Playing Phantom Square with Quantum Entanglement A Big Win

 

A joint team from the University of Science and Technology of China and Nanjing University used a hyperentanglement scheme for a traditional phantom square game and prepared photon pairs entangled in polarization and orbital angular momentum degrees of freedom, thus demonstrating in a resource-efficient way that quantum players can win all classical strategy queries simultaneously. The research results were published in Physical Review Letters under the title "Experimental Demonstration of Quantum Pseudosensing" [1].

 

 

01Quantum Magic Square and Quantum Pseudo-telepathy

 

Magic Square is a traditional Chinese game, which was commonly played in government offices and schools in the old days. It is a game in which natural numbers from one to several numbers are arranged in squares of several numbers each, so that the sum of several numbers in the same row, column and diagonal are equal.

 

By the 1990s, quantum physicists introduced the concept of quantum phantom squares.

 

There is a phenomenon in quantum mechanics called Quantum pseudo-telepathy: in certain Bayesian games with asymmetric information, players have access to shared physical systems in entangled quantum states and are able to execute strategies that depend on measurements on the entangled physical systems, in the same game as players who do not have access to the entangled quantum systems in the equilibrium, they are able to achieve higher expected gains in the equilibrium compared to what a player without access to the entangled quantum system could achieve in any mixed strategy Nash equilibrium of the same game.

 

The induction is usually used as a thought experiment to demonstrate the non-deterministic character of quantum mechanics, but now scientists have demonstrated it to us experimentally.

 

02Mermin-Peres phantom game: classical v.s. quantum

 

Mermin Peres Phantom Square game. The referee randomly sends x and y to Alice and Bob respectively. alice and Bob respond with three numbers (+1 or -1) in one row and one column respectively. If the overlapping entries are the same, they will win the game.

 

Two quantum physicists, Asher Peres and David Mermin, independently designed a game called the Mermin-Peres Phantom Square Game (MPMS) in 1990. It involves two participants (called Alice and Bob, in the tradition of quantum mechanical thought experiments) who must fill out a "phantom square": a three-by-three grid of numbers, with each grid element assigned a value of +1 or -1. In each round, the referee (Charlie) sends Alice a random In each round, the referee (Charlie) sends a random row to Alice and then a column to Bob (there are nine such combinations of rows and columns). Players must tell Charlie to place +1 or -1 values in their three grid spaces. As with any phantom-square challenge (such as Sudoku), the sum of each row and column must satisfy a specific constraint: here, the product of all entries in a row must equal +1, and the product of all columns must be -1. Alice and Bob win a round if they both specify the same value for the grid element where the columns and rows overlap.

 

Classically, it is not possible to win all rounds. Because even if Alice and Bob guess well each time, there will inevitably be one round for each completed square, and their assignments must conflict. The best they can do is to win eight out of every nine games.

 

Classic MPMS Phantom Square game

 

But now suppose that Alice and Bob can use a quantum strategy: instead of assigning +1 or -1 values to each grid element, they assign it a pair of quantum bits (qubits), each with +1 or -1 when measured. the value each player gives to a particular grid element is determined by measuring the two quantum bit values and finding the product of the pair of quantum bits. Now, the classical conflict can be avoided because Alice and Bob can derive different values from the same two quantum bits, depending on how they make their measurements. There is a special measurement strategy that ensures the winning criterion for any given round, i.e., that the product of Alice and Bob's three entries is +1 and -1, respectively, and that all nine permutations of rows and columns are satisfied.

 

However, this strategy has a drawback. In order to make the correct measurements, Alice and Bob need to know which of their three grid elements overlap with the other players they need to coordinate with. But in MPMS, this is not a problem, because they measure on the same three quantum bit pairs in sequence. This means that the pair that reaches Bob has the imprint of how Alice has already measured these quantum bits: they can pass information to each other.

 

Quantum MPMS Phantom Cube Game

 

02Quantum pseudo-telepathy: distribution using entanglement of information

 

In 1993, it was shown [2] that MPMS can be used to demonstrate a quantum phenomenon called "contextuality"; contextuality means that the result of a quantum measurement may depend on the way the measurement is made. A set of classical measurements in a system will give the same result regardless of the order in which they are made; however, this is not always the case for quantum measurements; in MPMS, intertextuality arises from the fact that a measurement of a given pair of quantum bits may give a different result depending on the measurements of two other pairs of quantum bits.

 

But what if we forbid any communication in MPMS, assign different quantum bit pairs to Alice and Bob, and say that they cannot negotiate about how to measure?

 

If each player makes a correct guess about the other player's behavior, it is guaranteed to win nine games out of nine. But in a study published in 2005 [3], a quantum team at the University of Montreal showed that players could use quantum principles to guarantee victory in each round, even without communication, in what is called quantum pseudo-telepathy.

