Understanding Hilbert Subspace for Two-Particle Entangled Systems

[bluecap]

Once again, I don't see anything in what you've quoted from the papers that says the environment interacts with the system directly; it interacts with the apparatus, and that interaction with the apparatus, since the apparatus is already entangled with the system, entangles the environment with the system.

I think when system is directly exposed to the environment. We can bypass apparatus which is only used if system is isolated to environment so apparatus acts as interface between them. In the same Zurek paper, the following is clear that he only deals with system and environment and omits any apparatus:

"These ideas can be made precise. The basic tool is the reduced density matrix ρS. It represents the state of the system that obtains from the composite state ΨSE of S and E by tracing out the environment E: ρS = TrE|ΨSEihΨSE| . (1) Evolution of ρS reveals preferred states: It is most predictable when the system starts in a pointer state. To quantify this one can use (von Neumann) entropy, HS = H(ρS) = −TrρS lgρS, as a function of time. Pointer states result in smallest entropy increase. By contrast, their superpositions produce entropy rapidly, at decoherence rates, especially when S is macroscopic."

So "when S is macroscopic" and directly exposed to environment. We can use pointer states directly on the system.. right?

 

[PeterDonis]

So "when S is macroscopic" and directly exposed to environment. We can use pointer states directly on the system.. right?

I'm not sure. For example, if you "measure" an apple's position by looking at it (and perhaps comparing it with a ruler or a grid of coordinates marked on the table it's sitting on), the "apparatus" is photons bouncing off the apple (and the ruler/grid/table, etc.) and coming into your eyes (and perhaps your eyes themselves). The "system" is the apple, and the "environment" is everything else. You are correct that both the system and the apparatus interact with the environment, but the environment interaction that counts is with the apparatus. Basically, multiple people can look at photons bouncing off the apple and agree on its position, because there are so many photons. But there are so many photons because the environment is basically an inexhaustible source of them, and the information that each person receives through their eyes gets stored in other environment degrees of freedom like their brains, the sound waves they emit when they describe what they see, etc. None of this involves any interaction with the system--the only interaction with the system that made any difference was the photons bouncing off the apple.

Zurek's formulation in the paper you link to here, which doesn't include an "apparatus", is unclear to me in an example like the above. Would he include the photons bouncing off the apple in the "system" or the "environment"? Neither one seems right to me; the photons seem like an "apparatus", which isn't the same as the system (since the photons don't stay localized at the apple), but isn't the same as the environment either (because other degrees of freedom don't interact with the apple, at least not in a way that makes any difference in the scenario above).

But as I said before, all this is an open area of research, so I'm not sure how much weight to put on any particular description of it.

 

[bluecap]

I'm not sure. For example, if you "measure" an apple's position by looking at it (and perhaps comparing it with a ruler or a grid of coordinates marked on the table it's sitting on), the "apparatus" is photons bouncing off the apple (and the ruler/grid/table, etc.) and coming into your eyes (and perhaps your eyes themselves). The "system" is the apple, and the "environment" is everything else. You are correct that both the system and the apparatus interact with the environment, but the environment interaction that counts is with the apparatus. Basically, multiple people can look at photons bouncing off the apple and agree on its position, because there are so many photons. But there are so many photons because the environment is basically an inexhaustible source of them, and the information that each person receives through their eyes gets stored in other environment degrees of freedom like their brains, the sound waves they emit when they describe what they see, etc. None of this involves any interaction with the system--the only interaction with the system that made any difference was the photons bouncing off the apple.

Zurek's formulation in the paper you link to here, which doesn't include an "apparatus", is unclear to me in an example like the above. Would he include the photons bouncing off the apple in the "system" or the "environment"? Neither one seems right to me; the photons seem like an "apparatus", which isn't the same as the system (since the photons don't stay localized at the apple), but isn't the same as the environment either (because other degrees of freedom don't interact with the apple, at least not in a way that makes any difference in the scenario above).

But as I said before, all this is an open area of research, so I'm not sure how much weight to put on any particular description of it.

