Both quantum entanglement and quantum state complexity are important in quantum information processing. They are usually highly correlated, i.e., roughly a state with a higher entanglement corresponds to a higher quantum state complexity or a complex state is usually highly entangled. But of course this correspondence is not exact. There are some highly entangled states that are not complex quantum states, for example quantum states represented by branching MERA.

On the other hand, if we use the geometric measure of entanglement (defined as the minimal distance to the nearest separable state w.r.t. a certain distance metric in Hilbert space) to justify the entanglement, then it seems it's very similar with the definition of quantum state complexity (the minimal distance to a simple product state). If we only consider pure states and choose the same distance metric for them, for example, the Fubini-Study distance or Bures distance, then they are really almost identical.

Of course, when we are talking about state complexity, it's better to use the more physically motivated 'quantum circuit complexity' to measure the distance. But still, this distance can also be used to define the geometric measure of entanglement(maybe it's not a perfect distance measure for entanglement).

Then what's the relationship between entanglement and quantum state complexity? Are they essentially two different distance measures on Hilbert space? What should be the optimal metrics for them?

Or, if entanglement and complexity are both distance measures on the Hilbert space, can we find a transformation between these two metrics?

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    $\begingroup$ can you provide some reference(s) for your use of "quantum state complexity"? $\endgroup$
    – glS
    Oct 29, 2018 at 15:20
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    $\begingroup$ @glS Literature updated. Mainly I follow the work of Nielsen at arxiv.org/abs/quant-ph/0701004. Other definitions are similar and the only difference is they use different metric to measure the distance between states, for example Bures distance or dynamic distance for mixed states. I think Nielsen's quantum circuit complexity is the most physical one. $\endgroup$
    – XXDD
    Oct 29, 2018 at 15:30
  • $\begingroup$ Are cat states far from separable states? Because they are very easy to prepare. $\endgroup$
    – user1813
    Oct 29, 2018 at 20:52
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    $\begingroup$ @AHusain Personally I believe that computational complexity has a physical meaning for example it's related with the quantum phase transition. Susskind argued that complexity=action. So physics does know the complexity, but we do not know which complexity metric physics uses and why physics prefers that complexity metric. $\endgroup$
    – XXDD
    Nov 5, 2018 at 14:41
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    $\begingroup$ This may not answer the question. But it will definitely add more fuel to the thoughts in this thread. simons.berkeley.edu/news/… $\endgroup$
    – Vivek G
    Jul 16, 2021 at 2:42