|x∗|2) =∥(I−H)x∥22.

Therefore, the statement holds.

We have the following quantum smooth support uncertainty principle.

###### Theorem  3.13 (The quantum L2 smooth support uncertainty principle).

Let be a von Neumann -bi-algebra. Suppose and are finite dimensional and . For any non-zero operator , we have

 S2ϵ(x)S2η(F(x))≥k(1−ϵ−η)2,∀ϵ,η∈[0,1], ϵ+η≤1.
###### Proof.

Take , then . Since the definition of is invariant under rescaling, we have that .

Let and be the polar decompositions, where and are the polar parts in and respectively. Let be the abelian von Neumann subalgebra of generated by and be the abelian von Neumann subalgebra of generated by . Let be the trace-preserving conditional expectation from onto and . Then is a linear operator from into such that . Let and be mutually orthogonal minimal projections in and such that and . The linear operator is a matrix with by .

By Proposition 3.12, we can find two positive operators in and in such that

 0≤H≤R(x) ,0≤K≤R(y), ∥(I−H)x∥2≤ϵ∥x∥2 ,∥(I−K)y∥2≤η∥y∥2, d(H)=S2ϵ(x) ,τ(K)=S2η(y).

By direct computations, we have

 H=n∑i=1d(eiH)d(ei)ei, K=m∑j=1τ(fjK)τ(fj)fj.

Let , then is a linear operator from into . For any , we have

 ∥˜Mv∥22 =m∑i=1τ(fi)∣∣∣τ(fiK)τ(fi)n∑j=1aijd(ejH)d(ej)vj∣∣∣2 ≤m∑i=1τ(fiK)2τ(fi)n∑j=1|aij|2d(ejH)2d(ej)3n∑j=1d(ej)|vj|2 =m∑i=1τ(fiK)2τ(fi)n∑j=1|aij|2d(ejH)2d(ej)3∥v∥22 ≤1km∑i=1τ(fiK)2τ(fi)n∑j=1d(ejH)2d(ej)∥v∥22 ≤d(H)τ(K)k∥v∥22.

The first inequality is true by the Cauchy-Schwartz inequality and the second one uses the fact that . This implies

 (3) ∥˜M∥2→2≤√d(H)τ(K)/√k=√S2ϵ(x)S2η(y)/√k.

For the lower bound of , we firstly observe that

 ∥M(I−H)|x∗|∥2 =∥ΦRV∗WRU(I−H)|x∗|∥2 ≤∥RV∗WRU(I−H)|x∗|∥2 =∥(I−H)|x∗|∥2.

Since is a contraction, so

 ∥KM(I−H)|x∗|∥2≤∥M(I−H)|x∗|∥2≤∥(I−H)|x∗|∥2.

Therefore, we have

 ∥M|x∗|−˜M|x∗|∥2 =∥M|x∗|−KMH|x∗|∥2 =∥(I−K)M|x∗|+KM(I−H)|x∗|∥2 ≤∥(I−K)|y∗|∥2+∥(I−H)|x∗|∥2 ≤(ϵ+η)∥|x∗|∥2.

This implies

 (4) ∥˜M|x∗|∥2≥∥M|x∗|∥2−(ϵ+η)∥|x∗|∥2=∥|y∗|∥2−(ϵ+η)∥|x∗|∥2=(1−ϵ−η)∥|x∗|∥2.

Finally, combining equations (3) and (4) we see that

 S2ϵ(x)S2η(F(x))≥k(1−ϵ−η)2.

This completes the proof of the theorem. ∎

###### Remark  3.14.

When is a -Hadamard matrix, A. Wigderson and Y. Wigderson proved the following results (See Theorems 3.17 and 3.20 in [22] ):

1. For any ,

 |supp1ϵ(x)||supp1η(F(x))|≥(1−ϵ)(1−η),∀ϵ,η∈[0,1];
2. If , then for any ,

 |supp2ϵ(x)||supp2η(F(x))|≥(1−ϵ−η)2,∀ϵ,η∈[0,1],ϵ+η≤1.

By Remark 3.7, we have

 |supp1ϵ(x)||supp1η(F(x))| ≥S1ϵ(x)S1η(F(x))≥(1−ϵ)(1−η), |supp2ϵ(x)||supp2η(F(x))| ≥S2ϵ(x)S2η(F(x))≥(1−ϵ−η)2.

So Theorems 3.9 and 3.13 imply Theorems 3.17 and 3.20 in [22].

When is a -Hadamard matrix, Theorems 3.9 and 3.13 are strictly stronger than Theorems 3.17 and 3.20 in [22]. We construct the following example.

###### Example  3.15.

Let and , . Take and . Then while . Let be the 1-transform, we have

 4 =|supp1ϵ(x)||supp1η(F(x))|>S1ϵ(x)S1η(F(x))=169, 4 =|supp2ϵ(x)||supp2η(F(x))|>S2ϵ(x