Dieter Zeh's MWI as Schrödinger equation + further assumptions

[gentzen]

TL;DR Summary

I decided to finally "find out" what Dieter Zeh had in mind, when he wrote that a little more than just a universal Schrödinger equation is required for explaining the stochastic nature of measurements

All the universe needs is the wave function and the Hamilton operator.

No, that is not enough, see Nothing happens in the Universe of the Everett Interpretation by Jan-Markus Schwindt. MWI proponents like Lev Vaidman agree with that assessment, and Dieter Zeh independently already remarked much earlier that "a little bit more" is needed, even so I never found out what "little addition" exactly he had in mind when he made that remark. (I read his remark in his "Physik ohne Realität: Tiefsinn oder Wahnsinn?")

That is not even a peer reviewed paper and I'm not talking about the Everett interpretation in the first place.

Sorry for derailing that discussion even further. My reference to Dieter Zeh's German book was unlucky, not just because it is not a peer reviewed paper, but also because I did not remember the exact place with the remark.

Since this has bothered me since a long time anyway, I now searched the book until I found that exact place in chapter "13 Wurzeln des Dekohärenzkonzepts in der Kernphysik". That chapter is an abriged translation of Roots and Fruits of Decoherence (Talk given at the Seminaire Poincare (Paris, November 2005)), and the relevant paragraph occurs already on the first page:

So let me first emphasize that by decoherence I do neither just mean the disappearance of spatial interference fringes in the statistical distribution of measurement results, nor do I claim that decoherence without additional assumptions is able to solve the infamous measurement problem by explaining the stochastic nature of measurements on the basis of a universal Schrödinger equation. Rather, I mean no more (and no less) than the dynamical dislocalization of quantum mechanical superpositions, which are defined in an abstract Hilbert space with a local basis (given by particle positions and/or spatial fields, for example). The ultimate nature of this Hilbert space basis (the stage for a universal wave function) can only be found in a unifying TOE (theory of everything), but does not have to be known for the general arguments.

I highlighted local basis, because sentences like "we rely on a single key principle: changes purely to the environment do not affect the probabilities one ought to assign to measurement outcomes in a local subsystem" from the abstract of arXiv:1405.7577 (a derivation of the Born rule by C. Sebens and S. Carroll) and Jan-Markus Schwindt's analysis both seem to hint that what is missing is a connection to a space-time structure. Or maybe a tensor product structure (for quantum computing), I guess the important part is just that talk about locality has to be meaningful. (Not so sure about talk of system and environment, probably less important, because it will remain somewhat arbitrary anyway.)

Another relevant paragraph mentioning assumptions occurs on page 5:

However, what I had in mind went beyond what is now called decoherence, since it was inspired by the above mentioned picture of an observer inside a closed quantum system. An external observer, who is part of the environment of the observed object, becomes entangled, too, with the property he is observing – just as the observer within the deformed nucleus is entangled with its orientation. He is thus part of a much bigger “nucleus” (or closed system): the quantum universe. So he “feels”, or can be aware, only of a definite value of the property he has measured (or separately of different values in different “Everett worlds”). All you have to assume is that his various quantum states which may exist as factor states in these different components of the global wave function are the true carriers of awareness. This is even plausible from a quite conventional point of view, since these decohered component states, which are a consequence of the Schrödinger equation, possesses all properties required to define observers, such as complexity and dynamical stability (memory). Indeed, these states are the same ones that would arise in appropriate collapse theories if they were, according to von Neumann’s motivation, constructed in order to re-establish a psycho-physical parallelism. But I do not see why such a modification, that just eliminates all “other” components from reality, should be required.

That sound like Zeh's Many Minds interpretation to me. I believe Zeh knew that it was not widely accepted, therefore I guess that this is not what he meant by "additional assumptions".

I also found paragraphs relevant to Zeh's opinion on derivations of the Born rule and the role of the initial condition of the universe in chapter "5 Physik ohne Realität: Tiefsinn oder Wahnsinn?" and chapter "18 Warum Quantenkosmologie?". Those can be found here, but there is no English version, nor "the slightest trace of peer review". The places in the "WebEssays" are page 16 "Alle Interpretationen postulieren die bornschen statistischen ... Versuche, sie aus der everettschen Interpretation abzuleiten, muß man als fehlgeschlagen ansehen." and page 3 "It is the initial condition of the universe that ... explains the origin of quasiclassical domains within quantum theory itself." Those places helped me to remember, why it was so difficult for me to be sure what exactly Dieter Zeh had in mind when he mentioned those "additional assumptions". (But in the meantime, at least I learned that Dieter Zeh had always warned about misinterpreting what can be explained by decoherence alone, so maybe it was just a mistake from my side to interpret his warning as "a remark about a hypothetical completion of MWI".)

