Is Energy Entanglement Possible?

[LarryS]

Can two particles be entangled in the energy property/observable? If so, what kind of experiment could verify that the two particles were energy-entangled?

Thanks in advance.

 

[rootone]

I am not an expert, but I think you need to define 'energy entanglement'.

 

[hilbert2]

Suppose we have two replicas of a two-state system with available states $\left| 1 \right>$ and $\left| 2 \right>$ corresponding to energies $E_1$ and $E_2$. Now the state of the combined system is something like:

$\left| \psi \right> = a \left| 1 \right>\left| 1 \right> + b\left| 1 \right>\left| 2 \right>+c\left| 2 \right>\left| 1 \right>+d\left| 2 \right>\left| 2 \right>$ .

An entangled state of this system could be something like $\frac{1}{\sqrt{2}}\left(\left| 1 \right>\left| 1 \right>+\left| 2 \right>\left| 2 \right>\right)$, where you can be certain that the both subsystems have equal energies.

 

[DrChinese]

Can two particles be entangled in the energy property/observable? If so, what kind of experiment could verify that the two particles were energy-entangled?

Since energy is conserved in entanglement generation (say of photon pairs), I would say there is entanglement on that basis. For photons pairs created via Parametric Down Conversion, for example: frequency is proportional to energy and so frequency is conserved. If you knew the input frequency fairly precisely (which you normally would), then the relationship between the 2 output photons would be well defined.

How would test you that? Usually you have an inequality of some type involving (partially) non-commuting observables. The experimental realization is using energy-time entanglement which I am not so familiar with. Here is a reference, I have not read it but one of the authors (Larsson) writes on this a lot and I have read some of his other work.

https://arxiv.org/abs/1103.6131
"In the spirit of Einstein, Podolsky, and Rosen it is possible to ask if the quantum-mechanical description (of this setup) can be considered complete. This question will be answered in detail in this paper, by delineating the quite complicated relation between energy-time entanglement experiments and Einstein-Podolsky-Rosen (EPR) elements of reality."

 

[newjerseyrunner]

As far as I understand, entanglement has no understood physical mechanism, it just "pops out" of the math. I like to explain entanglement using way simpler algebra.

Say you have a wave function f(x) which describes two particles. Now when you want to split them, you define one of the particles as g(x). You have now automatically defined the other particle as f(x) - g(x), there is nothing else it can be.

 

[DrChinese]

As far as I understand, entanglement has no understood physical mechanism, it just "pops out" of the math. I like to explain entanglement using way simpler algebra.

Say you have a wave function f(x) which describes two particles. Now when you want to split them, you define one of the particles as g(x). You have now automatically defined the other particle as f(x) - g(x), there is nothing else it can be.

Your comment has nothing to do with this thread.

Yes, it is true that entanglement is often associated with a conservation rule. However, the connections between entangled systems are more complicated than the algebra you present would imply. For one thing, not every system of 2 particles is entangled. For another, you have the 2 independent functions as being a sum when it might be a Product state. Entangled systems cannot be described by a Product state.

 

[Zafa Pi]

Entangled systems cannot be described by a Product state.

That's true. Thus if |ψ⟩ = √½(|00⟩ + |11⟩), which is an entangled state, then |ψ⟩⊗|ψ⟩ is not entangled. Kind of weird.

 

[LarryS]

Since energy is conserved in entanglement generation (say of photon pairs), I would say there is entanglement on that basis.

Conservation laws help experimenters prove that the two output particles are correlated but do not cause entanglement directly. Agree?

 

[atyy]

Suppose we have two replicas of a two-state system with available states $\left| 1 \right>$ and $\left| 2 \right>$ corresponding to energies $E_1$ and $E_2$. Now the state of the combined system is something like:

$\left| \psi \right> = a \left| 1 \right>\left| 1 \right> + b\left| 1 \right>\left| 2 \right>+c\left| 2 \right>\left| 1 \right>+d\left| 2 \right>\left| 2 \right>$ .

An entangled state of this system could be something like $\frac{1}{\sqrt{2}}\left(\left| 1 \right>\left| 1 \right>+\left| 2 \right>\left| 2 \right>\right)$, where you can be certain that the both subsystems have equal energies.

