This dissertation is motivated by the following fundamental questions: (a) are
there any exponential gaps between quantum and classical communication complexities?
(b) what is the role of entanglement in assisting quantum communications? (c)
how to characterize the nonlocality of quantum operations? We study four specific
problems below.
1. The communication complexity of the Hamming Distance problem. The
Hamming Distance problem is for two parties to determine whether or not the
Hamming distance between two n-bit strings is more than a given threshold. We
prove tighter quantum lower bounds in the general two-party, interactive communication
model. We also construct an efficient classical protocol in the more restricted
Simultaneous Message Passing model, improving previous results.
2. The Log-Equivalence Conjecture. A major open problem in communication
complexity is whether or not quantum protocols can be exponentially more efficient
than classical ones for computing a total Boolean function in the two-party, interactive
model. The answer is believed to be No. Razborov proved this conjecture
for the most general class of functions so far. We prove this conjecture for a broader
class of functions that we called block-composed functions. Our proof appears to be
the first demonstration of the dual approach of the polynomial method in proving
new results.
3. Classical simulations of bipartite quantum measurement. We define a new
ix
concept that measures the nonlocality of bipartite quantum operations. From this
measure, we derive an upper bound that shows the limitation of entanglement in
reducing communication costs.
4. The maximum tensor norm of bipartite superoperators. We define a maximum
tensor norm to quantify the nonlocality of bipartite superoperators. We show
that a bipartite physically realizable superoperator is bi-local if and only if its maximum
tensor norm is 1. Furthermore, the estimation of the maximum tensor norm
can also be used to prove quantum lower bounds on communication complexities.Ph.D.Computer Science & EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/58538/1/yufanzhu_1.pd