scholarly journals Non-local box complexity and secure function evaluation

2011 ◽  
Vol 11 (1&2) ◽  
pp. 40-69
Author(s):  
Marc Kaplan ◽  
Sophie Laplante ◽  
Iordanis Kerenidis ◽  
J\'er\'emie Roland
Keyword(s):  
Boolean Function ◽  
Lower Bounds ◽  
Communication Complexity ◽  
Upper And Lower Bounds ◽  
Entangled States ◽  
Function Evaluation ◽  
Secure Function Evaluation ◽  
Non Local ◽  
Maximally Entangled States ◽  
Non Locality

A non-local box is an abstract device into which Alice and Bob input bits $x$ and $y$ respectively and receive outputs $a$ and $b$, where $a,b$ are uniformly distributed and $a \oplus b = x \wedge y$. Such boxes have been central to the study of quantum or generalized non-locality, as well as the simulation of non-signaling distributions. In this paper, we start by studying how many non-local boxes Alice and Bob need in order to compute a Boolean function $f$. We provide tight upper and lower bounds in terms of the communication complexity of the function both in the deterministic and randomized case. We show that non-local box complexity has interesting applications to classical cryptography, in particular to secure function evaluation, and study the question posed by Beimel and Malkin \cite{BM} of how many Oblivious Transfer calls Alice and Bob need in order to securely compute a function $f$. We show that this question is related to the non-local box complexity of the function and conclude by greatly improving their bounds. Finally, another consequence of our results is that traceless two-outcome measurements on maximally entangled states can be simulated with 3 \nlbs, while no finite bound was previously known.

An anomaly of non-locality

2007 ◽  
Vol 7 (1&2) ◽  
pp. 157-170 ◽  
Author(s):  
A.A. Methot ◽  
V. Scarani
Keyword(s):  
Bell Inequalities ◽  
Present Knowledge ◽  
Quantum States ◽  
Entangled States ◽  
Low Dimensions ◽  
Local Correlations ◽  
Non Local ◽  
Entangled Quantum States ◽  
Maximally Entangled States ◽  
Non Locality

Ever since the work of Bell, it has been known that entangled quantum states can produce non-local correlations between the outcomes of separate measurements. However, for almost forty years, it has been assumed that the most non-local states would be the maximally entangled ones. Surprisingly it is not the case: non-maximally entangled states are generally more non-local than maximally entangled states for all the measures of non-locality proposed to date: Bell inequalities, the Kullback-Leibler distance, entanglement simulation with communication or with non-local boxes, the detection loophole and efficiency of cryptography. In fact, one can even find simple examples in low dimensions, confirming that it is not an artefact of a specifically constructed Hilbert space or topology. This anomaly shows that entanglement and non-locality are not only different concepts, but also truly different resources. We review the present knowledge on this anomaly, point out that Hardy's theorem has the same feature, and discuss the perspectives opened by these discoveries.

scholarly journals Parametrix construction for certain Lévy-type processes

2015 ◽  
Vol 0 (0) ◽  
Author(s):  
Victoria Knopova ◽  
Alexei Kulik
Keyword(s):  
Markov Process ◽  
Probability Density ◽  
Lower Bounds ◽  
Transition Probability ◽  
Local Operator ◽  
Upper And Lower Bounds ◽  
Transition Probability Density ◽  
Strong Markov Process ◽  
Non Local ◽  
Strong Markov

AbstractIn this paper, we show that a non-local operator of certain type extends to the generator of a strong Markov process, admitting the transition probability density. For this transition probability density we construct the intrinsic upper and lower bounds, and prove some smoothness properties. Some examples are provided.

scholarly journals Discrimination of Non-Local Correlations

Entropy ◽  
2019 ◽  
Vol 21 (2) ◽  
pp. 104 ◽  
Author(s):  
Alberto Montina ◽  
Stefan Wolf
Keyword(s):  
Communication Complexity ◽  
Simple Algorithm ◽  
Running Time ◽  
Numerical Tests ◽  
Quantum Non Locality ◽  
Local Correlations ◽  
Non Local ◽  
Np Complete ◽  
Very High ◽  
Non Locality

