You don't need a final answer at the quantum game show.So come on down says Adrian Cho,everyone's a winner.
If you want to get ahead then you should stab your best mate
in the back. At least that's what mathematicians would say. For decades they've
studied simple games in which players can either
cooperate or turn against one another, and they've found that logic often
demands ruthless betrayal-even if it sometimes means everyone loses. But
it doesn't have to be that way.
All this weirdness can make players pull together, even as
each strives to get the best deal. "Even though you still behave rationally
and selfishly, you don't end up in a nasty state," says Simon Benjamin, a
physicist at Oxford University.
To see why quantum mechanics changes
things so much, consider one version of a game called
the Prisoners' Dilemma. Suppose you and your
gang have been arrested for armed robbery. If you all stick together and
stonewall the police, you each get a couple of years in prison. But if you
snitch on the others, you go free while everyone else gets 20 years. And
if everyone turns on everyone else, you'll all get sentences nearly as long
as that. The precise options and consequences can be spelled out in a "pay-off
table" that displays the sentences to be handed down for each possible
combination of moves by you and your accomplices.
Of course, you don't need a mathematician to tell you that
you'll all probably be banged up for a long time. Each of you will soon realise
that, no matter what the others do, you can improve your lot by betraying
them. So however much you talk it over, everyone will grass on everyone else.
That qubit might be a single electron
whose tiny magnetic field can point either "up" or "down". Being
quantum-mechanical the electron can also be in a
"superposition" of these states, pointing both
up and down at the same time. The bizarre superposition of states is fragile
and persists only until someone tries to measure which way the particle's
field is actually pointing. When that happens the superposition "collapses"
to one brother of the states.
With this entanglement in place, any moves the players make
can change the relationships observed when the qubits are measured. If, say,
the first prisoner flips, the entanglement means the qubits are then put
into a superposition of "one up, two down" and "one down, two up".
Benjamin and Hayden used mathematical elbow grease and a bit
of computer help to examine the many possible combinations of states of the
three prisoners' qubits. The researchers then read off the prisoners' sentences
from the pay-off table. If the final state of the three qubits was a
superposition, then the sentences were a weighted average of two or more
outcomes from the table.
But while entanglement enforces a kind of unavoidable teamwork,
it isn't a necessary ingredient of every quantum game. Indeed, you can gain
huge advantages from superposition alone, says David Meyer, a mathematician
at the University of California, San Diego. Meyer has devised a contest between
two characters from the television show Star
Trek: The Next Generation.
Picard reckons his chances of winning are 50 per cent, so he's
not too surprised when he loses the first game. But then he loses the second
game, and the third, and the fourth. In fact, he loses every time. That's
because they're playing with a quantum coin-and only Q knows it.
'NOW THEY KNOW WHAT THEY'RE LOOKING FOR, RESEARCHERS MAY BE ABLE TO SPOT ATOMS ENGAGED IN QUANTUM CONTESTS'
Q can use his quantum powers to do more than win a coin-toss.
He can use the same principles to beat million-to-one odds. If Picard picks
a number in a specified range-say one to 1,000,000-Q can guess it every time
as long as Picard agrees to encode his choice in qubits that Q has already
put into a superposition, like his coin. The same routine of first making,
then finally undoing, a superposition of these qubits means that Picard will
leave telltale signs of his chosen number in the state of the qubits.
Moreover, Meyer's games do not require the bits to be entangled,
so they might even help answer one of the fundamental questions in quantum
computing: do the qubits in a quantum computer have to be entangled for it
to work at all? "The pick-a-number game is certainly a counter-example to
the statement that quantum speed-up comes from entanglement," he says.
But Elsert and colleagues did not allow for the full variety
of superpositions and entanglements that are possible in the quantum game.
And Benjamin and Hayden have shown that if more moves are allowed, then no
matter what the first prisoner does, the second prisoner can always make
a move that puts them in the clear and lands the other with the longest possible
sentence. "Anything you can do, I can undo, if there are just two qubits,"
Benjamin says. Of course, the first player can also undo whatever the second
attempts. So in the end neither player can anticipate what the other will
do and there can be no cooperation.
It's hard to imagine a quantum game show ever making it to
television, despite the fact that Flitney and Abbott's research was sponsored
by the South Australia Lotteries Commission and a major supplier of gaming
equipment. But other studies may prove surprisingly
practical. For example, researchers are developing
methods to use photons as qubits to encode and transmit
secret messages (New Scientist, 2 October 1999,
p 28). Meyer says you can think of such
"quantum cryptography" as a game played
by the sender, the receiver and a would-be spy. "From that perspective,"
he says, "quantum game theory describes something that people are trying
These applications may be a long way oft but physicists have
recently taken a first step towards them. Spurred by the work of Eisert and
colleagues, physicist Jiangfeng Du and colleagues at the University of Science
and Technology in Hefel, China, used nuclear magnetic resonance to
force two nuclei in a molecule to play Elsert's two-player version of the
Prisoner's Dilemma. They found the nuclei behaved as Eisert's team had
Researchers have seen viruses and bacteria playing classical games (Nature, vol 398, p 441), and now that they know what to look for, they may be able to spot atoms and electrons engaged in even funkier quantum contests. In some situations, atoms or electrons have to choose between two equally advantageous states-a dilemma formally known as "frustration". Quantum games might help frustrated particles resolve such dilemmas, and physicists believe that frustration is involved in some striking "emergent" phenomena, such as high-temperature superconductivity. It they can see the particles at play, it may help them understand and perhaps control the game. And all without a single stab in the back.
"Multiplayer quantum games" by Simon C. Benjamin and Patrick
M. Hayden, Physical Review A, vol 64, 030301(R) (2001)