Quantum Game Beats Classical Odds Thermodynamics Implications

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1 Entroy 05, 7, ; doi:0.3390/e77645 Article OPEN ACCESS entroy ISSN Quantum Game Beats Classical Odds Thermodynamics Imlications George Levy Entroic Power Cororation, 3980 Del Mar Meadows, San Diego, CA 930, USA; Tel.: ; ax: Academic Editor: Antonio M. Scarfone Received: August 05 / Acceted: 6 October 05 / Published: 9 November 05 Abstract: A quantum game is described maing use of coins embodied as entangled ermions in a otential energy well. It is shown that the odds are affected by the Pauli Exclusion Princile. They deend on the elevation in the energy well where the coins are selected, ranging from being a certainty of winning at the bottom of the well to being near classical at the to. These odds differ maredly from those in a classical game in which they are indeendent of elevation. The thermodynamics counterart of the quantum game is discussed. It is shown that the temerature of a Maxwellian gas column in a otential energy gradient is indeendent of elevation. However, the temerature of a ermion gas is shown to dro with elevation. The game and the gas column utilize the same comonents. When ermions are used, a shifting of odds is roduced in the game and a shifting of inetic energy is roduced in the thermodynamic exeriment, leading to a sontaneous temerature gradient. Keywords: quantum game; second law; entroy; ermi Dirac; Maxwell Boltzmann; ermion; Boson; temerature gradient; statistical mechanics; quantum mechanics. Introduction The idea that a quantum game strategy can beat a classical one is not new. litney and Abbott [] summarize a coin tossing game first described by Meyer [] as follows: Meyer considered the simle game enny fli that consists of the following: Alice reares a coin in the heads state, Bob, without nowing the state of the coin, can choose

2 Entroy 05, to either fli the coin or leave its state unaltered, and Alice, without nowing Bob s action, can do liewise. inally, Bob has a second turn at the coin. The coin is now examined and Bob wins if it shows heads. A classical coin clearly gives both layers an equal robability of success unless they utilize nowledge of the other s sychological bias, and such nowledge is beyond analysis by standard game theory. To quantize this game, we relace the coin by a two state quantum system such as a sin one-half article. Now Bob is given the ower to mae quantum moves while Alice is restricted to classical ones. Can Bob rofit from his increased strategic sace? Let 0> reresent the heads state and > the tails state. Alice initially reares the system in the 0> state. Bob can roceed by first alying the Hadamard oerator: H = () utting the system into the equal suerosition of the two states: ( 0 >+ > ). Now Alice can leave the coin alone or interchange the states 0 > and >, but if we suose this is done without causing the system to decohere either action will leave the system unaltered, a fact that can be exloited by Bob. In his second move he alies the Hadamard oerator again resulting in the ure state 0 > thus winning the game. Bob utilized a suerosition of states and the increased latitude allowed him by the ossibility of quantum oerators to mae Alice s strategy irrelevant, giving him a certainty of winning. A.P. litney and D. Abbott In Meyer s game, the coin is maniulated sequentially by the layers and the quantum layer (Bob) utilizes the Hadamard oerator to ensure that the outcome is always tail 0>. Meyer does not describe a real world imlementation for his game. This aer describes a hysical embodiment for a coin tossing game by reresenting a coin as a ermion inside a otential energy well and shows that a layer can maniulate the outcome of the game simly by changing her article s elevation in the well. In addition, the thermodynamics imlications of this game are also discussed in which the temerature of a non-maxwellian gas is also shown to deend on elevation.. Classical Coin Game Thought Exeriment Consider the following classical game layed by a layer, say Alice, against the House : The dealer tosses a large number of coins but their final states are hidden until the layer selects two coins. Before uncovering them, she calls her bet: matched or mismatched. as shown in igure. Matching coin combinations include (Head, Head) and (Tail, Tail). Mismatching coins include (Head, Tail) or (Tail, Head). In a classical game the odds are obviously equal since (Head, Head) = (Tail, Tail) = (Head, Tail) = (Tail, Head) = 0.5. The entroy corresonding to maing a bet is: S = 0.5log = 0.5log 4 = 0.5 log = 0.5shannons 0.5 ()