 

[Quantumental]

This post deserves a lot more attention than it has received thus far. I was privileged enough to have a couple of short interactions with Zeh before his passing (this was ~decade ago) and he seemed quite fervent that the original Everett + Decoherence was all you needed. The last paragraph you cited seems to support this. Maybe something is lost on me, but is he not merely restating that the brain states are defined by decoherence? I.E. no additional mind-body dualism (which tends to be the case with Many Minds Interpretation) is single out as preferred.

 

[Nullstein]

I'm not sure what the argument in this thread is or why you quoted my posts, but I was talking about time evolution in ordinary quantum mechanics and not the MWI. There, time evolution of a state is given by the Schrödinger equation. I don't think there is a working version of MWI at this point, because of the lack of a derivation of the Born rule.

 

[PeterDonis]

I was talking about time evolution in ordinary quantum mechanics and not the MWI.

What do you mean by "ordinary quantum mechanics"? Note that this is the QM interpretations forum, so the topic of discussion is not how the basic math of QM makes predictions (which is what I would normally take the term "ordinary quantum mechanics" to refer to) but how various interpretations account for what is going on in quantum systems and quantum processes.

There, time evolution of a state is given by the Schrödinger equation.

By "there", do you mean "ordinary quantum mechanics" (whatever that is) or the MWI? I suspect you mean the former, but I think most people would say the latter (the MWI) is the interpretation that says time evolution is the Schrodinger equation and nothing else. Other interpretations would say something else besides the Schrodinger equation has to be involved when measurements are made, i.e., that the system's state does not evolve in time solely according to the Schrodinger equation during a measurement.

I don't think there is a working version of MWI at this point, because of the lack of a derivation of the Born rule.

Proponents of the MWI would not agree with this (some would claim the Born Rule can be derived in the MWI, others would claim it doesn't matter), although some critics of it probably would.

 

[Nullstein]

What do you mean by "ordinary quantum mechanics"? Note that this is the QM interpretations forum, so the topic of discussion is not how the basic math of QM makes predictions (which is what I would normally take the term "ordinary quantum mechanics" to refer to) but how various interpretations account for what is going on in quantum systems and quantum processes.

Some interpretations such as MWI or BM are not really interpretations, but different theories. By ordinary QM I mean all the interpretations that share the same standard mathematical foundations as can be found in the standard textbooks.

By "there", do you mean "ordinary quantum mechanics" (whatever that is) or the MWI? I suspect you mean the former, but I think most people would say the latter (the MWI) is the interpretation that says time evolution is the Schrodinger equation and nothing else. Other interpretations would say something else besides the Schrodinger equation has to be involved when measurements are made, i.e., that the system's state does not evolve in time solely according to the Schrodinger equation during a measurement.

It's standard terminology to consider time evolution to mean unitary evolution. Unitary evolution is a physical process. The collapse of the wave function is subjective and depends on the observer (in ordinary QM), see e.g. Wigner's friend. It's not usually considered to be a physical process, except in certain interpretations such as objective collapse models. But that's anyway not what gentzen is talking about here. He's specifically talking about MWI where the lack of the Born rule has to be made up for by some additional ingredient that replaces the Born rule (not the collapse).

Proponents of the MWI would not agree with this (some would claim the Born Rule can be derived in the MWI, others would claim it doesn't matter), although some critics of it probably would.

I don't think they would disagree (exceptions prove the rule). The lack of a derivation of the Born rule is considered to be a major open problem in MWI. All proposed derivations of the Born rule have been shown to make some tacit assumptions that basically assume the Born rule from the outset.

 

[PeterDonis]

Some interpretations such as MWI or BM are not really interpretations, but different theories.

As long as they make the same testable predictions as standard QM, they're interpretations. If they make testable predictions that are different from those of standard QM, so that an experiment could potentially distinguish between them, then they are different theories.

If you want to claim that either the MWI or BM are different theories, you need to provide references that describe what testable predictions they make that are different from those of standard QM and how those predictions would be tested. I'm not aware of any such references, but if you are, please give them. If you can't, then your claim quoted above is personal speculation and is out of bounds here.

