I Is information lost in wavefunction collapse?

[MichPod]

Having in mind the idea that "information is not lost" (referring to the black hole information paradox), is not the same rule violated in the wave function collapse? I.e. during the decoherence process information is not lost as this process of entanglement of some object with its environment is reversible (in theory), but, supposing, only one "classical" alternative exists in the end (Many worlds interpretation aside), this looks for me like a loss of information. Or in other words it looks like an absolutely non-reversible process.

 

[romsofia]

What information is lost...?

 

[PeterDonis]

is not the same rule violated in the wave function collapse?

Wave function collapse is an interpretation of QM. It doesn't make any predictions that are different from standard QM. So this question can't really be resolved in standard QM, since standard QM also admits no-collapse interpretations.

At some point someone might figure out how to actually test whether wave function collapse happens as a real physical process--i.e., someone might develop a different theory from standard QM that includes an actual physical wave function collapse (some attempts have already been made at this, such as the GRW stochastic collapse theory) and we'll be able to run an experiment to test the theory. But we haven't (yet) reached that point. If we ever do, then we'll be able to actually test whether collapse occurs, and if so, what it does to information.

 

[MichPod]

Wave function collapse is an interpretation of QM. It doesn't make any predictions that are different from standard QM.

Actually I thought that the collapse is still a part of the standard QM (which is still the Copenhagen interpretation). If not, how the "Standard QM" describes expected results of "measurement"?

 

[MichPod]

What information is lost...?

I am sorry I am a layman and not knowledgeable enough to answer this question. I am just relying on my intuition. In my understanding lost of information in physics is somehow equivalent to time non-reversibility. So my question may be reformulated as whether the wave collapse is considered reversible, at least "in theory".

 

[PeterDonis]

I thought that the collapse is still a part of the standard QM (which is still the Copenhagen interpretation).

No, standard QM is not the Copenhagen interpretation. Standard QM is QM without any interpretation at all: just the math and the predictions for observable results.

how the "Standard QM" describes expected results of "measurement"?

In standard QM the term "measurement" does not have a precise meaning; it's basically "whatever works for a particular experiment". One of the main reasons that there are multiple interpretations of QM is that there are multiple ways of making more precise what a "measurement" is and what is going on "behind the scenes" during a measurement.

 

[MichPod]

Then if I rephrase my question asking whether what is calculated in the Standard QM looks as related to reversible/irreversible process, what may be the answer?

Let's for instance consider a circularly polarized photon incident on a linear polariser. According to the Standard QM it has 50% probability to be absorbed and 50% probability to pass the polariser. For a passed photon, for instance, can we consider this process potentially reversible?

 

[Star_Owl]

Having in mind the idea that "information is not lost" (referring to the black hole information paradox), is not the same rule violated in the wave function collapse? I.e. during the decoherence process information is not lost as this process of entanglement of some object with its environment is reversible (in theory), but, supposing, only one "classical" alternative exists in the end (Many worlds interpretation aside), this looks for me like a loss of information. Or in other words it looks like an absolutely non-reversible process.

Yes, information is lost after a measurement (the new wavefunction is given by a projection of the original one, and all other eigenstate are lost) and no, this is not contraddicting any axiom of QM, in opposite to the black hole information paradox case. This is beacuse information must not be lost in the evolution of a closed quantum system, like the black hole+radiated environment, but a measurement and wave function collapse involves the interference of an external agent, that is, the observer: the system is not closed.

 

[nrqed]

Yes, information is lost after a measurement (the new wavefunction is given by a projection of the original one, and all other eigenstate are lost) and no, this is not contraddicting any axiom of QM, in opposite to the black hole information paradox case. This is beacuse information must not be lost in the evolution of a closed quantum system, like the black hole+radiated environment, but a measurement and wave function collapse involves the interference of an external agent, that is, the observer: the system is not closed.

I will play the devil's advocate...

Now imagine I am inside a box with a Stern-Gerlach setup. Won't I observe a specific spin measurement, even if I am in a closed environment with the SG? I know that an interpretation is that I will exist in a superposition of states having observed the two possible results but I don't buy it. If I do observe one result, then what happened to the information?

 

[PeterDonis]

Let's for instance consider a circularly polarized photon incident on a linear polariser. According to the Standard QM it has 50% probability to be absorbed and 50% probability to pass the polariser. For a passed photon, for instance, can we consider this process potentially reversible?

