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There is a need to use open quantum systems in describing the reality since, in general, the real systems are often found correlated with the environment whose properties cannot be realized in closed quantum systems.

But how justified are we in imposing the condition that evolutions must always be linear? Leaving the mathematical complications the non-linearity would cause, could the non-linear models of evolution be more accurate to reality (at least in some instances)?

Mark Mitchison
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exp ikx
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1 Answers1

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First of all, let me point out that there are theories that propose nonlinear extensions to quantum mechanics (for instance Weinberg's nonlinear quantum mechanics). But there are very strong arguments against those approaches. Here are my favourites:

  • Experiments. If we could have nonlinear quantum evolution, this should be visible in experiments. However, that hasn't happened and the precision of experiments puts very sharp bounds on what type of nonlinearity might still be possible. I have listed a few experiments here: Experimental evidence of linearity of states
  • Faster than light communication. The non-cloning theorem from quantum mechanics stating that you cannot clone an unknown quantum state, follows easily from the linearity postulate of quantum mechanics. However, it is also necessary to prohibit faster-than-light communication. In other words: If you allow nonlinearities in your time evolution, then faster-than-light communication would be possible. This has been studied in various settings and degrees of sophistication - let me just mention the derivation in Weinberg's model of nonlinear qm by Polchinski (https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.66.397) and the more recent and more abstract derivations in https://arxiv.org/abs/1411.1768 and https://www.sciencedirect.com/science/article/pii/037596019090786N)
  • Immensely powerful quantum computers. There is an interesting argument by Abrams and Lloyd (https://arxiv.org/abs/quant-ph/9801041) saying that if quantum evolution was non-linear, then this could be exploited to solve NP complete problems and even #P complete problems in polynomial time - and this is not something that seems possible given anything we know.
Martin
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