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AIP conference proceedings

Quantum Tic-Tac-Toe as Metaphor for Quantum Physics

Allan Goff

Published Jan 1, 2004• 4 citations
Publisher

Abstract

Quantum Tic‐Tac‐Toe is presented as an abstract quantum system derived from the rules of Classical Tic‐Tac‐Toe. Abstract quantum systems can be constructed from classical systems by the addition of three types of rules; rules of Superposition, rules of Entanglement, and rules of Collapse. This is formally done for Quantum Tic‐Tac‐Toe. As a part of this construction it is shown that abstract quantum systems can be viewed as an ensemble of classical systems. That is, the state of a quantum game implies a set of simultaneous classical games. The number and evolution of the ensemble of classical games is driven by the superposition, entanglement, and collapse rules. Various aspects and play situations provide excellent metaphors for standard features of quantum mechanics. Several of the more significant metaphors are discussed, including a measurement mechanism, the correspondence principle, Everett’s Many Worlds Hypothesis, an ascertainity principle, and spooky action at a distance. Abstract quantum systems also show the consistency of backwards‐in‐time causality, and the influence on the present of both pasts and futures that never happened. The strongest logical argument against faster‐than‐light (FTL) phenomena is that since FTL implies backwards‐in‐time causality, temporal paradox is an unavoidable consequence of FTL; hence FTL is impossible. Since abstract quantum systems support backwards‐in‐time causality but avoid temporal paradox through pruning of the classical ensemble, it may be that quantum based FTL schemes are possible allowing backwards‐in‐time causality, but prohibiting temporal paradox.