Would this qualify as energy entanglement? https://arxiv.org/abs/quant-ph/0507189

 

[LarryS]

Suppose we have two replicas of a two-state system with available states $\left| 1 \right>$ and $\left| 2 \right>$ corresponding to energies $E_1$ and $E_2$. Now the state of the combined system is something like:

$\left| \psi \right> = a \left| 1 \right>\left| 1 \right> + b\left| 1 \right>\left| 2 \right>+c\left| 2 \right>\left| 1 \right>+d\left| 2 \right>\left| 2 \right>$ .

An entangled state of this system could be something like $\frac{1}{\sqrt{2}}\left(\left| 1 \right>\left| 1 \right>+\left| 2 \right>\left| 2 \right>\right)$, where you can be certain that the both subsystems have equal energies.

In your example, both subsystems would have equal probabilities, but the two energy eigenvalues need not be equal. Right?

 

[hilbert2]

In your example, both subsystems would have equal probabilities, but the two energy eigenvalues need not be equal. Right?

If you measure the energy of one of the subsystems and find out thst it's $E_1$, then you can be certain that the energy of the second subsystem is $E_1$, too.

 

[Nugatory]

In your example, both subsystems would have equal probabilities, but the two energy eigenvalues need not be equal. Right?

That is correct. Even the equal probabilities are not fundamental to entanglement: $\frac{1}{2}|1\rangle|1\rangle+\frac{\sqrt{3}}{2}|2 \rangle|2\rangle$ is an entangled state with different probabilties of the two possible outcomes. The essential property if the entangled state is that it can be written as a superposition of two states such that collapsing the superposition determines the value of both observables. (This can be rephrased to conform to your favorite interpretation, and note that the two observables must have have common eigenfunctions to superimpose).

 

[vanhees71]

Sure, concerning the above cited paper I didn't understand, why one could claim that the given state is not entangled. By definition it is. I also don't need two particles to have entanglement. The classic example is the Stern-Gerlach experiment, where after the magnets you have an entanglement between position and spin of one particle (take a neutron instead of the original version with an Ag atom).

 

[Demystifier]

That's true. Thus if |ψ⟩ = √½(|00⟩ + |11⟩), which is an entangled state, then |ψ⟩⊗|ψ⟩ is not entangled. Kind of weird.

|ψ⟩⊗|ψ⟩ is entangled. If we think of |ψ⟩ as a 2-particle state, then |ψ⟩⊗|ψ⟩ is a 4-particle state. It is entangled because it cannot be written as a product of four 1-particle states.

 

[LarryS]

All very interesting. It's almost as if the central players in entanglement are not individual particles but individual sets where each set is a complete eigenbasis (with corresponding eigenvalues) for some observable, for the same or different particle.

 

[Nugatory]

t's almost as if the central players in entanglement are not individual particles

As far as the math is concerned, there aren't two particles. There is a single quantum system with a single wave function evolving according to Schrodinger's equations. There are various observables on this system, such "the result at instrument A" and "the result at instrument B"; but it's residual classical thinking that leads us to sort these observables into groups and call one of the groups "the properties of particle A" and another "the properties of particle B".

 

[Zafa Pi]

In responding to my post #7: That's true. Thus if |ψ⟩ = √½(|00⟩ + |11⟩), which is an entangled state, then |ψ⟩⊗|ψ⟩ is not entangled. Kind of weird.

|ψ⟩⊗|ψ⟩ is entangled. If we think of |ψ⟩ as a 2-particle state, then |ψ⟩⊗|ψ⟩ is a 4-particle state. It is entangled because it cannot be written as a product of four 1-particle states.

This is great, some more conflict - a big part of what makes science fun, educational + prizes.
Well, I certainly agree that |ψ⟩⊗|ψ⟩ cannot be written as a product of four 1-particle states. However, several sources say a state in a tensor product space is entangled if it is not a (tensor) product. But |ψ⟩⊗|ψ⟩ sits as a product of |ψ⟩ with itself, and furthermore |ψ⟩ is a pure state, for what that's worth.

Suppose we have 4 photons P1, P2 from |ψ⟩, and another two P3, P4 from a different |ψ⟩. It seems that there will be no weird correlations when measuring both P1 and P3, as there would be in measuring both P1 an P2. This may not prove anything, but does lead me to think that |ψ⟩⊗|ψ⟩ is not entangled. Say, as opposed to the state √½(|0000⟩ + |1111⟩).

