Quantum world is probabilistic.
We established the above fact in the last post which introduced the idea of a quantum system that was spin and how different an experiment is in this small world. Another thing we saw was that this world is what we expect it to be, on average & not for individual experiments.
In this post I would like to share with you about what kind of questions are we allowed to ask and experiments we can actually think about doing. What can we know or confirm about a quantum system? Is its logic same as the classical or the normal world? Let us look into it.
Here's an example to understand how the normal logic in our classical world works. It is based on something we call propositions. What's that? It is just a statement which can have a truth value associated with it. It is either true or false.
It is raining here today here is an example. So is My age is 21. Both of them can be either true or false because if it is raining then the first one is true else no and similar reasoning works for the latter one. But are we really constrained to single statements?
Suppose you have two choices to make. You have a box and you wish to draw two balls out of your box without peeking inside. You are given the information that the box only contains red or blue balls. What do you get? Either you get a red and a blue ball or you get the balls of the same color - red or blue. Now, think of these two statements :
On the first try a red ball is drawn and a blue ball is drawn on the second try.
On the first try a red ball is drawn or a blue ball is drawn on the second try.
These two choices are propositions individually and the full statements are combinations of propositions. How have they been combined?
The and & the or which combine our little events of grabbing a particular colored ball are the basic elements of normal logic and the rules for combining propositions. They are used as connectives to combine two or more than two propositions together.
Let us continue with our example and see what and means. 'and' evaluates to true when both the statements are true simultaneously. If I draw a red ball on the first try and I draw a blue ball on the second try, the first statement evaluates to true.
What does or mean? I think what comes to your mind when you think of or is either this or that but not both. That is an exclusive or. You can have either one of the cookies not both. In logic, the case is a bit different. Here, 'or' evaluates to true if at least one of the two propositions evaluated to true. Moreover, it is also true when both of them are true. So I just need either a red ball on the first try or a blue ball on the second to get my second statement true (both work too). Some other examples can be -
Sun rises everyday or chocolate chip cookies are tasty is true as both the statements are true
Sun does not rise everyday or chocolate chip cookies are tasty is true as at least one i.e. the second statement, is true (what kind of a person doesn't like chocolate right!)
Now that we have established the notions of propositions and the elements of logic, let us try them on some experiments related with our strange little fella' - the spin.
Consider these two propositions below -
A : the spin in the z-direction is measured to be +1
B : the spin in the x-direction is measured to be +1
How could we test the truth value of the two individual propositions? Don't forget that we also had an apparatus with us which permitted us to measure our spin and had a 'this way up' sign for the orientation. For the purposes of this post consider z direction to be up and x direction to be right. Apparatus' 'this way up' sign lies in perfect line to the z direction and as you make it perfectly horizontal, that's your x direction. Considering we know how to orient our apparatus in the right directions, we define the two propositions below :
A or B : the spin in the z-direction is measured to be +1 or the spin in the x-direction is measured to be +1
A and B : the spin in the z-direction is measured to be +1 and the spin in the x-direction is measured to be +1
Can we evaluate them? Yes we can. We just need to evaluate them individually first. Use the apparatus to measure the spin first in the z direction. Record it and then use the apparatus to measure the spin in the x direction.
For the sake of an example, let us see what happens if we see spin as a normal 3-dimensional vector in the space. Suppose you have a spin prepared with you in some state. The following steps is how we go about evaluating our propositions :
Evaluate A : Bring the apparatus near the spin, orient in the positive z direction, measure it and record your result.
Evaluate B : Then, bring the apparatus near your spin, orient it along the positive x direction, measure it and again record your result.
If spin in z direction came out to be +1, we would simply say that A or B is true since at least one of the two statements evaluates as true. Now if spin in the x direction also came out to be +1, we could also conclude that A and B is true. I guess most of you would also say that B and A and B or A is also true then. You may think, well if I have a red ball as the first ball and blue ball as the second , I have a red and a blue ball. But even if the order of getting balls was reversed I would still end up with one red and one blue ball with me and that's perfectly right. This is called associativity and that is a given in the normal logic. Truth values do not change with subsequent experiments. The above reasoning would have worked nicely if our spin behaved classically. But we established that it doesn't and now we will see why the spin won't really go with the assumptions we made while evaluating propositions.
