Why does "control" mean "coherency" in Wigner's friend paradox?

[SEYED2001]

TL;DR Summary

I am an undergraduate student who has just finished his first term and I have a question on Wigner's friend paradox, in particular, the Brukner's article:
Brukner, Č. A No-Go Theorem for Observer-Independent Facts. Entropy 2018, 20, 350.

There we read:
"note that if Wigner did not know this phase due to the lack of control of it, he would describe the “spin + friend’s laboratory” in an incoherent mixture of the two possibilities".

Why is this the case? Given that the author has propoede neither a citation nor a proof for this proposition, I guess it is something trivial to an educated physicist. The problem is I am not an educated physicist :) so please help me understand why that statement is correct.

Thank you very much in advance

Seyed

 

[Arne]

Hello Seyed,

I'm not an expert on this topic, so I might not explain this correctly, but I will try to give my view on your question.

The state of the composite system is basically a Bell state. So to get this in a maybe more familiar form, I will replace $|z+\rangle$ and $|F_{z+}\rangle$ with $|\uparrow\rangle$, and I similarly $|z-\rangle$ and $|F_{z-}\rangle$ with $|\downarrow\rangle$
$$|\phi\rangle_{SF} = \frac{1}{\sqrt{2}} \left(|\uparrow\rangle_S |\uparrow\rangle_F + |\downarrow\rangle_S |\downarrow\rangle_F\right)$$
If Wigner does a measurement in the basis $\{|\uparrow\uparrow\rangle,|\uparrow\downarrow\rangle,|\downarrow\uparrow\rangle,|\downarrow\downarrow\rangle\}$ (which is basically just looking at the measurement result and asking the friend about the measurement result), then he will measure $|\uparrow\uparrow\rangle$ with 50% probability and $|\downarrow\downarrow\rangle$ with 50% probability. But if we add a phase $e^{i\alpha}$
$$|\phi\rangle_{SF} = \frac{1}{\sqrt{2}} \left(|\uparrow\uparrow\rangle + e^{i\alpha}|\downarrow\downarrow\rangle\right)$$
and measure in the same basis, then we still get $|\uparrow\uparrow\rangle$ with 50% probability and $|\downarrow\downarrow\rangle$ with 50% probability. So if we just can measure in this basis, we cannot distinguish this state from a incoherent mixture of 50% $|\uparrow\uparrow\rangle$ and 50% $|\downarrow\downarrow\rangle$. That just means that the state is already in $|\uparrow\uparrow\rangle$ or $|\downarrow\downarrow\rangle$ before measurement, just with a classical 50-50 probality. This means we have no information about the phase.

As stated further down in that paragraph, if we can measure in a different basis, like the Bell basis
$$\left\{\frac{1}{\sqrt{2}} \left(|\uparrow\uparrow\rangle \pm |\downarrow\downarrow\rangle\right), \frac{1}{\sqrt{2}} \left(|\uparrow\downarrow\rangle \pm |\downarrow\uparrow\rangle\right)\right\}$$
then we will measure $\frac{1}{\sqrt{2}} \left(|\uparrow\uparrow\rangle +|\downarrow\downarrow\rangle\right)$ with 100% probability, so we actually know the exact state, and with that also the phase.

I hope this helps a bit. Best wishes,

Arne

Edit: I just realized that I didn't really answer your question. So lack of control could in this context just mean, that we only can measure in the $\{|\uparrow\uparrow\rangle,|\uparrow\downarrow\rangle,|\downarrow\uparrow\rangle,|\downarrow\downarrow\rangle\}$ basis and therefore cannot distinguish between a coherent state and a probabilistic mixture, and do not have knowledge about the phase. If we can measure in another basis, we might actually be able to determine the state and with that the phase.

Edit 2: I maybe should also add, that this is a thought experiment to explore different interpretations of quantum mechanics and maybe reveal paradoxes in some of them. To my knowledge there is no way to actually do a Bell state measurement on a person + a QM system. On a quantum computer for example, one would apply operations to turn the Bell state measurement into a measurement in a basis that is actually available. But on a person + a QM system it is not really possible to perform these operations. And further, determining the exact state of a system with one type of measurement, the state of the system has to be in the basis that we measure. Usually to determine the exact state or even a probabilistic mixture (e.g. the complete density matrix), one has to perform a state tomography https://en.wikipedia.org/wiki/Quantum_tomography .

