After having a discussion with a quantum computing colleague, a question came up: is there any meaningful way to measure entanglement (or something related to it) in a solid-state many body system where you don't have the luxury of individual qubit control? Specifically, I am interested in the case of the solid-state, not cold atoms where you can do quite marvelous things at the single qubit level. I suppose there might be various entanglement witnesses out there, but I am wondering which among them is (in-principle) measurable in a real-life solid-state experiment.

To explain what I mean by "meaningful", please not that, as the comments indicate, I acknowledge that measuring the true degree of entanglement of a solid-state system is extremely difficult for fundamental reasons. It is near impossible to put a physical cat into a superposition of two states, for example. However, I would contend that in solid-state systems one is only interested in low-energy phenomena, so entanglement is only of interest for a small subset of the total degrees of freedom. For example, core electrons make up the majority of the degrees of freedom in a solid, but play almost no role in most properties. On the other hand, valence electrons are much fewer in number, but dictate most of the interesting chemical and physical properties. So, by "meaningful" measure of entanglement, I mean a method which allows you to ignore the inactive degrees of freedom that don't contribute anything to the physics of the solid-state system of interest.

As an example of the use of such an entanglement measurable, quantum phase transitions are quite interesting in the context of solid-state systems, and are thought to give rise to enormous entanglement among degrees of freedom as one nears the critical point. If one had an experimentally measurable quantity describing the entanglement, one should be able to identify one of the key properties of quantum phase transitions.

A somewhat related question that I posted several years ago about Quantum Fisher Information https://physics.stackexchange.com/questions/274891/interpreting-the-quantum-fisher-information

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    $\begingroup$ I don't know about entanglement measures for macroscopic systems, but it seems even detecting macroscopic superpositions is difficult see: arxiv.org/pdf/2009.07450.pdf $\endgroup$ – Condo Sep 18 at 13:21
  • $\begingroup$ Well I think that statement comes from a different viewpoint than what I am thinking. It is normal in solid-state systems to separate out degrees of freedom by energy/length scale. Yes it is very difficult (near impossible) to measure a macroscopic superposition of the center-of-mass position of bulk object, but it is much more straightfoward to put a solid into a superposition of different electronic/vibrational states. $\endgroup$ – user157879 Sep 19 at 4:43
  • $\begingroup$ Even quantum phase transitions only give rise to a tiny amount of entanglement, compared to what is possible. $\endgroup$ – Norbert Schuch Sep 19 at 20:11
  • $\begingroup$ @NorbertSchuch True, but it is a general fact that most solid-state phenomena (transistors, diodes, superconductivity) only involve a very tiny amount of electrons/degrees-of-freedom compared to the total possible. Despite this, you still are talking about way more DOF/entanglement compared to single atoms/qubits though. I think my question is still of practical importance and meaningful to ask in any case. $\endgroup$ – user157879 Sep 20 at 5:04
  • $\begingroup$ @user157879 There is quite a body of literature on entanglement and quantum phase transitions. And on entanglement in many-body systems. Your question is rather broad. What do you mean by "meaningful"? How is the short question you ask at the beginning related to the discussion you give afterwards? $\endgroup$ – Norbert Schuch 2 days ago

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