Carl A. Miller

Robust protocols for securely expanding randomness and distributing keys using untrusted quantum devices

Randomness is a vital resource for modern day information processing, especially for cryptography. A wide range of applications critically rely on abundant, high quality random numbers generated securely. Here we show how to expand a random seed at an exponential rate without trusting the underlying quantum devices. Our approach is secure against the most general adversaries, and has the following new features: tolerating a constant level of implementation imprecision, requiring only a unit size quantum memory per device component for the honest implementation, and allowing a large natural class of constructions. In conjunct with a recent work by Chung, Shi and Wu (QIP 2014), it leads to robust unbounded expansion using just 2 multi-part devices. It can also be adapted for distributing cryptographic keys securely. The proof begins with a known protocol and proceeds by showing that the Renyi divergence of the outputs of the protocol (for a specific bounding operator) decreases linearly as the protocol iterates. At the heart of the proof are a new uncertainty principle on quantum measurements, and a method for simulating trusted measurements with untrusted devices. A full version of this paper containing additional results developed after the conference submission is available as arXiv:1402.0489.

Universal Security for Randomness Expansion from the Spot-Checking Protocol

Colbeck [Ph.D. thesis, 2006] proposed using Bell inequality violations to generate certified random numbers. While full quantum-security proofs have been given, it remains a major open problem to identify the broadest class of Bell inequalities and lowest performance requirements to achieve such security. In this paper, working within the broad class of spot-checking protocols, we prove exactly which Bell inequality violations can be used to achieve full security. Our result greatly improves the known noise tolerance for secure randomness expansion: for the commonly used CHSH game, full security was only known with a noise tolerance of 1.5% [Miller and Shi, J. ACM, 63 (2016), 33], and we improve this to 10.3%. We also generalize our results beyond Bell inequalities and give the first security proof for randomness expansion based on Kochen--Specker inequalities. The central technical contribution of the paper is a new uncertainty principle for the Schatten norm, which is based on the uniform convexity ineq...

Matrix pencils and entanglement classification

Quantum entanglement plays a central role in quantum information processing. A main objective of the theory is to classify different types of entanglement according to their interconvertibility through manipulations that do not require additional entanglement to perform. While bipartite entanglement is well understood in this framework, the classification of entanglements among three or more subsystems is inherently much more difficult. In this paper, we study pure state entanglement in systems of dimension 2⊗m⊗n. Two states are considered equivalent if they can be reversibly converted from one to the other with a nonzero probability using only local quantum resources and classical communication (SLOCC). We introduce a connection between entanglement manipulations in these systems and the well-studied theory of matrix pencils. All previous attempts to study general SLOCC equivalence in such systems have relied on somewhat contrived techniques which fail to reveal the elegant structure of the problem that ...

Optimal Robust Self-Testing by Binary Nonlocal XOR Games

Self-testing a quantum apparatus means verifying the existence of a certain quantum state as well as the effect of the associated measuring devices based only on the statistics of the measurement outcomes. Robust (i.e., error-tolerant) self-testing quantum apparatuses are critical building blocks for quantum cryptographic protocols that rely on imperfect or untrusted devices. We devise a general scheme for proving optimal robust self-testing properties for tests based on nonlocal binary XOR games. We offer some simplified proofs of known results on self-testing, and also prove some new results.

Randomness in Nonlocal Games Between Mistrustful Players

Quantum information theory determines the maximum rates at which information can be transmitted through physical systems described by quantum mechanics. Here we consider the communication protocol known as quantum reading. Quantum reading is a protocol for retrieving the information stored in a digital memory by using a quantum probe, e.g., shining quantum states of light to read an optical memory. In a variety of situations using a quantum probe enhances the performances of the reading protocol in terms of fidelity, data density and energy dissipation. Here we review and characterize the quantum reading capacity of a memory model, defined as the maximum rate of reliable reading. We show that, like other quantities in quantum information theory, the quantum reading capacity is super-additive. Moreover, we determine conditions under which the use of an entangled ancilla improves the performance of quantum reading.If two quantum players at a nonlocal game G achieve a superclassical score, then their measurement outcomes must be at least partially random from the perspective of any third player. This is the basis for device-independent quantum cryptography. In this paper we address a related question: does a superclassical score at G guarantee that one player has created randomness from the perspective of the other player? We show that for complete-support games, the answer is yes: even if the second player is given the first players input at the conclusion of the game, he cannot perfectly recover her output. Thus some amount of local randomness (i.e., randomness possessed by only one player) is always obtained when randomness is certified from nonlocal games with quantum strategies. This is in contrast to non-signaling game strategies, which may produce global randomness without any local randomness. We discuss potential implications for cryptographic protocols between mistrustful parties.

