B How to understand unitarity in QM?

[zonde]

Quote lifted from a thread in the cosmology forum.

What does it mean to know the exact state of a QM system? QM predicts probabilities that particles will be in one of multiple states when the particles are observed, and when observed, not all properties of a particle are simultaneously knowable to an exact degree (eg position and momentum).

Does knowing the exact state mean I know the probability functions for each particle in a given system, or is it different than that?

I would say that the statement you quoted is rather incorrect. It would be better if instead of "state" he would talk about "quantum state (vector)" like this:

But quantum mechanics, as far as we know, has a property called unitarity: if I know the exact quantum state (vector) of the system at time T, then I can, given enough computer power, calculate the precise quantum state (vector) of the system at any other time, no matter what.

As I see it there is important distinction between (classical) state and quantum state (vector) as they are very different. Description of quantum state (vector) can be transformed into description of classical state only probabilistically via Born rule.

The way I see the difference can be illustrated with crude analogy - let's say that classical state of particles is like actual configuration of molecules in complex object, then quantum state (vector) would be like chemical composition of complex object.

 

[vanhees71]

Hmm. That's a bit of a cop out. For any given observable a sufficiently ingenious experimenter may be able to devise several technically different ways of realizing one of these ideal measurements. Granted it might take a considerable degree of ingenuity. Is your position that QM can not answer the question at all? So, on the assumption that I can construct an experiment to perform one of these idealized measurements, what does QM say about the state after the measurement?

I disagree that one actually needs the specific experimental details to even discuss this. I think QM tells us very nicely what the state is given a particular result - on the assumption that we can actually do one of these kinds of experiments.

On the other hand, is it even a reasonable assumption that these kinds of measurements even exist? I'm not sure - it's certainly beyond my ingenuity to think of a really clean-cut example (I don't consider the Stern-Gerlach set-up to be such a 'clean-cut' example).
Ah OK, I see what you mean by this now - the word 'component' threw me a bit and I thought you were talking about entangling different properties of a single system, which I couldn't make sense of. The magnetic field is acting like a beamsplitter. There's an entanglement between the path and the spin - in the same way we have an entanglement of the spatial mode (path) and polarization in a polarizing beamsplitter - which isn't a measurement either but a process by which we can filter different polarization states.

The real measurement is done when there is a subsequent measurement of the particles in one of the beam arms. So in the Stern-Gerlach case the absorption is acting rather like the technique for the production of heralded photons using parametric downconversion.

In the photon polarizing BS case you'd have a mixed state along the lines of $| 0 \rangle \langle 0| + | 1 \rangle \langle 1|$ in each timeslot in one of the arms before making a measurement on the other arm (an absorptive photodetection). After this 'heralding' measurement you'd have pure states in each timeslot (either a polarized photon or the vacuum).

A photon is a good example. In almost all photon detectors the photon gets absorbed, and for sure then you cannot describe its state as being the one you get by assuming a collapse. That's all what I meant. It's nothing deep but just looking at what experimentalists really do in their labs when they measure all kinds of (microscopic) observables.

 

[Simon Phoenix]

How do you do this actual physical experimental setup where you can encode ten bits of information onto a spin 1/2 particle?

The first thing is to get clear what is being described here. It's a communication channel. Normally we think of a communication channel as some transmission of information between two (or more) parties - that's just transmission in space. Storage and retrieval is just transmission in time. So there's a coding process, a transmission (either in space or time) and then a measurement. The critical thing we're trying to achieve is that the original information (the message) can be reconstructed from the measurement results.

Rather than talk about trying to code 10 bits on a single spin-1/2 particle it's probably easier to think about just trying to encode 1 bit.

So let's think of the obvious way first. Suppose we use the up and down eigenstates of spin-z for our encoding. That's just choosing $| + \rangle _Z$ or $| - \rangle _Z$, respectively.

But how are we to recover that information? We can make a measurement of spin-z and for each particle transmitted we'll get the right answer - we'll get the bit that has been encoded with unit probability (assuming perfect measurements and no other noise sources).

But suppose we decide to measure spin-x instead? Now each measurement will give us the result + or - with 50% probability. With this measurement we'll recover no information (about the message that has been encoded) whatsoever. It's like having a binary channel with 'perfect' noise. No information can be transmitted by such channels. In classical systems we usually tend to think in terms of the noise source being a characteristic of the transmission medium, rather than the measurement process. In QM, however, using a misaligned coding and measuring set-up will be equivalent to a noise source irrespective of how perfect our coding or measurement process is.

