Questions about information in QM

[Dmitry67]

Several questions.

1. Do different observers (with the same knowledge) in different inertial frames agree on the number of states of some specific system?

2. The same question for the different accelerated frames;

3. The most difficult one
Say, Alice observes some system X with N particles. She calculates the amount of information in that system. But Bob knows, that all particles in X are entangled with particles in system Y. Hence, for Bob system X contains less infromation, than for Alice. How it is possible?

If the number of states is physical, how can it depend on something subjective, as the knowledge of some observer?

 

[Bill_K]

Accelerated observers disagree on which state is the vacuum state, and whether or not particles are present. This is the basis of Hawking radiation.

 

[jambaugh]

3. is actually the easier issue. Your example is confusing potentially encoded information and actually encoded information.

You have a composite system which you can factor into subsystems X and Y. It takes a physical measurement or selection of a physical observable's value to entangle the component systems. So you're asserting that Bob has made an actual measurement on the composite Z=XY system.

Now Alice if she observed system X before Bob observed the entangling measurement then the composite system is no long in Alice's eigen-"state". (For Bob to entangle he needs causal access to Alices X part and here we assume this access is after Alice measured.) Likewise if Alice observes system X after Bob observed the entangling measurement the the composite system is no longer in Bob's eigen-"state". Alice has causally affected the composite by measuring a part of it after Bob.

If you are concerned with information content, after a maximal measurement the system is in a zero entropy mode, it is in 1 of 1 "states". If you're considering a partial measurement you may allow some subset of the total available number of modes. So let's start simple. Let system X be say 2 dimensional (1 qubit) and likewise with system Y. There are then 2x2 = 4 dimensions in the composite. Let's pick a basis for each half system with states 0 and 1 and so for the composite we have the factorable basis xy= 00, 01, 10, or 11.

Bob can measure a partial entanglement which correlates the two by measuring a $\langle 00| +\langle 11 |$ state, or a $\langle 00 | -\langle 11|$ state. He can make a partial measurement projecting out only these two states. He still sees two dimensions of potential encoding and has reduced the total 4 by eliminating the other two possibilities.

Alice can measure X in one of its two states, say x=0. She still sees two possibilities in the unobserved Y system y=0, y=1 and has eliminate the two cases where x=1.

Note that Bob, by projecting onto the span of superpositions of factorible basis elements has chosen a non-factorable basis. He cannot reduce his measurement action to indpendent measurements of X and Y separately and most importantly the projection operator he uses cannot commute with any measurement of X that Alice might perform.

But there is no disagreement as to how much information in the form of not-yet observed degrees of freedom either of them has available at any given instant just where that information is encoded. Bob's info is encoded in the composite in such a way that it is distributed among the parts X and Y. Both see a 4-dim composite. Alice sees only 2 dim available to her. Bob by insisting on a correlation between the two can only see 2 remaining available to him.

 

[Dmitry67]

Accelerated observers disagree on which state is the vacuum state, and whether or not particles are present. This is the basis of Hawking radiation.

... and Unruh effect, I know
But my question is different.

I was asking not about the state, but about the number of states. Even if they disagree on the particular state, may be they can agree on the number of states.

For example, I think different accelerated observers should agree that black hole saturates the Bekenstein bound. But do they always agree on the number of states?

 

[Dmitry67]

jambaugh, thank you. I understand it, you describe the situation using measurements. Again, your explanation makes sense.

But can we think about information without having any measurements? For example, the definition of Bekenstein bound is not based on any measurements (and in case of Black holes, such measurements are not even possible). Or are there any hidden assumptions?

Can we talk about 'pure' information, not related to any specific observer? Observer can't learn that information without measurement, but doesn't that information exist before the measurement?

 

[Jano L.]

Can we talk about 'pure' information, not related to any specific observer? Observer can't learn that information without measurement, but doesn't that information exist before the measurement?

Instead of 'information', we can use better expression 'physical state'. Then we can talk about state not related to any specific observer, such as number of electrons in an atom. It is common to think this state exists with no dependence upon measurement.

 

[Dmitry67]

I was able to answer my own question. Different accelerated observers should always agree on the number of states.

Sketch of Proof: Let's assume that there are 2 observers A and B, flying on different trajectories near each other. Because of the different acceleration A observes more states in system X than B.

Then we can imagine observer C, which is initially on the same trajectory as A, but then it joins B on his trajectory (this is why A and B must be close enough to each other). As B observes less states in X, observer C would observe that information was suddenly 'leaking' from X, and the number of states decreases.

