# What is the relation between uncertainty and information obtained from a measurement?

For ex. if a measurement gives a position with twice the uncertainty as another measurement, how much less information regarding position are you getting? In other words, if uncertainty doubles, is the information gained cut in half? What is the relationship? Also, does infinite uncertainty equate to 0 information?

Edit: David, I am referring to measurement in terms of the uncertainty principle. If you choose to increase the frequency of the photon used to locate an electron, the uncertainty of momentum will obviously increase. My question is, if uncertainty in an observable rises or falls between 2 measurements, what effect does that have on the information obtained for each observable? Put explicity, if you do a measurement that gives you a more precise particle location, doesnt that mean you are getting more information about that observable? If you tell me my seat in a hockey arena is in “section 104/row E/seat 16 “, isnt that more information than telling me my seat is “somewhere in the stadium”?

• How do you define information? – my2cts Aug 4 '18 at 19:12
• My information concept is based on the tenet information cannot be created or destroyed. – user21909 Aug 4 '18 at 21:28

## 1 Answer

"Uncertainty" is an ambiguous term. For a standard measurement in physics, there are two recognized components of error.

Accuracy is a measure of how close a measurement matches the "known" value. Obviously, the known or true value of a measurement isn't normally known, but for situations such as determining the concentration of a chemical in a chemistry experiment, you would deliberately mix a known concentration of this chemical, known as the "standard", and make a measurement of this concentration to ensure that your measurement device/method was giving you a true reading.

Precision is a measure of how many significant figures you can assign to a given measurement, and is related to the device that was used to make the measurement. For a meter stick measurement, where the meter stick is marked in centimeters and contains no millimeter marks, you can measure lengths to the nearest centimeter and estimate the fraction of a centimeter that is exceeded by the unknown length. Thus, a line that is 65.3 centimeters long would be reported as being 0.653 meters in length, where the least significant digit (e.g., "3") is recognized as an estimated value. You wouldn't report the length as 0.6530 meters, because the rules of precision require that you only estimate and report the first estimated digit in your answer. Likewise, if a different meter stick was marked in centimeters and millimeters, you would report the measured line as 0.6532 meters, where again, the least significant digit (e.g., "2") is recognized as an estimated value. Note that if precision is properly reported, it is easy to discern that the first measurement was taken with a device that was marked in centimeters while the second measurement was taken with a device that was marked in millimeters.

Regarding the "uncertainty" in the measurements described above, BOTH measurements give an answer within the precision of the device that was used. EACH answer is valid "information" in the context of accuracy and precision. Whether one answer can be considered as "better" or more valid than the other often depends on how precise an answer you need for a given application or experiment. This, of course, means that there is no strictly defined relation between "uncertainty" and "information", as both of these terms are ambiguous within the context of established physics norms.

• could you please respond to my edit of the question? Your answer isnt what I was after. – user21909 Aug 9 '18 at 0:51
• @user21909, I can only answer your questions that are "classical". For a quantum mechanical answer, I will have to defer to the experts on this site. While my answer wasn't what you were after, I'm hoping that it prompted you to rephrase in a specific enough way for someone else to provide the kind of answer that you actually are looking for. – David White Aug 9 '18 at 1:13