This is the problem with the scheme above - it's not the 'coding' that's the issue - it's the retrieval of the information.

Let's suppose we attempt to send 1 bit of information again, but this time we're going to use the state $| + \rangle _Z$ to be the "1" and the state $| - \rangle _X$ to be the "0". No problem with the coding here at all. We can send our message (real information) by coding according to this state preparation procedure. But what are we going to do to retrieve (read) that message?

One simple measurement we could try is simply to measure spin-z for every incident particle. We recover the "1" symbol ( we get the result +) perfectly every time the state $| + \rangle _Z$ is sent, but with this choice of measurement we're going to get the result + or - with 50% probability whenever we measure the particles prepared in the state $| - \rangle _X$. So there's going to be a 25% error rate overall.

That's not the best we can do - to find the optimum we'd have to delve into the POVM formalism and optimize over all possible quantum measurements that could be done, but whatever we do there's going to be an unavoidable error that leads to less than one bit per particle being transmitted.

It's the fact that in order to code more than 1 bit on to a spin-1/2 particle (using the spin states as our coding basis) we'd have to select eigenstates from non-commuting operators - and therein lies the problem - it's this that prevents us from retrieving more than 1 bit per particle using this kind of coding.

 

[Blue Scallop]

Lol. Vanhees and Bill are much more expert than I - I have a rather 'old fashioned' view of QM and I struggle a bit with the more modern (and logically sound) approaches.

Vanhees and Bill are statistical ensemblers. On first impression they seemed to be shut up and calculate types.. but then what if Many worlds, Bohmians, Objective Collapse, Transactional I., etc. were all wrong. What if there was no processes or physical stuff like wave functions or even objective reality. Have you heard of Qbism, where wave function is only a tool inside our mind. Maybe Vanhees and Bills are Qbists in disguised? There is one other possibility where only statistics are real.. What if our world is just output of some infinite processing that occurs in an AdS/CFT domain.. here in the bulk only statistics or ensembles make sense.. particles or field just output or pixels of the programming? then there is no Many worlds, no Bohmians, No Objective Collapse.. Do you agree with this paragraph analysis?

About your last message:

That's not the best we can do - to find the optimum we'd have to delve into the POVM formalism and optimize over all possible quantum measurements that could be done, but whatever we do there's going to be an unavoidable error that leads to less than one bit per particle being transmitted.

It's the fact that in order to code more than 1 bit on to a spin-1/2 particle (using the spin states as our coding basis) we'd have to select eigenstates from non-commuting operators - and therein lies the problem - it's this that prevents us from retrieving more than 1 bit per particle using this kind of coding.

Is it true that superposition of up and down in the spin 1/2 particles can produce any other angles? What do you mean selecting eigenstates from from non-commuting operators?

Can you give other examples of superposition where you can encode some data in the nonorthogonal states (is nonorthogonal states a standard usage term)?

Thanks.

I'm sorry - I don't really understand what you're asking here. Is there another way of phrasing your question? By 'unitary' here, I think you mean separable. To extract any real information you need to do a measurement of some kind.

 

[Simon Phoenix]

Is it true that superposition of up and down in the spin 1/2 particles can produce any other angles? What do you mean selecting eigenstates from from non-commuting operators?

Can you give other examples of superposition where you can encode some data in the nonorthogonal states (is nonorthogonal states a standard usage term)?

A general (pure) spin state of a spin-1/2 particle can be written as $$ | \psi \rangle = a |- \rangle + b e^{ -i \alpha } | + \rangle $$ with $a$ and $b$ real such that $a^2 + b^2 = 1$. This is equivalent to some state $| - \rangle _{\theta , \phi}$ which would be an eigenstate of the spin operator $\hat {\mathbf \sigma}_{\theta , \phi}$ for some direction specified by the angles $\theta , \phi$.

But all this fancy stuff is really saying is that an eigenstate of spin in a given direction can be expanded in a different basis - just like a 2-dimensional vector (an arrow on a sheet of paper, for example) can be expanded as a superposition of any other two non parallel vectors in the same vector space.

In the quantum case it is often convenient, but not necessary, to restrict this expansion to orthogonal basis states. For a single mode EM field it is sometimes useful, for example, to expand a state in a coherent state basis. This is an overcomplete non-orthogonal basis for the mode.