3 Entroy 05, Playing such a game, Alice has a 50% chance of winning. Can she imrove her odds by using a quantum strategy? igure. In a classical environment the selected coins are uncorrelated, and could be (Head, Head), (Head, Tail), (Tail, Head) or (Tail, Tail). Alice wins 50% of the time. 3. Quantum Coin Game Thought Exeriment Let us assume that coins are actually ermions with Head having, for examle, a hysical reresentation of sin u, and Tail, of sin down. urthermore, let s say that the coins are entangled in a otential energy well characterized by a number of discrete quantum states. In addition, let each coin ossess energy, the lowest value being the ground state. This coin arrangement is exemlified by the well-nown electronic configuration in an atom. To mae it more interesting, we can assume that the atom is not in the ground state, so that some low energy levels are unoccuied. The general descrition of such system follows the ermi Dirac distribution. Since the coins are ermions, their states are constrained by the Pauli Exclusion Princile which requires that they are different as shown in igure. igure. When the coins are ermions in the ground state and constrained in a otential energy well, the Pauli Exclusion Princile requires them to be different allowing Alice to win all the time. The lowest energy levels are the most crowded and include two anti-arallel states, both states being occuied, one coin being Head (sin u) and the other coin being Tail (sin down). or the sae of simlicity let us assume the ground state where the coins are erfectly entangled such that if both coins are selected from the ground state they form a mismatched air. At the to of the energy well, the states are less crowded than at the bottom and therefore it is ossible to find unoccuied states, or coins occuying anti-arallel states as well as arallel states. or examle

4 Entroy 05, two coins can be Head (sin u), or two coins can be Tail (sin down). or the sae of simlicity we shall also assume that a coin selected from the to level of the energy well is unentangled from the other coins such that when combined with another coin it would form a matched or mismatched air with equal robability. Therefore, Alice can increase her chances of winning, simly by selecting low energy coins and calling a mismatch. Selecting from the bottom of the energy well, the entroy for each lay is: S = log () = 0shannons (3) Note that all Alice needs to do is observe the coins, not remove them from the well. 4. Discussion of the Coin Exeriments One should note that if classical coins are used, the odds are even and indeendent of elevation. In his game, Meyer shifts the odds in favor of Bob by using a quantum coin being flied reeatedly and alying the Hadamard oerator to ensure that the outcome is always Tail. In contrast, the game outlined above utilizes a multilicity of quantum coins having an anti-symmetric wave function (ermions). The coins are located in a otential energy well, thereby subjected to the Pauli Exclusion Princile. The layer draws two coins, his goal being to get a mismatch. The odds are shifted in favor of Alice by selecting coins at a low elevation where the coins are entangled and have a higher robability for mismatch. The same conclusion would have been reached had the coins been Bosons and the winning combinations, matched airs. The requirements for such a game are: () Non-Maxwellian objects such as ermions or Bosons. () A otential energy gradient that crowds the objects and forces them into their characteristic non-maxwellian distribution. or examle, if the objects are ermions, their distribution is, according to Landauer formalism, the roduct of their density of state and the ermi Dirac distribution. The classical and quantum coin exeriments have a Thermodynamics counterart as exlained below. 5. Classical Gas Thought Exeriment The coin thought exeriment described above can be translated to the context of Thermodynamics. Consider a column of gas subjected to a force field and held in a container with erfectly reflecting and elastic walls, floor and ceiling. A game between Alice and the House is defined as follows. The House rovides Alice with the osition of each molecule and Alice selects two molecules, A and B. If A has more inetic energy than B, Alice wins, otherwise the House wins. The question for Alice is whether she can imrove her odds by selecting molecules at different elevation in the column. One could rehrase the game in more friendly thermodynamics terms. Is the temerature at the to and at the bottom of the otential energy gradient the same or different? One shall first consider the classical case shown in igure 3, which uses a Maxwellian gas such as air, and a force such as gravity.

5 Entroy 05, igure 3. Classical gas subjected to a force field and enclosed in a erfectly reflecting box. Is the temerature of the gas the same or different at the to and at the bottom? The molecules collide with each other thereby evolving a Maxwellian distribution. The distribution of the gas at ground level is given by the Maxwell Boltzmann distribution equation: f E ( E ) = ex T B π T B (4) also shown by the red curve at the bottom of igure 4 (see Aendix A for details). igure 4. To comare the temerature of a classical gas at the bottom and at the to of a gas column, the following stes are taen: () a distribution is defined at the bottom; () The distribution at the to is obtained by inserting otential energy, thereby denormalizing the distribution; (3) The distribution at the to is then renormalized and comared to the one at the bottom. No shift in the first moment indicates that the gas is isothermal with elevation in conformance with the Second Law. The equation for the distribution comrises three inds of terms. The first corresonds to a normalization factor such that the area under the curve is unity. The second term is the square root of inetic energy. The third term is the Maxwell term which is exonential with inetic energy. The strict exonential nature of this term is imortant as shall be exlained.

