Quantum Comparison Machines with One-Sided Error
Chiu Fan Lee
and Neil F. Johnson
Received July 11, 2002; accepted September 2, 2002
It is always possible to decide, with one-sided error, whether two quantum states are
the same under a speciﬁc unitary transformation. However we show here that it is
impossible to do so if the transformation is anti-linear and non-singular. This result
implies that unitary and anti-unitary operations exist on an unequal footing in
quantum information theory.
KEY WORDS: limitations of quantum operations; comparison machines.
The formalism of quantum mechanics has given rise to many theorems con-
cerning ‘‘impossible machines’’. For example, many quantum machines such
as quantum copiers
and quantum erasers
are impossible. However these
operations can be performed approximately.
Hardy have shown that approximate cloning can actually enhance the per-
formance of some computations.
It is therefore very important to under-
stand the limitations of operations which are ‘‘approximately impossible’’.
Here we consider quantum decision machines with a one-sided error
probability. These machines output the answer YES or NO, however the
probability of error is non-zero for one of the two possible outputs.
Although this concept sounds quite theoretical, practical examples of such
one-sided machines are not unfamiliar to physicists. In particular the
interaction-free measurement scheme introduced by Elitzur and Vaidman in
Ref. 6, is inherently a decision machine with one-sided error. In this article,
we study the fundamental limit of a subclass of quantum decision machines,
which we call comparison machines. We show that although it is always
1570-0755/02/0800-0253/0 # 2003 Plenum Publishing Corporation
Centre for Quantum Computation and Physics Department, Clarendon Laboratory, Oxford
University, Parks Road, Oxford OX1 3PU, U.K.
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Quantum Information Processing, Vol. 1, No. 4, August 2002 (# 2003)