Equivalence of Different Notions of Quantum Expansion Based on Schatten-p Norms

Core Concepts

All notions of quantum expansion based on Schatten-p norms are equivalent.

Abstract

The content discusses the problem of showing that certain notions of quantum expansion are equivalent. Quantum expansion is an analogue of the well-studied concept of graph expansion, which has many equivalent versions that play an important role in various fields.
The authors introduce two notions of quantum expansion: the quantum edge expansion (hQ) and the dimension edge expansion (hD). They show that these two notions are not equivalent by exhibiting a sequence of bistochastic tuples Bm such that inf hD(Bm) > 0 but inf hQ(Bm) = 0.
To further explore this, the authors consider a different expression of hQ in terms of the Schatten-2 norm, denoted as hS2. They generalize this to other Schatten-p norms, defining hSp as:
hSp(B) = min_{V < Cn, dim V ≤ n/2} Σ_i ||Bi|V⊥,V||_Sp / (d dim V)
The authors show that all these Sp notions of quantum expansion are equivalent, proving the following theorem:
Theorem 1:

hSp(B) ≤ d^((p-q)/2) * hSq(B)
hSp(B) ≥ [hSq(B)]^(p/q)

In particular, hSp and hSq are equivalent for every p, q ∈ [1, ∞).
The authors discuss how this result can be seen as a quantum analogue of the equivalence between certain notions of graph expansion based on ℓp norms, as shown by Matoušek. However, they believe the analogy is more notational than operational and does not help in constructing finite metric spaces that are hard to embed into Sp.
The authors propose an open question: What is the minimum distortion Dn,Sp needed to embed any n-point metric space into Sp?

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Key Insights Distilled From

All $S_p$ notions of quantum expansion are equivalent

by Fran... at arxiv.org 05-07-2024

https://arxiv.org/pdf/2405.03517.pdf

Deeper Inquiries

What are the potential applications of the equivalence between different notions of quantum expansion based on Schatten-p norms

The equivalence between different notions of quantum expansion based on Schatten-p norms opens up various potential applications in quantum information theory and quantum computing. One application could be in cryptography, where the ability to establish equivalence between these expansion notions can lead to the development of more secure quantum encryption protocols. By understanding how different Schatten-p norms behave in the context of quantum expansion, researchers can potentially enhance the security and efficiency of quantum cryptographic systems. Additionally, these equivalences could find applications in quantum learning theory, where the study of quantum expansion plays a crucial role in understanding the complexity of quantum algorithms and machine learning models.

Can the techniques used to prove the equivalence of Sp notions of expansion be extended to construct finite metric spaces that are hard to embed into Sp spaces

The techniques used to prove the equivalence of Sp notions of expansion could potentially be extended to construct finite metric spaces that are hard to embed into Sp spaces. By leveraging the insights gained from the quantum expansion equivalences, researchers could explore the construction of metric spaces that exhibit specific properties making them challenging to embed into Sp spaces with low distortion. This extension could involve adapting the mathematical frameworks and methodologies used in quantum expansion studies to the realm of metric embeddings, paving the way for the creation of novel constructions that push the boundaries of embedding theory.

How might the insights from this work on quantum expansion be leveraged to gain a better understanding of the properties and limitations of Sp spaces in the context of metric embeddings

Insights from the work on quantum expansion can provide a deeper understanding of the properties and limitations of Sp spaces in the context of metric embeddings. By studying the equivalence between different notions of quantum expansion based on Schatten-p norms, researchers can gain valuable insights into the behavior of Sp spaces when used for embedding metric structures. This understanding can lead to advancements in the theory of metric embeddings, shedding light on the optimal distortion levels required to embed metric spaces into Sp spaces efficiently. Furthermore, leveraging quantum expansion techniques can offer new perspectives on the relationship between Sp spaces and metric embeddings, potentially uncovering novel applications and theoretical results in the field.