 

[Demystifier]

"note that if Wigner did not know this phase due to the lack of control of it, he would describe the “spin + friend’s laboratory” in an incoherent mixture of the two possibilities".

I didn't read the paper, but it is a general feature of QM that if a relative phase is not known then one has to average over all possible phases, in which case the resulting state is a mixed density matrix. See e.g. my lecture
http://thphys.irb.hr/wiki/main/images/5/50/QFound3.pdf
especially pages 5 and 11.

 

[SEYED2001]

Hello Seyed,

I'm not an expert on this topic, so I might not explain this correctly, but I will try to give my view on your question.

The state of the composite system is basically a Bell state. So to get this in a maybe more familiar form, I will replace $|z+\rangle$ and $|F_{z+}\rangle$ with $|\uparrow\rangle$, and I similarly $|z-\rangle$ and $|F_{z-}\rangle$ with $|\downarrow\rangle$
$$|\phi\rangle_{SF} = \frac{1}{\sqrt{2}} \left(|\uparrow\rangle_S |\uparrow\rangle_F + |\downarrow\rangle_S |\downarrow\rangle_F\right)$$
If Wigner does a measurement in the basis $\{|\uparrow\uparrow\rangle,|\uparrow\downarrow\rangle,|\downarrow\uparrow\rangle,|\downarrow\downarrow\rangle\}$ (which is basically just looking at the measurement result and asking the friend about the measurement result), then he will measure $|\uparrow\uparrow\rangle$ with 50% probability and $|\downarrow\downarrow\rangle$ with 50% probability. But if we add a phase $e^{i\alpha}$
$$|\phi\rangle_{SF} = \frac{1}{\sqrt{2}} \left(|\uparrow\uparrow\rangle + e^{i\alpha}|\downarrow\downarrow\rangle\right)$$
and measure in the same basis, then we still get $|\uparrow\uparrow\rangle$ with 50% probability and $|\downarrow\downarrow\rangle$ with 50% probability. So if we just can measure in this basis, we cannot distinguish this state from a incoherent mixture of 50% $|\uparrow\uparrow\rangle$ and 50% $|\downarrow\downarrow\rangle$. That just means that the state is already in $|\uparrow\uparrow\rangle$ or $|\downarrow\downarrow\rangle$ before measurement, just with a classical 50-50 probality. This means we have no information about the phase.

As stated further down in that paragraph, if we can measure in a different basis, like the Bell basis
$$\left\{\frac{1}{\sqrt{2}} \left(|\uparrow\uparrow\rangle \pm |\downarrow\downarrow\rangle\right), \frac{1}{\sqrt{2}} \left(|\uparrow\downarrow\rangle \pm |\downarrow\uparrow\rangle\right)\right\}$$
then we will measure $\frac{1}{\sqrt{2}} \left(|\uparrow\uparrow\rangle +|\downarrow\downarrow\rangle\right)$ with 100% probability, so we actually know the exact state, and with that also the phase.

I hope this helps a bit. Best wishes,

Arne

Edit: I just realized that I didn't really answer your question. So lack of control could in this context just mean, that we only can measure in the $\{|\uparrow\uparrow\rangle,|\uparrow\downarrow\rangle,|\downarrow\uparrow\rangle,|\downarrow\downarrow\rangle\}$ basis and therefore cannot distinguish between a coherent state and a probabilistic mixture, and do not have knowledge about the phase. If we can measure in another basis, we might actually be able to determine the state and with that the phase.

Edit 2: I maybe should also add, that this is a thought experiment to explore different interpretations of quantum mechanics and maybe reveal paradoxes in some of them. To my knowledge there is no way to actually do a Bell state measurement on a person + a QM system. On a quantum computer for example, one would apply operations to turn the Bell state measurement into a measurement in a basis that is actually available. But on a person + a QM system it is not really possible to perform these operations. And further, determining the exact state of a system with one type of measurement, the state of the system has to be in the basis that we measure. Usually to determine the exact state or even a probabilistic mixture (e.g. the complete density matrix), one has to perform a state tomography https://en.wikipedia.org/wiki/Quantum_tomography .

