Wednesday, May 09, 2012

Is everything entangled?

Roman Buniy and Steve Hsu published a preprint named
Everything Is Entangled
which claims that... everything is entangled. Because of the cosmological evolution, everything evolves into an entangled state and the entanglement easily transgresses the cosmic horizon. However, you can't observe this entanglement too easily because the entanglement is "diluted" so a randomly chosen pair of nearby objects won't inherit too much of this entanglement.

Maybe accidentally, maybe because of the preprint, a user named "confused" asked the following question at the Physics Stack Exchange:
Given entanglement, why is it permissible to consider the quantum state of subsystems?

The user makes the following points:
Quantum entanglement is the norm, is it not? All that exists in reality is the wave function of the whole universe, true? So how come we can blithely talk about the quantum state of subsystems if everything is entangled? How is it even possible to consider subsystems in isolation? Anything less than the quantum state of the whole universe at once. Enlighten me.
I am afraid that pretty much everything that is being written about these issues in the preprint as well as the discussion thread at Physics SE is confused, indeed.

First, it is true that the entangled states are "generic" according to any continuous uniform measure on the space of states (or density matrices). The states of the form\[

\ket{\psi_1} \otimes\ket{\psi_2}

\] represent a measure-zero, infinitely "special" subset of all the pure states that describe a composite system, i.e. all states in the tensor product of the two Hilbert spaces. After all, the dimension of the set of states which are tensor products is roughly \(d_1+d_2\) while the dimension of the tensor product is \(d_1\cdot d_2\) which is larger if \(d_1,d_2\gt 2\). However, the vanishing of the measure doesn't mean that unentagled states are unimportant.

The key thing that all the folks seem to miss is that once we measure i.e. learn the value of any observable, we are learning that the state is an eigenstate of the corresponding operator with the measured eigenvalue. And whenever we measure two commuting quantities \(X,Y\), we know that the state of the system is an eigenstate of both of them. If we measure a complete commuting set of observables, we know the pure state completely.

In the misleading "materialist" interpretation of the wave function, every measurement "collapses" the state of the system into an eigenstate of the observables that were measured.

For example, if we measure the momentum and polarization of one photon in Boston and the same observables of another photon in San Francisco, it may indeed be true that their state was entangled to start with. The entanglement meant nothing else than a predicted correlation between various quantities we could measure using these two photons. But once we measure the values, we actually learn what the right momenta and polarization axes are. It's irrelevant that they were uncertain, correlated, or uncorrelated before the measurement. After the measurement, they're certain and uncorrelated. The entanglement simply disappears during the measurement.

The entanglement is nothing else than a correlation in the predicted sets of probabilities for various future measurements; it only exists if there's an uncertainty about the character of the future measurements and/or their results. But once a particular measurement yields particular results, the entanglement – or any property of the wave function or density matrix before the measurement that is linked to the measurement – becomes an irrelevant trivium about the history. The actual state-of-the-art state of the composite system is well-known and unentangled.

And when the measured quantities can be separated to two sets, \(A\) and \(B\), which contain observables that commute with each other (in the same set as well as the other set) and which make \(A\) fully describe one subsystem (it is a maximal set of commuting observables) and \(B\) describe another subsystem, then the resulting state is an eigenstate of all elements of \(A\) and \(B\) and it is inevitably a tensor product i.e. unentangled state of an eigenstate of elements of \(A\) and eigenstates of elements in \(B\). Because we're facing a similar situation all the time, whenever we actually measure local objects or systems fully, we have to deal with unentangled states all the time.

Buniy and Hsu also seem to be confused about the topics that have been covered hundreds of times on this blog. In particular, the right interpretation of the state is a subjective one. Consequently, all the properties of a state – e.g. its being entangled – are subjective as well. They depend on what the observer just knows at a given moment. Once he knows the detailed state of objects or observables, their previous entanglement becomes irrelevant.

One may also argue that the entanglement between observables \(X,Y\) whose measured values cannot be compared by any observer in the future (especially for causal reasons) is unphysical. In fact, the black hole complementarity uses this inability to operationally decide whether such quantities \(X,Y\) are entangled or not: it postulates that their entanglement is mandatory because the observables aren't quite independent from each other. The degrees of freedom describing the black hole interior are complicated functions of degrees of freedom that describe the exterior of the same black hole, the black hole complementarity principle postulates. This assumption is pretty much guaranteed to lead to no demonstrably wrong conclusions – no contradictions – exactly because the measurements inside and outside the black hole (with some extra constraints on the locations and times) cannot be compared by any observer in the future. A similar comment holds for cosmic horizons.