I'm trying to relate what you mentioned above about the scattering photons being apparatus to what this Maximilian book is saying. In page 74:
"For the simple example of the system-environment interaction given by (2.78), we immediately see that a system that starts out in either one of the states |psi1> and |psi2> will not get entangled with the environment: The final composite system-environment state will at all times remain in the separable product forms|psi1>|E1> and |psi2>|E2> respectively. Thus the environment can be regarded as carrying out a quantum nondemolition measurement on the system. i.e. a measurement that does not disturb the state of the system. In the case of the scattering of environment particles discussed above, |psi1> and |psi2> represent spatially well-localized states, and accordingly the environment-superselected preferred observable is the position of the system. Conversely, if the system is described by the superposition (2.70) of |psi1> and |psi2> , it will become maximally entangled with the environment, leading to decoherence in the {|psi1> , psi2> } basis. The system observable corresponding to the measurement of a superposition of positions would there be "difficult" to measure, in the sense that the spatial interference terms in the reduced matrix corresponding to a measurement of this observable decay rapidly due to the interaction with the environment."

How do you use the concept of maximal entanglement here (monogamy?). How come if the system is described by the superposition (2.70) of |psi1> and |psi2> will they be maximally entangled with the environment while if |psi1> and |psi2> are separated, it will remain in the separable product forms |psi1>|E1> and |psi2>|E2> respectively (the comment in page 71 described this "We motivated this form of the interaction by referring to the fact that the states |psi1> and |psi2> correspond to sufficiently distinct physical states of the system (such as different spatial positions and orientations of a body), and this difference is resolved by the environment continuously monitoring the system.".. and this refers to page 67 "Supposed the system is described by a coherent superposition of two quantum states |psi1> and |psi2> representing localization around two different positions x1 and x2 (in the case of the double-slit experiment, the corresponding position space wave functions psi1(x) and psi2(x) would represent the partial waves at the slits)."

Thanks!

 

[PeterDonis]

How do you use the concept of maximal entanglement here (monogamy?)

Maximal entanglement isn't the same thing as monogamy of entanglement, though they are related. Monogamy of entanglement basically says that a subsystem can't be maximally entangled with more than one other subsystem.

I'm not sure what you mean by "how do you use the concept".

How come if the system is described by the superposition (2.70) of |psi1> and |psi2> will they be maximally entangled with the environment while if |psi1> and |psi2> are separated, it will remain in the separable product forms |psi1>|E1> and |psi2>|E2> respectively

Because that's how the interaction with the environment works. It's just unitary evolution. It's no different in principle from putting a spin up or spin down electron through a vertically oriented Stern Gerlach device, vs. putting a spin left electron (equal superposition of spin up and spin down) through it. At least, that's the type of model that it looks like Maximilian is using. The only difference is that here the "system" (as well as the apparatus) is made up of many quantum particles, not just one.

 

[bluecap]

Maximal entanglement isn't the same thing as monogamy of entanglement, though they are related. Monogamy of entanglement basically says that a subsystem can't be maximally entangled with more than one other subsystem.

I'm not sure what you mean by "how do you use the concept".
Because that's how the interaction with the environment works. It's just unitary evolution. It's no different in principle from putting a spin up or spin down electron through a vertically oriented Stern Gerlach device, vs. putting a spin left electron (equal superposition of spin up and spin down) through it. At least, that's the type of model that it looks like Maximilian is using. The only difference is that here the "system" (as well as the apparatus) is made up of many quantum particles, not just one.

My question really was. If the system is described by the superposition of |psi1> and |psi2> , and it will become maximally entangled with the environment, and lead to decoherence in the {|psi1> , psi2> } basis. How come the system observable corresponding to the measurement of a superposition of positions would there be "difficult" to measure (in the sense that the spatial interference terms in the reduced matrix corresponding to a measurement of this observable decay rapidly due to the interaction with the environment). You earlier just described the photons reflecting off the object. But now we have to consider it directly interacting with the system. Do you have example of system being maximally entangled with the environment and the setup difficulty getting the preferred observable or basis? Contrast this with the case where the final composite system-environment state will at all times remain in the separable product forms|psi1>|E1> and |psi2>|E2>respectively. Here how come "the environment can be regarded as carrying out a quantum nondemolition measurement on the system. i.e. a measurement that does not disturb the state of the system.". In this message we are.not just describing photons reflecting from object but interacting with the object. Thanks!