Standard QM can't answer this question; both answers (reversible, not reversible) are consistent with the math and predictions of standard QM. Different interpretations will give different answers, but unless and until "different interpretations" turns into "different theories that make different predictions that can be tested by experiment", we have no way of resolving the issue.

 

[PeterDonis]

I know that an interpretation is that I will exist in a superposition of states having observed the two possible results but I don't buy it.

Then you've already answered the question: you believe in actual, physical, collapse, which is a non-reversible, non-unitary process and destroys information. But you have no way of showing by experiment that your belief is correct.

 

[Star_Owl]

I will play the devil's advocate...

Now imagine I am inside a box with a Stern-Gerlach setup. Won't I observe a specific spin measurement, even if I am in a closed environment with the SG? I know that an interpretation is that I will exist in a superposition of states having observed the two possible results but I don't buy it. If I do observe one result, then what happened to the information?

Measuring is considered an action where an external system affects the quantum one, and from this comes the projection or "collapse". Should both be quantum, either they are decoupled and so no information can be obtained from the ""observer"" (no way to see a collapse) or the observer will change sensibly beacuse of entanglement, which is not what happens in reality. You are not engangled with the spin particle.

Just to mention: I am thinking according to Copenhagen interpretation.

 

[MichPod]

Well, if we believe that information may be lost in the collapse (as an option), I wonder why we generally don't hear much about it. For instance, there exists an opinion (popularised by Sean Carroll currently) that the time arrow exists due to the low entropy of the Universe in the past, so that the time asymmetry may be just due to the observable growth of entropy. On the other side, nobody mentions that the wave collapse itself may be time-asymmetrical and so be somehow (hypothetically) a cause of the existence of the time arrow. If this option is not mentioned, then may be physicists do not consider collapse as irreversible at all?

Sorry for this kind of non-scientifical argument.

 

[PeterDonis]

if we believe that information may be lost in the collapse (as an option)

It's an option logically speaking, because we don't have any way of experimentally distinguishing different interpretations of QM. But that doesn't necessarily mean it's an option in the minds of physicists who work with QM. Many of them (Carroll is an example) appear to believe, for theoretical reasons, that the most fundamental dynamics of the universe is unitary and therefore no information is ever lost. This also implies that the dynamics is always, in principle, reversible (even if reversibility is not possible in practice because there are far too many degrees of freedom involved). That's why you don't see them talking about information loss as a realistic option.

 

[stevendaryl]

Well, if we believe that information may be lost in the collapse (as an option), I wonder why we generally don't hear much about it.

Well, in a measurement, there are (at least) two systems involved: the system being measured, and the measuring device/observer/environment/rest-of-the-universe. Since there is an interaction between these two systems, you wouldn't expect information to be conserved when you just look at one of the two. The other system is usually not studied with the same rigor (since it's basically impractical to treat a macroscopic system completely quantum-mechanically). So there is no way to rigorously demonstrate that information is lost. It's lost for all practical purposes, but maybe that's due to our treating the measurement device non-rigorously. I am not claiming that that solves the problem you bring up, but I think it explains why it's not as pressing a problem as the black hole information loss problem.

 

[MichPod]

Many of them (Carroll is an example) appear to believe, for theoretical reasons, that the most fundamental dynamics of the universe is unitary

Carroll personally support MWI interpretation as far as I know. Are there any other (mainstream) options/interpretation except for MWI to expect that the dynamics is always unitary and still have a wave function collapse or whatever else which looks like a "measurement" which "selects" one component from a superposition? That is, Carroll and MWI aside, why others do not consider this option of the measurement irreversibility seriously?

 

[PeterDonis]

Carroll personally support MWI interpretation as far as I know.

So do the other physicists I referred to.

Carroll and MWI aside, why others do not consider this option of the measurement irreversibility seriously?

You can't put "MWI aside" because MWI is a main reason most physicists don't take the irreversibility/information loss option seriously. More precisely, most physicists find unitarity to be a very strong theoretical requirement, and treating unitarity as a universal principle of dynamics in QM implies the MWI.

 

[atyy]

So do the other physicists I referred to.

Carroll does not teach MWI as correct. Who are the other physicists?

All the major textbooks use Copenhagen. Standard QM is the Copenhagen interpretation.

 

[PeterDonis]

Carroll does not teach MWI as correct.

