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.
Spice up game theory with a dash of quantum mechanics - the strange principles that govern the behaviour of molecules, atoms and subatomic particles-and lose-lose contests can become win-win. In a classical game, players must choose between cooperating and betraying. But, paradoxically, quantum mechanics allows players to do both at the same time. What's more, a spooky quantum connection called entanglement means each player's choice can secretly affect everyone else's.

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.
Such quantum games are not just esoteric exercises. They could form part of the longed- for quantum technologies of tomorrow, such as ultra-fast quantum computers. They might even help traders construct a crash resistant stock market. And quantum games could provide new insights into puzzling natural phenomena such as high-temperature superconductivity.

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.
But, Hayden and Benjamin say, this unpleasantness can be avoided if you play the game quantum-mechanically. They studied a three-player Prisoners' Dilemma game where each player had a "qubit". This is a quantum bit, rather like a computer's zero or one binary digit, but with an important difference: a qubit can be in two different states at the same time.

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.
Benjamin and Hayden imagined that the three players start with their qubits pointing down, representing their supposed solidarity-regardless of whether they intend to stick together They then allow their qubits to be entangled. This creates a link between them and puts the group in a superposition that will yield "all down" or "all up" if the qubits are measured. Only one of the qubits needs to be measured: if one is found to be pointing up, then entanglement means the other two instantaneously take on the "up" state, even if they've been carted off to the other side of the prison.

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".
Once the qubits are entangled, each player adjusts the state of their qubit according to their intended tactics, either leaving it as it is to remain true to the others, flipping it to betray them, or doing some combination of both at the same time. Finally, the entangling procedure is reversed, and the jailers measure the state of the qubits. It as above, the first player flips and the other two do nothing, then the measurement will show either the first player's electron pointing up and the other two pointing down, or theirs pointing down and the other two pointing 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.
In the classical game, selfishness makes the players betray each other, no matter how much they talk it over beforehand. All three go to jail for a long time. They can do the same in the quantum game, although it would not be the rational thing to do. They can do much better because superposition and entanglement provide far more appealing outcomes. And no one player can improve their own fate by changing their move alone. "The ultimate fates of the players are entwined in a way that they're not in a classical game," Hayden says.

Short stretch
The three do best when one does nothing, another betrays, and the third enters a half-and-half superposition. The prisoner who does nothing and the one who betrays both receive light sentences, and the one who does the combination goes free. Remarkably, even though the pay-off is not the same for all three players, there is no better option for any one of them to take in this quantum prison. So it's just a matter of deciding who plays which move, perhaps by drawing straws or cutting cards. Once that's determined, everyone does best to play along.

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.
In Meyer's game, Captain Jean-Luc Picard and Q, the omnipotent alien, agree to flip a coin to decide who gets control of the starship Enterprise. They put the coin heads-up in a box into which neither can see. First Q reaches into the box to manipulate the coin, then Picard reaches in to either flip it or leave it as it is. Finally, Q manipulates the coin once more. They then open the box, and if the coin shows tails, Picard wins.

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.
Q uses his first move to put the coin in a half-and-half superposition of heads and tails. Picard then either leaves the coin alone or flips it. But flipping the coin simply exchanges heads for tails and vice versa, leaving the coin in the same heads-and-tails state, much as interchanging black and white squares leaves a chequerboard a chequerboard. After Picard has made his completely ineffectual move, Q simply performs the inverse of his first move. This returns the coin to the original heads-up state.


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.
Meyer's games may sound like entertaining but useless quantum parlour tricks, but the lessons they teach us could be crucial in the emerging field of quantum computing. Indeed, musings about the guess-a-number scenario have already helped reveal a method for factoring very large numbers. This is the key step in cracking coded messages, one of the major tasks awaiting these machines. And in Meyer's games, Picard can only make classical moves, while Q's are quantum, so their contests also resemble the interactions between a non-quantum human and a quantum computer that can perform super-fast computations. Such games hint at how best to program a quantum computer. "Thinking about specific games and quantum strategies may lead to techniques for new quantum algorithms," Meyer says.

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.
Quantum moves don't always improve a game's outcome. It depends on how many players are involved, and what they are allowed to do. Two years ago, Jens Elsert of Imperial College, London, working with Martin Wilkens of the University of Pots-dam and Maciej Lewenstein of the University of Hanover, pioneered the quantum approach to prisoner games. Using just two prisoners, they showed that they could find a better solution to the dilemma than the back-stabbing scenario if both played a particular quantum move.

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.
Even if quantum mechanics isn't the answer to all such dilemmas, it could be useful. Researchers are applying it to games that simulate evolution, stock markets-and even game shows. Last year Adrian Flitney and Derek Abbott from Adelaide University investigated a quantum version of the Monty Hall game show. Here, a contestant called Bob chooses which of three doors he thinks is hiding the prize, and then Monty Hall, the host, stirs up the odds by opening one of the wrong doors and asking the contestant if he'd like to switch his choice. Classically-and counterintuitively-Bob's best bet is to switch, but that still only gives him a 66 per cent chance of finding the prize. If the contestant can use entanglement, however, he can win every time.

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 to do."
It may even be possible to use techniques from quantum cryptography to construct a quantum stock market in which traders encode their decisions to buy or sell in qubits. In such a market, entanglement might make traders cooperate and avoid crashes-the equivalent of everyone losing in game theory.

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 predicted.
Even if quantum games never prove technologically useful, these experiments might at least tell us about how the world works on the quantum level, Hayden says. "The more interesting possibility would be to look into nature and see it playing a quantum game," he says. "I don't think anyone has done that yet."

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.

Further Reading

"Multiplayer quantum games" by Simon C. Benjamin and Patrick M. Hayden, Physical Review A, vol 64, 030301(R) (2001)
"Quantum games and quantum strategies" by Jens Elsert, Martin Wilkens and Maciej Lewenstein, Physical Review Letters, vol 83, p 3077 (1999)
"Quantum strategies" by David Meyer, Physical Review Letters, vol 82, p 1052 (1999)
"Experimental realization of the quantum games on a quantum computer" by Jiangfeng Du and others,





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New Scientist 5/1/2002 File Info: Created 19/10/2002 Updated 17/12/2017 Page Address: