How ‘spooky’ is quantum physics? The answer could be incalculable

[Demystifier]

A new mathematical result seems to have very important implications for quantum mechanics.
https://www.nature.com/articles/d41...3C1Vfyn17TZshyCRrO8nX6D5Iq4CGahZ_DXgfeQ_Ii0Z8

The original paper is a very hard read:
https://arxiv.org/abs/2001.04383

 

[Demystifier]

Scott Aaronson on that:
https://www.scottaaronson.com/blog/...RqH6iwR0WHFh_vFPO63H7xM9avo_o#comment-1828159

In simple terms, essentially the result is that an infinite amount of entanglement cannot be approximated by a finite amount of entanglement. So I would say that it's just one more form of the general idea that some properties of quantum systems in infinite dimensional Hilbert spaces cannot be well modeled by finite dimensional Hilbert spaces.

 

[A. Neumaier]

some properties of quantum systems in infinite dimensional Hilbert spaces cannot be well modeled by finite dimensional Hilbert spaces.

This is not really surprising. To give the simplest example:

In finite dimensions, the trace of a commutator is always zero. But the commutator of position and momentum of a harmonic oscillator (which is modeled by an infinite-dimensional Hilbert space) is a multiple of the identity, and its trace is not even defined.

 

[PeterDonis]

an infinite amount of entanglement

Am I right in thinking, after briefly skimming Aaronson's post, that an infinite amount of entanglement requires a pair of infinite-sized quantum computers?

 

[chirhone]

https://www.physicsforums.com/threa...er-could-be-incalculable.983212/#post-6287422
https://www.nature.com/articles/d41...3C1Vfyn17TZshyCRrO8nX6D5Iq4CGahZ_DXgfeQ_Ii0Z8

"Wehner says, because all applications use quantum systems which are ‘finite’. In fact, it could be difficult to even conceive an experiment that could test quantum weirdness on an intrinsically ‘infinite’ system, she says.".

The classical world has no quantum weirdness (like spooky at a distance) because one needs so much isolation just to display quantum effects. But is there somehow still quantum weirdness embedded in the classical world, only hidden?

This especially pertains to the idea that non-locality being the natural form of this universe instead of being an exception. Can you give an example of how non-locality can be natural (or be hidden) in the classical world?

 

[bahamagreen]

This especially pertains to the idea that non-locality being the natural form of this universe instead of being an exception. Can you give an example of how non-locality can be natural (or be hidden) in the classical world?

Maybe Relativity? Points' neighborhoods in spacetime are within the hyperbolic bounds of light rays from them, the absolute magnitudes of spacetime intervals between them non- transitive (Minkowski).

Alice and Bob - null separated
Bob and Carol - null separated
Alice and Carol - may be indefinitely far apart

 

[PeterDonis]

Points' neighborhoods in spacetime are within the hyperbolic bounds of light rays from them

No, this is not correct. The neighborhood of an event includes points that are timelike, null, and spacelike separated from it; the light cone does not bound the neighborhood. The topology of spacetime is still $R^4$ even though the metric on it is not the one induced by that topology.

 

[hilbert2]

Am I right in thinking, after briefly skimming Aaronson's post, that an infinite amount of entanglement requires a pair of infinite-sized quantum computers?

Maybe one could create a quantum field configuration where there's a nonzero probability to find excitations (photons, fermions, etc.) of arbitrarily high frequency, such that there's a lot of entanglement in which modes are simultaneously excited. But the probability of finding the system having total energy greater than $E$ must approach zero when $E\rightarrow\infty$.

 

[Demystifier]

Maybe one could create a quantum field configuration where there's a nonzero probability to find excitations (photons, fermions, etc.) of arbitrarily high frequency, such that there's a lot of entanglement in which modes are simultaneously excited. But the probability of finding the system having total energy greater than $E$ must approach zero when $E\rightarrow\infty$.

It may be worthwhile to mention that even the vacuum (of free scalar field in Minkowski spacetime) contains an infinite amount of entanglement. It can be seen from computing the entanglement entropy (in a finite subregion), which turns out to be UV divergent.

 

[hilbert2]

It may be worthwhile to mention that even the vacuum (of free scalar field in Minkowski spacetime) contains an infinite amount of entanglement. It can be seen from computing the entanglement entropy (in a finite subregion), which turns out to be UV divergent.

I remember that sometime in early 2000s there was still a debate about how to properly quantify the degree of entanglement in quantum systems. Here's an example of a paper about that:

https://arxiv.org/pdf/quant-ph/0109081.pdf

The entanglement entropy seems to be a much older concept. Is there some reason why it's not useful for quantifying the entanglement in systems of a few qubits?

 

[Demystifier]

The entanglement entropy seems to be a much older concept. Is there some reason why it's not useful for quantifying the entanglement in systems of a few qubits?

The entanglement entropy involves a logarithm (of the density matrix), which is not easy to compute. But I think it's not a big problem for systems with only a few qubits.

