B How to understand unitarity in QM?

[Blue Scallop]

Anyway - we're going a bit off-topic with discussions of coding schemes. This thread is about the meaning and implications of unitarity in QM.

I'm going to hazard a guess where you're going with this. I get the impression (but may be very wrong on this) that you're thinking of coding 2 bits on a single spin-1/2 particle as somehow 'containing' all 4 spin directions in a single state - and then using one of these 'filter' type measurements to split this into 4 distinct paths. Is this what you had in mind? If so, that's not the right way to look at this at all.

Yes. Is it not possible to encode 4 spin directions in a single spin 1/2 particle? It's up, down, left and right spin. Up and down being orthogonal state and left and right being superposition nonorthogonal state. I'm not asking about decoding them.. just asking how you can encode it and if there is a way..

This is on topic because at the end of the paper it was mentioned "No matter how involved the demonstrations of
the inability to distinguish nonorthogonal quantum states become, the underlying principle is unitarity."

Don't worry. Let's transfer to another thread if I still can't understand your next message. Lol

 

[Simon Phoenix]

"No matter how involved the demonstrations of
the inability to distinguish nonorthogonal quantum states become, the underlying principle is unitarity."

Yes - that's because unitary transformations preserve overlaps. If I have two states $| \psi \rangle$ and $| \phi \rangle$ then their overlap is $\langle \psi | \phi \rangle$. This overlap is important - a zero overlap means the states are orthogonal and can be perfectly distinguished. Conversely a non-zero overlap means the states are non-orthogonal and cannot be perfectly distinguished - they can only be distinguished with a certain probability that depends on the squared magnitude of this overlap.

So if I have some unitary transformation, $\hat { \mathbf U }$, that is applied to the states $| \psi \rangle$ and $| \phi \rangle$ so that $\hat { \mathbf U } | \psi \rangle = | \psi ' \rangle$ and $\hat { \mathbf U } | \phi \rangle = | \phi ' \rangle$ then $$ \langle \psi ' | \phi ' \rangle = \langle \psi | \hat { \mathbf U } ^\dagger \hat { \mathbf U } | \phi \rangle = \langle \psi | \phi \rangle $$ where we have used the definition of unitarity here : $\hat { \mathbf U } ^\dagger = \hat { \mathbf U } ^{-1}$.

 

[Blue Scallop]

Yes - that's because unitary transformations preserve overlaps. If I have two states $| \psi \rangle$ and $| \phi \rangle$ then their overlap is $\langle \psi | \phi \rangle$. This overlap is important - a zero overlap means the states are orthogonal and can be perfectly distinguished. Conversely a non-zero overlap means the states are non-orthogonal and cannot be perfectly distinguished - they can only be distinguished with a certain probability that depends on the squared magnitude of this overlap.

So if I have some unitary transformation, $\hat { \mathbf U }$, that is applied to the states $| \psi \rangle$ and $| \phi \rangle$ so that $\hat { \mathbf U } | \psi \rangle = | \psi ' \rangle$ and $\hat { \mathbf U } | \phi \rangle = | \phi ' \rangle$ then $$ \langle \psi ' | \phi ' \rangle = \langle \psi | \hat { \mathbf U } ^\dagger \hat { \mathbf U } | \phi \rangle = \langle \psi | \phi \rangle $$ where we have used the definition of unitarity here : $\hat { \mathbf U } ^\dagger = \hat { \mathbf U } ^{-1}$.

In Wikipedia.. Unitarity is defined as "In quantum physics, unitarity is a restriction on the allowed evolution of quantum systems that ensures the sum of probabilities of all possible outcomes of any event always equals 1."

Decoherence is supposed to preserved unitarity in quantum system.
Yet measurements can destroy unitarity.

Can you just give a simple example how when a system is not measured, and no matter how complex.. unitarity is preserved such that sum of probabilities of all possible outcomes of any event always equals 1? Let's say your eyeglasses are entangled with the environment and your body.. what does it mean the sum of all possible outcomes of any event always equals 1?

 

[Boing3000]

what does it mean the sum of all possible outcomes of any event always equals 1?

Conservation of likelihood

 

[Blue Scallop]

Conservation of likelihood

Ok. Refer to this graph of wave functions in deterministic unitarity evolution...

3^2= 9%
7^2=49%
8^2=64%
-------------
122%

It's not unity or 100%. Why?

