What Superposition doesn't imply ...

[Alan Grayson]

It's easy to show that a Superposition does NOT imply that a system represented by a linear sum of a pure set of basis vectors, is in all of those states simultaneusly.This follows from the fact that the WF is an element of a vector space, a Hilbert space, and in vector spaces there is no unique set of basis vectors. IOW, any set of basis vectors can represent the WF of a system, and if we claim the system is in all states of some superposition, it must also be in all states of any other superposition. And every set of basis vectors is equivalent to, and indistinguishable from any other set of basis vectors. This shows that Schrodinger could have denied the usual interpretation of the WF as a superposition where the system it represented could be interpreted as being in all pure states in its sum simultaneously, without constructing his Cat experiment. He simply had to remind his colleagues that the set of basis vectors in a vector space is not unique. AG

[Russell Standish]

What you are talking about is known as the preferred basis
problem. This has been discussed on this list before.

My own take on this is that you can't ignore the observer. In any
physical situation, an observer chooses some measurement apparatus
(thereafter you can sweep the observer under the carpet, and focus on
the measurement apparatus). The measurement apparatus entangled with
the system under question has the dynamics that tensor product of
measuring apparatus state with that of the system evolves to be
diagonal in some basis, aka "einselection". And that is the origin of
the preferred basis.

In the multiverse, there will also be other observers choosing
different apparati eg ones that select a complementary basis (eg
momentum where the first chooses to measure position). These will have
a different set of preferred basis.

There is only a problem if you try to ignore the existence of
observers and measuring devices.

Cheers

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[Brent Meeker]

If it's in a pure state then that is single vector in Hilbert space.  So there is a basis
that includes that vector and then the state has a single component in that basis.
Of course there is no way to measure in that basis without already knowing what
what it is.

Brent

[Alan Grayson]

On Wednesday, January 15, 2025 at 5:15:35 PM UTC-7 Brent Meeker wrote:

On 1/15/2025 1:39 PM, Russell Standish wrote:

What you are talking about is known as the preferred basis problem. This has been discussed on this list before. My own take on this is that you can't ignore the observer. In any physical situation, an observer chooses some measurement apparatus (thereafter you can sweep the observer under the carpet, and focus on the measurement apparatus). The measurement apparatus entangled with the system under question has the dynamics that tensor product of measuring apparatus state with that of the system evolves to be diagonal in some basis, aka "einselection". And that is the origin of the preferred basis. In the multiverse, there will also be other observers choosing different apparati eg ones that select a complementary basis (eg momentum where the first chooses to measure position). These will have a different set of preferred basis. There is only a problem if you try to ignore the existence of observers and measuring devices. Cheers On Wed, Jan 15, 2025 at 11:58:33AM -0800, Alan Grayson wrote:

It's easy to show that a Superposition does NOT imply that a system represented by a linear sum of a pure set of basis vectors, is in all of those states simultaneusly.This follows from the fact that the WF is an element of a vector space, a Hilbert space, and in vector spaces there is no unique set of basis vectors. IOW, any set of basis vectors can represent the WF of a system, and if we claim the system is in all states of some superposition, it must also be in all states of any other superposition.

If it's in a pure state then that is single vector in Hilbert space.  So there is a basis
that includes that vector and then the state has a single component in that basis.
Of course there is no way to measure in that basis without already knowing what
what it is.

Brent

 

 Generally speaking, isn't a superposition a linear sum of pure states? AG

[Brent Meeker]

On 1/15/2025 4:55 PM, Alan Grayson wrote:

On Wednesday, January 15, 2025 at 5:15:35 PM UTC-7 Brent Meeker wrote:

On 1/15/2025 1:39 PM, Russell Standish wrote:

What you are talking about is known as the preferred basis problem. This has been discussed on this list before. My own take on this is that you can't ignore the observer. In any physical situation, an observer chooses some measurement apparatus (thereafter you can sweep the observer under the carpet, and focus on the measurement apparatus). The measurement apparatus entangled with the system under question has the dynamics that tensor product of measuring apparatus state with that of the system evolves to be diagonal in some basis, aka "einselection". And that is the origin of the preferred basis. In the multiverse, there will also be other observers choosing different apparati eg ones that select a complementary basis (eg momentum where the first chooses to measure position). These will have a different set of preferred basis. There is only a problem if you try to ignore the existence of observers and measuring devices. Cheers On Wed, Jan 15, 2025 at 11:58:33AM -0800, Alan Grayson wrote:

It's easy to show that a Superposition does NOT imply that a system represented by a linear sum of a pure set of basis vectors, is in all of those states simultaneusly.This follows from the fact that the WF is an element of a vector space, a Hilbert space, and in vector spaces there is no unique set of basis vectors. IOW, any set of basis vectors can represent the WF of a system, and if we claim the system is in all states of some superposition, it must also be in all states of any other superposition.

