Many constructions of randomness extractors are known to work in the presence of quantum side information, but there also exist extractors which do not [Gavinsky et al., STOC'07]. Here we find that spectral extractors with a bound on the second largest eigenvalue - considered as an operator on the Hilbert-Schmidt class - are quantum-proof. We then discuss fully quantum extractors and call constructions that also work in the presence of quantum correlations decoupling. As in the classical case we show that spectral extractors are decoupling. The drawback of classical and quantum spectral extractors is that they always have a long seed, whereas there exist classical extractors with exponentially smaller seed size. For the quantum case, we show that there exists an extractor with extremely short seed size d = O(log(1/ε)), where ε > 0 denotes the quality of the randomness. In contrast to the classical case this is independent of the input size and min-entropy and matches the simple lower bound d ≥ log(1/ε).
Many constructions of randomness extractors are known to work in the presence of quantum side information, but there also exist extractors which do not [Gavinsky et al., STOC'07]. Here we find that spectral extractors with a bound on the second largest eigenvalue - considered as an operator on the Hilbert-Schmidt class - are quantum-proof. We then discuss fully quantum extractors and call constructions that also work in the presence of quantum correlations decoupling. As in the classical case we show that spectral extractors are decoupling. The drawback of classical and quantum spectral extractors is that they always have a long seed, whereas there exist classical extractors with exponentially smaller seed size. For the quantum case, we show that there exists an extractor with extremely short seed size d = O(log(1/ε)), where ε > 0 denotes the quality of the randomness. In contrast to the classical case this is independent of the input size and min-entropy and matches the simple lower bound d ≥ log(1/ε).
Quantum-proof randomness extractors are an important building block for classical and quantum cryptography as well as device independent randomness amplification and expansion. Furthermore, they are also a useful tool in quantum Shannon theory. It is known that some extractor constructions are quantum-proof whereas others are provably not [Gavinsky et al., STOC'07]. We argue that the theory of operator spaces offers a natural framework for studying to what extent extractors are secure against quantum adversaries: we first phrase the definition of extractors as a bounded norm condition between normed spaces, and then show that the presence of quantum adversaries corresponds to a completely bounded norm condition between operator spaces. From this, we show that very high min-entropy extractors as well as extractors with small output are always (approximately) quantum-proof. We also study a generalization of extractors called randomness condensers. We phrase the definition of condensers as a bounded norm condition and the definition of quantum-proof condensers as a completely bounded norm condition. Seeing condensers as bipartite graphs, we then find that the bounded norm condition corresponds to an instance of a well-studied combinatorial problem, called bipartite densest subgraph. Furthermore, using the characterization in terms of operator spaces, we can associate to any condenser a Bell inequality (two-player game), such that classical and quantum strategies are in one-to-one correspondence with classical and quantum attacks on the condenser. Hence, we get for every quantum-proof condenser (which includes in particular quantum-proof extractors) a Bell inequality that cannot be violated by quantum mechanics.
Randomness extraction involves the processing of purely classical information and is therefore usually studied with in the framework of classical probability theory. However, such a classical treatment is generally too restrictive for applications where side information about the values taken by classical random variables may be represented by the state of a quantum system. This is particularly relevant in the context of cryptography, where an adversary may make use of quantum devices. Here, we show that the well-known construction paradigm for extractors proposed by Trevisan is sound in the presence of quantum side information. We exploit the modularity of this paradigm to give several concrete extractor constructions, which, e.g., extract all the conditional (smooth) min-entropy of the source using a seed of length polylogarithmic in the input, or only require the seed to be weakly random.
Randomness extraction involves the processing of purely classical information and is therefore usually studied in the framework of classical probability theory.However, such a classical treatment is generally too restrictive for applications, where side information about the values taken by classical random variables may be represented by the state of a quantum system.This is particularly relevant in the context of cryptography, where an adversary may make use of quantum devices.Here, we show that the well known construction paradigm for extractors proposed by hevisan is sound in the presence of quantum side information.We exploit the modularity of this paradigm to give several concrete extractor construc- tions, which.e.g, extract all the conditional (smooth) min-entropy of the source using a seed of length poly-logarithmic in the input, or only require the seed to be weakly random.
We establish an equivalence between matrix samplers and quantum-proof randomness extractors. The classical sampler--extractor equivalence is established by a seminal work of Zuckerman (Random Structures & Algorithms, 1997).
Using this equivalence, we construct explicit strong matrix samplers with optimal randomness overhead and near-optimal sample complexity via constructing quantum-proof extractors. Specifically for matrix dimension
D, accuracy
ε and failure probability
δ≤ε, our matrix sampler uses
m+Oα(log(D/δ)) random bits and
Oα(ε−2−αlog1+α(D/δ)) samples, for any constant
α>0. Compared with the matrix sampler of Xun and Zuckerman (CCC 2025), our construction has improvements in two aspects. Under the same optimal random seed length, we can replace the
polyD dependence in the number of samples by
O(log1−γD). Under the same scale of sample size, we can remove their
loglogD factor in the length of the random seed. Our underlying strong quantum-proof extractor has output length
n−Oα(a+log(1/η)) and seed length at most
(1+α)loga+(2+α)log(1/η)+Oα(1) for
n-bit sources with conditional min-entropy
n−a and error
η.
We also obtain strong quantum-proof extractors for every constant entropy rate, for any error as small as
2−Ω(n), having optimal seed length
O(log(n/η)) and output length
Ω(n), improving previous constructions.
Randomness extractors, widely used in classical and quantum cryptography and other fields of computer science, e.g., derandomization, are functions which generate almost uniform randomness from weak sources of randomness. In the quantum setting one must take into account the quantum side information held by an adversary which might be used to break the security of the extractor. In the case of seeded extractors the presence of quantum side information has been extensively studied. For multi-source extractors one can easily see that high conditional min-entropy is not sufficient to guarantee security against arbitrary side information, even in the classical case. Hence, the interesting question is under which models of (both quantum and classical) side information multi-source extractors remain secure. In this work we suggest a natural model of side information, which we call the Markov model, and prove that any multi-source extractor remains secure in the presence of quantum side information of this type (albeit with weaker parameters). This improves on previous results in which more restricted models were considered or the security of only some types of extractors was shown.