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.


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 k-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 k-Hadamard matrices, we introduce k-transforms between a pair of finite von Neumann algebras, and we call their combination a von Neumann k-bi-algebra.We introduce various smooth entropies and prove the corresponding uncertainty principles for von Neumann k-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 k-Hadamard matrices in [22].See Theorems 3.9, 3.13, 3.22 and 3.28.
The primary uncertainty principle for k-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 k-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 R.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 k-transforms and von Neumann k-bi-algebras with examples from quantum Fourier analysis.We prove some basic uncertainty principles for von Neumann k-bi-algebras.In Section 3, we prove uncertainty principles on von Neumann bi-algebras for smooth support and von Neumann entropy perturbed by p-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 R and this answers a conjecture proposed by A. Wigderson and Y. Wigderson in [22].

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 M is said to be finite if it has a faithful normal tracial positive linear functional τ M , see e.g.[13].We will call this linear functional as trace in the rest of the paper.We denote x p = τ M (|x| p ) 1 p , for p > 0. When 1 ≤ p < ∞, • p is called the p-norm.Moreover, x ∞ = x , the operator norm of x.It is clear that x p = x * p = |x| p for p > 0.
The following inequalities will be used frequently in the rest of the paper.
Notation 2.2.Suppose A and B are two finite von Neumann algebras with traces d and τ respectively.Let F : A → B be a linear map.For any 0 < p, q ≤ ∞, define Therefore, δF 1→∞ = 1 and the quintuple (P n,+ , P n,− , T r n,+ , T r n,− , δF ) is a von Neumann δ 2 -bi-algebra.
In [22], Wigderson and Wigderson proved the primary uncertainty principles (See Theorem 2.3 in [22]) for any k-Hadamard matrix A, 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 k-bi-algebras.When a von Neumann k-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 (A, B, d, τ, F ) be a von Neumann k-bi-algebra.For any x ∈ A, we have Proof.When 1 ≤ p, q ≤ ∞ and 1/p + 1/q = 1, we have that F * p→q = F p→q , because 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 Donoho-Stark uncertainty principle for arbitrary k-Hadamard matrices (See Theorem 3.2 in [22]).In this paper, we prove the Donoho-Stark uncertainty principle for von Neumann k-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 M be a finite von Neumann algebra with a trace τ M .For any x ∈ M, let R(x) be the range projection of x.The support S(x) of x is defined as τ M (R(x)).
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 k-bi-algebras.
Theorem 2.10 (Quantum Donoho-Stark uncertainty principle).Let (A, B, d, τ, F ) be a von Neuman k-bi-algebra.Then for any non-zero operator x ∈ A, we have Proof.We already have, from Theorem 2.8, that for any nonzero x ∈ A, Thus, all we need is to bound the 1-norm by the support of x, which can be implemented through Hölder's inequality, for any x ∈ A, Applying this bound to both x and F (x), we obtain the result.
Remark 2.11.Our theorem is a generalization of the Donoho-Stark uncertainty principle in [4] and some variations,

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 p-norms, 1 ≤ p ≤ ∞, 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.
is compact in the weak operator topology and the trace is normal, there exits an ). Remark 3.3.Take ǫ = 0, then (I − H)x = 0 and this implies HR(x) = R(x).In this case, S p 0 (x) = S(x).Besides Definition 3.1, there are three kinds of notions of the smooth support.
Proposition 3.5.For any x ∈ M, we have For any y ∈ M, we claim that If the claim holds, then f 3 (ǫ, p, x) ≤ f 1 (ǫ, p, x).Since R(R(y)x) ≤ R(y), the first inequality holds.Next, we prove the second inequality in the claim.It is enough to prove that For any normal state ρ on M, by the Cauchy-Schwartz inequality, we have Rearranging the above inequality, we obtain For any H ∈ M, 0 ≤ H ≤ I, we have The first inequality is true by Hölder's inequality.The second one uses the fact that |y * | ≤ y R(y), y ∈ M. The last inequality is due to R(x)H ≤ 1.So we have S p ǫ (x) ≤ f 2 (ǫ, 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 M = C n , n ∈ N * , and τ M be the counting measure.Let ǫ ∈ [0, 1] and p ∈ [1, ∞].For an operator x ∈ M, the (p, ǫ) support-size of x is defined to be When M is finite-dimensional and abelian and τ M is the counting measure, then Therefore, . When c > 1, replacing c by c −1 and ǫ by cǫ in Inequality (2), we have From the above discussions, S p ǫ (x) is continuous with respect to ǫ.
We have the following quantum L 1 smooth support uncertainty principle.
Theorem 3.9 (The quantum L 1 smooth support uncertainty principle).Let the quintuple (A, B, d, τ, F ) be a von Neumann k-bi-algebra and x ∈ A be a non-zero operator.For any ǫ, η ∈ [0, 1], we have Proof.Take a positive operator H in A such that By Hölder's inequality, we have Thus Repeating the above process for F (x), we obtain Multiplying these two inequalities, we have The second inequality uses Theorem 2.8, the quantum Wigderson-Wigderson uncertainty principle.
Applying Theorem 3.9 to the quantum Fourier transform on subfactor planar algebras, we obtain the following corollary.
Corollary 3.11.Suppose P • is an irreducible subfactor planar algebra with finite Jones index δ 2 .Let F be the Fourier transform from P n,± onto P n,∓ .Then for any non-zero n-box x ∈ P n,± , we have Proof.Let Φ be the trace-preserving conditional expectation from M onto N .For any and Note that any pure state ρ on N is multiplicative, so ρ(|Φ(y Take y = I − H, then Therefore, the statement holds.
We have the following quantum L 2 smooth support uncertainty principle.
Theorem 3.13 (The quantum L 2 smooth support uncertainty principle).Let (A, B, d, τ, F ) be a von Neumann k-bi-algebra.Suppose A and B are finite dimensional and F * F = kI.For any non-zero operator x ∈ A, we have Proof.Take W = F / √ k, then W * W = I.Since the definition of S 2 η is invariant under rescaling, we have that S

