While Schrödinger’s cat illustrates superposition, it may trigger some simple questions
- When we physically open the box, there is a 50% chance of seeing the cat either live or dead (clearly not both). So what does it even mean to say the cat is both live and dead when we haven’t looked inside, apart from the question evoking memory of equivalently vacuous babble on the same topic about trees falling in forests and not being heard etc?
- Even if we grant the cat is both live and dead before we open the box, what can we do with the information that the “cat is both live and dead”?
The following game will show how we can increase the odds of winning from 50% with a regular fair coin to 100% with a “quantum coin” by
- leveraging off information equivalent to “both live and dead cat state”, that is the quantum coin “in both heads and tails state” — and just a little cheating.
- The best part is even if our cheating is exposed by our opponent, the cheating will seem like an insignificant detail.
- Neither us or our opponent (or anyone else in the world for that matter) will have any satisfying common sense explanation for why we can always win, despite knowing how we can win when we cheated.
Regular coin game.
Lets first do the game with a regular fair coin.
- We place coin in a box.
- We declare our choice to our opponent — say heads.
- We shake the box and let the coin settle.
- We can choose to open box at this step and see if its heads or tails at this point, but it doesn’t matter — the rules of the game says we have to shake the box again one more time and let the coin settle.
- We now open the box to see if we won or not.
It is obvious that our chances of winning is 50%. Also, it doesn’t really matter if we opened the box and looked inside at step 4. If we played this game, say a 1000 times, regardless of us looking at the intermediate result at step 4, the chance of winning is always 50%. Figure 1 illustrates this.
Quantum coin game.
Lets now play the game with a quantum coin in the box. The quantum coin is similar to the regular coin — meaning it can be in two states, head or tail, and there is a 50% chance of it being in either of these states after an operation equivalent to shaking the box.
- We place quantum coin in a box.
- We declare our choice — heads or tails. Here is where we cheat. If the quantum coin is in head state, we declare tail as our bet to the opponent; if not we bet on head. Our opponent hopefully doesn’t notice it.
- We shake the box and let the coin settle.
- We do not open the box at this time. We just perform the second shaking as we did with regular coin.
- We now open the box to find the coin is in exactly the state we bet on.
We can repeat this game any number of times, and so long we cheat each time and our opponent doesn’t notice, we can win. Lets say, opponent notices that we are always looking at the quantum coin before the first coin toss.
- The opponent may suspect the quantum coin is not a fair coin like the regular one, and may ask us to open the box after the first shake in step 4.
- We can certainly allow that, but then we will tell opponent we won’t bet in that run.
- To our opponent’s bewilderment, assume we repeat the experiment a 1000 times each time opening the box after first shake — the result will always be roughly 50–50 exactly like a regular coin. So the quantum coin will look no different from a regular fair coin.
- Adding further dismay, the results of the second shake will also be exactly like that of a regular coin shake — heads will occur roughly 50% of the time and tails roughly 50% of the time — exactly like a regular coin toss.
So even though the opponent knows our looking into the quantum box before first shake has something to do with winning, it will not make sense how that can enable us to win, because opening the box after first shake showed the quantum coin behaved exactly like a fair coin.
We can now let in the opponent on our secret. We can confess even we don’t understand it too from a common sense perspective, except the following strategy always works
- if we see the quantum coin before first toss, we can always win so long as we don’t open the box after the first shake. Figure 2 illustrates this for head case and Figure 3 for tail case start
- We can also say, our chances of winning with a quantum coin becomes exactly like a regular coin toss once we open after first shake and observe its state, which is precisely why we didn’t bet when opponent asked us to open the box after first shake. This is shown in figure 4 below.
So to summarize the strangeness of the quantum coin toss
- Observing the quantum coin after first shake, clearly shows it is like a fair coin. We can even count the four paths from the root leading to the four outcomes regardless of what we start with at root — HH,HT,TH,TT — all occur with equal chances leading to 50% each for head and tail on second shake.
- However, when we don’t observe state after first shake, two of the four path outcomes disappear as shown in figures 2 and 3 and we always predictably get a tail or a head corresponding to the state of coin before first coin shake.
The mathematical explanation (without equations) for how two of the four branches disappear is quite simple.
- When it comes to quantum stuff, like quantum coins, we change the rules of calculating probability and allow for the paths of the tree in the figures for quantum coins to take negative values (we would never do that in daily life — imagine hearing someone say there is negative 10 percent chance of raining tomorrow). Given this relaxed rule, we can make some of the paths to cancel each other leading to a pruned tree with just heads or just tails. Figure 5 illustrates this
- But one may ask — how can branches of the tree cancel each other? When we played the game even with the quantum coin, the times we observed the coin after first shake clearly showed that each game run only traversed one path of the tree. That is, regardless of what we started with (heads or tails), we always took one of four paths in each run — HH, HT,TH,TT. So how can these paths cancel each other, when each path is different for each run?
- This is where superposition offers an interpretation. When we don’t observe the coin after the first toss, both the paths from the root are traversed simultaneously. The system is now in both states in parallel -or a superposition of both states. When we toss the coin again, all four paths are traversed simultaneously and given paths can have positive and negative values, they cancel, yielding a pruned tree giving only one outcome — heads or tails.
- The power of superposition is parallelism. Nature can compute in parallel so long as we don’t observe the intermediate steps, and we can construct ways to cleverly nudge a quantum system to perform different computations just like our box shaking, and also leverage off the cancelling effect (and other strange effects like entanglement, not discussed here) to prune computation paths to yield results, that would take a conventional computer much longer to do, since it has to evaluate all paths of the tree. Quantum computers make use of the same trick as our quantum coin. We won the game with a single qubit in a superposition of two states. If we had two qubits we could have a superposition of 4 states (00,01,10,11) or 4 parallel paths. If we had 30 qubits in superposition in our game we would be traversing 2³⁰ paths in parallel — thats approx the number of galaxies in universe. Any quantum system that is described by more than one state is essentially in superposition- so parallelism is inherent in quantum systems.
- A clarification on what we mean by observing the coin — all it means is some form of measuring its state — heads or tails. The act of measuring however disturbs the quantum coin causing it collapse out of its parallel state into either heads or tails.
- So in summary we started off with a quantum coin which was in just one state — heads or tails(not both) — we could even pick exactly what state we wanted to start with if we chose to. Then we shook it into a parallel state of being in both heads and tails state. Then we again shook it and the quantum coin system computed paths in parallel to generate a deterministic output of just one of the two states. The deterministic outcome was always based on what state we started with (heads always produced tails and vice versa). So we could use that to fact to win each time by being aware of the start state.
- This simple quantum coin toss illustrates two key concepts essential to quantum computing
- Parallelism or superposition. The ability for a quantum system to be in multiple states in parallel and hence traverse multiple computation paths in parallel
- Path culling or destructive interference. Without path culling, parallelism alone would be useless. We would have no way to extract the outcome of a result. By careful choice of nudging operations (transformations) we can weed out paths and narrow down to desired results.
This explanation was created based on Scott Aaronson’s blog. There is a book that is largely based on his blog Quantum Computing since Democritus: Scott Aaronson: Amazon.com: Books
For more details on what kind of “shaking operation” is performed on quantum coin etc, notes at the end of the answer to a related question may help How have physicists developed a mathematical model describing quantum phenomenon if we don’t really know what it means?
Originally published at www.quora.com.