Optimal entanglement-assisted one-shot classical communication

The one-shot success probability of a noisy classical channel for transmitting one classical bit is the optimal probability with which the bit can be successfully sent via a single use of the channel. Prevedel et al. [Phys. Rev. Lett. 106, 110505 (2011)] recently showed that for a specific channel, this quantity can be increased if the parties using the channel share an entangled quantum state. In this paper, we characterize the optimal entanglement-assisted protocols in terms of the radius of a set of operators associated with the channel. This characterization can be used to construct optimal entanglement-assisted protocols for a given classical channel and to prove the limits of such protocols. As an example, we show that the Prevedel et al. protocol is optimal for two-qubit entanglement. We also prove some tight upper bounds on the improvement that can be obtained from quantum and nonsignaling correlations.

Evasiveness of Graph Properties and Topological Fixed-Point Theorems

Many graph properties (e.g., connectedness, containing a complete subgraph) are known to be difficult to check. In a decision-tree model, the cost of an algorithm is measured by the number of edges in the graph that it queries. R. Karp conjectured in the early 1970s that all monotone graph properties are evasive—that is, any algorithm which computes a monotone graph property must check all edges in the worst case. This conjecture is unproven, but a lot of progress has been made. Starting with the work of Kahn, Saks, and Sturtevant in 1984, topological methods have been applied to prove partial results on the Karp conjecture. This text is a tutorial on these topological methods. I give a fully self-contained account of the central proofs from the paper of Kahn, Saks, and Sturtevant, with no prior knowledge of topology assumed. I also briefly survey some of the more recent results on evasiveness.

Pediatric pyogenic sacroiliitis and osteomyelitis

Pyogenic sacroiliitis accounts for 1–2% of all cases of septic arthritis with less than 200 cases reported in the English literature since the beginning of the twentieth century. Cultures of joint fluid usually grow Staphylococcus aureus. Prognosis is excellent; however, diagnosis may be difficult due to rarity of disease and non-specific signs, symptoms, and physical findings. Magnetic resonance imaging has been found to be the most useful imaging modality in diagnosis. Most reported cases required prolonged antimicrobial therapy of six to nine weeks. Presented here are two children with pyogenic sacroiliitis managed at a tertiary-care, university hospital and review of the literature on this relatively rare diagnosis.

Keyring models: An approach to steerability

If a measurement is made on one half of a bipartite system, then, conditioned on the outcome, the other half has a new reduced state. If these reduced states defy classical explanation-that is, if shared randomness cannot produce these reduced states for all possible measurements-the bipartite state is said to be steerable. Determining which states are steerable is a challenging problem even for low dimensions. In the case of two-qubit systems a criterion is known for T-states (that is, those with maximally mixed marginals) under projective measurements. In the current work we introduce the concept of keyring models-a special class of local hidden state models. When the measurements made correspond to real projectors, these allow us to study steerability beyond T-states. Using keyring models, we completely solve the steering problem for real projective measurements when the state arises from mixing a pure two-qubit state with uniform noise. We also give a partial solution in the case when the uniform noise is replaced by independent depolarizing channels.

Rigidity of the magic pentagram game

A game is rigid if a near-optimal score guarantees, under the sole assumption of the validity of quantum mechanics, that the players are using an approximately unique quantum strategy. Rigidity has a vital role in quantum cryptography as it permits a strictly classical user to trust behavior in the quantum realm. This property can be traced back as far as 1998 (Mayers and Yao) and has been proved for multiple classes of games. In this paper we prove ridigity for the magic pentagram game, a simple binary constraint satisfaction game involving two players, five clauses and ten variables. We show that all near-optimal strategies for the pentagram game are approximately equivalent to a unique strategy involving real Pauli measurements on three maximally-entangled qubit pairs.