 

[Simon Phoenix]

Can you just give a simple example how when a system is not measured, and no matter how complex.. unitarity is preserved such that sum of probabilities of all possible outcomes of any event always equals 1?

OK - let's consider a 2-level system (like a spin-1/2 particle, or a photon's polarization). We can express any pure state as a superposition of states in some basis so that $$ | \psi \rangle = \alpha |0 \rangle + \beta | 1 \rangle $$ where here $\alpha$ and $\beta$ are two complex numbers such that $| \alpha |^2 + | \beta |^2 = 1$. Now it's not too difficult to show that $| 0 \rangle \langle 0 | + | 1 \rangle \langle 1 | = \hat { \mathbf I}$, the identity operator.

Now let's consider performing a unitary transformation $\hat { \mathbf U }$ on our state $| \psi \rangle$ so that we obtain some new state $| \phi \rangle = \hat { \mathbf U } | \psi \rangle$. This new state can be expanded in any basis and in our original basis we're going to have $$ | \phi \rangle = \gamma |0 \rangle + \delta | 1 \rangle $$ The question is whether $| \gamma |^2 + | \delta |^2 = 1$?

Now we have that $\gamma = \langle 0 | \phi \rangle$ and $\delta = \langle 1 | \phi \rangle$ and after a tiny bit of algebra we can write $$ | \gamma |^2 + | \delta |^2 = \langle \psi | \left\{ \hat { \mathbf U } | 0 \rangle \langle 0 | \hat { \mathbf U } + \hat { \mathbf U } | 1 \rangle \langle 1 | \hat { \mathbf U } \right\} | \psi \rangle $$Now we know that unitary transformations preserve these overlaps. That means that if we apply a unitary transformation to an orthonormal basis we'll obtain another orthonormal basis. So if the states $| 0 \rangle$ and $| 1 \rangle$ get transformed to $| 0' \rangle$ and $| 1' \rangle$ under the unitary transformation this means that $| 0' \rangle \langle 0' | + | 1' \rangle \langle 1' | = \hat { \mathbf I}$, so that $$ \hat { \mathbf U } | 0 \rangle \langle 0 | \hat { \mathbf U } + \hat { \mathbf U } | 1 \rangle \langle 1 | \hat { \mathbf U } = | 0' \rangle \langle 0' | + | 1' \rangle \langle 1' | = \hat { \mathbf I } $$which means that $| \gamma |^2 + | \delta |^2 = 1$

And so we see that the sum of the probabilities is preserved under unitary transformation.

 

[Blue Scallop]

OK - let's consider a 2-level system (like a spin-1/2 particle, or a photon's polarization). We can express any pure state as a superposition of states in some basis so that $$ | \psi \rangle = \alpha |0 \rangle + \beta | 1 \rangle $$ where here $\alpha$ and $\beta$ are two complex numbers such that $| \alpha |^2 + | \beta |^2 = 1$. Now it's not too difficult to show that $| 0 \rangle \langle 0 | + | 1 \rangle \langle 1 | = \hat { \mathbf I}$, the identity operator.

Now let's consider performing a unitary transformation $\hat { \mathbf U }$ on our state $| \psi \rangle$ so that we obtain some new state $| \phi \rangle = \hat { \mathbf U } | \psi \rangle$. This new state can be expanded in any basis and in our original basis we're going to have $$ | \phi \rangle = \gamma |0 \rangle + \delta | 1 \rangle $$ The question is whether $| \gamma |^2 + | \delta |^2 = 1$?

Now we have that $\gamma = \langle 0 | \phi \rangle$ and $\delta = \langle 1 | \phi \rangle$ and after a tiny bit of algebra we can write $$ | \gamma |^2 + | \delta |^2 = \langle \psi | \left\{ \hat { \mathbf U } | 0 \rangle \langle 0 | \hat { \mathbf U } + \hat { \mathbf U } | 1 \rangle \langle 1 | \hat { \mathbf U } \right\} | \psi \rangle $$Now we know that unitary transformations preserve these overlaps. That means that if we apply a unitary transformation to an orthonormal basis we'll obtain another orthonormal basis. So if the states $| 0 \rangle$ and $| 1 \rangle$ get transformed to $| 0' \rangle$ and $| 1' \rangle$ under the unitary transformation this means that $| 0' \rangle \langle 0' | + | 1' \rangle \langle 1' | = \hat { \mathbf I}$, so that $$ \hat { \mathbf U } | 0 \rangle \langle 0 | \hat { \mathbf U } + \hat { \mathbf U } | 1 \rangle \langle 1 | \hat { \mathbf U } = | 0' \rangle \langle 0' | + | 1' \rangle \langle 1' | = \hat { \mathbf I } $$which means that $| \gamma |^2 + | \delta |^2 = 1$

And so we see that the sum of the probabilities is preserved under unitary transformation.

