Optimal bounds for parity-oblivious random access codes

Abstract

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 ...

Limited preparation contextuality in quantum theory and its relation to the Cirel'son bound

“…Recently, the authors of [37] have shown that evenparity-oblivious encodings are equivalent to the INDEX game, which implies 2 → 1 POM game is equivalent to the well known Bell-CHSH nonlocal game. Therefore, a quantum encoding of 2 → 1 POM with average success probability p Q exists only if a quantum strategy for playing the Bell-CHSH game with the same average success probability exists.…”

Can relativistic bit commitment lead to secure quantum oblivious transfer?

“…[45], we assume that all the security requirements (e.g., relativistic settings or experimental limitations) are already met, so that Bob does not have unlimited computational power to cheat within the BC stage. In this case, the validity of the no-go proofs of QOT [30][31][32][33][34][35][36][37][38] cannot be taken for granted, because all these proofs were derived without implying any limitation on the computational power of the cheater.…”

“…Intriguingly, the conclusions of some of the no-go proofs [31][32][33][34][35][36][37] remain valid, that unconditionally secure QOT is still impossible in this case. The key reason is that secure BC, being not a BCCC, cannot avoid the participant keeping the commitment at the quantum level instead of taking a fixed classical value.…”

“…Obviously, it indicates that QOT built upon QBC cannot be secure either. This stimulated the emergence of many other no-go proofs on quantum two-party secure computations including QOT [30][31][32][33][34][35][36][37][38].…”

“…On one hand, we will provide a cheating strategy in details, showing that some of the no-go proofs [31][32][33][34][35][36][37] remain valid even if QOT is based on secure BC. On the other hand, we found that some other no-go proofs [30,38] no longer work in such a QOT protocol, revealing that these proofs are not sufficiently general.…”