A Quantum Game

From wiki.quantumlab.net

Contents 1 Introduction 2 A 3 Vs. 3 Game 3 A Classical Strategy — Score 75% on Average 4 A Quantum Strategy — Score 100% Every Time 5 Objective vs. Subjective Probability 6 The Nature of Our World? Pick Any Two 6.1 Realism & Free Will 6.2 Realism & Locality 6.3 Locality & Free Will 7 References

Introduction

 The more success the quantum theory has, the sillier it looks.
 
             —Albert Einstein - 1921 Nobel Laureate

 If quantum mechanics hasn't profoundly shocked you, you haven't understood
 it yet.
 
             —Niels Bohr - 1922 Nobel Laureate

 The mathematical framework of quantum theory has passed countless
 successful tests and is now universally accepted as a consistent and
 accurate description of all atomic phenomena.
 
             —Erwin Schrödinger - 1933 Nobel Laureate

 I think I can safely say that no one understands quantum mechanics.
 
             —Richard Feynman - 1965 Nobel Laureate

Ever since quantum theory took its current form in the early 1930's, there has been controversy over what this physical model of reality says about the nature of our world beyond its unmatched accuracy in modeling our observations of the world. The issues we still face are not just problems of interest to physicists, but rather, these issues breach philosophical questions that would be of concern to anyone with interest in the fundamental nature of reality.

What follows is a description of a simple game^[1] played by two competing 3-player teams in which it will be shown that a team that uses a particular strategy based upon quantum principles will consistently beat a team that instead uses mainstream logic. When we take a closer look at the quantum phenomena that provide the winning strategy, we will see that basic assumptions of realism, locality, and free will are challenged, and that at least 1 of these 3 beliefs must be abandoned from any consistent worldview.

A 3 Vs. 3 Game

The game consists of 2 competing 3-player teams, each team playing 100 rounds. In each round, a team as a whole is either awarded 1 point or no points. At the end the team with the most points wins.

Prior to the beginning of the game, the teams start off at home base, a central meeting place where the teams can plan their strategy, and where the final scores will be tallied up when the game is over. When the game begins each of the 3 players on a team are split up and taken to 3 different pre-selected playing stations in space, possibly as distant as different stars. At each station, each player will meet with a game attendant. The 100 rounds will then commence, each round beginning at the same time, at 10-second intervals.

A round consists of:

1. The attendant will display a single sign to the player, with either an X or a Y on it. The 3 attendants as a whole are limited to showing the players either 3 X's, or 1 X and 2 Y's.
2. The player has 10 seconds to respond with either a thumbs up or a thumbs down.

Each player's 100 responses are recorded and brought back to home base.

In each round, a team will score 1 point in either case:

• When the attendants display 3 X's, the team responds with an odd number (1 or 3) of thumbs down.
• When the attendants display 1 X and 2 Y's, the team responds with an even number (0 or 2) of thumbs down.

Otherwise the team will not receive a point for that round.

Each of the 4 X and Y combinations (XXX, XYY, YXY, YYX) will occur 25 times each out of the 100 rounds. The game attendants will secretly plan ahead of time what order these combinations will be shown without letting the players know prior to the rounds in which they are shown.

There are no restrictions on what the players can do during each round, or what equipment they can bring with them, as long as they give their responses within the 10-second time requirement of each round. Due to the distances between each other, they cannot communicate with one another via radio signals in time for the communication to be received prior to giving an answer.

For example, one strategy would be: "If you are shown an X, give thumbs up. If Y, give thumbs down." This works in the cases when the attendants display XYY, YXY, or YYX to the players, since that team will collectively give an even number of 2 thumbs down. However when the attendants show XXX that team will give 0 thumbs down, and thus not get a point. On average, this strategy would give the team 75 points. Is there a perfect strategy?

A Classical Strategy — Score 75% on Average

Let us attempt to find a perfect strategy by reasoning as follows:

1. Since the members of a team are unable to communicate with each other during the rounds, it seems we have no choice but to plan ahead of time what responses each team member will give to an X or Y.
2. Each person will have 2 instructions assigned to them. For example, A_X = and A_Y = means person A will respond with thumbs up if shown X, or thumbs down if shown Y. Similarly, B will be assigned B_X and B_Y, and C will be assigned C_X and C_Y.
3. An optimal strategy will assign to each of these 6 instructions a thumbs up or down in order to score a point in each of the 4 combinations XXX, XYY, YXY, and YYX according to the scoring rules.