 

[PeterDonis]

My question really was

Instead of trying to answer this as you ask it, I'm going to try to rephrase it to simplify things.

The environment's interaction with the system picks out a "preferred" basis of states, which in this example are position eigenstates. (Technically those aren't actually states since they aren't normalizable, and you have to do a fair bit of mathematical work to make all this rigorous, but we'll ignore that here.)

If we use that basis, it is obvious that if the system is in a position eigenstate, measuring its position doesn't change its state; that is what he is talking about when he describes this case as a "quantum nondemolition" measurement.

If we use the position basis, it is also obvious that if the system is in a superposition of two position eigenstates, measuring its position, which is what the environment is doing (that's why the environment picks out the position basis), will entangle the system with the measuring device (which in this case is the environment), so you will end up with two branches, each corresponding to one of the position eigenstates and the corresponding "measured that position" state of the environment (i.e., the environment state that stores the information that that particular position was measured). The branches will then decohere.

 

[bluecap]

Instead of trying to answer this as you ask it, I'm going to try to rephrase it to simplify things.

The environment's interaction with the system picks out a "preferred" basis of states, which in this example are position eigenstates. (Technically those aren't actually states since they aren't normalizable, and you have to do a fair bit of mathematical work to make all this rigorous, but we'll ignore that here.)

If we use that basis, it is obvious that if the system is in a position eigenstate, measuring its position doesn't change its state; that is what he is talking about when he describes this case as a "quantum nondemolition" measurement.

If we use the position basis, it is also obvious that if the system is in a superposition of two position eigenstates, measuring its position, which is what the environment is doing (that's why the environment picks out the position basis), will entangle the system with the measuring device (which in this case is the environment), so you will end up with two branches, each corresponding to one of the position eigenstates and the corresponding "measured that position" state of the environment (i.e., the environment state that stores the information that that particular position was measured). The branches will then decohere.

Thanks..in an apple, how many percentage of the particles are in approx eigenstates of position and how many percentage are in superposition of approx position eigenstates? (I mentioned approx bec I'm aware that HUP disallows exact position of eigenstates). What must you do to the apple to change the percentage of each of the above (between approx eigenstates of position vs superpositions of approx eigenstates of position)?

Anyway I believe you now that the pointer states (or preferred states) are a result of joint system environment (or even apparatus) processes so observer can't perturb the pointer states directly. After 5 years. I finally understood it (thanks to you).. but it's bittersweet knowing the limitation of the model of Einselection and even quantum darwinism.

 

[vanhees71]

If we use the position basis, it is also obvious that if the system is in a superposition of two position eigenstates, measuring its position, which is what the environment is doing (that's why the environment picks out the position basis), will entangle the system with the measuring device (which in this case is the environment), so you will end up with two branches, each corresponding to one of the position eigenstates and the corresponding "measured that position" state of the environment (i.e., the environment state that stores the information that that particular position was measured). The branches will then decohere.

I thought you discuss about photons. How can you have a "position basis" then? Photons don't have a position. You cannot define a position operator for massless quanta with spin $\geq 1$!

 

[PeterDonis]

in an apple, how many percentage of the particles are in approx eigenstates of position and how many percentage are in superposition of approx position eigenstates?

The atoms are all entangled so they don't have definite states by themselves. There isn't really any difference from one atom to another, so it doesn't really make sense to ask "what percentage" are in one state (or one kind of entanglement) vs. another.

 

[PeterDonis]

I thought you discuss about photons.

We've been bouncing around among different examples. You're correct that there is no "position basis" for photons, but there is for, e.g, atoms in an apple (with the caveats I mentioned earlier).

 

[bluecap]

The atoms are all entangled so they don't have definite states by themselves. There isn't really any difference from one atom to another, so it doesn't really make sense to ask "what percentage" are in one state (or one kind of entanglement) vs. another.