At this point, our learnings from the last post would come to work. Again, the propositions A and B remain the same and we wish to find out the truth values of A or B and A and B. This is how we shall proceed.
We get a spin handed to us which was secretly prepared by some external agent in the +1 state along the z direction. Let us try testing for the proposition A or B. For the first proposition A, we use our apparatus & measure along the positive z direction. If we measure it as +1, we would conclude that A or B is true. Still if we keep on going, what would be the value for the x direction if we were to measure the spin in x? As we saw earlier, it would be randomly +1 or -1 with a 50% chance of being either one of them. So, although we have established that A or B came out as true, let us try to determine this through a different way.
We saw that the order of the experiments didn't matter classically. If A or B was true so was B or A. Is that so here too? We want to test if B or A is true for our strange little fella' (even if A or B came out as true). Let us experiment.
Our spin was secretly prepared in +1 state in the z direction but we measure first in the x direction to evaluate our proposition B. Let us say it comes out as -1. That's what was expected as it was going to be random. We now measure in the z direction. Think about this - Can we be sure to measure the spin as +1 in the z direction now, given that we have measured once in x direction? The answer is no. Even if we prepared the spin initially in the z direction, -
The act of measuring the spin in x direction leaves the spin orientation in a completely random sense for the z direction.
So, there is actually a 1 in 4 chance of getting a wrong answer i.e. the x and the z measurements come out as -1 both. This result comes even after the fact that the spin was secretly prepared to be in +1 state in the z direction and demonstrates that A or B is actually not equivalent to B or A. Let that sink in I guess!
Let us now think of about A and B. You again get the spin handed out to you, you measure the spin along z direction and see if it is +1 or not. If it is, then you go on to check for the measurement along the x direction. Note that the measurement initially along z is going to be +1 only for the newly handed out state; well because prepared in that. Now suppose you get the result +1 in x direction and you try to reason is A and B really true? Not really.
A fundamental thing in science is that if you can validate the experiment from later identical experiments, then only that experiment or , here, proposition is true. The sun rises everyday is true because well it does rise everyday. We have observed it enough and have confirmed the proposition through repeated observations. You may also think of it like this. If you have a blue ball in your left hand and if you look at a ball in your right hand 10 times, you are still going to have a blue ball in your left hand. Thus, you have a blue ball in your left hand evaluates as true.
Now, coming back to why A and B can't be validated is this. You measured +1 along x after you measured +1 along z. Right? And you were trying to say A and B is true. Think of this - if you measured your spin along the x direction, does its orientation stay the same along the z direction as you measured it? No, your subsequent measurements are not going to be able to confirm that result at a later point of time. Thus, the whole notions of and breaks down in quantum.
A and B is non-confirmable as the second experiment 'interferes' with the possibility of the first being true or false
This is where the true difference in the quantum domain arises.
Commutativity breaks down in quantum world. Simply put, order matters. If you say that A or B is true, you can't simply say that B or A is also true.
Another thing is that the notion of and is not meaningful here & those of you who know a little bit about physics may have read about the Heisenberg's Uncertainty Principle. It is a simple little result built from basic geometry of vectors and is quite really a proof of this ambiguity that arises in experiments. If that did not ring any bells, forget it!(until a later post at least!)
Coming to the end, I would like to present one of the key takeaway from this post-
"Any subsequent experiment on a system, even the act of measuring, interferes with the previously confirmed results as that experiment must have altered our system"
On the face of it, this looks trivial, but it really is not. Certainty is absent in quantum systems. We can only predict what to expect with a chance.
I hope certainty holds true for the frequency of my posts at least! Thanks for sticking around for yet another strange but hopefully exciting post. Till the next one, keep on thinking.
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