Crafting community online and offline landscapes of the indie craft movement

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U.l £1.0 0.0 � , '---05 (1.10 0.2;'1 i Cl OJ; (I,I)� ------1 .0 15 Time (C) N = 100, W (l.V 2.1) = 100 05 U) Tim . (c1) N = 200, W = 100 Figure . : Evolution of a random initial state of the rebit-ubit system with no SI: interaction strength s = 46 1. 2JJ 4. Investigation of the Generic Dynamics This is why we shall like to differentiate clearly between these two components while developing our analysis. Then, as given in Equation 2.23, the orthogonal evolution operator for our state becomes: which transforms our initial state with time in the following fashion: (4.5) What makes the problem non-trivial is that B does not commute with Stueckel­ berg's matrix J, vvhich means that: Vve would like to be able to derive some sort of expansion in power series of l/w, since the eigenvalues of the matrix B/w are what gets really small in our model with a fast rotating ubit. A straightforward power series expansion of the exponential gives us powers of w itself, which is just the opposite of what we want, since we typically need w to be large. \Ne must look for a different approach, therefore. Consider once again which is just a real anti-symmetric matrix. I-Ienee: S, Taylor-expanding in terms of it, we have: St -S�t� -S t 2! (I - ;!STSt2 + . . . ) + (Sf; - ;!STSSf;3 + . . . ) VSTSt) + ( ISTSf;) VSTS e S , = + = = or, finally, S sin ( + r ') ') + 3 3! cos + . \ (4.6) STS is a positive semi-definite real matrix, and, therefore, taking a Notice that square root of it is a well-defined operation. 4'7 o j 4. Investigation of the Generic Dynamics Now we make the substitution S = 0)J + B. Then: ST S = (wff + BT ) (wJ + B) = _ w J - wJB + BTwJ + BT B (4 . 7) = w - w(JB + BJ) - B where we have used the fact that since J and B are both anti-symmetric matrices, JT = -J and BT = -B. Recall from Chapter that every matrix can be expressed as a sum of a part that commutes with Stueckelberg's matrix and a part that anti-commutes with it. Thus, we shall denote the commuting part of B as: B+ = � (B - JBJ) and the anti-commuting part as: B - = -1 (B + JBJ) Then we can re-write Equation 4.7: (4.8) J -- or T = w' VSTS 2JB+ W B' -w2 ( 4.9) - Plugging in this expression in Equation 4.6 we obtain a nev·,l form of the evolution operator: e S I. ( B) = J +- w sin (w t / J1 - V1- 2J + � - 2JB + w- - B2 0;2- �:) + cos ( - J. --- ---) wt - 2JB+ W B2 w2 (4. 10 ) Although Equation . does not seem particularly illuminating, we are now in a position to make a reasonable approximation. We shall apply this approximation step-by-step, beginning by considering the factors outside of the sine and cosine. 48 4. Investigation of the Generic Dynamics Suppose we decide to keep there all terms of the order l/w and discard higher order terms. In particular, we have: + B B [...]... from bit" [12] Many of the attempts to reconstruct quantum mechanics from information principles, however, rely on the assumption of the complex number field The question of the role of the complex field in quantum theory has also been a long-standing issue among the scientific community The existence of alternatives to standard quantum mechanics, such as having the Hilbert space of quantum mechanics... other words, instead of postulating it from the very beginning, can we obtain Stueckelberg's superselection rule as a result of the generic dynamics of our real-vector-space system? The answers to these questions lie at the core of understanding the fundamental relationship between quantum theory defined over the real and the complex fields To investigate the issue at hand, we are going to imagine the. .. sharing of quantum information with the surroundings The loss of information to the environment is known as decoher-ence Decoherence causes pure states to become more mixed and it increases the entropy of mixed states The effective decoherence can be seen by starting with a density matrix describing the system and the environment together and tracing out the environment 2.2 Real-Vector- Space Quantum Theory... nature of objects Finally, real-vector-space quantum theory can potentially reveal a way of modify­ ing and further developing quantum mechanics, conceivably for the development of a theory of quantum gravity vVith all this in mind, we set out to build on Stueckelberg's work and explore the world of real-vector-space quantum theory After providing the reader with the necessary background for comparing the. .. cos( e) 16 3 ) and thus we notice that: D (rp) = 10)(01 0 I2 + 1 1) ( 1 1 0 e'Ph (3.4) Therefore, changing rp corresponds to a conditional rotation of the ubit In the Bloch sphere, all possible real states form just a solid circle in the xz-plane The rotation of the ubit can be seen as adding another circle, which through the interaction of the ubit with the real states, gives rise to the three-dimensional... (3) is actually saying that the only rule we need to impose is that it is impossible to extract information from the ubit This last claim gives us a physical interpretation of at least one of the fundamental aspects of quantum theory, the appearance of complex numbers, in terms of information 3.1 Preamble Before we approach the actual proof of Theorem 3.1, we will prove the following two lemmas: Lemma... equivalent to standard quantum mechanics The ad­ vantages of expressing quantum mechanics over the real field are multiple F irst, as Dyson noted, the time reversal operator is antilinear in standard quantum theory, whereas it is linear in the real quantum theory [2] This is the case, for example, for the fundamental CPT symmetry of quantum field theory [7] Dyson argued that the introduction of time reversal... 4 2 Background the proton in a uniform magnetic field [6] We shall assume the field points in the z-direction and is thus given by B = Bok The proton then experiences a torque, J.L x B, which causes it to precess around the z-axis Here J.L denotes the magnetic dipole of the charged particle and is given by J.L = ,,(S, where the proportionality constant, ,,(, is the gyromagnetic ratio The situation... simply have p = I'�)) ('l/JI The density matrix of the composite system RQ given in Equation 2.8, where R and Q are entangled, is again p(RQ) = IW(RQ)) (W(RQ)I We would like to be able to use the density matrix to describe the state of the subsystem Q: (2.10) This expression makes use of the partial trace of p(RQ) over the subsystem R Here we give a definition of partial trace of a density matrix, which... proceed to the proof of Theorem 3 l 19 3 Information-Theoretic Approach 3.2 3.2.1 Proof o f Theorem 3 1 St ueckelb erg 's rule i s equivalent t o commutativity of all op erators wit h rotat ion of t he ubit ( 1{::? 2 ) Without loss o f generality, consider a system o f a rebit A and the ubit U An operation on this system 'which results solely in a rotation of the ubit can be written in the following