 

This strategy involves entangling one of the two quantum bit pairs sent to Alice or Bob with the corresponding quantum bit used by the other player. The entangled particles have related properties, so that if Alice measures the value of her particle, it will also fix the value of Bob's particle. For example, two entangled quantum bit particles may be anti-correlated: if Alice's quantum bit is found to have a value of +1, then Bob's must be -1; until Alice's quantum bit is measured, there is no way to say which value it has: it may be +1 or -1, but Bob's quantum bit is always the opposite.

 

Importantly, the property of entanglement between a pair of particles is called "pseudo-telepathy", which means that it is not shared between any one particle, but between two particles: even if the particles are far apart, the entangled pair must be considered as a single, non-local object. In 2001, Adán Cabello, a quantum theorist at the University of Seville, proposed the same basic idea of winning quantum games in a game he called "all-or-nothing" [4], which was later shown to be equivalent to the non-local ("pseudo-telepathy") MPMS.

 

How to use quantum pseudo-telepathy to achieve victory?

 

Some researchers consider entanglement to be the most fundamental aspect of quantum mechanics, which implies a kind of information sharing between particles. This is the key to using entanglement for quantum pseudo-telepathy: Alice and Bob do not have to exchange information to coordinate their actions because the necessary information is already shared among the particle pairs. For example, in quantum computing, entanglement between quantum bits is often a resource that provides a shortcut to finding problems that cannot be solved by classical computers.

 

03Achieving the limits of quantum information sharing: intertextuality + nonlocality

 

Now, Chinese researchers have devised a new experiment: implementing a complete, consistent quantum pseudo-telepathy protocol and ensuring victory in every round of the two-player quantum race of the MPMS game. Winning the game requires players to coordinate their actions without exchanging information with each other. When used judiciously, quantum pseudo-telepathy allows players to win each round of the game; a perfect performance that would otherwise be impossible. The experiment, conducted using laser photons, explores the limits of how much quantum mechanics allows information to be shared between particles.

 

Ideally, Alice and Bob would prepare many sets of four quantum bits, each consisting of two pairs of entangled bits, before the game begins. alice would get one of these pairs, and Bob would get the other. However, making two pairs of entangled photons for each round of the game was a huge challenge, the researchers said. For one thing, in their device, producing even one pair of entangled photons is only a low-probability event, so producing two pairs at once is extremely unlikely. And detecting two entangled pairs at once, as required by a pseudo-telepathic MPMS, is almost impossible for this optical implementation.

 

The experimental team prepared single photon pairs and entangled them independently with two properties: their polarization state and a property called orbital angular momentum. The photons were contained in ultrashort laser pulses lasting only 150 femtoseconds and entangled by two so-called nonlinear optical crystals. A thin barium borate plate first splits a photon into two lower-energy, angular-momentum-dependent photons. Then, by feeding them into a crystal of yttrium-vanadium compound, they are also entangled by their polarization.

 

Experimental setup. (a) Generation of two-photon polarization states and OAM superentangled states. (b) Polarization interference prism (PIP). Implementation of a polarization OAM control non (CNOT) gate for single photons, with polarization as the control quantum bit and OAM as the target quantum bit. The HWP with WiFi symbol represents the HWP that changes the angle according to QRN (quantum random number).(c) The polarization measurement (d) reads the two-channel OAM by converting the OAM coherently to polarization through an exchange gate (inset).(e) The quantum random number sends 0, 1, and 2 to Alice and Bob, where 0, 1, and 2 represent the first, second, and third row or column in the game, respectively. The OAM is used as the target quantum bit.

 

To demonstrate a nearly 100 percent success rate, the researchers needed to improve their detection efficiency so that almost none of the entangled photons would go undetected. Even then, the theoretical limit could not be reached precisely in the experiment: but the researchers were able to show that they could win each round with a probability of 91.5% to 97%. This means that out of a total of 1,075,930 rounds, 1,009,610 reliably beat the classical eight out of nine limit.

 

"The pseudo-telepathic MPMS game exploits the strongest correlation between particles that quantum mechanics may offer." Kai Chen, a professor at the University of Science and Technology of China and one of the authors of the experiment, said, "Our experiment explores how extreme quantum correlations can be generated between particles. If these correlations were even stronger, they would imply faster-than-light information exchange, which a series of other independent experiments have shown to be impossible."

 

Adán Cabello said [5] that this work, besides being an experimental feat, shows a new problem of quantum rules made possible by the simultaneous mobilization of two sources of quantum dominance: one related to nonlocality and the other to intertextuality. "Studying these two effects simultaneously should allow physicists to explore the connections between them more rigorously. More importantly, each of these sources could, in principle, be used for different purposes in quantum processing, enhancing their versatility. "

 

For example, nonlocality could be used for secret communication [using quantum cryptography], while intertextuality could be used for quantum computing. In this case, for example, Alice could establish secure communication with Bob while performing computations with Charlie faster than the classical approach allows.

 

Reference links：

[1]https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.129.050402

[2]https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.65.803

[3]https://link.springer.com/article/10.1007/s10701-005-7353-4

[4]https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.87.010403

[5]https://www.scientificamerican.com/article/researchers-use-quantum-telepathy-to-win-an-impossible-game/