Wow, our future science advisor you are.
Well. Let's go conceptual for us without Einstein math abilities.

Many Worlds preserves unitarity because the wave function goes on and goes.
Bohmian mechanics preserves unitarity because the wave function also goes on and on but only one branch is manifested.
Copenhagen doesn't preserve unitarity because measurement collapses the wave function or destabilizes the unitarity.

Now for Many Worlds. If somehow there is a way to perceive all branches at once. Why does this violate unitarity? Can't we say that our probes would entangle with the systems and unitarity is preserved. Or what kind of probing all branches can be allowed conceptually that would still preserve unitarity?

 

[vanhees71]

Another way to put it: Anything defining a system within QT, i.e., the Hilbert space and operator algebra is invariant under unitary and antiunitary transformations, and according to Wigner's theorem any symmetry is realized either by a unitary or antiunitary transformation. Only discrete symmetry groups can be realized by antiunitary transformations. Since time-translation invariance, generated by the Hamiltonian of the system, is a continuous symmetry, it must be realized by unitary transformations.

 

[Simon Phoenix]

Anything defining a system within QT, i.e., the Hilbert space and operator algebra is invariant under unitary and antiunitary transformations, and according to Wigner's theorem any symmetry is realized either by a unitary or antiunitary transformation. Only discrete symmetry groups can be realized by antiunitary transformations. Since time-translation invariance, generated by the Hamiltonian of the system, is a continuous symmetry, it must be realized by unitary transformations.

I wish I could double like this - lovely answer

 

[Simon Phoenix]

Well. Let's go conceptual for us without Einstein math abilities.

I wish I could - but it's very difficult to answer the questions you're asking without some mathematical detail. To go beyond pop science accounts of QM there's really no way to do it (at least it's beyond my capabilities) and you're going to need to get to grips with at least some of the formalism - otherwise everything stays at the pop science level - and that's damnably difficult to get right for QM without being misleading in some way.

 

[Blue Scallop]

I wish I could - but it's very difficult to answer the questions you're asking without some mathematical detail. To go beyond pop science accounts of QM there's really no way to do it (at least it's beyond my capabilities) and you're going to need to get to grips with at least some of the formalism - otherwise everything stays at the pop science level - and that's damnably difficult to get right for QM without being misleading in some way.

Oh. I can visualize the maths and trace the unitarity evolution by imagination.. so don't worry. About the question what if somehow there is a way to perceive all branches at once. Why does this violate unitarity? I guess it's because of superposition (I heard this once). Have you heard of the many minds interpretation... note mind is something that even von Neumann or wigner was sure about.. so I'm just wondering if somehow the mind in many minds interpretation can be able to perceive all branches at once. Perhaps a physicist could discover an algorithm where unitarity would still be preserved even if you can scan all branches at once.. this is not impossible. Is it not. If you can refer me to the math of many worlds and how special Born-less procedure (remember Born rule is difficult in many worlds) is done to tap the branches, thanks for it.

 

[Boing3000]

About the question what if somehow there is a way to perceive all branches at once

There isn't, per definition of what "a branch" would mean.

Why does this violate unitarity?

If I understood vanhess71 post 68, the continuum of ALL "branches" wouldn't violate unitary.

 

[Blue Scallop]

unitary evolution conserves information.. so linking all the branches together create information... which is a violation of unitarity evolution. But can't we say the mind (think of von Neumann or Wigner or even Penrose you will not feel weird) introduce information to the system such that scanning all branches at once would be possible?

 

[Boing3000]

so linking all the branches together create information...

If your are referring to MWI, I don't think there is "linking of branches". All the infinities probabilities are described in the same Hilbert space, and sum up to 1. (and nobody can visualize that). The evolution of the system is not done by a "linking branches" operator.

But can't we say the mind (think of von Neumann or Wigner or even Penrose you will not feel weird) introduce information to the system such that scanning all branches at once would be possible?

"The mind" is not a QM entity described in that space, so I wouldn't bother too much about that.
But "the mind" most definitely can delve into some very simplified sub-set (few QBits) "possibilities". This is by no means equivalent to "scanning all branches at one".

