# Entanglement and superposition illustration

I found an example of how "the power" of superposition can be explained in simple words. Toss two coins. While they are still in the air, they can be thought of encoding 4 states. Wow! Cool! Only 2 coins, but they encode 4 states!

Now, when I want to demonstrate "the power" of entanglement I have a problem. If I entangle these 2 coins, the number of states becomes 2 for these 2 coins.

Is there an example that would demonstrate the importance of both entanglement and superposition?

I just don't want to say "just trust me entanglement can be used to speed up computations" without providing any intuition for why this can be true.

Is there an example that would demonstrate the importance of both entanglement and superposition?

The usual "quantum parallelization" example, (e.g., from Nielsen and Chuang chapter 1) is probably a good one.

I will summarize below, with some embelishments.

One assumes a unitary $$U_f$$ exists that can implement a function $$f$$ that takes $$n$$ bits to one bit. The action on a state $$|x\rangle_n|y\rangle$$ is: $$U_f|x\rangle|y\rangle = |x\rangle_n |y \oplus f(x)\rangle\;.$$

(Yawn, boring so far, right?)

OK, so now apply the same unitary $$U_f$$ to a superposition that is generated by hitting a single initial state $$|0\rangle_n$$ with a tensor product of n Hadamard gates (and leave the output state alone for now): $$U_f H^{\otimes n}|0\rangle_n|0\rangle = \frac{1}{\sqrt{2}^n}U_f\sum_{x=0}^{2^n-1}|x\rangle_n|0\rangle$$ $$=\frac{1}{\sqrt{2}^n}\sum_{x=0}^{2^n-1}|x\rangle_n|f(x)\rangle \;,$$ where, (Wow!), I now have "magically" "evaluated" that function for every single one of it's inputs. That resulting state is generally entangled.

I just don't want to say "just trust me entanglement can be used to speed up computations"...

Yeah, please don't say that. It's not even wrong.

For example, our "magical" "evaluation" of all of the $$2^n$$ function inputs "in parallel" is mostly illusory. There is no way to extract that data as a single computation. We can only measure the state and the result of the measurement will be one of the many (x, f(x)), which only tells us anything about a single function evaluation.

• Thank you very much for this example! Is there any way to somehow simplify it to make it more accessible for non-technical people? I'm thinking of some real-world analogy similar to the two coins toss example. Mar 4 at 2:00
• In my opinion, the two coins toss example does not really make sense. Unfortunately, quantum mechanics tends to not be intuitive in the normal sense. There is likely no great "real-world" analog (in the sense of an analogy with macroscopic classical phenomena).
– hft
Mar 4 at 2:18