Dear Arne

Thank you very much for your response! So I can understand somehow the relationship between control and coherency is via knowing the relative phase. However, I cannot yet understand how is it so. More accurately:

1. If lack of control means measuring in only the four basis you have mentioned (so you are limiting the number of basis we can use for measurement), then what about being limited to use only the two basis of |uu+-dd> and |ud+-du> (where u stands for up and d for down and +- is plus or minus sign)? This would give us knowledge on relative phase and so makes system to look coherent. So lack of control (as you have defined to be limited number of basis Wigner can use for meaurement) is not incompatible with coherency. If your definition of lack of control is NOT "having a limited set of basis to choose from", then my question is:
What is special about |uu>, |dd>, |ud> and |du> that |uu +- dd> and |ud +- du> don't have? In other words, why being limited to the set { |uu +- dd>, |ud +- du>} for measurement is not lack of control?

2. Does indistinguishibility of coherence and incoherence mean incoherence? I mean I can see your point that if a coherent system is 50% uu and 50% dd then we cannot tell of it is incoherent or coherent, but I cannot understand how you inferred that indistinguishibility is equivalent to identity. Can't there be a coherent system in a superposition of 50% uu and 50% dd?

Thank you very much once again for your time and consideration :)

Seyed

 

[SEYED2001]

I didn't read the paper, but it is a general feature of QM that if a relative phase is not known then one has to average over all possible phases, in which case the resulting state is a mixed density matrix. See e.g. my lecture
http://thphys.irb.hr/wiki/main/images/5/50/QFound3.pdf
especially pages 5 and 11.

Thank you so much for your response. I went through the slides and I could learn about density matrices which I didn't know before (I already knew about relative phase). However, I couldn't answer my question even with this new knowledge! Maybe because my question is based on ambiguity in what Brukner meant by "control" and how it is incompatible with incoherence.

 

[Arne]

Hello Seyed,

It might be best to answer your questions in the opposite order.

2. First coherence/incoherence: In my argumentation above it might have been better to talk about pure states and mixed states. A pure state means that the state of the system is well defined. In contrast a mixed state is a probabilistic mixture. Let's illustrate this with a very basic example, a system with two basis states |0> and |1>. A pure state would mean that the system is in one state. This could be |0>,|1> or any superposition of the two, for example the equal superposition state |+> = (|0>+|1>)/sqrt(2) . Now let's perform a measurement on this last state, and we will get |0> with 50% probability and |1> with 50% probability. The important detail is, that the system changes its state due to our measurement, but before our measurement it was in one well define state. Now the mixed state: For a mixed state, the system isn't actually in a different kind of state, it has more to do with our knowledge about the state. Assume we measured the state |+>, but we forgot to record and write down what the measurement result actually is. We then know that the system has to be in the state |0> or the state |1> but we don't know which one it is. We just know that it is in state |0> with 50% probability and |1> with 50% probability. Note, that if we measure the mixed state, and we can only measure in the basis {|0>,|1>}, we actually cannot distinguish if the system is in the pure state |+> or if it is in a mixed state, since both have the same outcome. We need a measurement in a different basis to be able to determine this.
Coherence has actually more to do what relation parts of the system have to each other, usually it's about the phase and how well defined it is. So coherence between two waves usually means that the phase difference between the two is constant. Think about for example laser light, it has a well defined frequence, so if you put two laser beams on top of each other, the phase difference will stay the same. In contrast, light from a light bulb is scrambled over different frequencies without a clearly defined phase, so we talk about incoherent light. (As a sidenote: A coherent state as defined in Quantum Optics is actually a special kind of state, which has special coherence properties. And also eigenstates of the harmonic oscillator are sometimes called coherent states, which is kind of related. But it might be best to not get into that right now.) In this case just think of a coherent state as a state with a well defined phase. A pure state usually has a well defined phase, while a mixed state is usually scrambled, so we call it incoherent.