 

[Blue Scallop]

If your are referring to MWI, I don't think there is "linking of branches". All the infinities probabilities are described in the same Hilbert space, and sum up to 1. (and nobody can visualize that). The evolution of the system is not done by a "linking branches" operator."The mind" is not a QM entity described in that space, so I wouldn't bother too much about that.
But "the mind" most definitely can delve into some very simplified sub-set (few QBits) "possibilities". This is by no means equivalent to "scanning all branches at one".

Ok. Let's not mention the mind as the group is not equipped to handle it.

Let's just focus on this unitarity conserving information thing. Is it like law of conservation of energy or the law of conservation of information where information can't be created nor destroyed? Then if you introduce new Hamiltonian to the system, then can't you add information to the system that can make you say couple two branches together? Ping expert Simon Phoenix and Vanhees. Need your expertise here and the math. Thanks!

 

[Blue Scallop]

Ok. Let's not mention the mind as the group is not equipped to handle it.

Let's just focus on this unitarity conserving information thing. Is it like law of conservation of energy or the law of conservation of information where information can't be created nor destroyed? Then if you introduce new Hamiltonian to the system, then can't you add information to the system that can make you say couple two branches together? Ping expert Simon Phoenix and Vanhees. Need your expertise here and the math. Thanks!

After googling a lot about Unitarity and understanding Simon formulas. I understood about it more. Unitarity just says that probability equals to 100% meaning when you have 50% in one slit, you have 50% in another slit. And so if you suddenly have 30% in one slit, you have 70% in one slit (or other setups). And overlap has to do with the vectors. But all this is assuming the wave function is a probabilistic medium per Born specification. Is there solid proof at all that this is the case. No. Because one can introduce Many worlds where the wave functions are not wave of probabilities but real. And here how can you say that unitarity is preserved? They now search for Lorentz violations, is there a similar search for unitarity violation? I think the black hole firewall stuff is one of those where they explore the possibility unitarity may not be true. So this means there is no solid proof unitarity is a law in Quantum mechanics, right?

 

[Boing3000]

Is there solid proof at all that this is the case. No.

In science only experiment count as proof. So as of now, it is the case.

Because one can introduce Many worlds where the wave functions are not wave of probabilities but real.

Real wave of probabilities, with the same mathematics. MWI does not introduce anything, it is an interpretation.

And here how can you say that unitarity is preserved?

For two reason
1) unitary is a core component of the mathematical coherence of the theory.
2) we flip coins, and as of now, it don't turn into whale & petunias

I think the black hole firewall stuff is one of those where they explore the possibility unitarity may not be true. So this means there is no solid proof unitarity is a law in Quantum mechanics, right?

Wrong. Unitary is by definition a part of QM.
But QM don't apply to Black Hole, there is no theory yet merging Quantum & Gravity

 

[Simon Phoenix]

Unitarity just says that probability equals to 100% meaning when you have 50% in one slit, you have 50% in another slit.

The unitary evolution is essential to allow us to interpret things probabilistically, yes, but you should also try to appreciate Vanhees' much deeper answer above (#68) - where symmetry is the key ingredient.

I wouldn't worry too much about trying to understand all of the various interpretations (unless you want to, of course) - they're all correct in the sense that they will all allow you to make the correct predictions about experiments. Just pick one that 'speaks to you' personally.

For me I stick to the axioms I presented earlier (with the necessary technical refinements, and extensions to cope with more generalized measurements). I know that if I use these axioms correctly I'm going to be able to calculate the right predictions for experiments. Of course the strict necessity of the 'projection postulate' is disputed as we've seen in this thread, but as a tool to aid calculation I personally find it extremely useful. Interpreting things like quantum key distribution or entanglement swapping is, for me, much cleaner and easier if we adopt this postulate. I don't think it's 'wrong' as such since the more general POVM formalism contains this as a special case, but if it bothers you, then as Vanhees and Bill have argued, it's probably not strictly necessary.

Ultimately you'll find extremely smart and capable physicists defending their own particular favourite interpretation with a great degree of passion and intellect - and currently there is no clear 'winner' in any experimental sense. So if the minimal ensemble interpretation is your thing - go with it. If you prefer Bohmian approaches, or the transactional interpretation, or MWI, or Copenhagen, or consistent histories - then they're all fine too. Until we have some experimental way to distinguish between them, they're all as good as each other. Nobody can tell you which of these is actually 'correct'

But unitary evolution is an essential feature of all of them as far as I can see.