I know that a macroscopic object like an apple has an enormous number of internal degrees of freedom and therefore an enormous number interactions inside it, so there is decoherence right inside the apple hence many states inside are constantly decohering. In other words, the environments and systems are right inside the apple. Therefore it seems a macroscopic object is not ideal thing for quantum darwinism? I mean.. Zurek focuses on fragments and invariance of the fragments and even deriving the born rule from it. But is it not the fragments that we see are only from the surface of the apple. How does quantum darwinism handle the fragments right inside the apple inside. In the book example of approx eigenstates of position and superposition of approx eigenstates of position. They are just simple system. In an apple, the states have so many interactions inside.. but you can't see the fragments inside the apple.. so does it mean for big object like an apple, it is useless for quantum darwism analysis?

Also when you said the atoms are all entangled so they don't have definite states by themselves. This is only completely true for diamond or crystals as they form one energy band.. is it not? The apple has holes inside them microscopically where some liquid moves.. therefore is it right to think they are not completely entangled as solid band.. but multiple entanglement areas or domains or portion inside the apple?

Also the surface cover membrane of the apple (red) is different from the inside (dirty white color meat). So they are not really fully entangled? Now about the photons hitting the apple surface.. I think it's more like it entangled with the approx eigenstates of position hence nothing happening instead of the photons entangling with the apple surface superposition of approx eigenstates position.. although it's possible we only have many mixed states from the superposition that has decohered already from the Cosmic Microwave Background Radiation or thermal phonons hitting the surface of the apple? If you are saying the entire surface of the apple is entangled.. then how do the photons or CMBR interacted with the single entangled apple surface?

Thank you!

 

[PeterDonis]

it seems a macroscopic object is not ideal thing for quantum darwinism?

Why not? The fact that we always observe macroscopic objects in classical states is exactly what quantum darwinism is trying to explain.

If you mean, it's hard to see all the details working in a macroscopic object, yes, that's true. Welcome to physics, where the interesting problems are not easy.

when you said the atoms are all entangled so they don't have definite states by themselves. This is only completely true for diamond or crystals as they form one energy band.. is it not?

No. Any macroscopic object will have its individual atoms entangled in all sorts of ways.

The apple has holes inside them microscopically where some liquid moves.. therefore is it right to think they are not completely entangled as solid band.. but multiple entanglement areas or domains or portion inside the apple?

No. The liquid moving inside the apple is still interacting strongly with the rest of the apple, so it's entangled with the rest of the apple.

the surface cover membrane of the apple (red) is different from the inside (dirty white color meat). So they are not really fully entangled?

What do you mean by "fully entangled"? If you mean maximally entangled, probably very few of the atoms anywhere in the apple are maximally entangled with other ones. Maximal entanglement usually requires a very precise preparation procedure.

The skin of the apple might be somewhat less entangled with the inside than either the skin or the inside are with themselves. But they're still interacting strongly, so they're still entangled. If they weren't entangled, the apple would not stay together; it would just fall apart (or the skin would just fall off, instead of having to be peeled).

how do the photons or CMBR interacted with the single entangled apple surface?

Um, by interacting with them? I don't understand why you would see a problem here. Photons interact with atoms in objects all the time; that's how we see the objects.

 

[bluecap]

We've been bouncing around among different examples. You're correct that there is no "position basis" for photons, but there is for, e.g, atoms in an apple (with the caveats I mentioned earlier).

According to Maximilian:

"System-environment interaction Hamiltonian frequently describe a scattering process of surrounding particles (photons, air molecules, etc.) interacting with the system under study. Since the force laws describing such processes typically depend on some power of distance (such as alpha r^-2) in Newton's or Coulombs' force law) the interaction Hamiltonian will usually commute with the position operator. According to the commutativity requirement (2.89), the pointer states will therefore be approximate eigenstates of position. The fact that position is typically the determinate property of our experience can thus be explained by referring to the dependency of most interactions on distance."

I'm trying to relate this to Vanheez71 statement: "I thought you discuss about photons. How can you have a "position basis" then? Photons don't have a position. You cannot define a position operator for massless quanta with spin ≥ 1 !"

