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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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  • $\begingroup$ @ Dan Yand In fact I am checking the problem of classical approximation of quantum state/evolution. I believe the essential answer should be to check a certain kind of complexity in the structure of the Hilbert space. Entanglement can also be regarded as a complex pattern on Hilbert space. But it seems it's not directly related with the operational complexity (as a quantum computation) to prepare it from a product state. So they have different metrics. I would like to check the difference of their metrics. $\endgroup$ – XXDD Oct 29 '18 at 2:08
  • $\begingroup$ @Dan Yand Should the metric be application specific? Or at least from the 'classical approximation' task's point of view, there should be an optimal metric to judge if a state can be classically approximated. I believe this task plays a key role in exploring the structure of the Hilbert space. $\endgroup$ – XXDD Oct 29 '18 at 2:18
  • $\begingroup$ @Dan Yand Thanks. By 'local observables that aren't too complicated', do you mean an equivalent basis transformation on the quantum state? So if it's not 'too complicated', then it will not change the entanglement too much? I would like to stay with a given computational basis and allow no non-local basis transformation so that the entanglement is not changed. We see simple quantum operation may lead to high entanglement, but the simple qantum operation sometimes can be classically simulated, then the high entanglement does not mean a strong non-classicalness. $\endgroup$ – XXDD Oct 29 '18 at 8:19
  • $\begingroup$ @Dan Yand If we choose a specific observable, then this is equivalent to first transform the state. For me, both the entanglement and complexity should be defined w.r.t. a fixed basis. $\endgroup$ – XXDD Oct 29 '18 at 9:27
  • $\begingroup$ cross-posted on quantumcomputing.SE $\endgroup$ – glS Nov 5 '18 at 13:51
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As you noted, it is true that there is some coincidence on geometric entanglement and quantum circuit complexity. But there are also several counterexamples so we should distinguish between them:

1) Haar random states and random physical states: it is known that true random state in a Hilbert space of n-qubits cannot be attainable from a product state with poly(N) gates (https://arxiv.org/abs/1102.1360). However, using O(N^3) random two-qubit gates, you can make a quantum state with such a large entanglement (https://arxiv.org/abs/1109.4391). Even though these studies used (Reyni) entanglement entropy for the entanglement measures, I am sure that geometric entanglement also shows similar behavior. Indeed, I have some numerical evidence for that (you may see e.g. https://doi.org/10.1103/PhysRevA.93.042314). So entanglement and circuit depth are different. You may also see https://arxiv.org/abs/1310.2702 in this context.

2) After applying $N$ random 2-qubit gates sequentially to a product state of $N$ qubits, the maximum overlap to a product states $e^{-aN}$. This also contrasts to the GHZ type of states that you need $N$ gates to prepare, but the maximum overlap to a product state is given by a constant.

So when they behave similarly? For a shallow depth (O(N^2)) quantum circuit with random unitary two-qubit gates, geometric entanglement grows as the number of gates increases. When the gates are special (such as CNOT) or the number of gates are too large, this coincidence breaks down.

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  • $\begingroup$ Thanks for the answer and literature. Entanglement and complexity can be regarded as two different metrics on Hilbert space, I am just wondering if there exists a transformation between them. It seems there is no simple answer to it. $\endgroup$ – XXDD Nov 7 '18 at 10:52

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