He certainly seems to think it's "probably correct":

http://www.preposterousuniverse.com...ion-of-quantum-mechanics-is-probably-correct/

Who are the other physicists?

Any physicist who takes the "information is not lost" side in the black hole information loss question. Which, as far as I can tell, is most physicists.

All the major textbooks use Copenhagen. Standard QM is the Copenhagen interpretation.

Let's please not get involved in an argument over what "Copenhagen interpretation" means. When I say "standard QM" I mean just the machinery that makes predictions, with no interpretation whatsoever over and above the predictions.

 

[atyy]

He certainly seems to think it's "probably correct":

http://www.preposterousuniverse.com...ion-of-quantum-mechanics-is-probably-correct/

Well, that is not the same as "correct", and shows that he still would not teach it as standard QM.

Any physicist who takes the "information is not lost" side in the black hole information loss question. Which, as far as I can tell, is most physicists.

Yes, but the reason is not that they support MWI. If that were correct, one would not be able to formulate the black hole information paradox in Copenhagen. However, the black hole information paradox can be formulated in Copenhagen.

Let's please not get involved in an argument over what "Copenhagen interpretation" means. When I say "standard QM" I mean just the machinery that makes predictions, with no interpretation whatsoever over and above the predictions.

I am taking Copenhagen to mean standard QM, as I believe the OP is also. What I am saying is:

1. MWI is not standard QM.
2. Standard QM does contain a postulate of non-unitary time evolution, which can be called state reduction or collapse.
3. The black hole information paradox is obtained in standard QM with state reduction, and it is a paradox because it appears that unitarity is lost before a measurement is made.

Here is an explanation of the information paradox showing why the mixed state in black hole evaporation is different from the mixed state in the usual thermal radiation from hot everday objects: http://qpt.physics.harvard.edu/simons/Polchinski.pdf

 

[Stephen Tashi]

I'm curious why knowing more about something would be called a "loss" of information.

If a experiment is performed involving a probabilistic phenomena and the experimenter learns the outcome, why isn't this a gain in information?

 

[Demystifier]

All the major textbooks use Copenhagen. Standard QM is the Copenhagen interpretation.

There is no Copenhagen interpretation. See http://lanl.arxiv.org/abs/1703.08341 Sec. 2.1.

 

[stevendaryl]

I'm curious why knowing more about something would be called a "loss" of information.

If a experiment is performed involving a probabilistic phenomena and the experimenter learns the outcome, why isn't this a gain in information?

But if you set up an electron in the spin state $\alpha |u\rangle + \beta |d\rangle$, where $|u\rangle$ and $|d\rangle$ are spin-up and spin-down relative to the z-axis, respectively, there is a lot of information in those coefficients $\alpha$ and $\beta$. When you measure the spin in the z-direction later, you get only a single bit of information. So it's a net loss of information.

 

[Stephen Tashi]

there is a lot of information in those coefficients $\alpha$ and $\beta$.

What definition of "information" is being used to make that statement?

 

[stevendaryl]

What definition of "information" is being used to make that statement?

Well, you can quantify information in terms of the number of bits necessary to specify a situation, but I was just using it in the informal sense. I have information about something if I can deduce something about it.

In deterministic classical physics, information is never lost, because complete knowledge of the state of the universe now allows me to retrodict the state of the universe yesterday. This theoretical reversibility doesn't do a whole lot of good, practically, because there is no way to know the current state of the universe in enough detail to retrodict everything about the past. But theoretically, there is no limits to retrodiction.

But if an electron is initially in a superposition of two states, and then I perform a measurement, there is (as far as anybody knows) no way, even theoretically, to retrodict what the initial superposition was. That information is gone forever. Or at least, QM doesn't specify where it has gone.

 

[Stephen Tashi]

Well, you can quantify information in terms of the number of bits necessary to specify a situation, but I was just using it in the informal sense.

Ok.

I don't understand whether the question in thread title can be formulated precisely - or whether any of the replies assume a particular formulation.

But if an electron is initially in a superposition of two states, and then I perform a measurement, there is (as far as anybody knows) no way, even theoretically, to retrodict what the initial superposition was.

Suppose you don't perform the measurement. If you try to retrodict what the superposition was ten years ago, how do you know that the current superposition wasn't a result of some intervening measurements?

 

[stevendaryl]

Suppose you don't perform the measurement. If you try to retrodict what the superposition was ten years ago, how do you know that the current superposition wasn't a result of some intervening measurements?

