Hurkyl said:
Alas, it has completely fallen out of my brain.
No problem, Hurkyl. Some 1000 threads back for both of us, you were explaining MWT and decoherence. The notion of a toy observer came up—I think for the first time—as a possible means to model decoherence in quantum mechanical observers.
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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 so-called 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 c-not 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 c-not gate, it’s not enough to say the value of T can be reacquired by acting a second c-not gate on T'. I think something has been left out of the popular description of a c-not 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 c-not 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 c-not as follows.
(c,t) \rightarrow (c',t')
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(0,0) e^{\delta} \rightarrow (0,0) e^{\delta}
(0,1) e^{\delta} \rightarrow (0,1) e^{\delta}
(1,0) e^{\delta} \rightarrow (1,1) e^{\delta + \phi}
(1,1) e^{\delta} \rightarrow (1,0) e^{\delta + \phi}
\delta is the relative phase of two qbits. \delta is an unphysical gauge, that we could just as well set to zero. \phi 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 c-not gate acting on the primed qubits would have the truth table:
(c',t') \rightarrow (c'',t'')
---------------------
(0,0) e^{\delta} \rightarrow (0,0) e^{\delta}
(0,1) e^{\delta} \rightarrow (0,1) e^{\delta}
(1,0) e^{\delta + \phi} \rightarrow (1,1) e^{\delta + 2\phi}
(1,1) e^{\delta + \phi} \rightarrow (1,0) e^{\delta + 2\phi}
If \phi = i \pi, two c-not 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 c-not, sent on separate paths encompasing a solenoid, then reentangled with a second c-not. 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.