I was hoping the examples I gave previously would be enough but I think you missed the main point. So let's consider what we would have to do to try to code just 2 bits of information on a single spin-1/2 particle.

To code 2 bits of information we're going to have to choose between 4 possible inputs which we can write as 00, 01, 10, and 11.

So we could choose the states $| - \rangle _Z$ and $| + \rangle _Z$ to represent the input symbols 00 and 11, respectively. But now what are we going to choose to represent the symbols 01 and 10? We've "used up" the eigenstates of spin-z. We're going to have to choose states from another basis - but these represent the eigenstates of spin in another direction. Let's pick the states $| - \rangle _X$ and $| + \rangle _X$ to represent the input symbols 01 and 10, respectively.

Now we have a problem because we've got 2 of our input symbols represented by eigenstates of $\hat {\mathbf \sigma}_Z$ and the other 2 input symbols represented by eigenstates of $\hat {\mathbf \sigma}_X$. These 2 operators don't commute - so there's no single measurement we can do that's going to allow us to perfectly distinguish between the 4 possible inputs 00,11, 01, and 10. For one thing it would violate the uncertainty principle if there was such a measurement.

The spin-z eigenstates are not orthogonal to the spin-x eigenstates - there's an overlap of squared magnitude 1/2.

If we can't perfectly distinguish between our possible input states then we can't recover the full 2 bits of information that has been coded. You can work out what you can recover from the example above (I'll leave that as an exercise for the reader). But really what we need is to work that out for all possible choices of input states and that's a more difficult calculation.

Anyway the upshot is that you can't recover more than 1 bit of information - even though we might have coded more than 1 bit.

 

[Blue Scallop]

A general (pure) spin state of a spin-1/2 particle can be written as $$ | \psi \rangle = a |- \rangle + b e^{ -i \alpha } | + \rangle $$ with $a$ and $b$ real such that $a^2 + b^2 = 1$. This is equivalent to some state $| - \rangle _{\theta , \phi}$ which would be an eigenstate of the spin operator $\hat {\mathbf \sigma}_{\theta , \phi}$ for some direction specified by the angles $\theta , \phi$.

But all this fancy stuff is really saying is that an eigenstate of spin in a given direction can be expanded in a different basis - just like a 2-dimensional vector (an arrow on a sheet of paper, for example) can be expanded as a superposition of any other two non parallel vectors in the same vector space.

In the quantum case it is often convenient, but not necessary, to restrict this expansion to orthogonal basis states. For a single mode EM field it is sometimes useful, for example, to expand a state in a coherent state basis. This is an overcomplete non-orthogonal basis for the mode.

I was hoping the examples I gave previously would be enough but I think you missed the main point. So let's consider what we would have to do to try to code just 2 bits of information on a single spin-1/2 particle.

To code 2 bits of information we're going to have to choose between 4 possible inputs which we can write as 00, 01, 10, and 11.

So we could choose the states $| - \rangle _Z$ and $| + \rangle _Z$ to represent the input symbols 00 and 11, respectively. But now what are we going to choose to represent the symbols 01 and 10? We've "used up" the eigenstates of spin-z. We're going to have to choose states from another basis - but these represent the eigenstates of spin in another direction. Let's pick the states $| - \rangle _X$ and $| + \rangle _X$ to represent the input symbols 01 and 10, respectively.

Now we have a problem because we've got 2 of our input symbols represented by eigenstates of $\hat {\mathbf \sigma}_Z$ and the other 2 input symbols represented by eigenstates of $\hat {\mathbf \sigma}_X$. These 2 operators don't commute - so there's no single measurement we can do that's going to allow us to perfectly distinguish between the 4 possible inputs 00,11, 01, and 10. For one thing it would violate the uncertainty principle if there was such a measurement.

The spin-z eigenstates are not orthogonal to the spin-x eigenstates - there's an overlap of squared magnitude 1/2.

If we can't perfectly distinguish between our possible input states then we can't recover the full 2 bits of information that has been coded. You can work out what you can recover from the example above (I'll leave that as an exercise for the reader). But really what we need is to work that out for all possible choices of input states and that's a more difficult calculation.

Anyway the upshot is that you can't recover more than 1 bit of information - even though we might have coded more than 1 bit.