Now consider first the Schrödinger equation. The position basis is implicit in the normal form of the Schrödinger equation. When expressed in that basis virtually all interactions turn out to be radial ie the V(x) in the Schrödinger equation.

Is it true that this is because the universe originally has chosen position preferred basis in the initial decomposition of the everettian universe. Let me summarize it (or elaborate):

There are 2 kinds of preferred basis. First is the original preferred basis that originally created the universe first subsystems (or decomposing systems and environment) in the everettian universe (without basis). The second preferred basis is from the pointer states (or preferred states) you get from present system-environment Einselection. So Demystifier once said that without the initial preferred basis decomposition, the radial thing interaction choosing position basis won't even work. Therefore can it be that even if the photon doesn't have position basis, the mere existence of photons already assume the universe has initially chosen the position basis as preferred (think of the Nothing Happens in Many Worlds discussions way back).

Or to make it more illustrative.. what if you suddenly remove the initial prefer basis that cause the initial system-environment (subsystem) decomposition in the everettian universe (originally without any basis), would distance suddenly disappear in our universe that our Earth would become a mere point? is the initial prefer basis choosing position the reason why particles are separate and there are photons in the surrounding? This means even if the photon doesn't have position, the fact we can see it have dynamics is because of the initial decomposition of the everettian universe which uses position as preferred?

 

[PeterDonis]

I'm trying to relate this to Vanheez71 statement:

The position eigenstates Maximilian is talking about are for the atoms in the object, not for the photons that bounce off the object.

consider first the Schrödinger equation

The Schrödinger equation can't be used to describe photons. It only works for non-relativistic particles (i.e., particles moving much slower than light).

The rest of your post just appears to build on the mistaken assumption that it is possible to give photons a position basis. It isn't. @vanhees71 is correct about that.

 

[bluecap]

Why not? The fact that we always observe macroscopic objects in classical states is exactly what quantum darwinism is trying to explain.

If you mean, it's hard to see all the details working in a macroscopic object, yes, that's true. Welcome to physics, where the interesting problems are not easy.

No. Any macroscopic object will have its individual atoms entangled in all sorts of ways.

We always heard that for entanglement experiment like Aspect experiment. It is very hard because you have to isolate it to avoid decoherence. Is the reason because you are trying to isolate the information? Contrast this with decoherence with object where entanglement occurs anywhere without isolation. Is this because you are not isolating or manipulating any single information. Just want to verify if this is correct thinking. So it means even if there are two apples 1 foot apart.. the apples are still entangled?? What is the minimum distance between any object is entangled?

No. The liquid moving inside the apple is still interacting strongly with the rest of the apple, so it's entangled with the rest of the apple.

Have you heard about this so called Decoherence Free Subspaces. Is there a way to do this inside the apple or other objects by manipulating with laser to cause a domain free from decoherence?

What do you mean by "fully entangled"? If you mean maximally entangled, probably very few of the atoms anywhere in the apple are maximally entangled with other ones. Maximal entanglement usually requires a very precise preparation procedure.

The skin of the apple might be somewhat less entangled with the inside than either the skin or the inside are with themselves. But they're still interacting strongly, so they're still entangled. If they weren't entangled, the apple would not stay together; it would just fall apart (or the skin would just fall off, instead of having to be peeled).

Just want to know why entanglement monogamy doesn't stop all the entanglement of the apple. You said "Monogamy of entanglement basically says that a subsystem can't be maximally entangled with more than one other subsystem." But yet all the apple is entangled with billions of entanglement.. what does it mean for the apple to be maximally entangled? Can you give example of how to cause two particles of the apple to be maximally entangled?

Um, by interacting with them? I don't understand why you would see a problem here. Photons interact with atoms in objects all the time; that's how we see the objects.

Thanks so much.

 

[bluecap]

The position eigenstates Maximilian is talking about are for the atoms in the object, not for the photons that bounce off the object.
The Schrödinger equation can't be used to describe photons. It only works for non-relativistic particles (i.e., particles moving much slower than light).

The rest of your post just appears to build on the mistaken assumption that it is possible to give photons a position basis. It isn't. @vanhees71 is correct about that.