 

[Blue Scallop]

The unitary evolution is essential to allow us to interpret things probabilistically, yes, but you should also try to appreciate Vanhees' much deeper answer above (#68) - where symmetry is the key ingredient.

I wouldn't worry too much about trying to understand all of the various interpretations (unless you want to, of course) - they're all correct in the sense that they will all allow you to make the correct predictions about experiments. Just pick one that 'speaks to you' personally.

For me I stick to the axioms I presented earlier (with the necessary technical refinements, and extensions to cope with more generalized measurements). I know that if I use these axioms correctly I'm going to be able to calculate the right predictions for experiments. Of course the strict necessity of the 'projection postulate' is disputed as we've seen in this thread, but as a tool to aid calculation I personally find it extremely useful. Interpreting things like quantum key distribution or entanglement swapping is, for me, much cleaner and easier if we adopt this postulate. I don't think it's 'wrong' as such since the more general POVM formalism contains this as a special case, but if it bothers you, then as Vanhees and Bill have argued, it's probably not strictly necessary.

Ultimately you'll find extremely smart and capable physicists defending their own particular favourite interpretation with a great degree of passion and intellect - and currently there is no clear 'winner' in any experimental sense. So if the minimal ensemble interpretation is your thing - go with it. If you prefer Bohmian approaches, or the transactional interpretation, or MWI, or Copenhagen, or consistent histories - then they're all fine too. Until we have some experimental way to distinguish between them, they're all as good as each other. Nobody can tell you which of these is actually 'correct'

But unitary evolution is an essential feature of all of them as far as I can see.

I think you have come across Zurek Quantum Darwinism. I think this makes more sense than others.. But someone said: "Unitarity is not directly related to the Born rule. Unitarity is a mathematical property, Born rule is a physical law." What can you say about Quantum Darwinism with regards to unitarity and the born rule? The quantum states in quantum Darwinism is the primitive.. Zurek derives the born rule without the born rule.. but yet Zurek quantum states automatically satisfied unitarity... What for you is the relationship between unitarity and the born rule?

And by the way, did you just take up physics undergraduate course (B.S. in physics) or are you a Ph.D? I wonder if an undergraduate physics major graduate can have the vastness of your knowledge or if it requires a Ph.D?

 

[RockyMarciano]

The unitary evolution is essential to allow us to interpret things probabilistically, yes.

This is clear and yet it is remarkable that we are still always faced with two different evolution types in QM, the continuous unitary evolution bewtween measurements, which is reversible, preserving probabilities and the measurement evolution(aka stochastic, probabilistic, acausal, irreversible, nonunitary, etc...) at each measuremnt event , and unitarity is anyway key to interpret the stochastic or nonunitary evolution of any event too and therefore its normalization of probabilities sum to 1, so it is hard not to interpret somewhat the circularity apparent in that unitarity of the theory is actually enforced by the unitary interpretation of the probability at every event. Of course this unitarity is ultimately postulated in the theory and it is essential for its consistency and this circularity is therefore granted.

The problem that always comes back is that measurements, that appear as the physical part, the empirical part with which predictions are checked, are because of the above necessarily left unexplained in its irreversible nonunitary essence that requires to undergo a new normalization after each measurement. In other words the unitarity postulate itself prevents to have any access to explain this irreversibility within the formalism, since it is actually absent of the formalism, it is a purely operational part that is integrated in the so called FAPP not formalized part of the physical practice of QM. All QM interpretations are required to respect the mathematical formalism, therefore there is no hope for them to address the irreversibility of measurements.

 

[vanhees71]

That you have to renormalize in many preparation procedures is very intuitive, because many preparations are in a sense done with filters. E.g., if you use a polarizer for em. waves/photons you usually get half the intensity of the incoming unpolarized light. So you get a renormalization factor of 2 by just using only half of the total incoming intensity.

 

[RockyMarciano]

That you have to renormalize in many preparation procedures is very intuitive, because many preparations are in a sense done with filters. E.g., if you use a polarizer for em. waves/photons you usually get half the intensity of the incoming unpolarized light. So you get a renormalization factor of 2 by just using only half of the total incoming intensity.

Yes, the filterings, the operational use of polarizers, nonlinear crystals, etc,... in preparations, or measurig apparati in measurements are all FAPP, are intuitive as motives for subsequent normalizations, but the irreversibility in this operational procedures of measurement and preparation is not part of the formalism, that follows the unitarity postulate.