If it's in a pure state then that is single vector in Hilbert space.  So there is a basis
that includes that vector and then the state has a single component in that basis.
Of course there is no way to measure in that basis without already knowing what
what it is.

Brent

 

 Generally speaking, isn't a superposition a linear sum of pure states? AG

Right.  And a linear sum of vectors is a vector.

Brent

And every set of basis vectors is equivalent to, and indistinguishable from any other set of basis vectors. This shows that Schrodinger could have denied the usual interpretation of the WF as a superposition where the system it represented could be interpreted as being in all pure states in its sum simultaneously, without constructing his Cat experiment. He simply had to remind his colleagues that the set of basis vectors in a vector space is not unique. AG -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-li...@googlegroups.com. To view this discussion visit https://groups.google.com/d/msgid/everything-list /39b8f674-1f27-4ff2-ad1e-637230c397bcn%40googlegroups.com.

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[Alan Grayson]

On Wednesday, January 15, 2025 at 6:44:27 PM UTC-7 Brent Meeker wrote:

On 1/15/2025 4:55 PM, Alan Grayson wrote:

On Wednesday, January 15, 2025 at 5:15:35 PM UTC-7 Brent Meeker wrote:

On 1/15/2025 1:39 PM, Russell Standish wrote:

What you are talking about is known as the preferred basis problem. This has been discussed on this list before. My own take on this is that you can't ignore the observer. In any physical situation, an observer chooses some measurement apparatus (thereafter you can sweep the observer under the carpet, and focus on the measurement apparatus). The measurement apparatus entangled with the system under question has the dynamics that tensor product of measuring apparatus state with that of the system evolves to be diagonal in some basis, aka "einselection". And that is the origin of the preferred basis. In the multiverse, there will also be other observers choosing different apparati eg ones that select a complementary basis (eg momentum where the first chooses to measure position). These will have a different set of preferred basis. There is only a problem if you try to ignore the existence of observers and measuring devices. Cheers On Wed, Jan 15, 2025 at 11:58:33AM -0800, Alan Grayson wrote:

It's easy to show that a Superposition does NOT imply that a system represented by a linear sum of a pure set of basis vectors, is in all of those states simultaneusly.This follows from the fact that the WF is an element of a vector space, a Hilbert space, and in vector spaces there is no unique set of basis vectors. IOW, any set of basis vectors can represent the WF of a system, and if we claim the system is in all states of some superposition, it must also be in all states of any other superposition.

If it's in a pure state then that is single vector in Hilbert space.  So there is a basis
that includes that vector and then the state has a single component in that basis.
Of course there is no way to measure in that basis without already knowing what
what it is.

Brent

 

 Generally speaking, isn't a superposition a linear sum of pure states? AG

Right.  And a linear sum of vectors is a vector.

Brent

If it can be proven what I've initially stated about a superposition, why is it necessary to consider entangement of experimenter and

apparatus, when the result follows directly from the properties of a vector space? AG 

[Alan Grayson]

On Wednesday, January 15, 2025 at 6:51:28 PM UTC-7 Alan Grayson wrote:

On Wednesday, January 15, 2025 at 6:44:27 PM UTC-7 Brent Meeker wrote:

On 1/15/2025 4:55 PM, Alan Grayson wrote:

On Wednesday, January 15, 2025 at 5:15:35 PM UTC-7 Brent Meeker wrote:

On 1/15/2025 1:39 PM, Russell Standish wrote:

What you are talking about is known as the preferred basis problem. This has been discussed on this list before. My own take on this is that you can't ignore the observer. In any physical situation, an observer chooses some measurement apparatus (thereafter you can sweep the observer under the carpet, and focus on the measurement apparatus). The measurement apparatus entangled with the system under question has the dynamics that tensor product of measuring apparatus state with that of the system evolves to be diagonal in some basis, aka "einselection". And that is the origin of the preferred basis. In the multiverse, there will also be other observers choosing different apparati eg ones that select a complementary basis (eg momentum where the first chooses to measure position). These will have a different set of preferred basis. There is only a problem if you try to ignore the existence of observers and measuring devices. Cheers On Wed, Jan 15, 2025 at 11:58:33AM -0800, Alan Grayson wrote:

It's easy to show that a Superposition does NOT imply that a system represented by a linear sum of a pure set of basis vectors, is in all of those states simultaneusly.This follows from the fact that the WF is an element of a vector space, a Hilbert space, and in vector spaces there is no unique set of basis vectors. IOW, any set of basis vectors can represent the WF of a system, and if we claim the system is in all states of some superposition, it must also be in all states of any other superposition.