But in the example we were talking about or in the book too. It is photons from the environment that either entangled with the approx. eigenstates of position that "doesn't do anything to the system" or entangling with the superposition of approx. eigenstates of positions causing decoherence. If it is not photons from environment. then what is? If it's photons and photons don't have positions.. perhaps the reason the position eigenstates is chosen is because the atoms or particles in the system can define a position operator and this is what caused the pointer states to be eigenstates of position instead of momentum or the self Hamiltonian (energy of the system)?

 

[bluecap]

The atoms are all entangled so they don't have definite states by themselves. There isn't really any difference from one atom to another, so it doesn't really make sense to ask "what percentage" are in one state (or one kind of entanglement) vs. another.

This is the vital part. You said above the atoms are all entangled so they don't have definite states by themselves. So does this mean the photons or whatever in the environment shouldn't interact with an atom only or a few because these don't have definite state? That's why I asked "how do the photons or CMBR interacted with the single entangled apple surface?" where you replied: "Um, by interacting with them? I don't understand why you would see a problem here. Photons interact with atoms in objects all the time; that's how we see the objects.". So do you mean when photons or others interact with an object where all the atoms are entangled. The photons or something interacts with the whole molecules as entangled object as same time (not with any of the atoms because the entangled parts don't have definite state)?

In short.. are you saying it's a Quantum Field Theory situation where the entire apple and photon need to be considered together.. meaning no photon interacts with any of the atom since any atom has no definite state?

Also this means you as Peter has all your atoms entangled in your body so they don't have definite state. But when I shake your hands.. how come I'm shaking the atoms and molecules of your hands only and not the entire entangled QFT thing in your whole body at once?

 

[PeterDonis]

It is very hard because you have to isolate it to avoid decoherence. Is the reason because you are trying to isolate the information?

No, it's because you are trying to isolate the system you want to measure from any interactions other than the specific interaction you are using to measure it.

So it means even if there are two apples 1 foot apart.. the apples are still entangled?

To some extent, because there are photons, air molecules, etc. bouncing between both apples; but the apples are entangled with each other to a much lesser extent than each apple is entangled with itself. That's why we can view the two apples as separate classical objects.

Btw, I feel like this answer should have been obvious. You might want to spend more time thinking these things through before you ask questions.

Have you heard about this so called Decoherence Free Subspaces

No. Do you have a reference?

Just want to know why entanglement monogamy doesn't stop all the entanglement of the apple.

Because the apple isn't maximally entangled with anyone thing.

I feel like this answer should also have been obvious. See my comment above about thinking things through.

Can you give example of how to cause two particles of the apple to be maximally entangled?

You should be able to get this from a modern QM textbook.

If it's photons and photons don't have positions.. perhaps the reason the position eigenstates is chosen is because the atoms or particles in the system can define a position operator

I already answered this. See the first sentence I wrote in post #74.

does this mean the photons or whatever in the environment shouldn't interact with an atom only or a few because these don't have definite state?

No. This answer should also be obvious; there are plenty of quantum experiments where one particle of an entangled pair is measured. That could not be done if it wasn't possible to interact with a particle that doesn't have a definite state by itself (because it's entangled with something else).

 

[bluecap]

No, it's because you are trying to isolate the system you want to measure from any interactions other than the specific interaction you are using to measure it.

To some extent, because there are photons, air molecules, etc. bouncing between both apples; but the apples are entangled with each other to a much lesser extent than each apple is entangled with itself. That's why we can view the two apples as separate classical objects.

Btw, I feel like this answer should have been obvious. You might want to spend more time thinking these things through before you ask questions.

No. Do you have a reference?

https://en.wikipedia.org/wiki/Decoherence-free_subspaces

But I think it's useless for apples.. it can only happen in let's say Bose-Einstein condensates...

I wonder if Penrose can apply it to his microtubules in the brain.. will read his stuff next week with all this basic background... anyway Maximilian book says:

"Pointer subspaces, or DFS, have attracted much interest over the past decade because of their relevance in quantum computing. These the basic idea is to encode the fragile quantum information stored in the quantum computer in such subspaces so as to naturally protect it from decoherence. We will describe this approach in more detail in Sec. 7.5 (see also the review article by Lidar and Whaley [101]. The ideas behind pointer subspaces have also been used to propose methods for taming decoherence in other areas of interest, for example, in the context of superposition states of macroscopically distinguishable states in Bose-Einstein condensates [102] (see Sect. 6.4.1).

