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The Topological Structure of the Unit Octonions and Games with Quantum Communication

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Title: The Topological Structure of the Unit Octonions and Games with Quantum Communication


1
The Topological Structure of the Unit Octonions
and Games with Quantum Communication
  • By
  • Aden Ahmed
  • Portland State University
  • April 2009

2
Overview
  • Quaternions
  • Octonions
  • Topological Features
  • Quantum Games

3
Quaternions
  • The quaternions H are an associative,
    noncommutative, normed division algebra over the
    reals.
  • The quaternions are spanned by the identity
    element 1 and three imaginary units
  • The multiplication is polynomial subject to

4
Quaternions
  • A general quaternion
  • Length of q
  • A unit quaternion has length 1
  • Set of unit quaternions is S3 C2
  • Quaternionic multiplication

5
Quaternions
  • There are other identifications of S3 that
    interest
  • us beyond that of the unit quaternions.
  • In particular the identification of S3 with the
    special
  • unitary group of 2x2 complex matrices. That
    is,
  • those matrices with orthonormal columns and
  • determinant 1.
  • where

6
Octonions
  • The octonions O are a non-associative,
    noncommutative, normed division algebra over the
    reals.
  • They are not obtained from H the way we obtain C
    from R2 or H from C2
  • C R Ri
  • H C Cj
  • They are spanned by 1 and 7 square roots of -1
    denoted by i1, i2, i3, i4, i5, i6, and i7.

7
Octonions
  • Given any two distinct ir, is, there is a third
    it, so that these three square roots of -1
    satisfy
  • ir2 is2 it2 ir is it -1
  • Thus any pair of basic square roots of -1
    determine a quaternionic subspace.
  • Up to order there are exactly seven such choices.
  • Thus 7 natural quaternionic subspaces all
    together.

8
Octonions
  • Any two such quaternionic subspaces intersect in
    a common copy of the complex
  • numbers.
  • Consider the 7 square roots of -1 as points,
    the quaternionic subspaces as lines these
    points are incident to.
  • We see that octonionic algebra satisfies the
    following two axioms
  • (1) Two points determine a line
  • (2) Two lines determine a point

9
Octonions
  • Not surprisingly, octonionic multiplication of
    the seven square roots of -1 is modeled along the
    7 point, 7 line projective plane shown
    below

10
Octonions
  • We can see that the octonions are not associative
  • Take i1(i5i7) i1i4 -i2
  • On the other hand (i1i5)i7 i6i7 i2
  • Like commutativity for the quaternions,
    associativity for the basic octonions sometimes
    fails up to a sign.

11
Octonions
  • A general octonion
  • Length of o
  • A unit octonion has length 1
  • Set of unit octonions is S7

12
A Posy of Spheres
  • For us the octonions will be of great use due to
    the existence within of a posy of 3-spheres.
    That is, a collection of 3-spheres that intersect
    in a common circle, generalizing the more usual
    algebraic topological notion of bouquet where
    the spheres intersect in a common point.
  • Further these 3-spheres will be identified with
    the special unitary matrices SU(2) and the common
    circle with the unitary matrices U(1), all
    respecting octonionic multiplication.

13
Bouquet vs Posy
A Bouquet of Spheres
A Posy of Spheres
14
Posy of Spheres
  • Our posy is given by the three copies of unit
    quaternions, meeting in a common copy of the unit
    complexes.

15
Posy of Spheres
  • Our quaternionic subspaces are thus
  • I a0 a1i1 b0i2 b1i4
  • II p0 p1i1 q0i5 q1i6
  • III e0 e1i1 f0i3 f1i7
  • Which meet in the complex space xo x1i1
  • Our posy is the set of unit length octonions
  • in this collection.

16
Games
  • Given
  • a set of players 1, 2, , n,
  • a set Si of pure strategies for each player, and
  • a set Oi of possible outcomes or payoffs for each
    player,
  • a game G is a function
  • which assigns to each n-tuple of strategic
    choices
  • an outcome to each player.

16
17
Definitions
  • An n player, m strategy game is a game with n
    players where each player has access to exactly m
    pure strategies, i.e.,
  • Without loss of generality, the Ois are all
    copies of R, the so called utility of the
    outcomes.
  • A strategy profile is an n-tuple (s1, s2,, sn)
    where each si ? Si is a pure strategy chosen by
    player i.

17
18
Special Strategies
  • The theory of games concerns the identification
    of
  • special strategies or strategic profiles.
  • For example, players may want to identify
  • a strategy which guarantees them an outcome
    having maximal utility.
  • a security strategy that is, a strategy that
    guarantees a specific lower bound to the utility
    received.
  • for a fixed n-1 tuple of opponents' strategies,
    rational players seek a best reply that is, a
    choice si ? Si of strategy that delivers a
    utility at least as great of any other strategic
    choice.

18
19
Nash Equilibrium
  • A Nash equilibrium (also called a solution or
    just an
  • equilibrium) is a strategy profile (s1, s2,,
    sn) such that
  • each si is a best reply to the n-1 tuple of
    opponents'
  • strategies.
  • The identification of Nash equilibria is a
    fundamental goal in the theory of games.

19
20
Octonionization
  • Thesis work concerns in part 3 player 2
    strategy games played under quantum
    mediated communication, i.e.
  • the quantum internet. The resulting games
    are huge extensions of the originals, with
    individual strategy
  • spaces enlarging to copies of SU(2) with
    computationally
  • complex payoff functions.
  • Now we identify the SU(2) strategy spaces of
    the players
  • with our posy of 3-spheres lying in the unit
    octonions.
  • Player Is strategies are identified with
  • I a0 a1i1 b0i2 b1i4, player
    IIs with
  • II p0 p1i1 q0i5 q1i6 and
    player IIIs with
  • III e0 e1i1 f0i3 f1i7.

21
Octonionization
  • From this identification we obtain an algebraic
    formulation of the payoff functions. Payoffs here
    are expressed as unit octonions. These octonions
    represent the probability distribution over the 8
    possible classical outcomes that arises from the
    quantum communication.
  • This payoff formula is computationally friendly.

21
22
Octonionization
  • Moreover, this formula is robust enough to prove
    the existence of a mixed quantum equilibrium
    where each player appeals to the uniform
    probability distribution over their pure quantum
    strategic choices.
  • Even more remarkable is that this equilibrium
    exists independently of the classical game played
    !
  • We conjecture that the formula is robust enough
    to classify all potential Nash equilibria for
    these games.
  • Applied to the Nash-Shapley poker model, our
    result is enough to show that certain players
    will be able to do better playing poker over the
    quantum internet than is possible in real
    life.
  • For details, see Steve Bleilers paper
    (Quantized Poker)
  • on the arxive.org.

22
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