What is the origin of the irreducible error $\propto e^{-N}$? In this (timestamped) lecture, the professor says that any measurement includes an irreducible error that scales as $e^{-N}$ where $N$ is the number of degrees of freedom in the measurement apparatus itself. To summarize the pertinent aspects of the video, the professor expresses that as $N$ increases for any measurement apparatus, the error decreases, and therefore the only way for the error to be zero is an infinitely massive system. I was simply trying to understand the mathematical origin and veracity of such an irreducible error that is said to be universal, i.e. why does there exist a measurement error that scales reciprocally with the exponential of the degrees of freedom? Further insights and potential references regarding the existence of this irreducible uncertainty in physical systems are appreciated.
 A: As Professor Arkani-Hamed says a few minutes later in the video, he is "being very rough here" (and in the related paper).  It might have been better if the video stated that the irreducible error goes as $\sim e^{-\Omega} \sim e^{\sim -N}$, where $\Omega$ is the system's total number of possible (microstates).
A system with $\Omega$ possible states cannot provide a measurement with a relative precision better than $1/\Omega$, since each different measurement value must correspond to a different state.  This is why a 14-bit ADC (Analog-to-Digital Converter) is $2^6$ times more precise than an 8-bit ADC which can only report results to $1$ part in $256$ $(2^8)$. More generally, an N-bit ADC output has $N$ binary degrees of freedom and $2^N$ possible states, so it can't provide a result with relative precision better than $1/2^N = 2^{-N} = e^{-N ln 2}$.
The number of microstates is not always simply dependent on the number of degrees of a system, but if a system has $N$ degrees of freedom, each with $M$ possible values, then $\Omega = M^N = e^{N \ln M}$ is "growing exponentially"  with respect to N. People may then sometimes write "$\sim e^{N}$ " as shorthand, ignoring any constant or logarithmic factors in the exponent.
Of course, each degree of freedom doesn't necessarily have the same range of possible value, e.g. an electron in free space has binary spin and continuous position, but each additional degree of freedom does multiply the number of possible states.
Of particular interest in the video is that the irreducible error of a physical system of area $A$ is $\sim e^{-Area/G_N}$.  If position provides the only degrees of freedom, this roughly follows from the assumption that length is quantized at the Planck Length $L_P \sim \sqrt{G_N}$, so the minimum unit of area is $\sim L_P^2 \sim G_N$.  For a classical black hole the irreducible error is  $1/\Omega = e^{-Area/(4 G_N)}$; this follows from inserting the Bekenstein-Hawking value for the entropy of a black hole $S=k_B A/(4G_N)$ into Boltzmann's entropy definition ($S = k_B \ln \Omega)$.  The number of degrees of freedom associated with a black hole is usually considered to be $N=Area/G_N$, so in the limit of a measurement system becoming a black hole, its irreducible error is (as stated in the video) $\sim e^{-N}$.