1. The basis |uu>,|dd>,|ud>,|du> is the standard basis that is already available to us. It is just: Look at the result of the experiment, and ask the friend what he thinks the result is. In this case we will get |uu> with 50% probability and |dd> with 50% probability, and we cannot distinguish between the pure state |uu>+|dd> and the mixed state |uu> with 50% probability and |dd> with 50% probability. We will also have no information about the phase, in case of the mixed state there wouldn't even be a well defined phase. If we want to get more information, we need to be able to measure in other bases. This is usually done by performing operations on the system, that turns the measurement in the desired basis into a measurement in a basis that we actually can do, here |uu>,|dd>,|ud>,|du>. But if we don't have control over the system, then we cannot perform these operations, thus we cannot measure in another basis and therefore get no information about the phase. But you are right, if our standard way to measure would be in the basis {|uu>+|dd>,|uu>-|dd>,|ud>+|du>,|ud>-|du>} and we again would not have the control to turn this into a measurement in another basis, then we would run into the same problem.

I hope this helps,

Arne

 

[SEYED2001]

Hello Seyed,

It might be best to answer your questions in the opposite order.

2. First coherence/incoherence: In my argumentation above it might have been better to talk about pure states and mixed states. A pure state means that the state of the system is well defined. In contrast a mixed state is a probabilistic mixture. Let's illustrate this with a very basic example, a system with two basis states |0> and |1>. A pure state would mean that the system is in one state. This could be |0>,|1> or any superposition of the two, for example the equal superposition state |+> = (|0>+|1>)/sqrt(2) . Now let's perform a measurement on this last state, and we will get |0> with 50% probability and |1> with 50% probability. The important detail is, that the system changes its state due to our measurement, but before our measurement it was in one well define state. Now the mixed state: For a mixed state, the system isn't actually in a different kind of state, it has more to do with our knowledge about the state. Assume we measured the state |+>, but we forgot to record and write down what the measurement result actually is. We then know that the system has to be in the state |0> or the state |1> but we don't know which one it is. We just know that it is in state |0> with 50% probability and |1> with 50% probability. Note, that if we measure the mixed state, and we can only measure in the basis {|0>,|1>}, we actually cannot distinguish if the system is in the pure state |+> or if it is in a mixed state, since both have the same outcome. We need a measurement in a different basis to be able to determine this.
Coherence has actually more to do what relation parts of the system have to each other, usually it's about the phase and how well defined it is. So coherence between two waves usually means that the phase difference between the two is constant. Think about for example laser light, it has a well defined frequence, so if you put two laser beams on top of each other, the phase difference will stay the same. In contrast, light from a light bulb is scrambled over different frequencies without a clearly defined phase, so we talk about incoherent light. (As a sidenote: A coherent state as defined in Quantum Optics is actually a special kind of state, which has special coherence properties. And also eigenstates of the harmonic oscillator are sometimes called coherent states, which is kind of related. But it might be best to not get into that right now.) In this case just think of a coherent state as a state with a well defined phase. A pure state usually has a well defined phase, while a mixed state is usually scrambled, so we call it incoherent.

1. The basis |uu>,|dd>,|ud>,|du> is the standard basis that is already available to us. It is just: Look at the result of the experiment, and ask the friend what he thinks the result is. In this case we will get |uu> with 50% probability and |dd> with 50% probability, and we cannot distinguish between the pure state |uu>+|dd> and the mixed state |uu> with 50% probability and |dd> with 50% probability. We will also have no information about the phase, in case of the mixed state there wouldn't even be a well defined phase. If we want to get more information, we need to be able to measure in other bases. This is usually done by performing operations on the system, that turns the measurement in the desired basis into a measurement in a basis that we actually can do, here |uu>,|dd>,|ud>,|du>. But if we don't have control over the system, then we cannot perform these operations, thus we cannot measure in another basis and therefore get no information about the phase. But you are right, if our standard way to measure would be in the basis {|uu>+|dd>,|uu>-|dd>,|ud>+|du>,|ud>-|du>} and we again would not have the control to turn this into a measurement in another basis, then we would run into the same problem.

I hope this helps,

Arne

Dear Arne

Thank you for your detailed, comprehensive answer one more time. So your answer indeed helped me get a better understanding, but they don't directly answer my very specific question. So maybe I have to be straight in what I think.

