arXiv subjects

Carl A. Miller

Publications and source records attributed to Carl A. Miller.

28 records · Page 2Linked to original sources

arXiv2016

Certifying the absence of quantum nonlocality

Quantum nonlocality is an inherently non-classical feature of quantum mechanics and manifests itself through violation of Bell inequalities for nonlocal games. We show that in a fairly general setting, a simple extension of a nonlocal game can certify instead the absence of quantum nonlocality. Through contraposition, our result implies that a super-classical performance for such a game ensures that a player's output is unpredictable to the other player. Previously such output unpredictability was known with respect to a third party.

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arXiv2016

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: cryptographic level of security, tolerating a constant level of imprecision in the devices, requiring only a unit size quantum memory per device component for the honest implementation, and allowing a large natural class of constructions for the protocol. In conjunct with a recent work by Chung, Shi and Wu, it also leads to robust unbounded expansion using just 2 multi-part devices. When adapted for distributing cryptographic keys, our method achieves, for the first time, exponential expansion combined with cryptographic security and noise tolerance. The proof 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.

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arXiv2015

Universal security for randomness expansion from the spot-checking protocol

Colbeck (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%, 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 inequality of Ball, Carlen, and Lieb (Inventiones mathematicae, 115:463-482, 1994).

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arXiv2013

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.

math.CO↗

arXiv2013

Optimal robust quantum self-testing by binary nonlocal XOR games

Self-testing a quantum device means verifying the existence of a certain quantum state as well as the effect of the associated measurements based only on the statistics of the measurement outcomes. Robust, i.e., error-tolerant, self-testing quantum devices are critical building blocks for quantum cryptographic protocols that rely on imperfect or untrusted quantum devices. We give a criterion which determines whether a given binary XOR game is robust self-testing with the asymptotically optimal error parameter. As an application, we prove that the celebrated CHSH game is an optimally robust self-test. We also prove the same for a family of tests recently proposed by Acin et al. (PRL 108:100402, 2012) for random number generation, thus extending the benefit of the latter tests to allow imperfect or untrusted quantum devices.

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arXiv2012

On 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 sent via a single use of the channel. Prevedel et al. (PRL 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. We completely 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 from the given classical channel and to prove the limit of such protocols. As an example, we show that the Prevedel et al. protocol is optimal for two-qubit entanglement. We also prove some simple upper bounds on the improvement that can be obtained from quantum and no-signaling correlations.

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arXiv2010

Deciding Unitary Equivalence Between Matrix Polynomials and Sets of Bipartite Quantum States

In this brief report, we consider the equivalence between two sets of $m+1$ bipartite quantum states under local unitary transformations. For pure states, this problem corresponds to the matrix algebra question of whether two degree $m$ matrix polynomials are unitarily equivalent; i.e. $UA_iV^\dagger=B_i$ for $0\leq i\leq m$ where $U$ and $V$ are unitary and $(A_i, B_i)$ are arbitrary pairs of rectangular matrices. We present a randomized polynomial-time algorithm that solves this problem with an arbitrarily high success probability and outputs transforming matrices $U$ and $V$.

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arXiv2009

Comment on "Matrix Pencils and Entanglement Classification"

In our earlier posting "Matrix Pencils and Entanglement Classification", arXiv:0911.1803, we gave a polynomial-time algorithm for deciding if two states in a space of dimension $2\otimes m\otimes n$ are SLOCC equivalent. In this note, we point out that a straightforward modification of the algorithm gives a simple enumeration of all SLOCC equivalence classes in the same space, with the class representatives expressed in the Kronecker canonical normal form of matrix pencils. Thus, two states are equivalent if and only if they have the same canonical form. As an example, we present representatives in canonical form for each of the 26 equivalence classes in $2\otimes 3\otimes n$ systems.

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arXiv2009

Matrix Pencils and Entanglement Classification

In this paper, we study pure state entanglement in systems of dimension $2\otimes m\otimes n$. 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 can be seen from the matrix pencil approach. Based on this method, we report the first polynomial-time algorithm for deciding when two $2\otimes m\otimes n$ states are SLOCC equivalent. Besides recovering the previously known 26 distinct SLOCC equivalence classes in $2\otimes 3\otimes n$ systems, we also determine the hierarchy between these classes.

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arXiv2009

Equicharacteristic etale cohomology in dimension one

The Grothendieck-Ogg-Shafarevich formula expresses the Euler characteristic of an etale sheaf on a curve in terms of local data. The purpose of this paper is to prove a version of the G-O-S formula which applies to equicharacteristic sheaves (a bound, rather than an equality). This follows a proposal of R. Pink. The basis for the result is the characteristic-p "Riemann-Hilbert" correspondence, which relates equicharacteristic etale sheaves to O_{F, X}-modules. In the paper we prove a version of this correspondence for curves, considering both local and global settings. In the process we define an invariant, the "minimal root index," which measures the local complexity of an O_{F, X}-module. This invariant provides the local terms for the main result.

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