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research
Two Results about Quantum Messages
Authors
Hartmut Klauck
Supartha Podder
Publication date
1 January 2014
Publisher
Doi
Cite
View
on
arXiv
Abstract
We show two results about the relationship between quantum and classical messages. Our first contribution is to show how to replace a quantum message in a one-way communication protocol by a deterministic message, establishing that for all partial Boolean functions
f
:
{
0
,
1
}
n
Γ
{
0
,
1
}
m
β
{
0
,
1
}
f:\{0,1\}^n\times\{0,1\}^m\to\{0,1\}
f
:
{
0
,
1
}
n
Γ
{
0
,
1
}
m
β
{
0
,
1
}
we have
D
A
β
B
(
f
)
β€
O
(
Q
A
β
B
,
β
(
f
)
β
m
)
D^{A\to B}(f)\leq O(Q^{A\to B,*}(f)\cdot m)
D
A
β
B
(
f
)
β€
O
(
Q
A
β
B
,
β
(
f
)
β
m
)
. This bound was previously known for total functions, while for partial functions this improves on results by Aaronson, in which either a log-factor on the right hand is present, or the left hand side is
R
A
β
B
(
f
)
R^{A\to B}(f)
R
A
β
B
(
f
)
, and in which also no entanglement is allowed. In our second contribution we investigate the power of quantum proofs over classical proofs. We give the first example of a scenario, where quantum proofs lead to exponential savings in computing a Boolean function. The previously only known separation between the power of quantum and classical proofs is in a setting where the input is also quantum. We exhibit a partial Boolean function
f
f
f
, such that there is a one-way quantum communication protocol receiving a quantum proof (i.e., a protocol of type QMA) that has cost
O
(
log
β‘
n
)
O(\log n)
O
(
lo
g
n
)
for
f
f
f
, whereas every one-way quantum protocol for
f
f
f
receiving a classical proof (protocol of type QCMA) requires communication
Ξ©
(
n
/
log
β‘
n
)
\Omega(\sqrt n/\log n)
Ξ©
(
n
β
/
lo
g
n
)
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