y). By direct computations, we have
Let M = KM H, then M is a linear operator from A 0 into B 0 .For any v ∈ A 0 , we have The first inequality is true by the Cauchy-Schwartz inequality and the second one uses the fact that |a ij | ≤ d(e j )/ √ k.This implies For the lower bound of M , we firstly observe that Since K is a contraction, so Therefore, we have

This implies
Finally, combining equations ( 3) and (4) we see that This completes the proof of the theorem.Remark 3.14.When F is a k-Hadamard matrix, A. Wigderson and Y. Wigderson proved the following results (See Theorems 3.17 and 3.20 in [22] ): (1) For any x ∈ M, 2 .So Theorems 3.9 and 3.13 imply Theorems 3.17 and 3.20 in [22].
When F is a k-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
Let F = I be the 1-transform, we have Applying Theorem 3.13 to the quantum Fourier transform on subfactor planar algebras, we obtain the following corollary.Corollary 3.16.Suppose P • is an irreducible subfactor planar algebra with finite Jones index δ 2 .Let F be the Fourier transform from P n,± onto P n,∓ .Then for any non-zero n-box x ∈ P n,± , we have Quantum Wigderson-Wigderson uncertainty principle.In this section, we prove the quantum Wigderson-Wigderson uncertainty principle for von Neumann k-bi-algebras for 1/p + 1/q = 1, and for quantum Fourier transform on subfactor planar algebras for any 0 < p, q ≤ ∞.
We prove the quantum Hausdorff-Young inequality for k-transforms using the standard interpolation method.Theorem 3.17.Let (A, B, d, τ, F ) be a von Neumann k-bi-algebra such that F * F = kI.For any x ∈ A, we have where 2 ≤ p ≤ ∞ and 1/p + 1/q = 1.
Proof.Note that Applying the Riesz-Thorin interpolation theorem ( [19], Theorem IX.17), we have that Then we have the following quantum Wigderson-Wigderson uncertainty principles for k-transforms.Theorem 3.18.Let (A, B, d, τ, F ) be a von Neumann k-bi-algebra such that F * F = kI.For any x ∈ A, we have where 2 ≤ p ≤ ∞ and 1/p + 1/q = 1.
Proof.By Theorem 3.17, we have For the adjoint operator F * , we have Applying the same process in Theorem 3.17 to F * , we also have Multiplying the above two inequalities, we obtain This completes the proof of the theorem.
Next, we introduce the quantum Wigderson-Wigderson uncertainty principle for quantum Fourier transform for any 0 < p, q ≤ ∞, based on the norm of quantum Fourier transform computed in [18].
Theorem 3.19 (The norm of quantum Fourier transform).Suppose P is an irreducible subfactor planar algebra.Let F be the Fourier transform from P 2,± onto P 2,∓ .Let x ∈ P 2,± be a 2-box and 0 < p, q ≤ ∞.Then We refer the readers to Appendix A for the specific definition of the function K( 1p , 1 q ) = F p→q .The following theorem follows immediately from Theorem 3.19.
Theorem 3.20 (The quantum Wigderson-Wigderson uncertainty principle for quantum Fourier transform).Let x ∈ P 2,± , we have Proof.By Theorem 3.19, we have Multiplying the above two equations, we can obtain the result.
We have the quantum Hirschman-Beckner uncertainty principle for von Neumann k-bi-algebras.Theorem 3.22.Let (A, B, d, τ, F ) be a von Neumann k-bi-algebra.Suppose A and B are finite-dimensional and F * F = kI.Let x be a non-zero element in A. Then we have In particular, since Proof.By Theorem 3.17, we have where 2 ≤ q ≤ ∞ and 1 p + 1 q = 1.Let f (q) = log F (x) q − log x p − 1 q log k, then f (q) ≤ 0 and f (2) = 0, which implies f ′ (2) ≤ 0. Let |F (x)| = n i=1 λ i e i be the spectral decomposition.We have Analogously, .
We have Since f ′ (2) ≤ 0, we obtain This completes the proof.A natural question is to consider the perturbations of the inequality in Theorem 3.22.We firstly consider the smooth von Neumann entropy.Proof.Let {e i } n i=1 and {f i } n i=1 be two orthonormal basis of C n such that |A|e i = λ i e i and Let E be the projection from C n onto span{e 1 , . . ., e j } and F be the projection from C n onto span{f j , . . ., f n }, then E ∧ F = ∅.Take a unit vector v in E ∧ F , we have By Lemma 3.26, we have Suppose A and B are finite-dimensional, let We have the quantum smooth Hirschman-Beckner uncertainty principle.
Proof.By Proposition 3.27, for any y ∈ A with x − y p ≤ ǫ, we have 1 τ (I)η.Adding the above two equations and applying Theorem 3.22, we obtain the result.
In [9], the minimizers of Hirschman-Beckner uncertainty principle on subfactor planar algebras were characterized as bi-shifts of biprojections (See Theorems 6.4 and 6.13 in [9]).So it is natural to ask the following inverse problem.Problem 3.31.Find a positive function C(ǫ, δ), for ǫ, δ > 0, such that lim ǫ→0 C(ǫ, δ) → 0, and for any 2-box x of any irreducible subfactor planar algebra with Jones index for some bi-shift of biprojection y. 4

