# I Do we need a reference frame in Quantum Hilbert space?

#### facenian

Consider the 4 maximally entangled states, eg |ud>-|du>. If a Martian chose them for basis states then he would see them as separable
I see a tautology here, |ud>-|du> is en entangled state because it cannot be put in the form $| a>|b>$ where a and b are vectors in their respective spaces. I you redefine |ud>-|du> as a new vector in the product space you have simply mathematically erased entanglement by redefining the Hilbert space that now, in a sense, is no longer a product space because now |ud>-|du> is by definition a single vector.
This way of looking at entanglement have no physical implications.

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#### Robert Shaw

No, that's false, read my post again, I said exactly the opposite. The entanglement is independent of the reference frame. Every observer in every reference frame will conclude that Alice and Bob's particles are entangled. See post #42 for a proof. If the Martians make a conclusion about Alice and Bob's subsystems, the don't see different subsystems. They see the same subsystems from from a different reference frame.

Entanglement is independent of the reference frame. A change of reference frame is induced by unitary transformation and as post #42 proves, unitary transformations won't change any physically detectable fact about the world. If you deny this, then please point out the mistake in the utterly trivial proof I have given.

Observables can be tailored to change the entanglement of any pure state
N. L. Harshman and Kedar S. Ranade
Phys. Rev. A 84, 012303 – Published 5 July 2011

Observables and entanglement in the two-body system
N. L. Harshman
AIP Conference Proceedings 1508, 386 (2012)

Entanglement or separability: The choice of how to factorize the algebra of a density matrix
Walter Thirring, Reinhold A. Bertlmann, Philipp Köhler, Heide Narnhofer
https://arxiv.org/abs/1106.3047

Factorization and entanglement in quantum systems

Jon Eakins and George Jaroszkiewicz

Published 17 December 2002 • Journal of Physics A: Mathematical and General,Volume 36, Number 2

#### rubi

Observables can be tailored to change the entanglement of any pure state
N. L. Harshman and Kedar S. Ranade
Phys. Rev. A 84, 012303 – Published 5 July 2011

Observables and entanglement in the two-body system
N. L. Harshman
AIP Conference Proceedings 1508, 386 (2012)

Entanglement or separability: The choice of how to factorize the algebra of a density matrix
Walter Thirring, Reinhold A. Bertlmann, Philipp Köhler, Heide Narnhofer
https://arxiv.org/abs/1106.3047

Factorization and entanglement in quantum systems

Jon Eakins and George Jaroszkiewicz

Published 17 December 2002 • Journal of Physics A: Mathematical and General,Volume 36, Number 2
All these references support my position, because they are concerned with different choices of subsystems. A change of reference frame, however, does not change the subsystem and hence does not affect the entanglement of the choosen subsystems. There are only four equal signs in my proof in post #42. If you disagree, then please explain, which one of them is incorrect.

#### Robert Shaw

Observables can be tailored to change the entanglement of any pure state
N. L. Harshman and Kedar S. Ranade
Phys. Rev. A 84, 012303 – Published 5 July 2011

Observables and entanglement in the two-body system
N. L. Harshman
AIP Conference Proceedings 1508, 386 (2012)

Entanglement or separability: The choice of how to factorize the algebra of a density matrix
Walter Thirring, Reinhold A. Bertlmann, Philipp Köhler, Heide Narnhofer
https://arxiv.org/abs/1106.3047

Factorization and entanglement in quantum systems

Jon Eakins and George Jaroszkiewicz

Published 17 December 2002 • Journal of Physics A: Mathematical and General,Volume 36, Number 2
You seem to be linking "reference frame" to "unitary transformation" in a way that makes no sense in the context of my question.

Of course a unitary transformation will trivially produce the result you cite. That is not the point I am making.

I am talking about two Earth observers and two Martian observers. Their observable operators are different.

When the Martians make observations on the same state as the Earthpersons they get different results and draw different conclusions about separability.

You seem to be making an assumption that the reduced states for Earth and Mars are the same but that is not correct in the case of the model we are discussing.

#### Robert Shaw

You seem to be linking "reference frame" to "unitary transformation" in a way that makes no sense in the context of my question.

Of course a unitary transformation will trivially produce the result you cite. That is not the point I am making.

I am talking about two Earth observers and two Martian observers. Their observable operators are different.

When the Martians make observations on the same state as the Earthpersons they get different results and draw different conclusions about separability.

