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ManyWorlds Theory 
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#73
Feb2009, 04:53 AM

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#74
Feb2009, 05:03 AM

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The best I obtained for an answer was an oblique hint that the phase of the control bit, C> was effected by the state of A>. Is this correct? 


#75
Feb2009, 04:34 PM

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If we focus (relative) state of the control bit, then the typical result is that the CNOT gate forces decoherence: after passing through the CNOT gate, an initial pure state p0> + q1> would be transformed into a mixed state of 0> and 1> with probabilities p^2 and q^2 respectively. The same would happen with any input mixture. The only exception is if the control and target lines were entangled in a way that would cause interference. So, as we would expect from a toy measurement, the object being 'measured' has decohered into a statistical mixture of the two outcomes. And as long as there's not entanglement involved, the actual state of the target line is completely irrelevant. (At least, if I've done the calculation correctly) 


#76
Feb2109, 02:41 AM

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 Maybe I should start over. To make sense of MWT we need to understand an observer as a quantum mechanical system. To understand this, we could construct a toy observer. To do this we need to model the nature of human or machine information processing. To do this we need: 1) An underlying principle of classical information processing. We need this because it will tell us that classical information processing requires that information is discarded. The socalled reversible classical gates are reversible only in principle. Even a not gate discards information. This will invoke decoherence as a sufficient, though not necessary, element of classical information processing. (more on this later.) and 2) A schema for building classical logic gates or neurons out of multiple quantum gates. Now, if we are to model gates, or neurons, or the neurons of neuronetworks out of quantum gates, we had better darned well understand quantum gates. I cannot believe that two quantum bits can interact, where qbit C changes qbit T, without qbit T changing qbit C. Say we have a cnot and C=1. What has happened to the information of the fomer state of T? If we are questioning the validity in the operation of a cnot gate, it’s not enough to say the value of T can be reacquired by acting a second cnot gate on T'. I think something has been left out of the popular description of a cnot gate. I will have to go out on a limb in the following, because the fact of the matter is, I don’t know if the following is true or not. Let me know. a) A quantum gate is reversible. b)There is a relative phase between two qbits. c) If the phase information is not preserved, the gate cannot be reversed. For simplicity, assume the inputs to a cnot are both pure states; either 1> or 0>. Just as in boolean logic, there are 4 possible outcomes. If not, reversibility is violated; quantum mechanics would not obey time reverse symmetry. d) I'm going to make a wild stab at the truth table of a cnot as follows. [tex](c,t) \rightarrow (c',t')[/tex]  [tex](0,0) e^{\delta} \rightarrow (0,0) e^{\delta}[/tex] [tex](0,1) e^{\delta} \rightarrow (0,1) e^{\delta}[/tex] [tex](1,0) e^{\delta} \rightarrow (1,1) e^{\delta + \phi}[/tex] [tex](1,1) e^{\delta} \rightarrow (1,0) e^{\delta + \phi}[/tex] [itex]\delta[/itex] is the relative phase of two qbits. [itex]\delta[/itex] is an unphysical gauge, that we could just as well set to zero. [itex]\phi[/itex] is the change in the relative phase of c and t. We should be free to attach the phase to either output bit, as long as we are consistant. A second cnot gate acting on the primed qubits would have the truth table: [tex](c',t') \rightarrow (c'',t'')[/tex]  [tex](0,0) e^{\delta} \rightarrow (0,0) e^{\delta}[/tex] [tex](0,1) e^{\delta} \rightarrow (0,1) e^{\delta}[/tex] [tex](1,0) e^{\delta + \phi} \rightarrow (1,1) e^{\delta + 2\phi} [/tex] [tex](1,1) e^{\delta + \phi} \rightarrow (1,0) e^{\delta + 2\phi} [/tex] If [itex]\phi = i \pi[/itex], two cnot gates in series will restore the primed states to their original unprimed states. So, have I told any fibs yet? The above contention is testable with two electrons (C,T), entangled in a cnot, sent on separate paths encompasing a solenoid, then reentangled with a second cnot. Varying the strength of the solenoid field should cause the resultant spin states (C'',T'') to vary. Sending both electrons around the same side of the solenoid should obtain (C'',T'')=(C,T), independent of the strength of the solenoid field. 