6 Entroy 05, The distribution at the to of the column can be generated by inserting otential energy exonential term adjacent to the inetic energy E to exress the total energy of molecules: E in the E f ( E, E ) unnormalized = ex T B π T B (5) Because it is exressed within an exonent, otential energy can also be factored out in its own exonential term as shown in the equation below: E f ( E, E ) unnormalized = ex ex T B π T B T B (6) One must note that inserting otential energy denormalizes the distribution, reducing its amlitude and resulting, as exected, in a lower density of molecules at higher elevation as shown at the to of igure 4. Of concern however, is not density but temerature. A change in temerature can be exressed by a shift in the first moment of the distribution which can be visually exressed by comaring the distributions at the bottom and at the to. This comarison, however, must be made after the distribution at the to is renormalized. Since otential energy is exressed exonentially, it can be factored out as indicated in Equation (6) and eliminated by the renormalization rocess. The resulting renormalized equation for the gas at the to is: f E ( E ) = ex T B π T B renormalized (7) which is identical to the distribution at the bottom Equation (4). The corresonding curves, thic red for the bottom and thin blue for the to are shown at the right of igure 4. These curves are identical, imlying that the first moments of both curves are the same, and that the average inetic energy and the temerature are invariant with elevation. As exected, the Maxwellian gas is isothermal with elevation in comliance with the Second Law. This exeriment corresonds to the coin selection exeriment in which the layer comes out even against the House (assuming no entrance or laying fee). 6. Quantum Gas Thought Exeriment The quantum version of the gas exeriment outline above requires that the molecules be non-maxwellian, for examle ermions. As shown in igure 5, these can be embodied in a thermoelectric material by electrons or holes which can be regarded as being in a gas hase (see Aendix B for details). The thermoelectric slab is subjected to an electric field, roduced for examle by insulated caacitor lates or a junction. The slab is comosed of a material with a high ZT, such that the carriers do not collide with heat honons. ZT is a dimensionless figure of merit used in the thermoelectric industry. It is a measure of the ability of a given material to roduce thermoelectric ower and is given by σst ZT =. It deends on the electrical conductivity σ, the Seebec coefficient S, the temerature λ T and the thermal conductivity λ.

7 Entroy 05, igure 5. A gas comrised of quantum articles (e.g., ermions) is subjected to a force field and enclosed in a slab of semiconductor. Is the temerature of the gas the same or different at the to and at the bottom? Because of the high ZT, electrons are mostly ballistic with resect to the lattice, but they still interact between themselves thereby evolving their own ermion distribution. A detailed descrition of the exeriment is given by the author in [3] and exerimental evidence is rovided that the henomenon described below does occur. In this exeriment as in the revious one, the question being ased is whether the temerature at the to and at the bottom of the otential energy gradient is the same or different. The same rocedure can be alied as before. irst the distribution at the bottom of the otential energy gradient is defined. In accordance with Landauer Boltzmann formalism, since the gas is comosed of ermions, their distribution is the roduct of the density of state (which we will assume to be arabolic for the sae of simlicity): 8π gc( Et) = m E 3 (8) h and of the ermi Dirac distribution: D ( E ) = t E + Ec + ex T B (9) Combining Equations (8) and (9), and normalizing yields the normalized ermion distribution: fd ( E ) = A E E + Ec + ex (0) T B where A is the normalization factor. The distribution at the to of the slab is obtained by inserting otential energy energy E in the ermi Dirac term, to exress the total energy of molecules: E next to inetic

8 Entroy 05, f ( E ) = A E D E + E + Ec + ex T B () The ste of inserting otential energy denormalizes the distribution which is shown as the thin blue curve at the to of igure 6. A change in temerature corresonds to a change in the first moment of the distribution. The change can be visually determined by comaring the original distribution shown at the bottom of igure 6 with the distribution at the to. However, this comarison must be made after the distribution at the to is renormalized. igure 6. As in the classical exeriment, the temeratures of the gas at the bottom and at the to of a otential energy gradient are comared using the following rocedure: () a distribution is generated at the bottom of the column () The distribution at the to is obtained by inserting otential energy, thereby denormalizing the distribution; (3) The distribution at the to is renormalized and comared to the one at the bottom. A shift in the first moment indicates that the gas is colder at the to of the columns than at the bottom. f ( E ) = A E D renormalized E + E + Ec + ex T B where A is the normalization factor. In contrast to the revious exeriment, the distribution is not a strict exonential function of otential energy. Therefore, otential energy cannot be eliminated by the renormalization rocess, resulting in the distribution at the to being sewed toward the left in comarison with the one at the bottom and indicating a shift of the first moment. This shift demonstrates a lower average inetic energy, and therefore a lower temerature with elevation. The ()

