Alice wants to communicate an arbitrary $x \in \{0 ,1\}^n$ to Bob. Alice and Bob communicatein rounds, in each round Alice (or Bob) applies a unitary transformation on his/her part andtransmits a qubit to the other side, until at the end Bob measures his state and tries to inferx. The protocol is successful if for every $x \in \{0,1\}^n$ Bob succeeds with probability 1. Let$n_A / n_B$ be the number of messages sent by Alice / Bob respectively.Now I want to show that for any successful protocol we must have $n_A \geq  \lceil n/2 \rceil $, and $n_A+n_B \geq n$, and I need to show that the same argument above follows also for p-successful protocols, i.e if there's a p-successful protocol then the ineqaulities above should also follow for this protocol; a p-successful protocol is one in which Bob succeeds with probability p.Any hints or prefereably bibliography where this or similar problem is disscused?Thanks.
Communicating a string of zeros and ones quantumly
quantum computing;quantum information
This paper proves the results you have described:Ashwin Nayak and Julia Salzman. Limits on the Ability of Quantum  States to Convey Classical Messages. Journal of the ACM, 53(1):  184-206, 2006.