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If I have a 3-bit ADC and a reference voltage of 8 V, any input voltage of exact increment of 1 V (e.g., 1 V, 2 V, 3 V, etc.) should not have any quantization error. With a 2 V DC input, the ADC outputs 010 with zero quantization error.
Why then do we still have $ \Delta^2 \over 12 $ quantization noise and do we still see the white noise as quantization noise?
Why then we still have $ \Delta^2 \over 12 $ quantization noise
You need to read more closely: this is the quantization noise for continuous-uniformly distributed signal amplitudes and equally spaced steps.
This signal model clearly does not apply to your example signal!
Note that your example signal only takes discrete values - and hence is already quantized. In that case, it's not quite fair to consider your ADC a quantizer; it might be more of something like a discrete value mapper.
Why can't quantization noise be zero?
A time-discrete quantized signal always has finite entropy¹, whereas using the same metric as for discrete sources, a value-continuous source has "infinite entropy". Assume we construct a continuous-valued source as ever increasing finer-grained quantized source, it becomes clear that there's a difference in the amount of information in the discrete- and the continuous-valued source.
That difference in information has to "go" somewhere. So, there needs to be an amount of information "lost" during quantization, and we can consider that equivalent to a noise that overlays the original signal.
¹ that's the expectation of the information of symbols coming out of that source, i.e. $$\mathbb E(I(x))=\mathbb E(-\log_2(\mathrm P(X))=-\sum_x \mathrm P(x)\log_2(\mathrm P(x))$$
Strictly speaking, there is no stochastic process that's injecting uniformly distributed random noise at the output of an ADC converter, on the interval $n \in \left (-\frac{LSB}{2}, \frac{LSB}{2} \right ]$. Strictly speaking there is no quantization noise at all.
Strictly speaking, quantization is a purely deterministic (i.e. -- no noise) nonlinear process, that turns a continuous signal into that typical stair-step quantization.
So, strictly speaking, one is in error to even think about quantization noise.
However, analyzing a system that has a quantization step in it "correctly", by treating that quantization as a pure deterministic nonlinearity is just not mathematically feasible. It's not just hard to do without resorting to simulation (and thus losing the opportunity to make generalized conclusions), it's impossible.
So we introduce quantization noise. It is a fictitious noise process, that allows us to treat the quantization step as a linear system with injected noise. We can analyze such systems precisely, in a way that does give us results that are both exact and generalized.
When we introduce quantization noise into an analysis, we are trading an exact representation of a system that we cannot analyze for an approximate representation of a system that we can analyze. As a bonus, in useful systems the system's quantization behavior is often so close to the quantization noise model that there's no practical difference in overall system behavior between quantization as a nonlinearity (the real system) and quantization as injected noise (the fiction).
And note: the model of quantization as a linear system plus injected white Gaussian noise is not always correct. There's a lot of middle ground, but there's two endpoints to this.
One such endpoint is a system that has noise equal to several LSB's going into the ADC, and whose ADC output is averaged or filtered before being used. An example of this is a radio front-end, where you have a high-bandwidth signal with high bandwidth noise that's being sampled then processed into a narrow-band signal. Such a system has quantization noise whose effect is so close to the injected white Gaussian model that you don't have to worry about it.
An opposite endpoint is a system that is very quiet and slowly varying, and is going into a fairly coarse ADC. In this case, you cannot treat the quantization noise as white Gaussian. An example of this, which I've personally dealt with, is a servo loop that had an old-style 8-bit ADC in the feedback path. You could watch the thing oscillate around one ADC count, trying to hit a target that was "between" LSBs.
When the white Gaussian approximation for quantization noise doesn't hold you have to use other approximations, or not use the quantization noise approximation. For the most part, the only cases I've dealt with where that white Gaussian approximation didn't hold, I could treat the quantization as a sinusoid with a magnitude of $\frac{2}{\pi}\ LSB$ at the worst possible frequency for my given system. This would (A) give me a pessimistic prediction; if my system "passed" with that noise it would work in real life, and (B) isn't too inaccurate for closed-loop control, where an oscillation often seeks out that worst-frequency case anyway.
It seems you don't want to follow the statistical approach to define the quantization error; but unfortunately you have to consider it that way.
In particular, the error power $\Delta^2/12$ signifies a uniform quantizer with a uniformly distributed input signal and uniform error distribution between $-\Delta/2$ and $\Delta/2$, $\Delta$ being the quantization step-size.
Therefore, the error power $\Delta^2/12$ does not occur for every quantized sample, rather it's the expected value (or the average) of the error powers for a sample set with sufficiently large number of quantized samples.
In this regard, there are samples with zero error as well samples with maximum error (which is $\pm \Delta/2$ for bounded (unsaturated) input signals). However the expected value of the error power (for a uniformly distributed error) happens to be $\Delta^2/12$.
Note that the expected value of the error is zero by the way...
Many mathematical models used for signal processing assume that systems will behave in linear fashion. Taking two signals, performing some process on them, and adding the outputs will yield the same result as adding the signals, feeding the sum into that same process, and taking its output as the final result. Such models won't work with quantizing systems.
On the other hand, if one models a quantizing step as a process which takes a signal and adds a "quantizing noise" component whose strength might be anywhere between zero and the largest possible difference between actual and sampled value, one can compute a worst-case difference between a linear system and a system where the inputs or outputs of various stages are quantized in various ways.
Note that unlike some other forms of noise which can be expected to have predictable spectral characteristics, quantization noise should be presumed to have whatever spectral characteristics will be maximally annoying because, in practice, it often will. On the other hand, if one can show that the amount of quantization noise at the output of a process will not exceed -57dB at the output, and that would be quiet enough that even the most annoying combination of spectral characteristics would still be tolerable, one can guarantee that quantization won't cause unacceptable degradation without having to precisely model the effects of quantization precisely.
In some circumstances, it may be desirable to avoid the "maximally annoying" combinations of spectral characteristics. That may be accomplished by adding certain forms of noise to an input signal before quantization. This will increase the total level of noise in the output, but ensure that much of that noise will have benign spectral characteristics. A signal which has -57dB of worst-spectral-trait spectral noise and 0dB of other noise may be more objectionable than one which has -50dB of spectrally-flat noise and -60dB of worst-spectral-trait noise. In this scenario, the quantization process will end up adding more noise than it otherwise would have done, but the presence of other noise may help obscure the spectrally-objectionable parts.