The "That's true." in my #7 post refers to the DrChinese post #6 where he says, "Entangled systems cannot be described by a Product state." I wonder what he would say?

OK, your turn.

 

[LarryS]

As far as the math is concerned, there aren't two particles. There is a single quantum system with a single wave function evolving according to Schrodinger's equations. There are various observables on this system, such "the result at instrument A" and "the result at instrument B"; but it's residual classical thinking that leads us to sort these observables into groups and call one of the groups "the properties of particle A" and another "the properties of particle B".

How does the Planck energy equation, E=hf, fit into the above picture? Does the "single quantum system with a single wave function" have a single frequency?

 

[Nugatory]

How does the Planck energy equation, E=hf, fit into the above picture?

It doesn't. That equation relates the frequency of an infinite monochromatic plane wave to the energy of the photons associated with it (and be warned that the word "associated" must not be taken too literally - a more accurate but still very simplified picture of the relationship can be found here). You won't find such waves anywhere; we study them because any real waveform can be written and analyzed as a mathematical superposition of these waves.

That equation also belongs to the "old" quantum mechanics that was largely abandoned after 1925 or thereabouts; it's not part of the modern treatment that we're using in this thread and that is needed to analyze entanglement situations. You'll see this equation used sometimes because under the right conditions (a single particle isolated well enough to be treated as a quantum system in its own right and prepared with a definite energy) you can still get good results out of it - but you have to be aware of its limitations.

Does the "single quantum system with a single wave function" have a single frequency?

It depends on whether the system is in an energy eigenstate or a superposition of energy eigenstates.

 

[Demystifier]

However, several sources say a state in a tensor product space is entangled if it is not a (tensor) product.

Either the sources are sloppy or you are a sloppy reader. Take for example wikipedia
https://en.wikipedia.org/wiki/Quantum_entanglement#Meaning_of_entanglement
which says (my bold):
"An entangled system is defined to be one whose quantum state cannot be factored as a product of states of its local constituents; that is to say, they are not individual particles but are an inseparable whole."

 

[vanhees71]

Hm, but entanglement needs not to be over longranged separated parts. You can have entanglement between properties of the same particle, e.g. space and spin of a particle after running through a Stern-Gerlach apparatus.

 

[Zafa Pi]

Either the sources are sloppy or you are a sloppy reader. Take for example wikipedia
https://en.wikipedia.org/wiki/Quantum_entanglement#Meaning_of_entanglement
which says (my bold):
"An entangled system is defined to be one whose quantum state cannot be factored as a product of states of its local constituents; that is to say, they are not individual particles but are an inseparable whole."

Rather than give definitions by others you might consider sloppy, I'll just admit I like the definition you provided from wikipedia.
I am also impressed that you thought I may be a sloppy reader, since just the other day I dripped some pasta sauce on a book I was reading.

So according to the definition |0⟩⊗|ψ⟩ (|ψ⟩ = √½(|00⟩ + |11⟩)) must be an entangled state.
However, the opening sentence of the very same reference says, "Quantum entanglement is a physical phenomenon that occurs when pairs or groups of particles are generated or interact in ways such that the quantum state of each particle cannot be described independently of the others,". (my underline).
Yet you will notice that the particle with state |0⟩ is described independently of the others.
A little sloppiness here?

 

[Demystifier]

So according to the definition |0⟩⊗|ψ⟩ (|ψ⟩ = √½(|00⟩ + |11⟩)) must be an entangled state.
However, the opening sentence of the very same reference says, "Quantum entanglement is a physical phenomenon that occurs when pairs or groups of particles are generated or interact in ways such that the quantum state of each particle cannot be described independently of the others,". (my underline).
Yet you will notice that the particle with state |0⟩ is described independently of the others.
A little sloppiness here?

This looks like a fine subtlety of english language. Consider the sentence "Each politician is not honest". Does it mean that all politicians are dishonest? Or does it mean that at least one politician is dishonest? The problem is actually what does the "not" refers to. Does it mean "For each politician it is true that he is not honest"? Or does it mean "It is not true that each politician is honest"?

 

[zonde]

Either the sources are sloppy or you are a sloppy reader. Take for example wikipedia
https://en.wikipedia.org/wiki/Quantum_entanglement#Meaning_of_entanglement
which says (my bold):
"An entangled system is defined to be one whose quantum state cannot be factored as a product of states of its local constituents; that is to say, they are not individual particles but are an inseparable whole."