It's standard terminology to consider time evolution to mean unitary evolution.

Maybe to you. I don't think it is to everyone. If you want to be sure you are clear, then if you mean specifically unitary evolution, you should use that term.

Unitary evolution is a physical process.

For some interpretations, yes. Unitary evolution applies to the wave function/state vector. Not all interpretations of QM consider the wave function/state vector to be the physical state of the system; for those that don't, unitary evolution is not a physical process.

The collapse of the wave function is subjective

In some intepretations, yes. But not in all. There are objective collapse interpretations.

and depends on the observer (in ordinary QM)

You can't help yourself to terms like "ordinary QM" based on your personal opinion. Really you shouldn't be using such idiosyncratic terms at all. Particularly not in this forum, where we are discussing QM interpretations and the subforum guidelines require you to be explicit about which interpretations you are using, and to provide references giving the basis for your understanding of those interpretations when requested.

that's anyway not what gentzen is talking about here.

I agree (see below), but you were the one who brought up the items I asked about. If you are going to stick to the thread topic, then stick to it.

He's specifically talking about MWI where the lack of the Born rule has to be made up for by some additional ingredient that replaces the Born rule (not the collapse).

Yes, I agree with this.

The lack of a derivation of the Born rule is considered to be a major open problem in MWI.

By some, yes. Not by all MWI proponents. (I personally do think it's a major open problem.)

 

[gentzen]

This post deserves a lot more attention than it has received thus far.

You mean, because Dieter Zeh would have deserved more attention? But for this post, who could I have expected to answer? Nullstein? Or rather, who do I hope will answer? tom.stoer! But then, why do I ask it here, and not on http://www.quanten.de/forum ? Because I fear that he wouldn't answer. And because of Nullstein. There are good reasons why he received a 2021 award.

he seemed quite fervent that the original Everett + Decoherence was all you needed. The last paragraph you cited seems to support this. Maybe something is lost on me,

Let me try to guess some background why Zeh hinted at "additional assumptions": At the time when Zeh wrote this, he was well aware of many attempted derivations of the Born rule (and similar MWI based attempted resolutions of the measurement problem), and he got the impression that those either were circular, or made some additional assumptions. He also had his own favorite additional assumption, but didn't want to focus on it. But the many worlds (or many minds) aspect is not what worried him, so he is a fervent supporter of Everett in this respect.

Let me try to guess what Zeh's own favorite additional assumption could be: There is some reasonable local basis (but we have to wait for a unifying TOE before we learn what it is), and the initial state of the universe was extemely simple (=lowest possible entropy) in that local basis. This is a true additional assumption, there is some observational evidence hinting that it contains some grains of truth, but far from sufficient for proving such an extremely bold assumption. And Zeh knew this. For example, David Wallace questions the strength of that evidence, by pointing out that: of course the entropy today is bigger than the entropy near the beginning of our observable universe, that is just what the second law of thermodynamics tells us to expect.

 

[gentzen]

I'm not sure what the argument in this thread is or why you quoted my posts,

This thread is about what "little addition" exactly Dieter Zeh could have had in mind, or whether I simply misinterpreted him, and his reference to "additional assumptions" meant something completely else, something much more trivial. My arguments here only revolve around that question.

I quoted your posts, because they are the context why I decided to finally try to make some progress on that question, and also because my reaction to you illustrate my efforts in the past to make sense of Dieter Zeh's remark.

And also, because those "additional assumptions" are the "more physical" part of the riddle, but your observations seem to contain hints how to crack the "more randomness and information theory related" part of the riddle. So from my perspective, it seems to make sense to clarify the difference between these two aspects of the riddle.

but I was talking about time evolution in ordinary quantum mechanics and not the MWI. There, time evolution of a state is given by the Schrödinger equation. I don't think there is a working version of MWI at this point, because of the lack of a derivation of the Born rule.

Indeed, you were not talking about MWI. (There is that lack of a derivation of the Born rule, but that doesn't invalidate MWI. It only hints that MWI might be just as incomplete as consistent histories.) When jbergman in post #42 quoted the entire passage to which I reacted, I noticed that you could have applied that same argument to a classical computer, that all its 0s and 1s only "are something that humans have ascribed meaning to". But still, some of those pattern of 0s and 1s are simpler than others, not in a totally absolute sense, but still in a sense made precise by Kolmogorov complexity and related concepts. There is a sort of locality here, which is independent of the antropomorphic aspects. And the same is true for quantum mechanics and the Born rule, even in the orthodox Copenhagen interpretation.

