We implemented the protocol of entanglement assisted orientation in the space proposed by Brukner et al (quant-ph/0603167). We used min-max principle to evaluate the optimal entangled state and the optimal direction of polarization measurements which violate the classical bound.
We study the problem of state redistribution both in the classical (shared randomness assisted) and quantum (entanglement assisted) one-shot settings and provide new upper bounds on the communication required. Our classical bounds are in terms of smooth-min and max relative entropies and the quantum bounds are in terms of max relative entropy and Renyi relative entropy of order 21. We also consider a special case of this problem in the classical setting, previously studied by Braverman and Rao (2011). We show that their upper bound is optimal. In addition we provide an alternate protocol achieving a priory different looking upper bound. However, we show that our upper bound is essentially the same as their upper bound and hence also optimal. For the quantum case, we show that our achievability result is upper bounded by the achievability result obtained in Berta, Christandl, Touchette (2016).
Shared entanglement is a resource available to parties communicating over a quantum channel, much akin to public coins in classical communication protocols. Whereas shared randomness does not help in the transmission of information, or significantly reduce the classical complexity of computing functions (as compared to private-coin protocols), shared entanglement leads to startling phenomena such as ``quantum teleportation'' and ``superdense coding.'' The problem of characterising the power of prior entanglement has puzzled many researchers. In this paper, we revisit the problem of transmitting classical bits over an entanglement-assisted quantum channel. We derive a new, optimal bound on the number of quantum bits required for this task, for any given probability of error. All known lower bounds in the setting of bounded error entanglement-assisted communication are based on sophisticated information theoretic arguments. In contrast, our result is derived from first principles, using a simple linear algebraic technique.
Following initial work by Gregoratti and Werner [J. Mod. Optics 50, 913-933, 2003 and quant-ph/0403092] and Hayden and King [quant-ph/0409026], we study the problem of the capacity of a quantum channel assisted by a "friendly (channel) environment" that can locally measure and communicate classical messages to the receiver. Previous work [quant-ph/0505038] has yielded a capacity formula for the quantum capacity under this kind of help from the environment. Here we study the problem of the environment-assisted classical capacity, which exhibits a somewhat richer structure (at least, it seems to be the harder problem). There are several, presumably inequivalent, models of the permitted local operations and classical communications between receiver and environment: one-way, arbitrary, separable and PPT POVMs. In all these models, the task of decoding a message amounts to discriminating a set of possibly entangled states between the two receivers, by a class of operations under some sort of locality constraint. After introducing the operational capacities outlined above, we show that a lower bound on the environment-assisted classical capacity is always half the logarithm of the input space dimension. Then we develop a few techniques to prove the existence of channels which meet this lower bound up to terms of much smaller order, even when PPT decoding measurements are allowed (assuming a certain superadditivity conjecture).