9 Entroy 05, reason for this result is the non-strictly exonential nature of the ermi Dirac distribution which revents the cancellation of otential energy by renormalization. igure 7 summarizes the result of the classical and quantum gas exeriment. It shows the energy distribution of the gas at two elevations (thic red: ground level; thin blue: elevation above ground) for a Maxwellian gas (a) and for a non-maxwellian gas (b). The calculator software for generating these curves is available at [4]. (a) (b) igure 7. (a) Normalized energy distribution for a Maxwellian gas at two elevations, (red curve for ground level and blue curve for elevation above ground). The two distributions are suerimosed indicating that their first moment is identical, and that the gas is isothermal with elevation. These distributions were calculated using the Maxwell Boltzmann distribution; (b) Normalized energy distribution for a ermion gas at two elevations, (red curve for ground level and blue curve for elevation above ground). The distribution above ground is sewed to the left comared to the one at ground level, indicating a smaller first moment and a dro in temerature with elevation. These distributions were calculated as the roduct of a arabolic density of state and the ermi Dirac distribution. 7. Discussion of the Gas Exeriments The thermodynamics counterart of the quantum game above is fully described by the author in [3] and reroduced in art in this aer. He shows that when a gas comrised of non-maxwellian articles (ermions or Bosons) is subjected to an external force (otential energy gradient), the temerature of the gas dros with elevation. This thermodynamic effect is Quantic in nature and can be used to lay the Quantum game described above. In contrast, when the gas is Maxwellian, elevation has no effect on temerature the gas is isothermal in accordance with the Second Law. This corresonds to the classical game described above in which Alice s chances of winning are 50% not including any fees she may have to ay the House for laying the game, which would reduce her exected winnings. 8. Conclusions A coin tossing game is described in which a layer lays against the House by selecting two coins. Before uncovering them, she calls a match or a mismatch. The classical game utilizes coins that follow rules of Classical Physics. When the layer lays classically, she comes out even. The quantum game uses coins embodied as ermions and subjected to a otential energy gradient in a otential well. It is shown that a quantum strategy can beat classical odds. According to the Pauli Exclusion Princile no two ermions can occuy the same state and therefore, selecting ermions from

10 Entroy 05, the bottom of the energy well is the best strategy to achieve a mismatch. The odds of winning the game are shown to deend on the elevation of the ermions being selected ranging from being a certainty of winning at the bottom of the well to being near classical at the to. These odds differ maredly from those in a classical game in which they are indeendent of elevation. The thermodynamics counterart of the quantum game is discussed in which a Maxwellian gas and a non-maxwellian gas are subjected to a force. Temerature is shown to be indeendent of elevation for the Maxwellian gas in accordance with the Second Law, but dros with elevation for the non-maxwellian gas. The game and the ermion gas column utilize the same comonents: ermions are subjected to a force field, and develo their own characteristic distribution. A shifting of odds is roduced in the game and a shifting of inetic energy is roduced in the thermodynamic exeriment: the ermion gas column develos a sontaneous temerature gradient. Obviously such a henomenon is not allowed in classical Physics but is ermitted in Quantum Mechanics when the articles follow a non-maxwellian distribution. The three laws of Thermodynamics can facetiously be described as You can t win, You can t get even, and You can t get out of the game. The game is rigged by the rules of Classical Mechanics. The best strategy then, is to lay according to different rules, in other words, to cheat. Quantum Mechanics allows you to do just that. Conflicts of Interest The author declares no conflicts of interest. Aendix A. Derivation of Maxwell Boltzmann Distribution in the Presence of an External orce ield The first ste is to exress the Maxwell Boltzmann distribution when an external force field is alied on the gas. In the absence of a force each comonent of the velocity vector has a normal distribution: f f / m mv x ( vx ) = ex πt B T B / mvy m ( vy ) = ex T B T π B (3) (4) f / m mv z ( vz ) = ex πt B T B If a force is alied in the z direction the gas acquires otential energy E Equation (5) becomes: mvz mgz m f (, ) ex vz z = πt B T B / (5) = mgz and (6)

