THERE was a time when I could be certain of beating my kids at games. Whether it was chess, cards or my personal favourite, , I could sit down confident in my superior abilities. Sadly, those days are gone. My children are teenagers now, and the occasions when I manage to outwit them are dismayingly rare. But this holiday season, I have cooked up a plan to get my own back.
My kids have the advantage of nimble-minded youth. But I am the only one in the family familiar with quantum physics: a famously strange world where, as this year’s Nobel prizewinners in physics showed, objects can seem to be in instant communication over any distance via a phenomenon called entanglement. I have now discovered a handful of games where my knowledge of quantum theory is going to give me the upper hand. They may be a little more exotic than snakes and ladders. They may even require a quantum computer to play properly. But did I mention that I really, really want to beat my kids?
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My first discovery is a puzzle set by mathematician Leonhard Euler in 1779. He imagined a group of 36 army officers, each assigned to one of six ranks and six differently coloured regiments. Could these officers be arranged in a 6Ă—6 grid so that no regiment or rank is repeated in any row or column?
I will watch my family struggle, before smugly announcing that it is impossible. In the 1960s, mathematicians showed that, while there are solutions for similar games involving 3 × 3, 4 × 4 and 5 × 5 grids, there is – strangely – none for 6 × 6. But then I will say that I can do it anyway thanks to my quantum know-how.
The credit goes to Suhail Rather at the Indian Institute of Technology Madras and his colleagues, who showed earlier this year that Euler’s puzzle can be solved – providing the officers are . Such soldiers can be placed in “quantum superpositions” of their possible states, which means an officer can be, say, partially a red colonel and partially a blue lieutenant. That is, unless we actually look at one, whereupon this act of observation “collapses” the officer’s quantum state to one thing or the other.
Now the officers in each row and column aren’t so much different in terms of rank and regiment, as mutually exclusive in terms of their superposition states of rank and regiment. The officers also need to be entangled – meaning that collapsing the state of one officer by measurement affects how another collapses into a complementary state in just the right way to satisfy the rules.
To demonstrate this game for real, we would need a quantum computer to encode the information about the soldiers. I don’t have one handy at home, but we could log into a cloud-based quantum computer, such as those offered by or . I suspect my family will think all this is cheating. But, hey, the whole world is fundamentally quantum in the end, so what’s the problem?
After trouncing the family at Euler’s square, how about a simpler game like noughts and crosses (also known as tic-tac-toe)? This involves two players alternately drawing Xs or Os on a 3 × 3 grid. To win, one player must place three of the same mark in a horizontal, vertical or diagonal row.
Quantum cunning
The traditional version generally ends in a stalemate, but with quantum rules there is much more room for cunning strategy. On each go, players of make entries on a pair of spaces that are then in superposition on the grid. If you like, the player’s entry is “delocalised”. This means we don’t know which of the two grid spaces it will end up in until a player elects to “measure” one of those grid spaces, determining whether the entry is there or not. You can even make an entry into a space that the other player has already used, knowing that, on measurement, only one entry or the other will be permitted. Again, we would have to set this all up on a quantum computer.
In effect, each entry sets up a new parallel entangled game, while measurements gradually collapse the grid to just one of them. Part of the strategy lies in choosing when to measure and collapse a grid space, whereupon its entry is fixed and no further entries are allowed there. My familiarity with quantum mechanics should give me a head start on my family. Though, if mastering other new technologies is any guide, I might only have about half an hour before the kids overtake me.
For the last game, I am stepping things up a notch. My kids aren’t going to know what’s hit them, because this time I am making use of my secret weapon: quantum pseudotelepathy.
The Mermin-Peres magic square was devised independently in 1990 by quantum physicists and , and, like noughts and crosses, it uses a 3 Ă— 3 grid that two players fill out in successive moves. However, for this game, the players assign values of +1 or -1 to each grid space, instead of Xs and Os, and rather than competing, they are trying to collaborate to win together without conferring.
A referee assigns one player a random row and the other a random column to fill out. The rule is that the product of every column (found by multiplying together the three numbers in it) has to equal +1 and the product of every row must equal -1. The players win a round if they both assign the same value to the square where their column and row overlap.
There are nine possible row and column combinations, but it isn’t possible for the players to win all nine of these rounds – there will always be a space where what each player needs conflicts. Once again, I will let my family struggle with this fiendish game before showing them how it can be played to perfection with a quantum twist.
The key is to fill in each square with quantum bits (qubits), which can take the value +1, -1 or a superposition of both. Players now fill each space with a pair of qubits, and the product of the pair gives the value of that entry: for example, if they measure to be -1 and 1, the value is -1 Ă— 1 = -1. When resolving what these values are, we only measure one row and one column at a time. This means that different classical outcomes can be obtained from identical quantum entries, depending on whether we measure the entry as part of a row or column. For instance, the top right entry may resolve as -1 when measured as part of its row, but as +1 when measured as part of a column. This actually illustrates a fundamental principle of quantum mechanics called contextuality. It is now possible for the two players to assign the same states to their two qubits for all nine grid elements without any contradiction, while still obeying the rules.
So far, the two players can only win all nine rounds if they make lucky guesses about what each other will do. But there is a way to ensure success – this is where quantum pseudotelepathy comes in. The rules may forbid players from conferring, but they don’t rule out entangling the players’ qubits. If we did this, it means that the qubits can effectively share information, so that the state of one affects the state of the other. Set this up just right and, though the players don’t confer, the qubits themselves will influence each other so that they snap into just the right state to win every time.
After all that mental exertion, I think I will be ready for my turkey. And perhaps after dinner I should give my kids a fair shot at winning. Anyone for a game of Exploding Quantum Kittens?