I have concluded from your two previous responses that:

Prop. 1: Lack of control can be defined as"being limited to the set {|uu>, |ud>, |du>, |dd>} for choice of measurement basis.
Prop. 2: " not knowing if the system is incoherent or coherent" is equivalent to "system is incoherent".

So if prop. 1 and 2 are both true then we can conclude porp. 3: lack of control makes system to be incoherent i.e. a mixed "state" of states. So prop 3 is just what Brukner has written in his article that I can't understand.

So, what I can understand is why assuming the truth of prop. 1 and 2 lead us to prop. 3 :)
Then what I don't understand is why prop. 1 and 2 are true :(

Prop. 1 is strange because simply another basis set can be used which allows Wigner to know if the system is coherent or not (even when he is not allowed to use any other basis and hence by definition lacks control). This means lack of control doesn't necessarily mean incoherency of the system, since the system maybe coherent and this coherency is detectable by Wigner.

Prop. 2 is strange because we are basically saying if S=A or S=B then S=A! Here S is system, A means "coherent" and B means "incoherent". If we don't know S is A or B, then how come we can say it is B?! Isn't this a paradox? (because at one side we are supposed not to know if S is coherent or not but on the other side we claim S is not coherent.

I really hope I could exactly show you the black spots in my head where I cannot exactly understand and also hope you kindly answer my questions one more time :)

Thank you very much for your time and considering my question

Kind regards,
Seyed

 

[PeterDonis]

Brukner, Č. A No-Go Theorem for Observer-Independent Facts. Entropy 2018, 20, 350.

The arxiv preprint of that paper is here:

https://arxiv.org/abs/1804.00749

 

[PeterDonis]

There we read:
"note that if Wigner did not know this phase due to the lack of control of it, he would describe the “spin + friend’s laboratory” in an incoherent mixture of the two possibilities".

For reference, this is near the top of p. 3 in the preprint, in the paragraph after equation (1).

 

[PeterDonis]

Why is this the case?

Note this statement from the previous sentence:

the particular phase (here ”+”) between the two amplitudes in Eq. (1) is specified by the measurement interaction in control of Wigner

In other words, Wigner has to choose a particular measurement to perform on the entire system inside the box, which includes his friend and the quantum system his friend measured. The assumption of the scenario is that the quantum operator describing particular measurement Wigner chooses to perform on that system has a basis of eigenstates which include the state described in equation (1).

In the sentence you originally quoted, just after the one quoted above, Bruckner is simply saying that if Wigner were not able to precisely control the measurement interaction he uses, he would not be able to guarantee that his measurement operator had that particular basis of eigenstates. And without that level of control, the further arguments Bruckner intends to make in the rest of the paper would not go through.

 

[Demystifier]

Thank you so much for your response. I went through the slides and I could learn about density matrices which I didn't know before (I already knew about relative phase). However, I couldn't answer my question even with this new knowledge! Maybe because my question is based on ambiguity in what Brukner meant by "control" and how it is incompatible with incoherence.

Well, just replace "control" with "measure". (One cannot control something without ability to measure it.) Does it make more sense then?

 

[Arne]

Dear Arne

Thank you for your detailed, comprehensive answer one more time. So your answer indeed helped me get a better understanding, but they don't directly answer my very specific question. So maybe I have to be straight in what I think.

I have concluded from your two previous responses that:

Prop. 1: Lack of control can be defined as"being limited to the set {|uu>, |ud>, |du>, |dd>} for choice of measurement basis.
Prop. 2: " not knowing if the system is incoherent or coherent" is equivalent to "system is incoherent".

So if prop. 1 and 2 are both true then we can conclude porp. 3: lack of control makes system to be incoherent i.e. a mixed "state" of states. So prop 3 is just what Brukner has written in his article that I can't understand.

So, what I can understand is why assuming the truth of prop. 1 and 2 lead us to prop. 3 :)
Then what I don't understand is why prop. 1 and 2 are true :(

Prop. 1 is strange because simply another basis set can be used which allows Wigner to know if the system is coherent or not (even when he is not allowed to use any other basis and hence by definition lacks control). This means lack of control doesn't necessarily mean incoherency of the system, since the system maybe coherent and this coherency is detectable by Wigner.