Because the apple isn't maximally entangled with anyone thing.

I feel like this answer should also have been obvious. See my comment above about thinking things through.
You should be able to get this from a modern QM textbook.
I already answered this. See the first sentence I wrote in post #74.
No. This answer should also be obvious; there are plenty of quantum experiments where one particle of an entangled pair is measured. That could not be done if it wasn't possible to interact with a particle that doesn't have a definite state by itself (because it's entangled with something else).

Consider $\vert \Psi' \rangle \vert E \rangle = \left( a \vert + \rangle \vert U \rangle + b \vert - \rangle \vert D \rangle \right) \vert E \rangle \rightarrow a \vert + \rangle \vert U \rangle \vert E_U \rangle + b \vert - \rangle \vert D \rangle \vert E_D \rangle$

I thought when something in the environment entangled with the system... it's the entire quantum state.. not just one term of it.. you are saying above that one term can be entangled with? I thought you that when something is entangled with a new thing.. it's new terms added.. for example:

$\Psi_0 = \left( a_1 \vert u_1 \rangle + b_1 \vert d_1 \rangle \right) \left( a_2 \vert u_2 \rangle + b_2 \vert d_2 \rangle \right) \vert R_1, R_2 \rangle \vert O_{R1}, O_{R2} \rangle$
$\rightarrow \Psi_1 = \left( a_2 \vert u_2 \rangle + b_2 \vert d_2 \rangle \right) \left( a_1 \vert u_1 \rangle \vert U_1, R_2 \rangle \vert O_{U1}, O_{R2} \rangle + b_1 \vert d_1 \rangle \vert D_1, R_2 \rangle \vert O_{D1}, O_{R2} \rangle \right)$
$\rightarrow \Psi_2 = a_1 a_2 \vert u_1 \rangle \vert u_2 \rangle \vert U_1, U_2 \rangle \vert O_{U1}, O_{U2} \rangle + a_1 b_2 \vert u_1 \rangle \vert d_2 \rangle \vert U_1, D_2 \rangle \vert O_{U1}, O_{D2} \rangle \\ + b_1 a_2 \vert d_1 \rangle \vert u_2 \rangle \vert D_1, U_2 \rangle \vert O_{D1}, O_{U2} \rangle + b_1 b_2 \vert d_1 \rangle \vert d_2 \rangle \vert D_1, D_2 \rangle \vert O_{D1}, O_{D2} \rangle$

Each time a measurement happens, it creates another entanglement. So after two measurements, we have an entangled state containing four terms, one corresponding to each possible combination of the results of the two measurements.

How is this compatible with what you just said above that only one particle of an entangled pair is measured? How would the math of what you described looks like. You mean putting the system into mixed state? Then making new entanglement? But how would the photon know whether it should be entangled with the entire system or only one part of it (which doesn't even have any state)?

 

[PeterDonis]

you are saying above that one term can be entangled with?

No.

How is this compatible with what you just said above that only one particle of an entangled pair is measured?

The math you wrote down is for measurement of a single particle that's not entangled with anything before the measurement. You need to look up the math for a measurement of one particle of an entangled pair; this is a common case in quantum computing so it should be easy to find examples.

 

[bluecap]

No.
The math you wrote down is for measurement of a single particle that's not entangled with anything before the measurement. You need to look up the math for a measurement of one particle of an entangled pair; this is a common case in quantum computing so it should be easy to find examples.

Is it the math of reduced density matrix and improper mixed states? I'm familiar with the math of these. If it's not. Please just mention me what is the math called when one particle of an entangled pair can be measured without any collapse or born rule applied or breaking the entanglement so I can look for the particular math or concept. Thanks.

 

[PeterDonis]

Is it the math of reduced density matrix and improper mixed states?

No. Look up, for example, Bell state measurements for quantum "teleportation".