. An answer to a conjecture of Wigderson and Wigderson
The famous Heisenberg uncertainty principle in [6] could be mathematically formulated in terms of Schwarz functions on R, (see e.g.[14,21] and Theorem 4.9 in [22]), as follows: Theorem 4.1 (Heisenberg's uncertainty principle).Let S(R) be the space of Schwartz functions.For any f ∈ S(R), where f (ξ) = R f (x)e −2πixξ dx is the Fourier transform of f .
Moreover, they proved the conjecture for q = ∞ in Theorem 4.12 in [22].
In the following theorem, we verify Conjecture 1 for q > 2 and disprove Conjecture 1 for 1 < q < 2.More precisely, Theorem 4.3.
(2) If q > 2, then the image of F q is R >0 .
To prove Theorem 4.3, we firstly prove a technical lemma.
Proof.We define then h n (λ) is a continuous function for any n ≥ 1.Thus h n (λ) can take all real values between F p,q (f n ) and F p,q (g n ).The result follows immediately from the assumptions.
When 1 p + 1 q < 1, then the image of F p,q is all of R >0 .

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 M be a finite von Neumann algebera with a trace τ M .Let ǫ ∈ [0, 1] and p ∈ [1, ∞].For any element x ∈ M, we define the (p, ǫ) smooth support to be

When p = 2 in
Definition 3.1, we are able to choose a positive contraction H in the abelian *-subalgebra generated by |x * | such that the (2, ǫ) support-size is exactly the trace of H.More precisely, we have Proposition 3.12.Suppose M is a finite von Neumann algebra with a trace τ M .Let x ∈ M, and let N be the abelian von Neumann subalgebra generated by |x * | in M. For any ǫ ∈ [0, 1], we have

2 η
(W (x)) = S 2 η (F (x)).Let x = |x * |U and y = W (x) = |y * |V be the polar decompositions, where U and V are the polar parts in A and B respectively.Let A 0 be the abelian von Neumann subalgebra of A generated by |x * | and B 0 be the abelian von Neumann subalgebra of B generated by |y * |.Let Φ be the trace-preserving conditional expectation from B onto B 0 and M = ΦR V * W R U .Then M is a linear operator from A 0 into B 0 such that M |x * | = |y * |.Let {e i } n i=1 and {f j } m j=1 be mutually orthogonal minimal projections in A 0 and B 0 such that n i=1 e i = I A and m j=1 f j = I B .The linear operator M is a

3. 3 .
Quantum Hirschman-Beckner uncertainty principle.In this subsection, we will prove the quantum (smooth) Hirschman-Beckner uncertainty principle (See Theorems 3.22 and 3.28) for von Neumann k-bi-algebras.For classical Hirschman-Beckner uncertainty principle [7, 1], the Shannon entropy is used to describe the uncertainty principle on R. For finite von Neumann algebras, we would like to use von Neumann entropy instead of Shannon entropy.Definition 3.21.Let M be a finite von Neumann algebra with a positive trace τ M .The von Neumann entropy of |x| 2 ∈ M is defined as follows