Based on this general approach, there are 2⁶ = 64 possible strategies, each strategy assigning thumbs up or down to the 6 variables A_X, A_Y, B_X, B_Y, C_X, and C_Y. However we will see that none of these 64 strategies will give a perfect score. How do we see this? Count the total number of thumbs down in each of the 4 cases of XXX, XYY, YXY, and YYX. We'll count them in 2 different ways and get contradictory results:

	odd's even's even's even's
even's	odd's

• The first way is to directly total them in the first column. Each of the 6 variables A_X, A_Y, B_X, B_Y, C_X, and C_Y show up twice. Which ever of these are thumbs down are counted twice and thus the total is an even number.
• The second way is to look at the requirements of a perfect score. Out of A_XB_XC_X we want an odd number of thumbs down. Out of A_XB_YC_Y, A_YB_XC_Y, and A_YB_YC_X we want an even number of thumbs down for each. Odd + even + even + even = odd, so counting the number of thumbs down on the chart this way gives an odd number as the total.

Based on this contradiction we have no choice but to conclude that our original assumption is false. There does not exist an assignment of thumbs up or down to the 6 variables A_X, A_Y, B_X, B_Y, C_X, and C_Y for a team to get a perfect score.

It appears that a perfect strategy does not exist.

Or does it...?

A Quantum Strategy — Score 100% Every Time

What follows is a calculation to show how a team using a particular quantum strategy can score 100 points. The purpose of this section is not to give general readers a lesson in the fundamentals of quantum physics and how this result follows, but rather, to give readers a glimpse of the internal mathematics of quantum theory that lead to these bizarre predictions. For the sake of space there will necessarily be gaps in the explanation where some things will need to be taken for granted. But don't worry, even physics professors have learned to simply take some things for granted in quantum theory.

First, the team agrees upon three mutually perpendicular +X, +Y, and +Z directions and maintain these directions throughout the course of the game (e.g. relative to external galaxies.) Next the team entangles the spin states of 3 spin-½ particles together forming a GHZ state (named after Greenberger, Horne, and Zeilinger^[2]) . These 3 particles are distributed to each team member, and are used in the same round. When a team member is shown an X, they measure their particle's spin state in the +X direction. When shown a Y, they measure in the +Y direction. If their measurement results in spin up, they give a thumbs up. If spin down, then give thumbs down.

Since the 3 particles are entangled, we can view the collective measurements performed by the individual players as combined measurements of the whole 3-particle system. By using the following equalities relating the spin states along different axes for single-particles states

we substitute these into the GHZ state to see the possible outcomes in each of the 4 cases XXX, XYY, YXY, YYX:

	.

From this we see that if the game attendants show XXX to the team, they will measure with equal probability , , , or . In any of these cases, there are an odd number of down arrows, leading to an odd number of thumbs down, and thus they will receive a point. If the attendants show 1 X and 2 Y's, the team will measure with equal probability , , , or . Likewise, they will always show an even number of thumbs down, and thus receive a point. In total, the team will end the game with 100 points.

Objective vs. Subjective Probability

Prior to getting to what these results mean about our world, let's take a moment to put these issues into context.

A central issue ever since the early days of quantum theory has been the meaning of these quantum probabilities. Prior to this, within the framework of Newtonian physics, all probabilities arose out of our ignorance of the complete state of a physical system. For example, Alice can flip a coin, catch it in her hand and know with 100% certainty what it is, but Bob from whom the coin in concealed can only say with 50-50 certainty heads or tails. It is Bob's ignorance of the complete state of the Alice-coin system that gives rise to his probability. We call this a subjective probability. In a Newtonian model in which all particles move under the law of F=ma and there is no limit to the precision with which we can know the positions and speeds of all particles and the forces acting on them, all probabilities are subjective, since in principle the state evolution and our observations can be predicted insofar as we have knowledge of the system. In contrast as the branches of statistical and thermal physics arose that included probabilities in them, many physicists then and still do consider these fields to be non-fundamental (but important nonetheless) for the reason that they include subjective probabilities which allow us to effectively deal with multi-particle systems like we do on a day-to-day basis. However even these statistical methods still have F=ma as the underlying deterministic law of motion that transform the lack of detailed atom-by-atom knowledge of a system into useful statistical summaries of the system.

Quantum physics on the other hand includes probabilities as a fundamental feature. For example, when we write as the spin state of a particle, this implies that if we measure the spin state in the +Z direction, we will with 100% certainty get spin up as a result. When we measure in the +X direction, we will get spin up half the time, and spin down the other half. To calculate this we take the coefficient of the possible resulting state and square the magnitude to get the probability. The coefficient of is 1, hence 100% certainty of getting spin up when measured in the +Z direction, but the coefficient of is hence we will get spin up of the time when measured in the +X direction.

Many physicists, including Einstein, felt that these were again subjective probabilities like those found in statistical mechanics. As such, he called quantum theory an incomplete theory — it is incomplete until we find the true mechanism behind these probabilities, so that we can make precise predictions in principle, and handle our incomplete knowledge of a system separately.