Prop. 2 is strange because we are basically saying if S=A or S=B then S=A! Here S is system, A means "coherent" and B means "incoherent". If we don't know S is A or B, then how come we can say it is B?! Isn't this a paradox? (because at one side we are supposed not to know if S is coherent or not but on the other side we claim S is not coherent.

I really hope I could exactly show you the black spots in my head where I cannot exactly understand and also hope you kindly answer my questions one more time :)

Thank you very much for your time and considering my question

Kind regards,
Seyed

Dear Seyed,

I'm sorry if my answers confuse you. I'm not sure how well your understanding in quantum mechanics is, so I tried to explain it as simple as possilbe, but as always in quantum mechanics, a simple explanation might actually be not very precise. For the sake of understanding, it might be best if you for now just forget about the phrases "coherent" and "incoherent" and think more about having a well defined phase or not.

The very short answer for both of your statements would be: 1. Yes that's true, 2. no that's false. For an explanation for 2. let's first take a step back and make some basic things clear.

First: It might be important to point out, that usually for any experiment in quantum mechanics, not only one measurement is done, but the experiment is repeated many times in order to get statistics. Do to the probabilistic nature of measurements in quantum mechanics, one single measurement is almost always useless. Only with a large number ob measurements we gain inside into the state of the system. Otherwise (at least if we have such a simple system with just two states) a mixed state doesn't really make sense. If we just look at one outcome, the system always has a well defined state, even if we don't know it.

Second: It is also important to note that the thought experiment is not that much conserned with what happens when Wigner does his measurement but when his friend does his measurement on a system that is in a superposition. Depending on how a measurement in quantum mechnics works, the friend will either collapse the state of the system to a single state in the basis he is measuring, or he will get entangled with the system and then the friend is in a superposition himself. This is basically the Schrödinger's cat thought experiment. Wigner expands this by now performing a measurement on the composite system, in order to get insight which of two possibilties might be more accurate. The paper you are citing uses this to examine if there actually is something like an objective information.

Now let's have a detailed look at how we would describe the two cases. Assume that the system is in a superposition |+> = |0> + |1> (for brevity I'm neglecting the 1/sqrt(2)). In the first case when Wigner's friend measures the system, the state collapses to either |0> or |1> with a 50-50 probability (remember: we would do this many times to actually see, that we get this probability). This, in any case, would also be exactly what Wigner's friend would describe, since he would not be able to perceive his own entanglement. Wigner would describe the composite system (including the friend) as a mixed state, 50% of the time the system is in |00> und else it would be in |11>, where the second first number stand for the outcome of the experiment and the second for the knowledge of the friend about the outcome. Since this is a probabilistic mixture, there is no well defined phase. Remeber, that probabilistic mixture needs, that we perform the same experiment over and over again, to get an ensemble of measurements, with that we get some statistics.

In contrast, if due to the measurement the friend becomes entangled, with the system, then Wigner would describe the system as some superposition of the states |00> and |11>, not just a probabilistic chance of having one of the two. Brukner now assumes, that the phase in this superposition is just 1, i. e. the state is|00>+|11>. This is described in: " where the particular phase (here ”+”) between the two amplitudes in Eq. (1) is specified by the measurement interaction in control of Wigner ".

Now to clarify your point 2: Not having control does not mean that there isn't a well defined phase. It just means, that we cannot find out if there is a well defined phase. So if we can only measure in one basis, then the system looks like a mixed state, even if it actually is a pure state. We need to be able to measure in a different basis to distinguish the two. This is basically what Brukner writes in the rest of the paragraph: " (Note that if Wigner would not know this phase due to lack of control of it, he would describe the ”spin + friend’s laboratory” in an incoherent mixture of the two possibilities.) Wigner can verify his state assignment (1), for example, by performing a Bell state measurement in the basis."

Regards, Arne

 

[SEYED2001]

Thank you all for helping me!

I can now Kind of see what the sentence actually means and why it is there.

Thank you once again
Seyed

 

[Demystifier]

Thank you all for helping me!

I can now Kind of see what the sentence actually means and why it is there.

Thank you once again
Seyed

Good, now you can explain the Brukner's paper to us.