Sorry for derailing that discussion even further.

And because of Nullstein. There are good reasons why he received a 2021 award.

It often happens to me that discussions with other people are inspiring for me, and that I come to conclusion that the people who inspired me would not be willing to subscribe. I really didn't want to derail that discussion, or put you under even more pressure. And I also don't want to "misuse" you or anybody else as an excuse to "make my own points".

 

[PeterDonis]

I highlighted local basis,

That phrase actually seems misleading to me, because, unless the quantum system under consideration is a single particle, the "basis" of the Hilbert space is not "local", since the wave function is not a function of a single spatial position. For example, if the quantum system is two particles, the wave function is a function of two positions (the position of particle 1 and the position of particle 2). This type of function is not "local" in any ordinary meaning of that word.

 

[PeterDonis]

Indeed, you were not talking about MWI.

He said that, but that statement is confusing to me, because the first statement of his that you quoted in the OP of this thread, "All the universe needs is the wave function and the Hamilton operator.", only makes sense if it is talking about the MWI, because the MWI is the only interpretation that says that.

 

[PeterDonis]

the relevant paragraph occurs already on the first page

The phrase "the stochastic nature of measurements" in that paragraph seems misleading to me, since he is talking about the MWI, and in the MWI, measurements are not stochastic: everything is completely deterministic, since the wave function is all of reality and the wave function evolves deterministically via the Schrodinger Equation and that's it.

To me, the key statement is in the second quote you give: "All you have to assume is that his various quantum states which may exist as factor states in these different components of the global wave function are the true carriers of awareness." For concreteness, if an experimenter measures the spin of a qubit, along an axis where there are equal amplitudes for either result, then the resulting entangled state is

$$
\frac{1}{\sqrt{2}} \left( \ket{\text{up}} \ket{\text{measured up}} + \ket{\text{down}} \ket{\text{measured down}} \right)
$$

and the "factor states" referred to are $\ket{\text{measured up}}$ and $\ket{\text{measured down}}$. Ordinarily, when talking about an entangled state, we would say that neither subsystem (here the subsystems are the qubit and the experimenter) has a definite state at all; but in the MWI, we say that there are two "branches" or "worlds", and the experimenter is in the "measured up" state in one and the "measured down" state in the other. In other words, we interpret the "factor states" as being actual states of awareness of the experimenter, not just terms in an entangled state in which the experimenter subsystem has no definite state at all.

This difference in how we interpret entangled states seems to me to be the "additional assumption" Zeh was talking about. But other things he says, as I have noted, seem misleading to me (and aren't really necessary anyway if what I have said just now about the "additional assumption" is correct).

 

[gentzen]

Indeed, you were not talking about MWI.

He said that, but that statement is confusing to me, because the first statement of his that you quoted in the OP of this thread, "All the universe needs is the wave function and the Hamilton operator.", only makes sense if it is talking about the MWI, because the MWI is the only interpretation that says that.

He intended to talk about orthodox QM, minus the antropocentric aspects (or agentocentric aspects as was later clarified). His concrete examples shows that he is not interested in MWI type arguments here, but in questions of how and where "meaning" gets assigned.

The difference to the MWI position is that he just tries to highlight those aspects, while MWI holds that those aspects can be explained/discussed away, if we accept consequences like the existence of those splitting many worlds.

 

[gentzen]

That phrase actually seems misleading to me, because, unless the quantum system under consideration is a single particle, the "basis" of the Hilbert space is not "local", since the wave function is not a function of a single spatial position.

I guess what is meant by "local basis" is a (tensor) product (or determinant for Fermions, or permanent for Bosons) of well localized functions. I guess that as long as the function is well localized, being a function of a single spatial position is not overly important. So a high-dimensional wavefunction of some atomic nucleus (with respect to its constituent quarks) would still be considered local.

That spatial localization is important (for explanations based on decoherence), because the physical interactions are spatially local. Therefore the computational basis in a quantum computer, which is the tensor product basis of the bases on each qubit, is a local basis, because the operations that a quantum computer can apply to those qubits are local with respect to those qubits (namely only the one- and two-qubit operations are needed, and currently implemented and benchmarked).

