In practice, smoothness interacts with measurement: we can usually measure the higher-order bits without measuring lower-order bits, but we can’t easily measure the lower-order bits without the higher-order bits. Imagine, for instance, trying to design a thermometer which measures the fifth bit of temperature but not the four highest-order bits. Probably we’d build a thermometer which measured them all, and then threw away the first four bits! Fundamentally, it’s because of the informational asymmetry: higher-order bits affect everything, but lower-order bits mostly don’t affect higher-order bits much, so long as our functions are smooth. So, measurement in general will favor higher-order bits.
There are examples of measuring lower-order bits without measuring higher-order bits. If something is valuable to measure, there’s a good chance that someone has figured out a way to measure it. Here is the most common example of this that I am familiar with:
When dealing with lasers, it is often useful to pass the laser through a beam splitter, so part of the beam travels along one path and part of the beam travels along a different path. These two beams are often brought back together later. The combination might have either constructive or destructive interference. It has constructive interference if the difference in path lengths is an integer multiple of the wavelength, and destructive interference if the difference in path length is a half integer multiple of the wavelength. This allows you to measure changes in differences in path lengths, without knowing how many wavelengths either path length is.
One place this is used is in LIGO. LIGO is an interferometer with two multiple kilometer long arms. It measures extremely small ( $ 10^{-19} $ m) changes in the difference between the two arm lengths caused by passing gravitational waves.
There are examples of measuring lower-order bits without measuring higher-order bits. If something is valuable to measure, there’s a good chance that someone has figured out a way to measure it. Here is the most common example of this that I am familiar with:
When dealing with lasers, it is often useful to pass the laser through a beam splitter, so part of the beam travels along one path and part of the beam travels along a different path. These two beams are often brought back together later. The combination might have either constructive or destructive interference. It has constructive interference if the difference in path lengths is an integer multiple of the wavelength, and destructive interference if the difference in path length is a half integer multiple of the wavelength. This allows you to measure changes in differences in path lengths, without knowing how many wavelengths either path length is.
One place this is used is in LIGO. LIGO is an interferometer with two multiple kilometer long arms. It measures extremely small ( $ 10^{-19} $ m) changes in the difference between the two arm lengths caused by passing gravitational waves.
Beautiful example😌