QUANTUM COMPUTING - Bio-Inspired and Nanoscale Integrated Computing - page 88

the volume would appear to be no more that the number of qubits. This
may be true, but there are a number of substantial issues that need to be
carefully considered. Recall the observation operation both provides a
measurement of a qubit with a resulting state reduction. However, the QC
literature has not yet carefully considered the volume bounds for the
observation operation and as we shall see, it is not yet at all clear what the
volume required. In spite of major works on the mathematical and physical
foundations of quantum observation, the precise nature of quantum state
reduction via a strong quantum measurement remains somewhat of a
mystery. Two distinct approaches to the mathematical and physical
foundations of observation have been developed:
(a) the Copenhagen Formulation, where the observation is simply an as-
sumed basic operation and is considered to be done by a macroscopic
measuring device, and
(b) the von Neumann Formulation [100, Chapter 4: Macroscopic Mea-
surement], which views the measuring apparatus as well as the quan-
tum system measured as both part of a quantum system. Hence the
evolution of the system (and resulting experimental predictions) can
be distinct from that predicted by the Copenhagen formulation of
observation (which does not take this into account since the measur-
ing apparatus is assumed in their formulation to be very large).
See Cerf and Adami [18] for a comparison the Copenhagen and von Neumann
formulations and see Hay, Peres [278] for an example of this difference. In
summary, the Copenhagen and the von Neumann formulations for observation
differ in the assumed context (macroscopic or microscopic measurement appara-
tus). (Note: Attempts to rectify the difference between the Copenhagen and the
von Neumann formulation for observation are given in Hay, Peres [278] and in
Zurek [279], but it appears not yet resolved.) The Copenhagen formulation for
observation is generally used in the context of quantum physics experiments which
use macroscopic measurement apparatus. However, the Copenhagen formulation
does not seem to be applicable in the context of a microscopic measurement
apparatus, which is so small that it is subject to quantum effects (and thus is within
a unitary quantum system). So the Copenhagen formulation for observation may
not appropriate for molecular size QC. Although the von Neumann formulation
of observation is not relevant to the vast majority of physics experiments (since
their experiments generally use large measuring apparatus and small number of
degrees of freedom (qubits)), it appears to be appropriate for molecular size QC. It
is possible that the volume for quantum observation apparatus grows very quickly
with the number of qubits in the von Neumann formulation. In particular, no one
has proved a upper bound on the volume for quantum observation (as a function
of the number of qubits) assuming the von Neumann formulation. See the
Appendix for a further discussion of the problem of determining volume bounds
for the observation operation in QC.