I'm not sure I understand the question. Are you saying that the information about the superposition might be encoded in the state of whatever device put the electron into a superposition in the first place? That might be a resolution to the information loss problem in quantum measurements, but it's not a part of standard quantum mechanics. A system can start in an arbitrary state, and after a measurement, the details of that initial state are (apparently) forever inaccessible.

On the other hand, if you have a mechanism that can reliably place an electron into a superposition of states, then many repeated measurements can reveal the coefficients (up to an undetectable phase). But for a one-off state, there is no way to know what the state was. Measurement seems to destroy that information.

 

[Stephen Tashi]

I'm not sure I understand the question. Are you saying that the information about the superposition might be encoded in the state of whatever device put the electron into a superposition in the first place?

A good idea, but my thinking isn't that sophisticated. I'm only saying that the ability to retrodict doesn't seem to be a reliable indicator of whether information is conserved or lost - because you can't actually retrodict the history of a physical system without assuming there has been no "outside interference". (That's true even in classical deterministic physics.)

To get a technical definition that relates information loss to retrodiction loss, we could pursue defining an "instaneous" retrodiction that retrodicts the prior state of the system to an "infinitely less different" previous time such that no outside interference could have intervened.

 

[PeterDonis]

the reason is not that they support MWI. If that were correct, one would not be able to formulate the black hole information paradox in Copenhagen.

Huh? "Supporting MWI" does not mean believing that the black hole information paradox can only be formulated under the MWI. Indeed, the whole point of "supporting MWI" with regard to black holes and information is that there is no paradox at all under the MWI, since everything is always unitary. Only under a collapse interpretation is there a paradox at all.

MWI is not standard QM.

No, but it's an interpretation of standard QM.

Standard QM does contain a postulate of non-unitary time evolution, which can be called state reduction or collapse.

This can't be right, since MWI is an interpretation of standard QM, and has entirely unitary time evolution with no collapse.

 

[Demystifier]

Huh? "Supporting MWI" does not mean believing that the black hole information paradox can only be formulated under the MWI. Indeed, the whole point of "supporting MWI" with regard to black holes and information is that there is no paradox at all under the MWI, since everything is always unitary. Only under a collapse interpretation is there a paradox at all.

Even though collapse and apparent disappearance of information by black hole evaporation both violate unitarity, those two processes are not directly related. They violate unitarity in very different ways.
In a collapse, a pure state evolves (jumps) into another pure state.
By black hole evaporation, a pure state evolves into a mixed state.

To answer the initial question, I would say that in a collapse the information is not really lost, but replaced by new information. For an analogy, suppose that someone burns your old phone book and gives you the new updated edition. Would you say that you lost the information in this process? No, you just updated it.

 

[atyy]

Huh? "Supporting MWI" does not mean believing that the black hole information paradox can only be formulated under the MWI. Indeed, the whole point of "supporting MWI" with regard to black holes and information is that there is no paradox at all under the MWI, since everything is always unitary. Only under a collapse interpretation is there a paradox at all.
No, but it's an interpretation of standard QM.
This can't be right, since MWI is an interpretation of standard QM, and has entirely unitary time evolution with no collapse.

All of this is wrong.

MWI is not a solution to the black hole information paradox, in any sense that Copenhagen is not.

The Carroll post you put in support of MWI states "These are the serious issues for EQM ..." and "But even given the real challenges of the preferred-basis issue and the probability issue, I think EQM is way ahead of any proposed alternative."

Standard QM has collapse - see the texts by Dirac, Landau and Lifshitz, Cohen-Tannoudji et al, Weinberg, Sakurai, Griffiths.

 

[PeterDonis]

Standard QM has collapse

Again, this can't be right, since MWI is an interpretation of standard QM and it doesn't have collapse.

see the texts by Dirac, Landau and Lifshitz, Cohen-Tannoudji et al, Weinberg, Sakurai, Griffiths.

Do any of these texts claim that MWI is not a valid interpretation?

 

[PeterDonis]

MWI is not a solution to the black hole information paradox

"Everything is always unitary" is a solution, or at least a claimed solution; and that implies the MWI.

 

[atyy]

"Everything is always unitary" is a solution, or at least a claimed solution; and that implies the MWI.