One might be asking themselves at this point, why even have this quantum formalism at all? Are we saying that Bob should write his knowledge of Alice's coin as ? Are we just calling our pet cat Felis silvestris catus? Without taking too great of a detour, the answer is that this quantum formalism has a large overlap with a mathematical structure known as a complex Hilbert space. Within this framework a great number of previously inexplicable phenomena was accounted for, perhaps the greatest of which was the structure of the hydrogen atom as being a flagship of quantum theory. Newtonian physics, which actually predicts the collapse of otherwise stable atoms all around us by predicting electrons spiraling into the nucleus, simply could not be adapted into these realms. It is within quantum theory's rigid mathematical framework with an extreme economy of principles with respect to the enormity of physical phenomena that provides the foundation from which we extrapolate into thought experiments and games such as the above.

So do these quantum probabilities mask a deeper deterministic theory yet to be found? Or are the probabilities inherent to quantum physics truly objective probabilities — that the state is really in an objectively undetermined spin state with respect to the X axis — a quantum superposition of and — where either result is not only unknown, but objectively not predetermined to be found until a measurement along the X axis is made?

In essence, the question here is, are the probabilities in quantum physics subjective or objective? If they are objective, then there lies fundamental indeterminacy in the world. (Einstein opposed this possibility in his comment, "God does not play dice with the universe.") If they are subjective, then there should exist hidden variables that ultimately determine the outcome of quantum measurements, whose ignorance of which give rise to the statistical probabilities we find in quantum theory.

The Nature of Our World? Pick Any Two

What do these results say about our world? In this section we will first take an unstructured approach to interpreting the results so far, followed by a more systematic overview and conclusion.

If the probabilities in quantum theory are only subjective, in that there exist hidden variables that determine the outcomes of measurements before they occur, then each of the 3 particles used by the quantum team should have a pair of hidden variables associated with them that determine the outcome in each spin measurement on axes +X or +Y that is made on them. Particle A would have hidden variables A_X and A_Y both set to either or . Particle B would have B_X and B_Y, and particle C would have C_X and C_Y. However finding values for these hidden variables in order to achieve a perfect score is precisely what the first team above tried to do, but this proved to be impossible. None of the 64 possible combinations give a perfect score.

Does this mean we have proved that hidden variables don't exist, and that the probabilities in quantum theory are truly objective, fundamental components of the physical system, contrary to Einstein's desire for a deeper deterministic theory? Not quite; there is still the possibility that these entangled particles are remotely influencing each other when measurements on them are made. Due to the time and distances involved, any such influence must exceed the bounds of the speed of light. We will go into this in greater detail shortly.

But now, have we proven that either hidden variables don't exist, or that there exist super-luminal influences in the universe? Not quite; all along we have been assuming we have free will. What if the game attendants who put together the XXX, XYY, YXY, and YYX combinations in the 100 rounds are influenced by a greater physical law that synchronizes these measurement choices with the hidden variables in the particles so as to produce the statistical outcomes consistent with what we calculate using the present quantum theory?

Does this mean either hidden variables don't exist, or there are super-luminal effects, or we don't have free will? Not quite; one last possibility is perhaps the most literal interpretation of quantum theory. Basically quantum theory has two parts: (1) A deterministic equation (the Schrödinger equation) that describes how a system evolves in time, and (2) a probabilistic rule that gives the probabilities of what we will measure, upon which the system then collapses into that state when the measurement is made. When considering the above game, we cannot take an omniscient point of view — we are after all a physical system ourselves, and must witness the game from somewhere within it. Say we remain at home base as an observer. We do not directly witness the game being played, but rather will come to hear of the results after the game is over. This in itself is our measurement of the spin triplets. Our measurements on the spin states did not occur until we heard what the players responded with, as far as thumbs up or down. We made the measurement from home base, using the players as the measuring instruments, and until we learned of the results, the spin states, the players' thumbs and their subsequent histories remained in a quantum superposition until we collapsed them by our "measurement."

Let us now take a more organized survey of the various interpretations we just ran through. Using common definitions from the literature:

• Realism - The belief that physical systems are generally in the states that we are familiar with on a day-to-day basis independently of our awareness of them. Even though the quantum formalism may represent states as being in quantum superpositions of observable states, the realist believes that the actual state of the system is really in one observable state or another. A realist believes that the electron spins in the game already had an existing value of up or down prior to the players' measurements on them, whereas a non-realist may not.
• Locality - The belief that physical systems can influence each other only via local effects and not remote effects. All causes and effects that we are commonly aware of are local, e.g. only adjacent dominoes knock each other down, instead of knocking each other down with large gaps of empty space between them. Speaking to someone on a cell phone vibrates a mechanism in the phone that turns into electrical signals that get transmitted via electromagnetic signals that spread out in 3-dimensional space like ripples in a pond, that hit a cellular receiving station that transmits the signals through the phone system to the recipient, etc. Every step of the way each cause and effect relationship between physical components happen within microscopic distances of each other.
• Free Will - The belief that the apparent choices we make are truly choices, and that under identical circumstances we could have made an alternate choice.