When I read papers such as one by Buniy and Hsu, I constantly see the wrong assumption written everything in between the lines – and sometimes inside the lines – that the wave function is an objective wave and one may objectively discuss its properties. Moreover, they really deny that the state vector should be updated when an observable is changed. But that's exactly what you should do. The state vector is a collection of complex numbers that describe the probabilistic knowledge about a physical system available to an observer and when the observer measures an observable, the state instantly changes because the state is his knowledge and the knowledge changes!

Whenever some regions' causal separation is just temporary, i.e. whenever they're guaranteed to return to contact in the future, it must be possible to talk about the observables that may be measured in both regions and about their correlation. But whenever it's not the case, the discussions about trans-horizon entanglement etc. may easily become unphysical. Don't get me wrong: if you had a crisp mathematical description that would force you to make a particular conclusion about the trans-horizon correlations, it could make sense to talk about it. But if you don't have any crisp mathematical description of the type, there exists no physical justification why a physicist should be able to answer the question whether the observables behind each others' horizon continue to be correlated or entangled once the observers in these two regions make their measurement. The answer can't be obtained by a well-defined operational procedure. So you don't have to "admit" that they have to be correlated. Instead, you may say that the entanglement is gone as soon as observers in these two regions make their first observations. After all, the unentangled Ansatz is natural for separated subsystems and subsystems separated by a horizon are as separated as they can be.

You're allowed to assume that they're not correlated and you're allowed to assume that they are correlated. Much like the "axiom of choice" or its negation in the axiomatic systems of set theory, none of these two assumptions may lead to contradictions with facts you may actually prove by experiments (and extend by calculations and logical thinking). So it's an unphysical question – a matter of subjective preference – what you think about the state of the observables behind your cosmic horizon. Only correlations that may be measured need to have "unique, calculable values" in a complete physical theory. Talking about such unobservable correlations independently of a physical theory is unphysical.


  1. Lubos,

    If you really want to talk about these things - I know you are capable of adapting to new ideas- but your objections to the speculatrions refered to is a matter of philosophy and I think on a higher level- the idea of the axiom of choice is of course in your application a matter of deep and expert insight.

    If you want something on this topic I have asserted now for half a century and to really critique the physicality of things and this cosmic flow or design so to speak I am posting these concerns as old as the ancient of Greece and the Buddhist in as close to an analysis we have gotten.

    The quantum theory although part of the picture is not enough to explain things to those who want a more tangeable reason for mechanisms- and the entanglement question does not answer things but merely suggests a paradox outside the research or thoughts- what have you added here but that your view is of a closed and centered universe where you assume things vanish on some scales? This paradox is not better or makes sense anymore more than the ideas of those who say we can transcend these ideas of entanglement. Intuitive, but philosophic with confused subjective grounding of errors.

    I am not saying your ideas cannot be the grounding case for it can be seen that way as one of several core theories of everything.

    But how does this connect to your understanding of the possible limits to scales of our experiments in your earlier post?

    Well, our thoughts are not necessarily entangled in some mystical or even the blogsphere I presume... but in any case as it seems we are talking to ourselves or around each other- Einstein said he believed the moon there those times he could not see it- well he did not say those who believe the moon is there when we have never seen it are crackpot physicists-

    I accept your honor that my positions in your statements are beyond crackpot as that is the case when we exceed what is obeserverable and perhaps beyond even cardinality.

    Even those who take offence with what should be neutral, honest and objective as science hope you add your intellect to the new physics.

    Global warming is a great metaphor for such views of mystical or physical connections of things and I must say those who organize things here are rather blind to the realities we face are are as dishonest as scientist who fudge their data- they are annoying but they can vote (and here are losing)
    So can this issue be resolved by philosophy or not if science cannot decisively now do it?

    The PeSla

  2. Prove I am not a robot? well Lubos not sure the comment got thru so I will append it to my last intended post for reference.

    The PeSla

  3. I've really enjoyed this series of posts you've done clarifying (resurrecting?) the Copenhagen interpretation of QM. As someone with almost no background in physics or higher mathematics, I’ve been able to form a qualitative picture of it that makes more sense to me than most of the hype out there. My understanding of the wave function is that if we know some property of a particle, then we can apply a wave calculation to tell us what the probability is that that property might be when measured at some point in the future. We cannot know exactly what it will be because that property exists in a superposition of all the possible states, and it doesn’t “choose” a particular one until it is “forced” to with something like a measurement. If I have this correct, the one thing I haven’t been able to grok yet is how this measurement forces an entangled, correlated particle to also choose when they are out of contact with each other as in EPR.

  4. Maybe sorcery is real! By measuring something here and now we predetermine the outcome of another measurement elsewhere and later. Abracadabra!

  5. Would be fair to present authors response Steve Hsu: Entanglement and Decoherence

    Great blog, Lubos!