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Alice wants to communicate an arbitrary $x \in \{0 ,1\}^n$ to Bob. Alice and Bob communicate in rounds, in each round Alice (or Bob) applies a unitary transformation on his/her part and transmits a qubit to the other side, until at the end Bob measures his state and tries to infer x. 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.

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1 Answer 1

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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.

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