The phrase "the stochastic nature of measurements" in that paragraph seems misleading to me, since he is talking about the MWI, and in the MWI, measurements are not stochastic: everything is completely deterministic, ...

Well, now that you point it out, that passage sounds to me like passages that discuss the relationship between (nearly) diagonal density matrices and stochastic ensembles. A specific stochastic ensemble of orthogonal wavefunctions is one way to get a diagonal density matrix, and Zeh is one of those who point out that a diagonal density matrix by itself is not enough to conclude that there was a stochastic ensemble of wavefunctions:

(But in the meantime, at least I learned that Dieter Zeh had always warned about misinterpreting what can be explained by decoherence alone, so maybe it was just a mistake from my side to interpret his warning as "a remark about a hypothetical completion of MWI".)

So maybe I really just misinterpreted a warning for something else. Or that place was not the correct one which I tried to find again, especially since I remember talk about some minor assumption:

"a little bit more" is needed, even so I never found out what "little addition" exactly he had in mind

I now searched the book again, for words like "little", "minor", "small", "weak", "klein", or "schwach". Nothing. Then I searched his webpage, and found the following in the short article There are no Quantum Jumps, nor are there Particles!

(3) In contrast to an assertion by Everett, the law of quantum probabilities cannot be derived without further (though weak) assumptions about the selection of ‘our’ world component.¹⁸ (4) The most important underivable assumption in a kinematically nonlocal (i.e., nonseparable) quantum world seems to be the locality of the ultimate (subjective) observer in spacetime (required in some vague but essential form). Quantum decoherence is meaningful (or ‘relevant’) only with respect to local parts of the nonlocal quantum world.

This fits better, because it contains the word "weak" and explicitly emphasizes "locality", and it is plausible that I read it too while reading the book, because of its catchy title and its short length.

To me, the key statement is in the second quote you give: "All you have to assume is that his various quantum states which may exist as factor states in these different components of the global wave function are the true carriers of awareness."

This could indeed fit better than my guess, because this is a weak assumption from Zeh's point of view. Assuming special initial conditions might be plausible, but it is hardly a weak assumption, not even for Zeh. And the "factor states" could imply the type of "locality" that seems to be so important for Zeh.

 

[PeterDonis]

He intended to talk about orthodox QM

What is "orthodox QM"? This is the interpretations forum; we aren't talking about the basic math of QM but about interpretations. There is no "orthodox" interpretation of QM.

minus the antropocentric aspects (or agentocentric aspects as was later clarified).

Again, if that was the intent, the statement is very confusing to me, because, as I said, it only makes sense if one is talking about the MWI. You don't need "agents" in order for measurements to take place, which brings in more than the wave function and the Schrodinger Equation on any interpretation other than the MWI. But I think that issue has been beaten enough at this point.

 

[PeterDonis]

a high-dimensional wavefunction of some atomic nucleus (with respect to its constituent quarks) would still be considered local.

But that's not the kind of case that raises issues. The kind of case that raises issues is an entangled system that is not local in this sense, such as two entangled particles that are spatially separated by a large distance (e.g., a pair of entangled photons separated by kilometers, as in some of the most recent Bell-type experiments). The wave function of such a system is a function of two positions that are widely separated.

That spatial localization is important (for explanations based on decoherence), because the physical interactions are spatially local.

This is true, yes: for example, a measurement made on just one of a pair of entangled particles involves local interactions with just that particle.

 

[gentzen]

What is "orthodox QM"?

I meant some typical unspecified Copenhagen interpretation, with probabilities and expectation values, but no deep discussions about ensembles or preparation procedures.

But that's not the kind of case that raises issues. The kind of case that raises issues is an entangled system that is not local in this sense, such as two entangled particles that are spatially separated by a large distance (e.g., a pair of entangled photons separated by kilometers, as in some of the most recent Bell-type experiments). The wave function of such a system is a function of two positions that are widely separated.

And a local basis ensures that such an obviously large distance entangled state is actually a clearly entangled state in that basis. That is the point why that type of "locality" seems to be so important for Zeh.

 

[PeterDonis]

a local basis ensures that such an obviously large distance entangled state is actually a clearly entangled state in that basis.

I don't understand. Whether or not a state is entangled is basis independent.

 

[gentzen]

I don't understand. Whether or not a state is entangled is basis independent.

