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 bialgebra. We introduce various smooth entropies and prove the corresponding uncertainty principles for von Neumann bialgebras. 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 WigdersonWigderson uncertainty principle. We prove the WigdersonWigderson uncertainty principle for von Neumann bialgebras in Theorems 2.8 and for subfactors in Theorem 3.19. In [22], A. Wigderson and Y. Wigderson proposed a conjecture on the WigdersonWigderson 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 bialgebras with examples from quantum Fourier analysis. We prove some basic uncertainty principles for von Neumann bialgebras. In Section 3, we prove uncertainty principles on von Neumann bialgebras for smooth support and von Neumann entropy perturbed by norms. We prove WigdersonWigderson uncertainty principles on von Neumann bialgebras, with a better constant in the case of subfactors. In Section 4, we provide a bound for WigdersonWigderson 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 bialgebras and ktransforms
In this section, we recall some basic definitions and results about von Neumann algebras. We introduce von Neumann bialgebras 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

, where , ;

, where , ;

, 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
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 bialgebra.
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 bialgebra , such that and are finitedimensional 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 finitedimensional von Neumann algebras with traces and respectively, and is commutative, and is unitary with respect to norms. By the quantum HausdorffYoung inequality , (Theorem 4.5 in [16]), we have that is a von Neumann bialgebra.
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 HausdorffYoung inequality, (Theorem 4.8 and Theorem 7.3 in [9]), we have that for any , and ,
Therefore, and the quintuple is a von Neumann bialgebra.
Remark 2.7.
The quantum HausdorffYoung 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 bialgebra.
In [22], Wigderson and Wigderson proved the primary uncertainty principles (See Theorem 2.3 in [22]) for any Hadamard matrix ,
(1) 
which is the fundamental result of that paper. We call the inequality as WigdersonWigderson uncertainty principle. In this paper, we prove the following quantum version of WigdersonWigderson uncertainty principle for von Neumann bialgebras. When a von Neumann bialgebra is obtained from Example 2.4, then our theorem implies Theorem 2.3 in [22].
Theorem 2.8 (The quantum WigdersonWigderson uncertainty principle).
Let be a von Neumann bialgebra. For any , we have
Proof.
When and , we have that , because
This implies that . Then for any , we have
Multiplying the above two inequalities, we obtain
This completes the proof of the theorem. ∎
Using the primary uncertainty principle, A. Wigderson and Y. Wigderson further prove the DonohoStark uncertainty principle for arbitrary Hadamard matrices (See Theorem 3.2 in [22]). In this paper, we prove the DonohoStark uncertainty principle for von Neumann bialgebras using the quantum WigdersonWigderson 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 DonohoStark 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 DonohoStark uncertainty principles from these quantum symmetries to von Neumann bialgebras.
Theorem 2.10 (Quantum DonohoStark uncertainty principle).
Let be a von Neuman bialgebra. Then for any nonzero operator , we have
Proof.
We already have, from Theorem 2.8, that for any nonzero ,
Thus, all we need is to bound the 1norm by the support of , which can be implemented through Hölder’s inequality, for any ,
Applying this bound to both and , we obtain the result.
∎
3. Quantum smooth uncertainty principles
In this section, we prove a series of smooth uncertainty principles for von Neumann bialgebras. We firstly prove the quantum smooth support uncertainty principles in §3.1. Then we proceed to prove quantum WigdersonWigderson 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 HirschmanBecker 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
where is the range projection of .
Remark 3.2.
Since the set
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
Proposition 3.5.
For any , we have
Proof.
It is clear that .
For any , we claim that
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 operatormonotone function, it reduces to prove . For any normal state on , by the CauchySchwartz inequality, we have
Therefore,
Rearranging the above inequality, we obtain
Thus,
The claim holds and we have .
For any , , we have
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
In summary, the statement holds. ∎
In [22], A. Wigderson and Y. Wigderson introduced the following smooth support for the finitedimensional and abelian case.
Definition 3.6.
(See Definition 3.15 in [22]) Let , , and be the counting measure. Let and . For an operator , the supportsize of is defined to be
Remark 3.7.
When is finitedimensional 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
Let , then . Moreover, we have
Therefore,
(2) 
So
When , replacing by and by in Inequality (2), we have
So
From the above discussions, is continuous with respect to .
∎
We have the following quantum smooth support uncertainty principle.
Theorem 3.9 (The quantum smooth support uncertainty principle).
Let the quintuple be a von Neumann bialgebra and be a nonzero operator. For any , we have
Proof.
Take a positive operator in such that
By Hölder’s inequality, we have
Thus
Repeating the above process for , we obtain
Multiplying these two inequalities, we have
The second inequality uses Theorem 2.8, the quantum WigdersonWigderson uncertainty principle. ∎
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 nonzero box , we have
When in Definition 3.1, we are able to choose a positive contraction in the abelian *subalgebra generated by such that the supportsize 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
Proof.
Let be the tracepreserving conditional expectation from onto . For any , . Take , then
and and .
Note that any pure state on is multiplicative, so , for any . Moreover. is a state on , by the CauchySchwartz inequality, . So , and therefore .
Take , then
Therefore, the statement holds.
∎
We have the following quantum smooth support uncertainty principle.
Theorem 3.13 (The quantum smooth support uncertainty principle).
Let be a von Neumann bialgebra. Suppose and are finite dimensional and . For any nonzero operator , we have
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 tracepreserving 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
By direct computations, we have
Let , then is a linear operator from into . For any , we have
The first inequality is true by the CauchySchwartz inequality and the second one uses the fact that . This implies
(3) 
Remark 3.14.
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 1transform, we have
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