Thanks for your indepth explanations. But I'd better share the following paper to know the context of my questions:

https://arxiv.org/pdf/quant-ph/9601025.pdf

quoting from page 2
"The quantum realization of a bit is a two-state quantum system—for example, a spin- 1/2
particle. A spin- 1/2 particle can be used to send one bit of classical information—and no more
than one bit—encoded in two orthogonal states, e.g., spin “down” for 0 and spin “up” for 1.
Thus it is convenient to denote the state of spin “down” by |0i and the state of spin “up” by
|1i. The difference between classical and quantum two-state systems is that quantum-mechanical
superposition gives a quantum two-state system other possible states, not available to a classical
two-state system: any linear combination of |0i and |1i is also a possible state. For a spin- 1/2
particle these states are in one-to-one correspondence with directions of the particle’s spin.

The crucial distinction between quantum and classical information appears when one attempts
to use these other states as alternatives for transmitting information. Suppose we attempt
to encode ten bits of information onto a spin- 1/2 particle, by preparing it so that its spin points in
one of 2^10 possible directions. Then we send the particle to you. Can you read out the ten bits?
Of course not. Quantum theory forbids any measurement to distinguish all 1024 possibilities."

Simon. In your messages. you were focusing on the first paragraph or orthogonal states and I understood it. I was simply asking now whether the author meant in the second paragraph as encoding the 10 bits by having 1024 angles in one spin-1/2 and trying to encode the information in one angle so that means when a spin 1/2 particle with one nonorthogonal angle say 43 degrees was send from you to me.. it's like sending 10 bits of information because of the 1024 angles? Is this what the second paragraph was describing? I need confirmation if it is.

But in superposition of spin up and spin down.. can the math really put it in a certain angle like 43 degrees? I know generally it is not possible to read nonorthogonal states... but for now just understanding the second paragraph..

 

[Simon Phoenix]

Take the paragraph above and rewrite it :

"The crucial distinction between quantum and classical information appears when one attempts
to use these other states as alternatives for transmitting information. Suppose we attempt
to encode 2 bits of information onto a spin- 1/2 particle, by preparing it so that its spin points in
one of 2^2 possible directions. Then we send the particle to you. Can you read out the 2 bits?
Of course not. Quantum theory forbids any measurement to distinguish all 4 possibilities"

I've just given you an example of an attempt to encode 2 bits using 4 possible spin directions.

Remember that it makes no sense to talk about the information encoded in a single transmission without knowledge of the associated coding scheme. So in the above case we're only able to send up to 2 bits of information per transmission because we're selecting from one of 4 possible inputs to the channel (the 4 possible spin directions).

The coding isn't the problem - we can indeed code 2 bits per transmission quite successfully per spin-1/2 particle. The problem is that we can never recover the full 2 bits at the receiver end of the channel - and in fact the best we can do is just recover 1 bit - even though we have coded 2 bits into each transmission.

 

[Blue Scallop]

Take the paragraph above and rewrite it :

"The crucial distinction between quantum and classical information appears when one attempts
to use these other states as alternatives for transmitting information. Suppose we attempt
to encode 2 bits of information onto a spin- 1/2 particle, by preparing it so that its spin points in
one of 2^2 possible directions. Then we send the particle to you. Can you read out the 2 bits?
Of course not. Quantum theory forbids any measurement to distinguish all 4 possibilities"

I've just given you an example of an attempt to encode 2 bits using 4 possible spin directions.

Remember that it makes no sense to talk about the information encoded in a single transmission without knowledge of the associated coding scheme. So in the above case we're only able to send up to 2 bits of information per transmission because we're selecting from one of 4 possible inputs to the channel (the 4 possible spin directions).

The coding isn't the problem - we can indeed code 2 bits per transmission quite successfully per spin-1/2 particle. The problem is that we can never recover the full 2 bits at the receiver end of the channel - and in fact the best we can do is just recover 1 bit - even though we have coded 2 bits into each transmission.

Again you were focusing on the first paragraph or orthogonal states and I understood it. Please address the second paragraph about encoding the 10 bits by having 1024 angles in one spin-1/2 and trying to encode the information in one angle so that means when a spin 1/2 particle with one nonorthogonal angle say 43 degrees was send from you to me.. it's like sending 10 bits of information because of the 1024 angles? Is this what the second paragraph was describing? I need confirmation if it is.

Maybe you meant since there was no way to distinguish the 1024 angles in one spin 1/2 particle and no coding scheme for it to be possible. It is not possible to encode it by choosing one angle??

 

[Simon Phoenix]

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.

 

[Simon Phoenix]

Please address the second paragraph

I did - you just don't yet understand why my reply does so

 

[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.