#77
Feb2109, 06:46 AM

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Symbolically, the joint state, expressed as a vector in Hilbert space in the 01 basis, is of the form a 00> + b 01> + c 10> + d 11>If b=c=d=0, for example, then we can interpret a as being the (unphysical) overall phase of the state vector, but there is nothing here expressing the idea of a relative phase between the two bits. We could use the partial trace to extract the single particle states out of this joint state  but the calculation for that I know involves using density matrices, and thus completely obliterates all information about the overall phase. (as well as destroying information about entanglement) 


#78
Feb2209, 02:06 AM

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Two particles of the same species have a relative phase. Either that, or I've misunderstood something. I'm referencing Mark P. Silverman, on the correlations of two electron interferometry, if that helps. But things are worse than I though. The quantum gates described in most web sites, including Wikipedia, ignore the gate apparatus. The unitary operations are so schematical that entanglement with the apparatus enabling the operation is ignored. I have managed to discover that in an NMR (Nuc. Mag. Resonance) Hadamard gate, for instance, the qubit is entangled with a radio frequency source. Nowhere does this show up in the truthtable. 


#79
Feb2209, 10:35 AM

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If you're referring to the fact I don't know what you mean by 'relative phase' here, then basically, one of the following is true: A. You are talking about something that isn't a 2qubit systemI guess I should point out that the place I am familiar seeing 'relative phase' is when talking about the components of a quantum state relative to some basis. e.g. for the single qubit state a0>+b1>, both the magnitude and the phase of the value (b/a) are physically significant. I'm somewhat skeptical of your description though: I don't think the action of the gate on the qubit state can be unitary if it results in an entanglement with the gate apparatus. However, there is no problem introducing a temporary entanglement while the apparatus is operating, as long as it gets wiped out by the end of the operation. I confess, however, I don't really know anything about the engineering of quantum gates. 


#80
Feb2209, 02:32 PM

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Re: engineering of quantum gates.
This is what it comes down to. I really won't be convinced of many things without a description of the experiental apparatus that manages a universal gate set. 


#81
Dec3010, 05:24 AM

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Many Worlds, like any other interpretation is not forced on us like, e.g. the Schroedinger equation is. In fact none of the (carefully thought out) interpretations is. Consider the following experiment. A fat coin is tossed. (It's fat so that the probability of it landing on its edge is not very small.) Spinning in the air, it is in a superposition of three states: heads, tails and edge. (There are hidden variables here of course, namely the initial conditions of the flip.) In a universe the coin lands on its edge. The people in this universe have never seen such a thing before. In the same universe the coin is flipped again. In another universe the coin lands on its edge and the people in this universe have the memory of two edges in a row. This goes on through many universes and the people in the "edge" universes begin to suspect that the laws of classical physics are just not applicable to this experiment for them. As time goes on and many flips have happened, in the "edge" universes the results of the flips are seen as just plain miraculous. If you extend this reasoning, you get an infinite number of universes where the laws of physics are wantonly broken. To me, this is unacceptable and a definite byebye to MWI. Comments?



#82
Dec3010, 05:56 PM

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#83
Dec3110, 05:24 AM

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Hurkyl How does "Classical physics permit[s] the laws of physics to be 'wantonly broken'[?]"



#84
Dec3110, 08:01 AM

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#85
Dec3110, 08:18 AM

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So, with that in mind we can explore your question regarding, if there's an infinite number of universes, then why hasn't at least one of these universes collided with our universe? Well, maybe one has, and we just haven't noticed the effects of it in our sector of our universe yet. Do you see how silly this sort of speculation can get? Forget it. Don't be a fan. Just study physics. 


#86
Dec3110, 08:19 AM

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#87
Dec3110, 08:31 AM

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#88
Dec3110, 08:51 AM

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You're right, of course. A disturbing thing about MWI, though, is that these improbable outcomes occur in some universe every time the experiment is done, i.e. for Classical physics these improbable events *could* occur where for MWI they *do* occur. I don't see how the equivalent of the Born rule can exist for MWI, or maybe what it would even mean. An outside observer would see that the pathological universes are of measure zero. The problem is that first of all, for Everett there is no such observer and second of all how can a measure be defined over the universes so that one could make statements about zero measure? Insights?



#89
Dec3110, 09:28 AM

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However, that was the phrase ccrummer described the outcome of repeated edge landings. 


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