No, it is basis dependent. Take for example a quantum computer, and a state where the 2nd qubit is entangled with the 5th qubit, but nothing else is entangled, like for $|010111\rangle+|000101\rangle$. Now take the 2nd qubit and the 5th qubit together as a four dimensional subsystem, and use a basis for that subsystem which includes that given state (or rather its part in the subsystem) as a basis vector. You can still use a tensor basis of two two-dimensional subsystems (of that four dimensional subsystem) if you like. In this basis, the state is no longer entangled.

 

[PeterDonis]

No, it is basis dependent.

No, it's not. This is basic QM. Your understanding of basic QM and what "entangled" means is flawed. See below.

Take for example a quantum computer, and a state where the 2nd qubit is entangled with the 5th qubit, but nothing else is entangled, like for $|010111\rangle+|000101\rangle$.

So to get rid of the extraneous qubits, we have the state (ignoring normalization, which is irrelevant for this question) $\ket{11} + \ket{00}$. This is an entangled state, yes: to make it even clearer, we can write it as $\ket{1}_2 \ket{1}_5 + \ket{0}_2 \ket{0}_5$, to make explicit the subspaces in which each ket "lives".

Now take the 2nd qubit and the 5th qubit together as a four dimensional subsystem, and use a basis for that subsystem which includes that given state (or rather its part in the subsystem) as a basis vector.

Sure, you can do that, but that doesn't stop the state from being entangled. Entangled means it is impossible to factor the state into a product state of a single ket in the "qubit 2" 2-d subspace and a single ket in the "qubit 5" 2-d subspace. That has nothing whatever to do with what basis you choose in the 4-d Hilbert space of the two qubits together.

If you seriously do not see how the state being entangled is basis independent, then please write down in your next post in this thread, explicitly, the basis in which you think the state is not entangled, and what the state looks like in that basis.

 

[gentzen]

Sure, you can do that, but that doesn't stop the state from being entangled. Entangled means it is impossible to factor the state into a product state of a single ket in the "qubit 2" 2-d subspace and a single ket in the "qubit 5" 2-d subspace. That has nothing whatever to do with what basis you choose in the 4-d Hilbert space of the two qubits together.

This "entangled means" depends on the "qubit 2" and "qubit 5" subspaces. Those subspaces themselves depend on a tensor-product structure on the Hilbert space. If you use a basis with a tensor product structure, you can consider that basis to define (or allow) such a tensor-product structure.

If you seriously do not see how the state being entangled is basis independent, then please write down in your next post in this thread, explicitly, the basis in which you think the state is not entangled, and what the state looks like in that basis.

My basis is
$\ket{0}_{2'} \ket{0}_{5'} := \ket{1}_2 \ket{1}_5 + \ket{0}_2 \ket{0}_5$
$\ket{1}_{2'} \ket{0}_{5'} :=\ket{1}_2 \ket{1}_5 - \ket{0}_2 \ket{0}_5$
$\ket{0}_{2'} \ket{1}_{5'} :=\ket{0}_2 \ket{0}_5 + \ket{1}_2 \ket{1}_5$
$\ket{1}_{2'} \ket{1}_{5'} :=\ket{0}_2 \ket{0}_5 - \ket{1}_2 \ket{1}_5$
which defines new "qubit 2' " and "qubit 5' " subspaces. The state in this basis looks like $\ket{0}_{2} \ket{0}_{5'}$, more precisely it is $\ket{0}_1\ket{0}_{2'}\ket{0}_3\ket{1}_4\ket{0}_{5'}\ket{1}_6$, or just $\ket{000101}$.

 

[PeterDonis]

My basis is
$\ket{0}_{2'} \ket{0}_{5'} := \ket{1}_2 \ket{1}_5 + \ket{0}_2 \ket{0}_5$
$\ket{1}_{2'} \ket{0}_{5'} :=\ket{1}_2 \ket{1}_5 - \ket{0}_2 \ket{0}_5$
$\ket{0}_{2'} \ket{1}_{5'} :=\ket{0}_2 \ket{0}_5 + \ket{1}_2 \ket{1}_5$
$\ket{1}_{2'} \ket{1}_{5'} :=\ket{0}_2 \ket{0}_5 - \ket{1}_2 \ket{1}_5$

None of this makes any sense as "redefining subspaces" (see below). The four states you list can of course be used as a basis for the 4-d Hilbert space of the combined system, but that doesn't change anything about the "qubit 2" and "qubit 5" subspaces or whether the state is entangled.

which defines new "qubit 2' " and "qubit 5' " subspaces.