You seem to be making an assumption that the reduced states for Earth and Mars are the same but that is not correct in the case of the model we are discussing.
In your terminology, you are assuming that subsystems for Earth and Mars are the same. That assumption is neither necessary nor correct.

#### rubi

You seem to be linking "reference frame" to "unitary transformation" in a way that makes no sense in the context of my question.
A change of reference frame is always induced by a unitary transformation. It absolutely is relevant to your question and you should make an effort to understand its relevance. It seems to me that you haven't understood how reference frames are dealt with in quantum theory.

I am talking about two Earth observers and two Martian observers. Their observable operators are different.
If they are not related by a unitary transformation, then they concern different subsystems and then trivially, the notion of entanglement changes as I have already said in my post #8. However, it has absolutely nothing to do with reference frames. A choice of reference frame still does not influence entanglement in the slightest.

You seem to be making an assumption that the reduced states for Earth and Mars are the same but that is not correct in the case of the model we are discussing.
If the same subsystems are considered, then the Martians will obtain the same reduced states (up to unitary equivalence). It's trivial that different subsystems needn't be entangled. Just because Alice is entangled to Bob, it doesn't mean Eve is entangled to Charlie. Not everything is entangled with everything. So what? However, Eve and Charlie will still consider Alice and Bob to be entangled.

In your terminology, you are assuming that subsystems for Earth and Mars are the same. That assumption is neither necessary nor correct.
It has nothing to do with Earth and Mars. Alice is one subsystem and Bob is another subsystem. It doesn't matter whether you observe those subsystems from Earth or Mars.

#### facenian

You seem to be linking "reference frame" to "unitary transformation" in a way that makes no sense in the context of my question.
A change of reference frame is always induced by a unitary transformation. It absolutely is relevant to your question and you should make an effort to understand its relevance. It seems to me that you haven't understood how reference frames are dealt with in quantum theory
I believe rubi is right, you can pick an arbitrary observer but this does not mean you can select an arbitrary sate, different observers are constrained by unitary transformations by Wigner's theorem.

#### kith

I think you are talking past each because Robert partly uses the terminology in a nonstandard way. Wenn Robert says "reference frame", he refers to what is more commonly called "tensor product factorization of the Hilbert space". (I think everybody here agrees that for a given factorization, the amount of entanglement of a state doesn't depend on the basis)

The second reference in my post #34 gives an example where different factorizations seem to be at least conceptually meaningful: we can look at the hydrogen atom as a two-particle system which consists of an electron and a proton or we can look at it as a center-of-mass degree of freedom and an "effective" electron in a potential. In the first factorization, a typical state is entangled, in the second, it is not (the COM movement is independent of what happens internally in the absence of external fields).

But in Robert's example, what does it mean to use the wildly different factorization of the Martians? It would yield strange non-local subsystems (two degrees of freedem each of which somehow includes half of Alice's particle and half of Bob's particle). Robert sidesteppes this important physical point by saying that the Earthians and the Martians "observe the same state" (of the combined system). But neither the Earthians nor the Martians observe states, they observe the behavior of subsystems.

So what's missing is how could measurements on the subsystems which are consistent with the Martian factorization be implemented? As far as I can see, the cited references don't claim that this is possible but instead note that what they do is mostly of theoretical interest and may lead to better calculation techniques.

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#### kith

Also to be clear: if spatially localized observers like Alice and Bob don't have access to such factorizations in principle, this would imply that the Martians themselves have to be non-local entities. Which would kind of put this example into the realm of esoterism. ;-)

#### Robert Shaw

Also to be clear: if spatially localized observers like Alice and Bob don't have access to such factorizations in principle, this would imply that the Martians themselves have to be non-local entities. Which would kind of put this example into the realm of esoterism. ;-)
Localisation does not exist in a qubit model. Alice and Bob are not spatial. To make them so would require a much more complex model, with position variables as well as Up/Down. Same for two qubit models.

Quantum information theory studies such simple models because they lead to significant discoveries. I would not dismiss them as esoteric.

They are certainly idealisations. But everything in physics is an idealisation. Engineers try to model reality, and do a pretty good job.That's why I set up the simple toy model. It's what we physicists use, we seldom study highly complex multiparameter models.

Separability is not problem-free in the way you suggest. Even in classical physics 6N dynamic variables could either be N objects with 6 parameters or 6 objects with N parameters. "Particle" has no universally accepted definition in physics.

There is much to discover concerning the problem of separability. It certainly isn't fully explored.