We now consider in turn the nature of the universe when any two of these principles are assumed to be true. In particular we will see that the truth of any two implies the third is false.

Realism & Free Will

Let us assume the principle of realism, and that we have free will. Realism implies that the electron spins in the game are determined prior to measurement. But how can this be, since we have already shown that no determination of A_X, A_Y, B_X, B_Y, C_X and C_Y can lead to a score of 100? Though we said that the players could not communicate with each other, we did not rule out the possibility that the particles themselves could influence each other during the rounds. It is possible that via some unknown law of physics each particle can somehow detect how the other two particles are going to be measured and what the measurement outcomes are. Because the three particles are space-like separated during the same round, this influence that the particles exert on one another must exceed the speed of light. Not only that, but different frames of reference will see different players as the first one to make the measurement, as there is no absolute sense in which there is a "first" measurement made out of the three. This phenomenon is what Einstein and 2 colleagues described as "spooky action at a distance" when discussing these seemingly non-local effects. This conclusion that physics includes non-local phenomena are what many physicists including Einstein have been led to believe.

Realism & Locality

This scenario is similar to the previous (realism & free will) except now we cannot resort to a non-local explanation. Realism still dictates that the electron spins are determined prior to the measurement. How did free will enter into the previous case? The players have agreed to a strategy that requires no choices to be made on their part. Who in this game is making any choices? The answer is the game attendants. It is possible that a "conservation of information" law influences both the decisions by the game attendants as well as the outcomes of the spin measurements in such a way as to produce statistical distributions which experimentally coincide with the predictions of quantum theory.

Does this really seem so far-fetched? A projectile that is launched into space and explodes, no matter how complicated the intricacies of the explosion, will still obey the conservation of momentum, even if the explosion happened over billions of years and sentient life evolved amongst the debris. Nothing that the pieces could do, including the sentient life forms, would violate the law of the conservation of momentum. Perhaps there is a deterministic conservation law that is at play here?

Though this is a logical possibility, it is quite a stretch. The game attendents may order the XXX, XYY, YXY, YYX signs based upon any source of information. This implies that no matter what source of this information, be it the attendents' direct choices, flipping coins, time of atomic decays, or detections of high-energy particles from outer-space that are used to determine the order of the X and Y signs that are displayed to the players, these seemingly random sources of information would be fundamentally tied into the states of the electron spins prior to the start of the games in order for the spin measurements to come out as they do in order to score 100.

The old familiar Newtonian sense of physical determinism would be restored, but at the cost of creating conspiracy-like relationships between disparate physical systems in order to explain the statistical distributions that are measured, and leaving us with the task of finding what these mysterious underlying relationships are.

Locality & Free Will

This scenario is the simplest in terms of a physical model (the model being the present quantum theory), yet the most unintuitive out of the three scenarios and perhaps the most philosophically unsettling for many readers. Simply stated, we must accept that any interaction we have with the world, including the game players themselves, are measurements. This means that until we learn (measure) how a player responded to a game attendant in a particular round, their response is in a quantum superposition of thumbs up and down. The superposition of their electron spin states in that round interacted with their measuring device which put the measuring device into a quantum superposition, which led them to respond in a superposition of thumbs up and down, which resulted in a brain state in that player of a superposition of two memories of reporting thumbs up and down. This quantum superposition in their brain state remained until we met with them and asked how they responded in that particular round. This act of verbalizing (or writing, drawing, etc.) which memory they have of that round and us experiencing their communication is our act of measurement on both their electron and how they responded in that round. When we come to learn (measure) of how each of the players responded in a particular round, we are ultimately making a measurement at that point in time on the original GHZ spin triplet state, of which there are only four possibilities all of which provide a winning combination for that round.

Notice that at each step in this scenario locality remained intact.

When the player makes a measurement of their electron, doesn't the superposition state collapse at that point into one state or the other? Yes, if we are taking the viewpoint of the player. No, if we are taking the viewpoint of someone at home base. An omniscient viewpoint does not exist. One must select a point of view within the system and describe matters from that perspective.

This interpretation is also known as the Many Worlds Interpretation, Relative State Interpretation, and Correlation Interpretation of quantum mechanics.

References

1. ↑ This game was first described in a paper Variations on the Theme of the Greenberger-Horne-Zeilinger Proof by Lev Vaidman.
2. ↑ D. M. Greenberger, M. A. Horne, and A. Zeilinger, in Bell's Theorem, Quantum Theory and Conceptions of the Universe Kluwer Academic, Dordrecht, 1989.