11 research outputs found
Quantum communication complexity of symmetric predicates
We completely (that is, up to a logarithmic factor) characterize the
bounded-error quantum communication complexity of every predicate
depending only on (). Namely, for a predicate
on let \ell_0(D)\df \max\{\ell : 1\leq\ell\leq n/2\land
D(\ell)\not\equiv D(\ell-1)\} and \ell_1(D)\df \max\{n-\ell : n/2\leq\ell <
n\land D(\ell)\not\equiv D(\ell+1)\}. Then the bounded-error quantum
communication complexity of is equal (again, up to a
logarithmic factor) to . In particular, the
complexity of the set disjointness predicate is . This result
holds both in the model with prior entanglement and without it.Comment: 20 page
Tensor Norms and the Classical Communication Complexity of Nonlocal Quantum Measurement
We initiate the study of quantifying nonlocalness of a bipartite measurement
by the minimum amount of classical communication required to simulate the
measurement. We derive general upper bounds, which are expressed in terms of
certain tensor norms of the measurement operator. As applications, we show that
(a) If the amount of communication is constant, quantum and classical
communication protocols with unlimited amount of shared entanglement or shared
randomness compute the same set of functions; (b) A local hidden variable model
needs only a constant amount of communication to create, within an arbitrarily
small statistical distance, a distribution resulted from local measurements of
an entangled quantum state, as long as the number of measurement outcomes is
constant.Comment: A preliminary version of this paper appears as part of an article in
Proceedings of the the 37th ACM Symposium on Theory of Computing (STOC 2005),
460--467, 200
Quantum Communication Complexity and Nonlocality of Bipartite Quantum Operations.
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