How can you get entangled ensembles if QM requires monogamy of entanglement?

If I understand properly, QM says a particle can only be entangled with one other particle. How then can you get entangled ensembles of particles, required for example in various theories of decoherence or in notions of fast scrambling in black holes or in tensor networks? I'm sure I'm missing something, and those examples are not necessarily related, but it seems they all require distributed entanglement in some form or other.

• QM does not say that. Consider a state like $\frac{1}{\sqrt{2}} \left ( \left | 000 \right > + \left | 111 \right > \right )$. Commented Jun 24, 2021 at 19:25
• monogamy of entanglement is indeed a thing, but it doesn't say that "you can only be entangled with one another particle". Rather, the statement is that if a particle is maximally entangled with another particle, then it cannot be entangled with something else. The GHZ example does not contradict this, as the partial states are not maximally entangled. See en.wikipedia.org/wiki/Monogamy_of_entanglement
– glS
Commented Jun 30, 2021 at 9:16
• >Thanks. This helps. Commented Jul 1, 2021 at 16:59

Nowhere does QM say that a particle can only be entangled with one other particle.

QM says a particle can only be entangled with one other particle

This is not correct.

These are the postulates of quantum mechanics, chosen to fit the data as far as observations and measurements of the present day.

Of these the wavefunction postulate is crucial for the definition of ".entanglement"

It is one of the postulates of quantum mechanics that for a physical system consisting of a particle there is an associated wavefunction .

For two bodies there exist solutions of Schrodinger or Dirac or Klein Gordon equations with a wavefunction.

The quantum mechanical many body problem is approached with quantum field theory, a theoretical level and calculational method above the simple solution of QM equations, but based on the postulates , including the wavefunction, of basic quantum mechanics. One gets an approximation to a collective wavefunction, for the system under study, and the particles there are "entangled" in principle, by the interactions, energy and momentum conservation and all the quantum numbers involved

Or for a large number of atoms in principle there exist quantum mechanical models that approximate the wavefunction of large systems of particles , as for example in the band theory of solids.

• Great answer!Btw we need more materials scientists on this site!An addition to what you said the wavefunctions of electrons in solids are not 'free'they are bound to the rules of molecular orbital chemistry. Commented Jun 24, 2021 at 19:42