DOI
10.1063/1.1649685

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Quantum Tic‐Tac‐Toe is presented as an abstract quantum system derived from the rules of Classical Tic‐Tac‐Toe. Abstract quantum systems can be constructed from classical systems by the addition of three types of rules; rules of Superposition, rules of Entanglement, and rules of Collapse. This is formally done for Quantum Tic‐Tac‐Toe. As a part of this construction it is shown that abstract quantum systems can be viewed as an ensemble of classical systems. That is, the state of a quantum game implies a set of simultaneous classical games. The number and evolution of the ensemble of classical games is driven by the superposition, entanglement, and collapse rules. Various aspects and play situations provide excellent metaphors for standard features of quantum mechanics. Several of the more significant metaphors are discussed, including a measurement mechanism, the correspondence principle, Everett’s Many Worlds Hypothesis, an ascertainity principle, and spooky action at a distance. Abstract quantum systems also show the consistency of backwards‐in‐time causality, and the influence on the present of both pasts and futures that never happened. The strongest logical argument against faster‐than‐light (FTL) phenomena is that since FTL implies backwards‐in‐time causality, temporal paradox is an unavoidable consequence of FTL; hence FTL is impossible. Since abstract quantum systems support backwards‐in‐time causality but avoid temporal paradox through pruning of the classical ensemble, it may be that quantum based FTL schemes are possible allowing backwards‐in‐time causality, but prohibiting temporal paradox.
Quantum tic-tac-toe was developed as a metaphor for the counterintuitive nature of superposition exhibited by quantum systems. It offers a way of introducing quantum physics without advanced mathematics, provides a conceptual foundation for understanding the meaning of quantum mechanics, and is fun to play. A single superposition rule is added to the child’s game of classical tic-tac-toe. Each move consists of a pair of marks subscripted by the number of the move (“spooky” marks) that must be placed in different squares. When a measurement occurs, one spooky mark becomes real and the other disappears. Quantum tic-tac-toe illustrates a number of quantum principles including states, superposition, collapse, nonlocality, entanglement, the correspondence principle, interference, and decoherence. The game can be played on paper or on a white board. A Web-based version provides a refereed playing board to facilitate the mechanics of play, making it ideal for classrooms with a computer projector.
This paper surveys a research effort to explore the potential for quantum nonlocality to support superluminal signaling. The emphasis is on the issue of temporal paradox. Through the metaphor of Quantum Tic-Tac-Toe, it is shown how superposition leads to a Many Worlds variant called the Classical Ensemble. As a quantum system evolves, superposition branches off new classical realities, but contradictory ones are pruned from the ensemble. The pruning results in nonseparable states and thus to entangling of the quantum system. When these entanglements become cyclic, they introduce nonlinearities into the quantum system that result in self-collapse. Since cyclic entanglements are in general self-referential in only one basis, collapse occurs only in the self-referential basis. Cyclic entanglements are thus a viable candidate as the measurement mechanism. Collapse is made consistent with relativity by introducing the symmetric spacetime interval. The resulting closed loops in spacetime entangle the near future with the recent past so collapse of quantum systems offers a potential means for achieving superluminal signaling (FTL) and something very like time travel. The pruning in the classical ensemble acts as the censor mechanism that prevents temporal paradox while allowing backwards-in-time causality. This removes the logical objections to FTL and time travel.
The learning of quantum mechanics is contingent upon an understanding of the physical significance of the mathematics that one must perform. Concepts such as normalization, superposition, interference, probability amplitude, and entanglement can prove challenging for the beginning student. Several class activities that use a nonclassical version of tic-tac-toe are described to introduce several topics in an undergraduate quantum mechanics course. Quantum tic-tac-toe (QTTT) is a quantum analogue of classical tic-tac-toe (CTTT) and can be used to demonstrate the use of superposition in movement, qualitative (and later quantitative) displays of entanglement, and state collapse due to observation. QTTT can be used to aid student understanding in several other topics with the aid of proper discussion.
We propose a quantum version of Tic-Tac-Toe which accurately reflects the inherent probabilistic nature of the measurement principle in quantum mechanics. We then formulate a quantum strategy which allows a quantum player to consistently win over a classical player, with a certain probability. This result can be seen as another proof of the superior computational power of a quantum system with respect to a classical one. Our investigation also reveals that the non-determinism and complexity introduced by the principles of quantum mechanics into even the most simple games make brute-force strategies considerably more difficult to implement. Consequently, games in which machines have gained the upper hand over humans may be made fair again by upgrading them to a quantum level.
*Special Relativity places severe constraints on any possibility for faster-than-light (FTL) phenomenon. There are four hopes for FTL; warp drive via General Relativity Theory (GRT), wormholes via GRT, tachyons, and quantum mechanics. This talk focuses on why quantum mechanics makes the list. The nonlocal nature of matter demonstrates correlations between distant events that cannot be explained by current concepts of causality. For two classical events to be correlated, either one causes the other, or both possess a common cause in their past light cones. Neither kind of causality can explain the correlations observed in entangled quantum systems. Reality seems to support a third mysterious type of causality, with an essential spacelike nature, but direct spacelike causality should be impossible as it would lead to temporal paradox. The two concepts of classical causality, timelike causes and common cause, are presented along with a review of the EPR experiment showing that something like spacelike causality exists between the distant parts of an entangled quantum system. Bell’s Inequality shows that the EPR correlations cannot be attributed to common causes. Gisin’s EPR-Relativity experiment shows that timelike causes cannot explain the EPR correlations either. The metaphor of Quantum Tic-Tac-Toe is used to show a third type of causality applicable only to quantum situations. This third type of causality is an indirect spacelike cause acting through multiple classical realities, permitting FTL but prohibiting temporal paradox. I. Introduction HIS paper considers the issue of causality in entangled quantum systems. The Background section reviews the two classical notions of causality, EPR type entanglements, and Special Relativity. The Common Cause section reviews Bell’s Inequality theorem and shows how this type of causality cannot explain the correlations seen in entangled systems. The Einsteinian Causality section reviews Gisin’s experiment demonstrating that time-like causality cannot explain these correlations either. Gisin’s conclusion is that quantum systems support a new mysterious 3 rd type of causality. The Quantum Causality section introduces a situation in Quantum Tic-Tac-Toe that exhibits this same kind of causality
In this paper, we perform a minimalistic quantization of the classical game of tic-tac-toe, by allowing superpositions of classical moves. In order for the quantum game to reduce properly to the classical game, we require legal quantum moves to be orthogonal to all previous moves. We also admit interference effects, by squaring the sum of amplitudes over all moves by a player to compute his or her occupation level of a given site. A player wins when the sum of occupations along any of the eight straight lines we can draw in the 3 × 3 grid is greater than or equal to 3. We play the quantum tic-tac-toe first randomly, and then deterministically, to explore the impact different opening moves, end games and different combinations of offensive and defensive strategies have on the outcome of the game. In contrast to the classical tic-tac-toe, the deterministic quantum game does not always end in a draw. In contrast also to most classical two-player games of no chance, it is possible for player 2 to win. More interestingly, we find that player 1 enjoys an overwhelming quantum advantage when he opens with a quantum move, but loses this advantage when he opens with a classical move. We also find the quantum blocking move, which consists of a weighted superposition of moves that the opponent could use to win the game, to be very effective in denying the opponent his or her victory. We then speculate what implications these results might have on quantum information transfer and portfolio optimization.