If it's in a pure state then that is single vector in Hilbert space.  So there is a basis
that includes that vector and then the state has a single component in that basis.
Of course there is no way to measure in that basis without already knowing what
what it is.

Brent

 

 Generally speaking, isn't a superposition a linear sum of pure states? AG

Right.  And a linear sum of vectors is a vector.

Brent

If it can be proven what I've initially stated about a superposition, why is it necessary to consider entangement of experimenter and

apparatus, when the result follows directly from the properties of a vector space? AG 

Another question I have about superposition is this; if the wf for some system is written in a momentum basis, do other momentum bases 

exist for expressing the wf? If so, is the set of momentum bases infinite? AG

[John Clark]

On Wed, Jan 15, 2025 at 2:58 PM Alan Grayson <agrays...@gmail.com> wrote:

> It's easy to show that a Superposition does NOT imply that a system represented by a linear sum of a pure set of basis vectors, is in all of those states simultaneusly.This follows from the fact that the WF is an element of a vector space, a Hilbert space, and in vector spaces there is no unique set of basis vectors.

I would argue that the preferred basis issue crops up in ALL quantum interpretations, not just Many Worlds, because they all involve a quantum wave function that is evolving unitarily. To be useful a basis state must have a consistent history and it must have "you" in it. Many Worlds defines basis states as those states that are stable under environmental decoherence. So the wave function consists of [{ (a live cat) + ("you" observing a live cat) + ( the environment) } ]   +   [ { (a dead cat) + ("you" observing a dead cat) + ( the environment) } ]. I don't claim that is unique, you could define a basis state in many other ways but that is the only way that enables you to get a useful result that can be checked by experimentation.  As a practical matter we don't care about submicroscopic changes to the environment, but we do care about whether we see a dead cat or a live cat. 

 John K Clark    See what's on my new list at  Extropolis

lcd

 

 

[Brent Meeker]

On 1/16/2025 3:04 AM, Alan Grayson wrote:

On Wednesday, January 15, 2025 at 6:51:28 PM UTC-7 Alan Grayson wrote:

On Wednesday, January 15, 2025 at 6:44:27 PM UTC-7 Brent Meeker wrote:

On 1/15/2025 4:55 PM, Alan Grayson wrote:

On Wednesday, January 15, 2025 at 5:15:35 PM UTC-7 Brent Meeker wrote:

On 1/15/2025 1:39 PM, Russell Standish wrote:

What you are talking about is known as the preferred basis problem. This has been discussed on this list before. My own take on this is that you can't ignore the observer. In any physical situation, an observer chooses some measurement apparatus (thereafter you can sweep the observer under the carpet, and focus on the measurement apparatus). The measurement apparatus entangled with the system under question has the dynamics that tensor product of measuring apparatus state with that of the system evolves to be diagonal in some basis, aka "einselection". And that is the origin of the preferred basis. In the multiverse, there will also be other observers choosing different apparati eg ones that select a complementary basis (eg momentum where the first chooses to measure position). These will have a different set of preferred basis. There is only a problem if you try to ignore the existence of observers and measuring devices. Cheers On Wed, Jan 15, 2025 at 11:58:33AM -0800, Alan Grayson wrote:

It's easy to show that a Superposition does NOT imply that a system represented by a linear sum of a pure set of basis vectors, is in all of those states simultaneusly.This follows from the fact that the WF is an element of a vector space, a Hilbert space, and in vector spaces there is no unique set of basis vectors. IOW, any set of basis vectors can represent the WF of a system, and if we claim the system is in all states of some superposition, it must also be in all states of any other superposition.

If it's in a pure state then that is single vector in Hilbert space.  So there is a basis
that includes that vector and then the state has a single component in that basis.
Of course there is no way to measure in that basis without already knowing what
what it is.

Brent

 

 Generally speaking, isn't a superposition a linear sum of pure states? AG

Right.  And a linear sum of vectors is a vector.

Brent

If it can be proven what I've initially stated about a superposition, why is it necessary to consider entangement of experimenter and

apparatus, when the result follows directly from the properties of a vector space? AG 

Another question I have about superposition is this; if the wf for some system is written in a momentum basis, do other momentum bases 

exist for expressing the wf? If so, is the set of momentum bases infinite? AG

It's Hilbert space.  For finite dimensions it's just a vector space.  Forget about bases, vectors are things that at independent of bases.  Bases are arbitrary.

Brent

[Alan Grayson]

Yes, of course, vectors are independent of bases, but I asked the question to determine if my conclusion about superposition, as described in the initial post on this thread, is correct.  For my conclusion to be true, for any wf representing a system as a superposition, there must be many bases which can also represent the same wf. So, for example, for a wf which can be expressed as a superposition of momentum states, are there other bases for which the same wf can be so expressed? I've never seen this done, and I wonder if it can be done. TY, AG