In view of the importance of quantum non-locality in cryptography, quantum computation, and communication complexity, it is crucial to decide whether a given correlation exhibits non-locality or not. As proved by Pitowski, this problem is NP-complete, and is thus computationally intractable unless NP is equal to P. In this paper, we first prove that the Euclidean distance of given correlations from the local polytope can be computed in polynomial time with arbitrary fixed error, granted the access to a certain oracle; namely, given a fixed error, we derive two upper bounds on the running time. The first bound is linear in the number of measurements. The second bound scales with the number of measurements to the sixth power. The former holds only for a very high number of measurements, and is never observed in the performed numerical tests. We, then, introduce a simple algorithm for simulating the oracle. In all of the considered numerical tests, the simulation of the oracle contributes with a multiplicative factor to the overall running time and, thus, does not affect the sixth-power law of the oracle-assisted algorithm.

scholarly journals Complexity and Probability of Some Boolean Formulas

1998 ◽  
Vol 7 (4) ◽  
pp. 451-463 ◽  
Author(s):  
PETR SAVICKÝ
Keyword(s):  
Boolean Function ◽  
Lower Bounds ◽  
Absolute Constant ◽  
Upper And Lower Bounds ◽  
Probabilistic Distribution ◽  
Boolean Formulas ◽  
Exponentially Small

For any Boolean function f, let L(f) be its formula size complexity in the basis {∧, [oplus ] 1}. For every n and every k[les ]n/2, we describe a probabilistic distribution on formulas in the basis {∧, [oplus ] 1} in some given set of n variables and of size at most [lscr ](k)=4k. Let pn,k(f) be the probability that the formula chosen from the distribution computes the function f. For every function f with L(f)[les ][lscr ](k)α, where α=log4(3/2), we have pn,k(f)>0. Moreover, for every function f, if pn,k(f)>0, thenformula herewhere c>1 is an absolute constant. Although the upper and lower bounds are exponentially small in [lscr ](k), they are quasi-polynomially related whenever [lscr ](k)[ges ]lnΩ(1)n. The construction is a step towards developing a model appropriate for investigation of the properties of a typical (random) Boolean function of some given complexity.

“Non-locality”

2017 ◽  
Author(s):  
Richard Healey
Keyword(s):  
Quantum Theory ◽  
Quantum Entanglement ◽  
Quantum States ◽  
Entangled States ◽  
Entangled Quantum States ◽  
Action At A Distance ◽  
Insight Into ◽  
Non Locality

Quantum entanglement is popularly believed to give rise to spooky action at a distance of a kind that Einstein decisively rejected. Indeed, important recent experiments on systems assigned entangled states have been claimed to refute Einstein by exhibiting such spooky action. After reviewing two considerations in favor of this view I argue that quantum theory can be used to explain puzzling correlations correctly predicted by assignment of entangled quantum states with no such instantaneous action at a distance. We owe both considerations in favor of the view to arguments of John Bell. I present simplified forms of these arguments as well as a game that provides insight into the situation. The argument I give in response turns on a prescriptive view of quantum states that differs both from Dirac’s (as stated in Chapter 2) and Einstein’s.

scholarly journals On the Generalized Distance Energy of Graphs

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 17 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Connected Graph ◽  
Distance Matrix ◽  
Wiener Index ◽  
Diagonal Matrix ◽  
Upper And Lower Bounds ◽  
Star Graph ◽  
Extremal Graphs ◽  
Generalized Distance

The generalized distance matrix D α ( G ) of a connected graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 , D ( G ) is the distance matrix and T r ( G ) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α ( G ) . Some new upper and lower bounds for the generalized distance energy E D α ( G ) of G are established based on parameters including the Wiener index W ( G ) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.

scholarly journals Hadamard k-fractional inequalities of Fejér type for GA-s-convex mappings and applications

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Hui Lei ◽  
Gou Hu ◽  
Zhi-Jie Cao ◽  
Ting-Song Du
Keyword(s):  
Lower Bounds ◽  
Hypergeometric Functions ◽  
Upper And Lower Bounds ◽  
Weighted Inequalities ◽  
Convex Mappings ◽  
Special Means

Abstract The main aim of this paper is to establish some Fejér-type inequalities involving hypergeometric functions in terms of GA-s-convexity. For this purpose, we construct a Hadamard k-fractional identity related to geometrically symmetric mappings. Moreover, we give the upper and lower bounds for the weighted inequalities via products of two different mappings. Some applications of the presented results to special means are also provided.

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