When know it is impossible. Hence, A and B must observe the same number of states, it is independent of their acceleration. Unruh effect changes particle contents (virtual vs real) but not the number of states (apparently the same number is distributed differently across different particle content)

 

[Jazzdude]

You keep talking about the "number of states". What is that supposed to mean? Number of states in what? A quantum system is in one state, always. Do you mean the number of possible eigenstates of an observed system? That's trivially conserved under unitary transformations, and observer changes are unitary (as all meaningful transformations in quantum theory).

So, please clarify what you mean.

 

[jambaugh]

Instead of 'information', we can use better expression 'physical state'. Then we can talk about state not related to any specific observer, such as number of electrons in an atom. It is common to think this state exists with no dependence upon measurement.

This makes the idea of "state" non-operational which makes it worse, not better. There are two things we can do with information (which we can encode in a physical system) we can read the "signal" (by measuring system observables) or we can "throw it away" by dynamically transmitting the information into an entropy dump...

...
But can we think about information without having any measurements? ...

No... and yes. The information is only meaningful as information if it is physically encodable in a physically measurable way. But no we needn't actually measure the information's carrier system to say the information is there...

For example, the definition of Bekenstein bound is not based on any measurements (and in case of Black holes, such measurements are not even possible). Or are there any hidden assumptions?

One can in principle measure the information content of a black hole in the form of the random noise as it evaporates, or in the black hole's state before and after the information has been dropped in. Consider encoding a signal in an electron, say using its momentum up to some resolution and its spin. Now take a black hole and place it in the electron's path, and measure the momentum and spin of the black hole before and after the consumption. The difference gives you back the information encoded in the electron. In a more sophisticated case, let a "perfect" black hole absorb an electron encoding a signal. The change in the BH is not just a matter of increasing mass. It will no longer be a perfect sphere but will have various minute deviations in its higher moments. These of course will dissipate as gravity waves over time but the information is still there, either in the surface configuration or in the configuration of gravity waves spreading outward at the speed of light.

It is no different from "destroying" the information in any thermodynamic way. You are mixing the existing information in a random way with the environment. You mix with incoming light signals coming from outside one's earlier past light-cone so its is information you did not have prior access to and also you are sending information outward at the speed of light (say from a candle's flame) so that it is forever beyond your future light-cone and you can't catch up to it and measure it.

The event-horizon of a BH is no different than that of a light-cone in that the information crosses and someone can potentially measure it but you can never get to it again.

Can we talk about 'pure' information, not related to any specific observer? Observer can't learn that information without measurement, but doesn't that information exist before the measurement?

We can talk about the information carrying capacity of a physical system without it having been encoded with known information. This is exactly what entropy is. I'm not sure what you mean by "pure" information though. At the quantum level ALL information is relative to the choice of commuting observables one utilizes to encode it. If I encode in X and you observe in P then by definition you have destroyed my encoded info and I have a priori randomized yours. But we will find that the system as a whole (when we've been careful to isolate it so we both agree which system we're talking about) encodes a specific amount of information we can describe as its maximum possible entropy. But note that isn't just say "an electron" we must in defining our system either limit ourselves to "the spin of the electron" or specify enough environment to say what position or momentum means. If say we're talking "an electron in a box" then it is the box, not which of us is looking at the electron which bounds the amount of encodable information. It is likewise the motion of the box not our motions which defines the amount of information as seen by different inertial observers.

I know I'm hitting on different points here in a bit of an uncorrelated way. Let me make one or two more...

One entangles two particles by encoding information into their correlation (a set of observables for the composite system) rather than encoding the information in the pair as two separate systems. If I then toss one of them into a black hole the I cannot recover that information EVER unless I choose to take the other particle and dive into the BH after the first.

Likewise and more completely, I can send the two particles (say they're photons) traveling outward in opposite directions at speed c and then it is forever impossible to, after the fact, recover that information. Take the BH case, the particle not tossed into the BH is now, for us outside, in a fundamentally randomized "state". We can say absolutely nothing significant about what we will see if we measure it. The particle's entropy is fundamentally non-zero.

Turn this around. A composite system of two particles can have zero total entropy, you make a sharp measurement of the whole. If that measurement is an entangling measurement then the entropy of each component of the whole has a non-zero entropy. One could view the entropy of any system, viewed as a part of a bigger whole, as a measure of its entanglement with the environment. One could then (on less solid physical footing) speculate that the entropy of the Universe as a whole (including interiors of BH event horizons) is exactly zero. We just see non-zero entropy in systems because we are only looking at a part of the Universe, in fact only have access to part of the universe.

Anyway these are some observations I've made over the years as I've contemplated the subject.