V. Crisscross Entanglement and the NonLinear Schrödinger Equation

Now, what happens if we have so many entangled electrons that the chain of particles crosses itself (as shown in figure 12)? That is to say, suppose the entanglement is sufficiently extensive that we no longer need the Stern-Gerlach apparatus. The tail of the chain of entangled particles (labeled “B” in figure 12) can effectively serve as the macroscopic magnet in the Stern-Gerlach apparatus while the front (labeled “A” in figure 12) acts as our chain of electrons passing through. Some especially interesting things now begin to happen.



Figure 12: System of entangled electrons that crosses itself. The region of the entangled chain near end “B” acts like the Stern-Gerlach magnetic field, and the electrons near end “A” play the traditional role of spinning particles in the experiment. The entire chain is one entangled system so it can simultaneously be in states forming various virtual Stern-Gerlach apparatuses as well as different system states. For example, if all the electron spins near end “B” are aligned pointing up, the magnetic field will be pointing up as in a traditional Stern-Gerlach apparatus. However, if the electron spins near end “B” are an even mixture of up and down spins then no Stern-Gerlach magnetic field would be present and the electron chain at end “A” would pass straight on through.


In the 3rd person quantum description the whole chain is entangled as one system and so will be described by a single wave function. We can expect all the features of the three-electron case, elucidated earlier, to still hold – such as robust quantum computing power, a potentially vast superposition of states, and a means to store the results of measurements in a form of memory while still maintaining entanglement – but, additionally, a complex non-linearity now emerges. The wave function interacts with itself. That is, the end of the entangled chain at “A” (in figure 12) will interact with a magnetic field induced by the chain at end “B” (in figure 12). This means that an NLSE is created, and, if the time dependence, too, is nonlinear, then the system is not functioning merely as a typically quantum computer, but, potentially has the power to solve any NP problem in polynomial time.


The system, in general, may exist in a vast superposition of states, but two broad categories of states are of special interest for characterizing the system. First, those states where substantially all the electrons near end “B” have spin up, and second, those states where the electron spins near end “B” are evenly mixed up and down. In the former case, illustrated in (figure 13 – top), the electron chain at end “A” experiences a magnetic field generated by end “B” similar to the external magnetic field of the Stern-Gerlach apparatus. With the magnetic field turned “on” in this case, the beam at end “A” is split depending on its spin state. In the latter case, illustrated in (figure 13 – bottom), the electron chain at end “A” experiences no magnetic field because the electron spins at end “B” mostly cancel each other out. The chain at end “A” is not split because the magnetic field is effectively turned “off”. Each state has a coefficient, \alpha_k , and so the likelihood that the system will have the magnetic field turned “on”, as in the first category, will be related to the aggregate probability associated with these coefficients, and, likewise, for the latter category with the field “off”. Beyond these two categories there are other interesting states the system could be in – for instance with substantially all spins pointing down, effectively creating an inverted Stern-Gerlach apparatus. We merely call out these two as being illustrative of the fact that measurement of this system reveals a complex nonlinear dependence on itself. Measurement of the system does not mean just measuring the state of the electron spins near end “A”, it means measuring whether the system induced a magnetic field or not as well.


Figure 13: Two broad categories of the electron system are remarkable. (Top) The category of states that have substantially all of the spins at end B pointing up – more or less replicating the magnetic field of the Stern-Gerlach apparatus, and (bottom) the category of states that have end B in a roughly evenly mixed distribution of spins up and down – effectively producing no net magnetic field.


Especially interesting, however, is the 1st person perspective: the system no longer has choice forced upon it like the prior cases we’ve examined. In this configuration, the system can choose whether to subject itself to making a choice or not. That is, by choosing the state of the electrons near end “B”, the system is choosing whether to use a Stern-Gerlach-like magnetic field to split itself at end “A”. Something resembling real free will has emerged: the ability not just to make a choice, but to choose what choices to make, and, even, whether to make a choice at all! To the system, each state in its quantum superposition feels preferable in relation to the \alpha_k coefficient of that state. The system prefers to turn “on” the magnetic field and “think” about some choice in direct relation to the aggregate \alpha_k coefficients that apply to the “all spins up” category of states as shown in (figure 13 – top). Similarly, the system prefers to ignore “thinking” about a particular choice in relation to the aggregate \alpha_k that correspond to the “spins evenly mixed” category of states as shown in (figure 13 – bottom). The system at once follows the laws of quantum mechanics, and has the freedom to choose what choices to make at the same time.


Figure 13 B: Breather interactions of the NLSE from “Breather interactions, higher-order rogue waves and nonlinear tunneling for a derivative nonlinear Schrödinger equation in inhomogeneous nonlinear optics and plasmas” by L. Wang et. al.


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