QUANTUM COMPUTING - Bio-Inspired and Nanoscale Integrated Computing - page 90

the von Neumann formulation of quantum measurement in the case
where the measuring apparatus is so small that it is subject to quantum
effects.

That is, there needs to be given, in addition to the experimental description
(which is only established for n 0 qubits):
(iii) A mathematical analysis of the quantum effects (in the context of a closed
unitary system) involving the measuring apparatus as the number n of qubits grows
large. In particular, bounds on the errors (or fidelity) of the measurement need to
be determined as a function of the size of the measuring apparatus.
Without this crucial final element, the proof is certainly not complete. Since
the observation operation is not reversible, such a proof (in the context of a closed
unitary system) seem unlikely to be obtainable.

3.12.2. Possible Experimental Demonstrations of Measurement
Another approach would be to test a proposed small-volume apparatus for
observation on n qubits for moderate size n (say, in the range of a few hundred,
which is required for a nontrivial factoring computation). But the experimental
evidence of the volume bounds for observation is unclear, since the QC experi-
ments have not yet been scaled to large or even moderate numbers (say dozens) of
qubits, and there are few if any physics experiments for this case. (Shnirman,
Schoen [120] describe the use of a single-electron transistor to perform quantum
measurements; D'Helon, Milburn [280] describe quantum measurements with
quantum computers; and Ozawa [281] describes methods for nondestructive
(known as nondemolition) quantum measurements of certain quantum

computations.)
Hence, at this time that there appears to be neither a mathematical proof nor an
experimental demonstration (for even a moderately large number of qubits n) that
observation can be done in small volume (in a closed quantum system). Thus there is
no evidence (either mathematical or experimental) at this time that QC using
measurement scales to large numbers of qubits with small volume.
We first consider a number of related questions concerning measurement and
quantum state reduction:

3.12.3. Is a Quantum Observation Instantaneous?
It appears not. Brune et al. [282] describe the progressive decoherence of the meter
in a quantum measurement.

3.12.4. Is an Observation Always Reversible?
It appears the answer is no (in a narrow mathematical sense of a state reduction),
yes (for small closed state spaces), and no (in a practical sense for entanglements in
a large state space).