11 Entroy 05, The robability distribution of the vector v at a given elevation z can be seen as the joint robability of three indeendent robabilities f ( v ), f ( v ) and f ( v, z ) along each degree of freedom. Hence: x y f ( v, z) = f ( vx, vy, vz, z) = f ( vx ) f ( vy ) f ( vz, z) (7) Therefore, substituting Equations (3), (4) and (6) into Equation (7) roduces: m mv ( x + vy + vz) mgz (, ) x y z = ex ex x y z πt B T B T B f v z dv dv dv dv dv dv To obtain the distribution for the inetic energy E, the velocity comonents are exressed in sherical coordinates. A velocity volume element is given by: Integrating θ and φ, this volume element becomes: z (8) dv xdvydvz = v sin θdθdφ dv (9) π π (0) dv sin 4 xdvydvz = v dv dφ θdθ= πv dv 0 0 Substituting Equation (0) into Equation (8) roduces: m mv mgz f (, vzdv ) = ex ex 4πvdv πt B T B T B () Since de = mvdv = medv and defining the otential energy as E = mgz results in: E f ( E, E ) = ex ex T B π T B T B () Normalizing Equation () yields: f ( E, E ) normalized = 0 E ex ex T B π T B T B E E ex ex de T B π T B T B (3) Rearranging yields: f ( E, E ) normalized = E ex ex T B π T B T B E ex ex de T T π T 0 B B B (4) Since E ex de 0 =, we get: T B π T B

12 Entroy 05, E f( E, E ) normalized = ex T B π T (5) B Hence the distribution is indeendent of otential energy E and of elevation in comliance with the Second Law as shown in the igure 7a. Aendix B. Derivation of the ermion Distribution in the Presence of an External orce ield Consider a ermion gas embodied as electrons confined to an n-doed semiconductor slab as shown in igure 5. The thermally significant electrons are those that haen to be in the conduction band and that are made available by the ermi Dirac distribution. Such electrons must have a ositive inetic energy E > 0 and a total energy E t greater than the otential energy of the band s bottom E c, i.e., E > E. In the conduction band, the electron s density of states is given by [,] by: t c 8π gc( Et) = m E 3 t Ec (6) h Exressing inetic energy as E = Et Ec and substituting into Equation (6) roduces: The ermi Dirac distribution of the carriers can be exressed as: and using E = Et Ec we get: D 8π gc( Et) = m E 3 (7) h D ( Et ) = Et E + ex T B ( E ) = E + Ec + ex T B In the resence of an electrical field, an electrical otential energy E is induced in the carriers, and the ermi Dirac distribution is modified accordingly: D ( E, E ) = E + Ec + E + ex (30) T B In accordance with Landauer Boltzmann formalism, the electron density ne (, E ) is the roduct of the carrier density of states and the ermi distribution: fermions( E, E) = n( E, E) = gc( E) D( E, E) (3) Substituting Equations (7) and (30) into Equation (3) yields: t (8) (9)

13 Entroy 05, π fermions ( E, E ) = m E 3 h E + E + Ec + ex T B (3) Normalizing yields: f ( E, E ) ermions normalized = 0 8π m E 3 h E + E + Ec + ex T B 8π 3 h + ex T B m E de E + E + Ec (33) and cancelling constant terms roduces: f ( E, E ) ermions normalized = 0 E E E + Ec + E + ex T B de E + Ec + E + ex T B Please note that, unlie in the Maxwell Boltzmann case (Aendix A), the otential energy term E is not readily cancelled causing the ermion distribution with E > 0 not to be identical with the distribution with E = 0. Since the areas under the distributions are equal to unity, and the distribution are not congruent, the distributions must be sewed with resect to each other as shown in igure 7b. This figure has been generated using a calculator rogram. It shows that the distributions have different first moments imlying different temeratures, in violation of Clausius formulation of the Second Law. (34) References. litney, A.P.; Abbott, D. An introduction to quantum game theory. luct. Noise Lett. 00,, doi:0.4/s Meyer, D.A. Quantum Strategies. Phys. Rev. Lett. 999, 8, doi:0.03/physrevlett Levy, G.S. Anomalous Temerature Gradient in Non-Maxwellian Gases. In Ceramics for Energy Conversion, Storage, and Distribution Systems, Proceedings of the th International Conference on Ceramic Materials & Comonents for Energy & Environmental Alications, Vancouver, BC, Canada, 4 9 June 05; Wiley: Hoboen, NJ, USA, 06; Volume 55, in ress. 4. Levy, G.S. Non-Maxwellian Distribution Calculator. Available online: htt:// onmaxwelliancalculatorsetu.zi?attredirects=0&d= (accessed on 8 October 05). 05 by the authors; licensee MDPI, Basel, Switzerland. This article is an oen access article distributed under the terms and conditions of the Creative Commons Attribution license (htt://creativecommons.org/licenses/by/4.0/).