Yes, but it does not imply MWI. The usual approach, eg, AdS/CFT to try to solve the paradox would also solve it for Copenhagen. The interpretations have nothing to do with the paradox. Introducing different degrees of freedom is the usual approach.

 

[atyy]

Do any of these texts claim that MWI is not a valid interpretation?

I edited my reply above before seeing your reply. Carroll states that MWI has serious issues, as does David Deutsch. If even supporters of MWI still think there are major problems with MWI, then it cannot be considered textbook physics.

 

[PeterDonis]

Carroll states that MWI has serious issues

But he also says, as you quote, that it is "way ahead of any proposed alternative".

If even supporters of MWI still think there are major problems with MWI, then it cannot be considered textbook physics.

By this reasoning, no interpretation of QM can be considered "textbook physics". Is that your position?

 

[PeterDonis]

The usual approach, eg, AdS/CFT to try to solve the paradox would also solve it for Copenhagen.

Unless "everything is always unitary" is consistent with what you mean by "Copenhagen", I don't see how this could be true. And if "everything is always unitary" is consistent with what you mean by "Copenhagen", then I am very confused as to what you mean by "Copenhagen".

 

[Nugatory]

If even supporters of MWI still think there are major problems with MWI, then it cannot be considered textbook physics.

All interpretations have major issues - that's why we can spend so much time arguing about them, and also why pointing out the issues cannot settle these arguments.

It would be a bad thing if this thread were to degenerate into another form of "your interpretation is uglier than mine".

 

[Stephen Tashi]

What is the technical definition for "information" in the context of this thread?

 

[romsofia]

I had a similar question months ago, (https://www.physicsforums.com/threads/where-does-the-energy-go.930637/)Even though it wasn't mentioned in the thread, if you take the collapse interpretation to be true, I believe you are essentially saying that your information (states) are not coupled to the environment until a measurement is taken place. So no information is loss even in the Copenhagen interpretation because as far the environment knows, there was really only one option?

If your information is coupled to the environment, then in theory, you would be able to measure the energy output of each state via some gravitational wave (I think Davies worked out some hand-wavy calculations in the 60/70s, i'll send references for anyone interested)... but maybe that's not for this thread :).

There are so many ifs when dealing with which is why...

What is the technical definition for "information" in the context of this thread?

I think to make any movement in a thread like this, this is the right way to go. Otherwise we will all just talk around each other using ambiguous words. Math triumphs. As far as I'm aware, information in QM is referred to as states! But what states are we considering? An energy operator is different than a position operator, and then some other people started talking about bits! If we start talking about bits, then there is already a reason how THAT information loss is handled! If we start to think as information as bits, then why not just invoke Launder's principle and call it a day?

Just to clarify, as far as I'm aware Launder's principle might not apply to quantum systems, but I'm not an expert in quantum computing nor computing in general!

 

[atyy]

Unless "everything is always unitary" is consistent with what you mean by "Copenhagen", I don't see how this could be true. And if "everything is always unitary" is consistent with what you mean by "Copenhagen", then I am very confused as to what you mean by "Copenhagen".

Well it depends on what one means by "everything". For the information paradox, there is a reasonable definition of everything. In Copenhagen, everything is unitary between measurements.

 

[atyy]

By this reasoning, no interpretation of QM can be considered "textbook physics". Is that your position?

Yes, except for Copenhagen or whatever one wishes to call what is in the textbooks.

 

[atyy]

It would be a bad thing if this thread were to degenerate into another form of "your interpretation is uglier than mine".

It has nothing to do with ugliness, but correctness. It is not correct, in a thread which mentions standard QM, and makes sense within standard QM, to tell the OP that his question doesn't make sense, by bringing in speculative approaches to the measurement problem as if it is settled physics - here I use speculative in the sense that string theory is speculative and not settled physics (although I do think it is the leading approach to quantum gravity).

 

[atyy]

It would be a bad thing if this thread were to degenerate into another form of "your interpretation is uglier than mine".

To add to my comment above, the OP is not a question about interpretations. Bringing in interpretations as Peter Donis did is irrelevant to the OP. The point is that the non-unitary evolution of collapse, and that of the information paradox are not related as I tried to say in post #20, and as Demystifier says clearly in post #30.

 

[stevendaryl]

Even though collapse and apparent disappearance of information by black hole evaporation both violate unitarity, those two processes are not directly related. They violate unitarity in very different ways.
In a collapse, a pure state evolves (jumps) into another pure state.
By black hole evaporation, a pure state evolves into a mixed state.