 

[DennisN]

It may be worthwhile to mention that even the vacuum (of free scalar field in Minkowski spacetime) contains an infinite amount of entanglement.

Interesting! And this is because the Hilbert space is infinite? And if so, does that mean that any volume (big or small so to say) of space contains/can contain an infinite amount of entanglement?

 

[Demystifier]

Interesting! And this is because the Hilbert space is infinite?

Yes.

And if so, does that mean that any volume (big or small so to say) of space contains/can contain an infinite amount of entanglement?

Yes. It stems from the continuum nature of space, i.e. from the UV divergence.

Discrete theories of space may not have this problem. A field theory on the lattice may also have an infinite-dimensional Hilbert space because the value of the field (at a given space point) may take an arbitrary value from the continuum, so it may have an infinite amount of entanglement in some state, but at least it will not have an infinite amount of entanglement in the ground state.

 

[atyy]

A field theory on the lattice may also have an infinite-dimensional Hilbert space because the value of the field (at a given space point) may take an arbitrary value from the continuum, so it may have an infinite amount of entanglement in some state, but at least it will not have an infinite amount of entanglement in the ground state.

How does one get this result?

 

[chirhone]

Yes.Yes. It stems from the continuum nature of space, i.e. from the UV divergence.

Discrete theories of space may not have this problem. A field theory on the lattice may also have an infinite-dimensional Hilbert space because the value of the field (at a given space point) may take an arbitrary value from the continuum, so it may have an infinite amount of entanglement in some state, but at least it will not have an infinite amount of entanglement in the ground state.

Can you give a practical example of how non-locality can exist amidst this classical world and have effects on matter?

I thought having classical world means all the quantum effects were gone from decoherence.

 

[Demystifier]

Can you give a practical example of how non-locality can exist amidst this classical world and have effects on matter?

I thought having classical world means all the quantum effects were gone from decoherence.

I'm not sure what do you mean by non-locality in this context.

 

[Demystifier]

How does one get this result?

Which result exactly? I've made several claims so I'm not sure which of those needs clarification.

 

[chirhone]

I'm not sure what do you mean by non-locality in this context.

Entanglement everywhere even edge to edge of a galaxy is non-locality bohmian-wise (not MWI where there is no non-locality present but shuffling statistics only).

I was asking what possible effects if edge to edge of a galaxy non-locality exist. Or is there absolutely no effects in this classical world because decoherence prevails?

 

[Demystifier]

I was asking what possible effects if edge to edge of a galaxy non-locality exist. Or is there absolutely no effects in this classical world because decoherence prevails?

For practical purposes, those effects are negligible due to decoherence.

 

[atyy]

Which result exactly? I've made several claims so I'm not sure which of those needs clarification.

I didn't understand why you said that the ground state of an infinite lattice system does not have infinite entanglement.

I also didn't understand why a non-ground state of an infinite lattice system might have infinite entanglement.

 

[Demystifier]

I didn't understand why you said that the ground state of an infinite lattice system does not have infinite entanglement.

I also didn't understand why a non-ground state of an infinite lattice system might have infinite entanglement.

First, I didn't say that the lattice is infinite. Second, even if it is (in the IR divergence sense), I say that the ground state entanglement entropy associated with a finite region has a finite entanglement entropy. That's because this entanglement entropy is proportional to the area of the region's boundary, which is finite. See e.g. https://arxiv.org/abs/0808.3773

Concerning non-ground states, in an infinite-dimensional Hilbert space there are always states with infinite entanglement entropy. For instance, the maximally mixed state in the Hilbert space ${\cal H}$ has von Neumann entropy
$$S={\rm log \;dim}{\cal H}$$

 

[QLogic]

In simple terms, essentially the result is that an infinite amount of entanglement cannot be approximated by a finite amount of entanglement

It's a bit more than that. It says that the analogue of the Tsirelson bound is uncomputable for infinite dimensional quantum systems. Or alternatively if you could harness all of the entanglement in a region of spacetime you could solve the halting problem. Of course you can't actually harvest it.

 

[Demystifier]

It's a bit more than that. It says that the analogue of the Tsirelson bound is uncomputable for infinite dimensional quantum systems. Or alternatively if you could harness all of the entanglement in a region of spacetime you could solve the halting problem. Of course you can't actually harvest it.

Can you suggest some source to learn more about entanglement harvesting in general?

 

[QLogic]

Can you suggest some source to learn more about entanglement harvesting in general?

I wish I could recommend a textbook but it's such a new field there isn't any.

The references in this paper provide a nice review of the literature, as does the paper itself:
https://arxiv.org/abs/1808.01764

This is a simplified worked example:
https://arxiv.org/abs/1808.01764

Currently the protocols yield poor harvesting, so this is an interesting engineering problem to improve on this.

 

[atyy]

The shape of MIP* = RE
Henry Yuen
https://quantumfrontiers.com/2020/03/01/the-shape-of-mip-re/