QUANTUM COMPUTING - Bio-Inspired and Nanoscale Integrated Computing - page 91

By the strict mathematical definition of the state reduction due to observa-
tion, an observation is not reversible in general. Under what conditions is a
measurement reversible in the strict mathematical sense? That is, when can
we measure classical information from a quantum source (yielding a set of
pure states with their probabilities with a reduction of quantum entropy), but
later be able to reverse this process to regenerate the entangled source state?
Bennett et al. [104] show that this is possible in the very special case where
the source states can be partitioned into two or more mutually orthogonal
subsets. (Other necessary and sufficient conditions for measurements to
be reversible have been proved in Bennett et al. [104]; Chuang, Yamamoto
[283] describe how to regenerate a qubit if it has observable error.)
There is experimental evidence that the physical execution of some reduc-
tions via measurement are in fact reversible (at least in very small closed
systems). Mabuchi, Zoller [284] have observed inversions of quantum jumps
in very small quantum-optical systems under continuous observation, and
Ueda [285] compares the notions of mathematical and physical reversibility.
On the other hand, in the case of entanglements in a large state space, even
if a measurement is in principle reversible in a closed system due the
reversible nature of the diffusion process, the likelihood of such a reverse to
the original state, within a moderate (say polynomial in n) time duration,
appears to drop exponentially with the number of qubits n. Gottfield [286]
and Diosi, Lukacs [287] explain quantum state vector reduction via strong
measurement as a physical process, e.g, state diffusion into the atoms of the
measurement apparatus. (Also see Pearle [288, 289].) This diffusion due to
reduction may be modeled by a system similar to a rapidly mixing Markov
system in probability theory, which seems to provide a very low (dropping
exponentially with n) likelihood for reversibility within a polynomial time
duration. (Others have modeled measurement by a nonlinear interactions
with the environment, which are irreversible.)

3.12.5. Should You Avoid Observation Operations?
An alternative approach is to completely avoid observation operations on the
basis that the observation operation is not actually essential to many quantum
computations. (This seems somewhat surprising, given the extensive use of the
observation operation in the QC literature for both algorithms and quantum error
correction.) Bernstein, Vazirani [50] (by showing that any given observation
operation can be delayed to future steps by use of the using XOR operation)
proved that all observation operations can be delayed to the final step of a
quantum computation. For a small e

W

0, let some particular qubit (of the linear
superposition of basis states) be e -near classic if had the qubit been observed, the
measured value would be a fixed value (either be 0 or 1) with e probability.
Suppose the output of a QC consists of the observation of a subset S of the qubits;
the resulting reduced superposition will be termed the output superposition.