Random access coding is an information task that has been extensively studied and found many applications in quantum information. In this scenario, Alice receives an n-bit string x, and wishes to encode x into a quantum state r x , such that Bob, when receiving the state r x , can choose any bit Î i n [ ] and recover the input bit x i with high probability. Here we study two variants: parity-oblivious random access codes (RACs), where we impose the cryptographic property that Bob cannot infer any information about the parity of any subset of bits of the input apart from the single bits x i ; and evenparity-oblivious RACs, where Bob cannot infer any information about the parity of any even-size subset of bits of the input. In this paper, we provide the optimal bounds for parity-oblivious quantum RACs and show that they are asymptotically better than the optimal classical ones. Our results provide a large non-contextuality inequality violation and resolve the main open problem in a work of Spekkens et al (2009 Phys. Rev. Lett.102 010401). Second, we provide the optimal bounds for evenparity-oblivious RACs by proving their equivalence to a non-local game and by providing tight bounds for the success probability of the non-local game via semidefinite programming. In the case of evenparity-oblivious RACs, the cryptographic property holds also in the device independent model. IntroductionQuantum information theory studies how information is encoded in quantum mechanical systems and how it can be transmitted through quantum channels. A main question is whether quantum information is more powerful than classical information. A celebrated result by Holevo [Hol73] shows that quantum information cannot be used to compress classical information. In high level, in order to transmit n uniformly random classical bits, one needs to transmit no less than n quantum bits. This might imply that quantum information is no more powerful than classical information. This however is wrong in many situations. In the model of communication complexity, one can show that transmitting quantum information may result in exponential savings on the communication needed to solve specific problems [BCWdW01, BJK04, GKK + 08, Raz99, RK11].One specific information task that has been extensively studied in quantum information is the notion of random access codes (RACs) [ANTV02, ANTV99, Nay99]. In this scenario, Alice receives an n-bit string x, drawn from the uniform distribution, and wishes to encode x into a quantum state r x , such that Bob, when receiving the state r x , can choose any bit Î i n [ ] and recover the input bit x i with high probability by performing some general quantum operation on r x .RACs have been used in various situations in quantum information and computation, including in communication complexity, non-locality, extractors and device-independent cryptography [BARdW08, DV10, INRY07, LPY + , PZ10]. Even though this task seems easier than transmitting the entire input string x, it is known that the length of quantum RACs must be ...
Oblivious transfer is a cryptographic primitive where Alice has two bits and Bob wishes to learn some function of them. Ideally, Alice should not learn Bob's desired function choice and Bob should not learn any more than what is logically implied by the function value. While decent quantum protocols for this task are known, many become completely insecure if an adversary were to control the quantum devices used in the implementation of the protocol. In this work we give a fully device-independent quantum protocol for XOR oblivious transfer.
The Fourier-Entropy Influence (FEI) Conjecture states that for any Boolean function f : {+1, −1} n → {+1, −1}, the Fourier entropy of f is at most its influence up to a universal constant factor. While the FEI conjecture has been proved for many classes of Boolean functions, it is still not known whether it holds for the class of Linear Threshold Functions. A natural question is: Does the FEI conjecture hold for a "random" linear threshold function? In this paper, we answer this question in the affirmative. We consider two natural distributions on the weights defining a linear threshold function, namely uniform distribution on [−1, 1] and Normal distribution.
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