You can't define new "qubit 2" and "qubit 5" subspaces. Those subspaces are defined by the degrees of freedom that physically correspond to qubit 2 and qubit 5. You can't change them.

Are you getting this from a reference somewhere?

 

[gentzen]

Are you getting this from a reference somewhere?

I wanted to use a Bell basis here, but I got it wrong:

My basis is
$\ket{0}_{2'} \ket{0}_{5'} := \ket{1}_2 \ket{1}_5 + \ket{0}_2 \ket{0}_5$
$\ket{1}_{2'} \ket{0}_{5'} :=\ket{1}_2 \ket{1}_5 - \ket{0}_2 \ket{0}_5$
$\ket{0}_{2'} \ket{1}_{5'} :=\ket{0}_2 \ket{0}_5 + \ket{1}_2 \ket{1}_5$
$\ket{1}_{2'} \ket{1}_{5'} :=\ket{0}_2 \ket{0}_5 - \ket{1}_2 \ket{1}_5$

That is simply not a basis. A correct Bell basis would have been:
$\ket{0}_{2'} \ket{0}_{5'} := \ket{1}_2 \ket{1}_5 + \ket{0}_2 \ket{0}_5$
$\ket{1}_{2'} \ket{0}_{5'} :=\ket{1}_2 \ket{1}_5 - \ket{0}_2 \ket{0}_5$
$\ket{0}_{2'} \ket{1}_{5'} :=\ket{0}_2 \ket{1}_5 + \ket{1}_2 \ket{0}_5$
$\ket{1}_{2'} \ket{1}_{5'} :=\ket{0}_2 \ket{1}_5 - \ket{1}_2 \ket{0}_5$

Using 2 and 5 instead of two neighboring qubits was my own modification, to hint at some non-locality and make it "rub against the grain". The idea was that you can feel that such a basis change is sometimes somehow wrong. The challenge is that it is not always wrong: logical qubits use highly entangled states! However, not all (quantum) error correcting codes are equally useful, namely the one- and two-qubit operations must also be implementable in an error protected and efficient way. I learned the idea to use quantum computing to make better sense of the Copenhagen interpretation from Mermin's Copenhagen Computation: How I Learned to Stop Worrying and Love Bohr. I learned quantum error correction intially from Mermin's book Quantum Computer Science: An Introduction.

Are you getting this from a reference somewhere?

Let me interpret your question in a different way: You found my attempts to explain what Zeh means by "local basis" and "factor states" rather confusing, and not helpful. You guess it would be more helpful, if I just gave some references from which I learned that stuff, instead of trying to explain it myself. I certainly believe that there are references explaining this much better than me, but I would have to spend some time to find good ones.

 

[PeterDonis]

I wanted to use a Bell basis here,

Which is fine, but it still doesn't do what you are claiming.

Let me interpret your question in a different way

You are getting close to a misinformation warning. I did not ask you to "interpret" my question "in a different way". I asked you for a reference to back up your claim that whether or not a state is entangled is basis dependent. That claim contradicts every QM reference I am aware of. Either provide a reference to back up the claim or stop making it.

You found my attempts to explain what Zeh means by "local basis" and "factor states" rather confusing, and not helpful. You guess it would be more helpful, if I just gave some references from which I learned that stuff, instead of trying to explain it myself.

No, that's not at all what I was asking. I restated what I was asking above. It has nothing to do with Zeh. It has to do with a claim you have made that contradicts every QM reference I am aware of, and I am asking you to either back it up or stop making it. (And if you think Zeh himself is making it, please give a specific reference--book or paper, page, explicit quote--that you are basing that on.)

 

[PeterDonis]

Using 2 and 5 instead of two neighboring qubits was my own modification, to hint at some non-locality and make it "rub against the grain". The idea was that you can feel that such a basis change is sometimes somehow wrong.

I have no idea what you mean by this; it seems like word salad to me. But it is not what I asked for in any case. Based on what you have posted in this thread, I do not think your understanding of this topic is anywhere good enough for you to be attempting your own "modifications" of anything in this subject area.

