Stephanie Wehner

Entropic uncertainty relations—a survey

Uncertainty relations play a central role in quantum mechanics. Entropic uncertainty relations in particular have gained significant importance within quantum information, providing the foundation for the security of many quantum cryptographic protocols. Yet, little is known about entropic uncertainty relations with more than two measurement settings. In the present survey, we review known results and open questions.

The Uncertainty Principle Determines the Nonlocality of Quantum Mechanics

Quantum Connection A system that is quantum mechanically entangled with another distant system can be predicted by measuring the distant system. This form of “action-at-a-distance,” or nonlocality, seemingly contradicts Heisenbergs uncertainty principle, which is one of the fundamental aspects of quantum mechanics. Oppenheim and Wehner (p. 1072) show that the degree of nonlocality in quantum mechanics is actually determined by the uncertainty principle. The unexpected connection between nonlocality and uncertainty holds true for other physical theories besides quantum mechanics. The two central elements of quantum theory, once assumed to be distinct concepts, are shown to be linked. Two central concepts of quantum mechanics are Heisenberg’s uncertainty principle and a subtle form of nonlocality that Einstein famously called “spooky action at a distance.” These two fundamental features have thus far been distinct concepts. We show that they are inextricably and quantitatively linked: Quantum mechanics cannot be more nonlocal with measurements that respect the uncertainty principle. In fact, the link between uncertainty and nonlocality holds for all physical theories. More specifically, the degree of nonlocality of any theory is determined by two factors: the strength of the uncertainty principle and the strength of a property called “steering,” which determines which states can be prepared at one location given a measurement at another.

Entropic uncertainty relations and their applications

Heisenberg’s uncertainty principle forms a fundamental element of quantum mechanics. Uncertainty relations in terms of entropies were initially proposed to deal with conceptual shortcomings in the original formulation of the uncertainty principle and, hence, play an important role in quantum foundations. More recently, entropic uncertainty relations have emerged as the central ingredient in the security analysis of almost all quantum cryptographic protocols, such as quantum key distribution and two-party quantum cryptography. This review surveys entropic uncertainty relations that capture Heisenberg’s idea that the results of incompatible measurements are impossible to predict, covering both finite- and infinite-dimensional measurements. These ideas are then extended to incorporate quantum correlations between the observed object and its environment, allowing for a variety of recent, more general formulations of the uncertainty principle. Finally, various applications are discussed, ranging from entanglement witnessing to wave-particle duality to quantum cryptography.

Improved lower bounds for locally decodable codes and private information retrieval

We prove new lower bounds for locally decodable codes and private information retrieval. We show that a 2-query LDC encoding n-bit strings over an l-bit alphabet, where the decoder only uses b bits of each queried position, needs code length

Analyzing worms and network traffic using compression

m=exp\left(\Omega\left(\frac{n}{2^{b}{\sum_{i=0}^{b}}(^{l}_{i})}\right)\right)

Unconditional Security From Noisy Quantum Storage

Similarly, a 2-server PIR scheme with an n-bit database and t-bit queries, where the user only needs b bits from each of the two l-bit answers, unknown to the servers, satisfies

The Quantum Moment Problem and Bounds on Entangled Multi-prover Games

t=\Omega \left(\frac{n}{2^{b}\sum_{i=0}^{b}(^{l}_{i})}\right)

Distinguishability of Quantum States Under Restricted Families of Measurements with an Application to Quantum Data Hiding

This implies that several known PIR schemes are close to optimal. Our results generalize those of Goldreich et al. [8], who proved roughly the same bounds for linear LDCs and PIRs. Like earlier work by Kerenidis and de Wolf [12], our classical bounds are proved using quantum computational techniques. In particular, we give a tight analysis of how well a 2-input function can be computed from a quantum superposition of both inputs.

Lower bound on the dimension of a quantum system given measured data

Internet worms have become a widespread threat to system and network operations. In order to fight them more efficiently, it is necessary to analyze newly discovered worms and attack patterns. This paper shows how techniques based on Kolmogorov Complexity can help in the analysis of internet worms and network traffic. Using compression, different species of worms can be clustered by type. This allows us to determine whether an unknown worm binary could in fact be a later version of an existing worm in an extremely simple, automated, manner. This may become a useful tool in the initial analysis of malicious binaries. Furthermore, compression can also be useful to distinguish different types of network traffic and can thus help to detect traffic anomalies: Certain anomalies may be detected by looking at the compressibility of a network session alone. We furthermore show how to use compression to detect malicious network sessions that are very similar to known intrusion attempts. This technique could become a useful tool to detect new variations of an attack and thus help to prevent IDS evasion. We provide two new plugins for Snort which demonstrate both approaches.

Experimental Bit Commitment Based on Quantum Communication and Special Relativity

We consider the implementation of two-party cryptographic primitives based on the sole assumption that no large-scale reliable quantum storage is available to the cheating party. We construct novel protocols for oblivious transfer and bit commitment, and prove that realistic noise levels provide security even against the most general attack. Such unconditional results were previously only known in the so-called bounded-storage model which is a special case of our setting. Our protocols can be implemented with present-day hardware used for quantum key distribution. In particular, no quantum storage is required for the honest parties.