I'm not sure I understand the distinction you are making. The way I understand "mixed state" in quantum mechanics, there are two different sources of mixed states:

1. If you don't know what the state of a system is, then you can represent it as a mixed state, where the probabilities reflect your subjective uncertainty about what the pure state is.
2. A pure state involving two subsystems (the system of interest and the environment, say) can be treated as a mixed state of just one of the subsystems, by a kind of averaging over the system that you're not interested in.

When people say that a black hole turns a pure state into a mixed state, I'm not exactly sure what notion of "mixed state" is meant. But if it is #1, then it seems to me equivalent to a measurement collapsing the wave function, but you don't know what the measurement result was.

 

[PeterDonis]

To add to my comment above, the OP is not a question about interpretations. Bringing in interpretations as Peter Donis did is irrelevant to the OP. The point is that the non-unitary evolution of collapse, and that of the information paradox are not related as I tried to say in post #20, and as Demystifier says clearly in post #30.

I agree with the point you and @Demystifier make that these two things (collapse vs. BH information paradox) are different. Are you saying that that, in itself, is a sufficient answer to the question in the OP? If so, I would like the OP to say whether he agrees with that.

 

[PeterDonis]

It is not correct, in a thread which mentions standard QM, and makes sense within standard QM, to tell the OP that his question doesn't make sense, by bringing in speculative approaches to the measurement problem as if it is settled physics

But in "standard QM", the OP's question can't be answered, because standard QM allows both kinds of interpretations: interpretations in which information is not lost in "wave function collapse" (because "collapse" is not a real process but just a calculational rule, no real non-unitary processes ever happen--for example, the MWI), and interpretations in which information is lost in collapse, because collapse is a real, non-unitary process.

 

[PeterDonis]

it depends on what one means by "everything"

Yes, which is exactly why the answer to the OP's question must be interpretation dependent: some interpretations, like the MWI, mean by "everything" literally everything--nothing non-unitary ever happens, anywhere in the universe. Whereas other interpretations only interpret "everything" to mean "everything between measurements" (and some go on to claim that during a measurement, an actual non-unitary process, wave function collapse, happens, while others are agnostic about this).

 

[Demystifier]

I'm not sure I understand the distinction you are making. The way I understand "mixed state" in quantum mechanics, there are two different sources of mixed states:

1. If you don't know what the state of a system is, then you can represent it as a mixed state, where the probabilities reflect your subjective uncertainty about what the pure state is.
2. A pure state involving two subsystems (the system of interest and the environment, say) can be treated as a mixed state of just one of the subsystems, by a kind of averaging over the system that you're not interested in.

When people say that a black hole turns a pure state into a mixed state, I'm not exactly sure what notion of "mixed state" is meant. But if it is #1, then it seems to me equivalent to a measurement collapsing the wave function, but you don't know what the measurement result was.

It is neither #1 nor #2. It is

3. Initially you have a pure state involving two entangled subsystems, one inside the black hole and the other outside of the black hole. So initially it corresponds to your 2. But then the inside subsystem gets destroyed in the black hole singularity, so what remains is only the outside subsystem, which is in a mixed state but no longer entangled with anything.

 

[stevendaryl]

It is neither #1 nor #2. It is

3. Initially you have a pure state involving two entangled subsystems, one inside the black hole and the other outside of the black hole. So initially it corresponds to your 2. But then the inside subsystem gets destroyed in the black hole singularity, so what remains is only the outside subsystem, which is in a mixed state but no longer entangled with anything.

Thanks. So that really is something new.

So the idea is that you create an EPR pair---an electron and positron with entangled anticorrelated spins. You throw the positron into a black hole, which then vanishes in a puff of Hawking radiation. Now, you still have the electron, but the electron by itself was not in a pure state, it was in an entangled state. So how do you describe it now that its entangled partner no longer exists? A mixed state.

Now that I say it out loud, it occurs to me that in the case of spin entanglement, you might still have the electron entangled, rather than in mixed state. When the positron falls into the black hole, it imparts a tiny bit of angular moment to the black hole. When the black hole evaporates, that angular momentum is distributed among the particles produced by the Hawking radiation. So in that particular case, it seems that the electron's spin would be entangled with the resulting Hawking radiation. So I think to really illustrate the information loss, you would need some property of a pair of particles that is nonconserved?