 

[PeterDonis]

That is simply not a basis. A correct Bell basis would have been:
$\ket{0}_{2'} \ket{0}_{5'} := \ket{1}_2 \ket{1}_5 + \ket{0}_2 \ket{0}_5$
$\ket{1}_{2'} \ket{0}_{5'} :=\ket{1}_2 \ket{1}_5 - \ket{0}_2 \ket{0}_5$
$\ket{0}_{2'} \ket{1}_{5'} :=\ket{0}_2 \ket{1}_5 + \ket{1}_2 \ket{0}_5$
$\ket{1}_{2'} \ket{1}_{5'} :=\ket{0}_2 \ket{1}_5 - \ket{1}_2 \ket{0}_5$

The RHS of these are fine. The LHS are wrong. You can't just wave your hands and relabel things like that. You have to actually compute what the Bell states look like when you choose a new basis for the qubit 2 and qubit 5 subspaces.

For example, suppose we choose a new basis for the qubit subspaces of (again ignoring normalization) $\ket{+} = \ket{0} + \ket{1}$ and $\ket{-} = \ket{0} - \ket{1}$. These easily invert to (once more ignoring normalization) $\ket{0} = \ket{+} + \ket{-}$ and $\ket{1} = \ket{+} - \ket{-}$. Then we need to plug those expressions into the Bell states to get their expressions in the new basis; for example, the fourth state you wrote down, the singlet state, becomes:

$$
\ket{0}_2 \ket{1}_5 - \ket{1}_2 \ket{0}_5 = \left( \ket{+}_2 + \ket{-}_2 \right) \left( \ket{+}_5 - \ket{-}_5 \right) - \left( \ket{+}_2 - \ket{-}_2 \right) \left( \ket{+}_5 + \ket{-}_5 \right)
$$

This can easily be shown to simplify to (ignoring normalization):

$$
\ket{-}_2 \ket{+}_5 - \ket{+}_2 \ket{-}_5
$$

Which, as is obvious, is still an entangled state (as it must be, since whether or not the state is entangled is basis independent). Similar calculations for the other Bell states will show them to also be entangled in the new basis.

 

[gentzen]

You are getting close to a misinformation warning.

Understood. I will try nevertheless to clarify our misunderstanding, after all we both invested some time into this already.

I asked you for a reference to back up your claim that whether or not a state is entangled is basis dependent.

I guess the following statement is fine, and clarifies one way in which the entanglement of a state is not basis dependent:

If we are given a Hilbert space $\mathcal H := \mathcal H_1 \otimes \mathcal H_2$ as the tensor product of the Hilbert spaces $\mathcal H_1$ and $\mathcal H_2$, then the entanglement of a state $\ket{\psi} \in \mathcal H$ with respect to $\mathcal H_1$ and $\mathcal H_2$ is independent of the basis of $\mathcal H_1$ and $\mathcal H_2$.

This can easily be shown to simplify to (ignoring normalization):

$$
\ket{-}_2 \ket{+}_5 - \ket{+}_2 \ket{-}_5
$$

Which, as is obvious, is still an entangled state (as it must be, since whether or not the state is entangled is basis independent).

This is fine, and I claim that this is exactly the situation covered by the statement above. You only changed the basis of $\mathcal H_1$ and $\mathcal H_2$, but not the tensor product structure on $\mathcal H$.So let me now come to the part where we disagree: "The LHS are wrong. You can't just wave your hands and relabel things like that." At the core, my (rather confusing, and not helpful) claim is that you can use a basis to define a tensor product structure on $\mathcal H$. Such an artificially defined tensor product structure risks to be physically meaningless. But since my goal is to clarify our misunderstanding, I have to go on nevertheless. My reasoning is that I can define a tensor product structure on $\mathcal H$ by defining an isomorphism to $\mathbb C^{n_1} \otimes \mathbb C^{n_2}$, which has a preferred basis. Mapping a basis of $\mathcal H$ to that preferred basis defines an isomorphism, because a morphism between Hilbert spaces is uniquely defined by the images of a basis.

 

[PeterDonis]

one way in which the entanglement of a state is not basis dependent:

You are still wrong, since you are implying that there are other ways in which entanglement is basis dependent. There aren't. "Entanglement" has only one meaning, and it is basis independent.

At the core, my (rather confusing, and not helpful) claim is that you can use a basis to define a tensor product structure on .

You can't. The tensor product structure of the Hilbert space is also basis independent.

You need to learn basic QM and stop making stuff up.

 

[PeterDonis]

The thread discussion has run its course. Thread closed.