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 , such that Bob, when receiving the state , can choose any bit and recover the input bit xi 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 xi; and even-parity-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 even-parity-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 even-parity-oblivious RACs, the cryptographic property holds also in the device independent model.
Highlights
Quantum 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
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
Summary
Quantum information theory studies how information is encoded in quantum mechanical systems and how it can be transmitted through quantum channels. One specific information task that has been extensively studied in quantum information is the notion of random access codes (RACs) [Nay, ANTV99, ANTV02] In this scenario, Alice receives an n-bit string x, drawn from the uniform distribution, and wishes to encode x into a quantum state ρx, such that Bob, when receiving the state ρx, can choose any bit i ∈ [n] and recover the input bit xi with high probability by performing some general quantum operation on ρx. RACs have been used in various situations in quantum information and computation, including in communication complexity, non-locality, extractors and device-independent cryptography [BARdW08, INRY07, PZ10, DV10, LPY+12] Even though this task seems easier than transmitting the entire input string x, it is known that the length of quantum encodings must be at least Ω(n) [Nay99]. A well-known example shows the advantages of quantum RACs by using a single qubit to encode two uniformly random classical bits.
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