Quantum state smoothing cannot be assumed classical even when the filtering and retrofiltering are classical
arxiv(2023)
摘要
State smoothing is a technique to estimate a state at a particular time,
conditioned on information obtained both before (past) and after (future) that
time. For a classical system, the smoothed state is a normalized product of the
filtered state (a state conditioned only on the past measurement
information and the initial preparation) and the retrofiltered
effect (depending only on the future measurement information). For the
quantum case, whilst there are well-established analogues of the filtered state
(ρ_ F) and retrofiltered effect (Ê_ R), their product
does not, in general, provide a valid quantum state for smoothing. However,
this procedure does seem to work when ρ_ F and Ê_ R are
mutually diagonalizable. This fact has been used to obtain smoothed quantum
states – more pure than the filtered states – in a number of experiments on
continuously monitored quantum systems, in cavity QED and atomic systems. In
this paper we show that there is an implicit assumption underlying this
technique: that if all the information were known to the observer, the true
system state would be one of the diagonal basis states. This assumption does
not necessarily hold, as the missing information is quantum information. It
could be known to the observer only if it were turned into a classical
measurement record, but then its nature depends on the choice of measurement.
We show by a simple model that, depending on that measurement choice, the
smoothed quantum state can: agree with that from the classical method; disagree
with it but still be co-diagonal with it; or not even be co-diagonal with it.
That is, just because filtering and retrofiltering appear classical does not
mean classical smoothing theory is applicable in quantum experiments.
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