Physicist: The Uncertainty Principle is a little subtle. Most folk are introduced to it as “the more precisely you measure the position of a particle, the less precisely you can measure its momentum, and vice versa”. That makes it sound like an engineering problem; something we can get around by, for example, trying harder.
But the Uncertainty Principle is a statement about what things are actually like in reality, and the weird limitations on how well we can do measurements is just a symptom. In a very fundamental, profound, and physical sense, everything is a little uncertain.
To be clear, you can measure the position and momentum of a single particle (or many particles or whatever quantum system you prefer) very precisely, and there’s nothing stopping you from getting very precise results from both measurements. The problem is that those precise measurements don’t really tell you much about the actual quantum state you’re looking at. You can prepare a series of identical quantum states and measure each of them in exactly the same way, but because quantum states are generally a combination of many different states together (what folk in the quantum biz call a “superposition of states”) you generally don’t get the same result over and over.
An electron in an atomic orbital is in a state with one amount of energy, but many positions. The electron is kinda “smeared out” around the atom. So if you measure the energy you get a definite result, but if you measure the position you could get any result within a range of positions (a small range, what with atoms being small). You can picture this as being like a musical chord and either asking “what chord was played?” or “what note was played?”. For example, the C chord is composed of the C, E, and G notes. If you do a chord measurement (this is not an actual thing, but bear with me), you get a definite answer: C. If you do a note measurement (again, more quantum metaphor than mechanics), then you get one of three results: C, E, or G.
The randomness of the Uncertainty Principle has the same root cause: a single quantum state being composed of many different states at the same time. Like the chord example, a position state is made up of a range of many momentum states. Unlike the chord example, the reverse is also true; a momentum state is composed of many position states. Unintuitively, the fewer position states something is in (the more specific the position) the more momentum states it is also in. Unfortunately, because of this fact (that position is describable in terms of momentum and vice versa) you can directly derive the uncertainty principle mathematically. In other words, assuming that every experiment ever designed to refute the basic physics didn’t all fail accidentally, the Uncertainty Principle is built into the universe and no cleverness or engineering will overcome it.
When you prepare many particles (or any other quantum system) in identical quantum states, measure them one after another, and write down the results, you’ll find that there’s some spread in where they show up as well as their momenta. You can prepare states with very little spread in their position or with very little spread in their momenta (so called “squeeze states”), so the Uncertainty Principle isn’t as simple as “everything is random and unpredictable”, it’s about pairs of measurements applied to “conjugate variables” (position and momentum are the classic example).
The Uncertainty Principle says that if you look at the spread (standard deviation) of those two measurements for many copies of any given state and multiply those spreads together, their product is always greater than some minimum amount. Explicitly, if and are how spread out the position and momentum are, then . This, it’s worth noting, is a really tiny lower bound. If you’re certain that a brick is in a box, then (were you so compelled) you could nail down its velocity to within around 0.000000000000000000000000000000002 meters per second, which is arguably fairly certain.
Entanglement doesn’t do much to change the picture. With entangled pairs of particles, a measurement on one mirrors a measurement on the other. You can entangle any property (energy, polarization, delectability, hell even existence), including position and momentum. In a dangerously succinct nutshell, entanglement basically/sorta gives you two chances to make a measurement on a quantum state. Assume particles A and B are position-entangled. If you measure the position of A you’ll be able to say “ah, it’s over here” and if you measure the position of B you’ll be able to say “it sure is”. The two measurements, although otherwise random, agree.
But what you’d really like to see is a precise position measurement on A and a precise momentum measurement on B. It turns out: that’s fine. Once again, that spread of results shows up. If A shows up in some region, there is a corresponding set of momentum states that are compatible with that (if a certain “chord is played” there is a particular “set of notes” involved) and when you measure B, you’ll see one of those.
So entanglement does give you another chance to measure a quantum state with as much precision as you might desire, but… it doesn’t really change anything. The Uncertainty Principle doesn’t say “you can’t simultaneously measure position and momentum with nigh perfect precision!”, it says “it doesn’t matter if you do!”.
The giant microscope picture is from here.