Product of two entangled states is not "inseparable whole" either as it can be separated into two entangled states. "Inseparable whole" of 4-particle state would be GHZ state.

 

[Demystifier]

Perhaps wikipedia is not the most reliable source. So let me take a definition from Nielsen and Chuang "Quantum Computation and Quantum Information" which is a kind of Bible for that stuff. At the top of page 96 it says:
"We say that a state of a composite system having this property (that it can’t be written as a product of states of its component systems) is an entangled state."
At the bottom of page 93, composite system is defined as a "system made up of two (or more) distinct physical systems".

I think it is quite clear that |0⟩⊗|ψ⟩ (|ψ⟩ = √½(|00⟩ + |11⟩)) is entangled by that definition.

 

[pranj5]

Prof. Masahiro Hotta of Tohoku Univeristy of Japan has published a few papers on this matter.

 

[DrChinese]

Prof. Masahiro Hotta of Tohoku Univeristy of Japan has published a few papers on this matter.

His papers would not currently meet the standards set for this forum. I think you and I have discussed this subject previously. If you want to discuss those further, it should be in a separate thread. The topic of those papers is not relevant to the OP question.

 

[Zafa Pi]

Product of two entangled states is not "inseparable whole" either as it can be separated into two entangled states. "Inseparable whole" of 4-particle state would be GHZ state.

Good observation. It is no surprise wikipedia is a bit sloppy in this very wordy article.
I think that the GHZ state is √½(|000⟩ + |111⟩), 3 particles, rather than √½(|0000⟩ + |1111⟩), 4 particles.

 

[Zafa Pi]

Perhaps wikipedia is not the most reliable source. So let me take a definition from Nielsen and Chuang "Quantum Computation and Quantum Information" which is a kind of Bible for that stuff. At the top of page 96 it says:
"We say that a state of a composite system having this property (that it can’t be written as a product of states of its component systems) is an entangled state."
At the bottom of page 93, composite system is defined as a "system made up of two (or more) distinct physical systems".

I think it is quite clear that |0⟩⊗|ψ⟩ (|ψ⟩ = √½(|00⟩ + |11⟩)) is entangled by that definition.

I was disappointed at your response in post #23 in defending Wikipedia sloppiness, you know full well that "each member of an ensemble X has property P" is the same as "every member of X has property P" in math or science.
On the other hand I was delighted to see you recommend N & C's book, it's my favorite, and I've cited it many times. Nonetheless, it has a number of errors and places where it is a bit fuzzy. One such place I noted several years ago was at the bottom of page 93, where he doesn't define a "distinct physical system". Could |ψ⟩ = √½(|00⟩ + |11⟩) be a distinct physical system or does he mean an individual particle?

I would like to propose the following: If a state |ζ⟩ representing multiple particles can't be factored it is fully entangled. If |ζ⟩ can be factored (to lowest terms) but one of its factors also represents multiple particles but it can't be factored then we say |ζ⟩ is partially entangled. We say |ζ⟩ is entangled if it is either fully or partially entangled.

So the GHZ state and |ψ⟩ are fully entangled, while |0⟩⊗|ψ⟩ and |ψ⟩⊗|ψ⟩ are partially entangled. All of them are entangled.

 

[Demystifier]

Could |ψ⟩ = √½(|00⟩ + |11⟩) be a distinct physical system

I am pretty sure that N&C would say that it consists of two disctinct physical systems.

 

[zonde]

I would like to propose the following: If a state |ζ⟩ representing multiple particles can't be factored it is fully entangled. If |ζ⟩ can be factored (to lowest terms) but one of its factors also represents multiple particles but it can't be factored then we say |ζ⟩ is partially entangled. We say |ζ⟩ is entangled if it is either fully or partially entangled.

Term "partial entanglement" is already taken. It refers to state vector $|\psi\rangle=a|00\rangle+b|11\rangle$ where |a| and |b| are different.

 

[Zafa Pi]

Term "partial entanglement" is already taken. It refers to state vector $|\psi\rangle=a|00\rangle+b|11\rangle$ where |a| and |b| are different.

Too bad for me. But good looking out. How about sortta entangled?

I am pretty sure that N&C would say that it consists of two disctinct physical systems.

Given what we've been discussing I am pretty sure you're right. But at the time I read it, it wasn't clear to me at all.