Dense Quantum Coding and a Lower Bound for 1-way Quantum Automata

We consider the possibility of encoding m classical bits into much fewer n quantum bits so that an arbitrary bit from the original m bits can be recovered with a good probability, and we show that non-trivial quantum encodings exist that have no classical counterparts. On the other hand, we show that quantum encodings cannot be much more succint as compared to classical encodings, and we provide a lower bound on such quantum encodings. Finally, using this lower bound, we prove an exponential lower bound on the size of 1-way quantum finite automata for a family of languages accepted by linear sized deterministic finite automata.Comment: 12 pages, 3 figures. Defines random access codes, gives upper and lower bounds for the number of bits required for such (possibly quantum) codes. Derives the size lower bound for quantum finite automata of the earlier version of the paper using these result

Optimal lower bounds for quantum automata and random access codes

Consider the finite regular language L_n = {w0 : w \in {0,1}^*, |w| \le n}. It was shown by Ambainis, Nayak, Ta-Shma and Vazirani that while this language is accepted by a deterministic finite automaton of size O(n), any one-way quantum finite automaton (QFA) for it has size 2^{Omega(n/log n)}. This was based on the fact that the evolution of a QFA is required to be reversible. When arbitrary intermediate measurements are allowed, this intuition breaks down. Nonetheless, we show a 2^{Omega(n)} lower bound for such QFA for L_n, thus also improving the previous bound. The improved bound is obtained by simple entropy arguments based on Holevo's theorem. This method also allows us to obtain an asymptotically optimal (1-H(p))n bound for the dense quantum codes (random access codes) introduced by Ambainis et al. We then turn to Holevo's theorem, and show that in typical situations, it may be replaced by a tighter and more transparent in-probability bound.Comment: 8 pages, 1 figure, Latex2e. Extensive modifications have been made to increase clarity. To appear in FOCS'9

Detection loophole attacks on semi-device-independent quantum and classical protocols

Semi-device-independent quantum protocols realize information tasks - e.g. secure key distribution, random access coding, and randomness generation - in a scenario where no assumption on the internal working of the devices used in the protocol is made, except their dimension. These protocols offer two main advantages: first, their implementation is often less demanding than fully-device-independent protocols. Second, they are more secure than their device-dependent counterparts. Their classical analogous is represented by random access codes, which provide a general framework for describing one-sided classical communication tasks. We discuss conditions under which detection inefficiencies can be exploited by a malicious provider to fake the performance of semi-device-independent quantum and classical protocols - and how to prevent it.Comment: 13 pages, 1 figure, published versio