# Re: Measuring a system in a superposition of states vs in a mixed state

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On Sunday, November 18, 2018 at 12:19:20 PM UTC, Bruno Marchal wrote:
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> On 16 Nov 2018, at 15:38, agrays...@gmail.com <javascript:> wrote:
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> On Friday, November 16, 2018 at 10:14:32 AM UTC, scerir wrote:
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>> Il 16 novembre 2018 alle 10.19 agrays...@gmail.com ha scritto:
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>>
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>> On Thursday, November 15, 2018 at 2:14:48 PM UTC, scerir wrote:
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>> Il 15 novembre 2018 alle 14.29 agrays...@gmail.com ha scritto:
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>> On Thursday, November 15, 2018 at 8:04:53 AM UTC, scerir wrote:
>>
>> Imagine a spin-1/2 particle described by the state psi = sqrt(1/2)
>> [(s+)_z + (s-)_z] .
>>
>> If the x-component of spin is measured by passing the spin-1/2 particle
>> through a Stern-Gerlach with its field oriented along the x-axis, the
>> particle will ALWAYS emerge 'up'.
>>
>>
>> *Why?  Won't the measured value be along the x axis in both directions,
>> in effect Up or Dn? AG*
>>
>> "Hence we must conclude that the system described by the |+>x state is
>> not the
>> same as a mixture of atoms in the |+> and !-> states. This means that
>> each atom in the
>> beam is in a state that itself is a combination of the |+> and |->
>> states. A superposition
>> state is often called a coherent superposition since the relative phase
>> of the two terms is
>> important."
>>
>> .see pages 18-19 here *https://tinyurl.com/ybm56whu
>> <https://tinyurl.com/ybm56whu>*
>>
>>
>> *Try answering in your own words. When the SG device is oriented along
>> the x axis, now effectively the z-axix IIUC, and we're dealing with
>> superpositions, the outcomes will be 50-50 plus and minus. Therefore,
>> unless I am making some error, what you stated above is incorrect. AG *
>>
>> sqrt(1/2) [(s+)_z +(s-)_z]  is a superposition, but since sqrt(1/2)
>> [(s+)_z +(s-)_z]  =  (s+)_x the particle will always emerge 'up'
>>
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> I'll probably get back to on the foregoing. In the meantime, consider
> this; I claim one can never MEASURE Up + Dn or Up - Dn with a SG apparatus
> regardless of how many other instruments one uses to create a composite
> measuring apparatus (Bruno's claim IIUC). The reason is simple. We know
> that the spin operator
>
>
> Which one?
>```
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*Good question. AG*

There are spin operator for each direction in space. The superposition of
> up and down is a precise pure state, with precise eigenvalues, when
> measuring state in the complementary directions.
>

*As I wrote earlier, based on scerir's superpositions on different axes,
and simulation, I now think that Up + Dn and Up - Dn can be measured along
the x axis but not along the z axis (which I was focused on). You were
probably correct about x axis measurements, but perhaps were not clear
enough. You were not explicit that measurements along the x axis is a
different SG experiment from along z axis. I thought you meant do them in
succession, not as separate experiments. Also introducing an infinity of
universes seems extraneous and confusing for a solution to this problem. AG
*

>
> has exactly two eigenstates, each with probability of .5. We can write
> them down. We also know that every quantum measurement gives up an
> eigenvalue of some eigenstate. Therefore, if there existed an Up + Dn or Up
> - Dn eigenstate, it would have to have probability ZERO since the Up and Dn
> eigenstates have probabilities which sum to unity. Do you agree or not, and
> if not, why? TIA, AG
>
>
> You add the probabilities, but you need to add the amplitudes of
> probabilities instead, and then take their square. You simply dismiss the
> quantum formalism, it seems to me.
>

*I did not; an incorrect inference on your part. I*
* never mentioned Born's rule (it wasn't necessary), from which one cannot
infer I am criticizing QM itself. AG *

The states constituted a vector space: the sum (superposition) of
> orthogonal states are pure state, after a change of base, and I did give
> you the corresponding operator. You are not criticising an interpretation
> of QM, but QM itself.
>

> Bruno
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>
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>>
>>
>>
>> In fact (s+)_z = sqrt(1/2) [(s+)_x + (s-)_x]
>>
>> and (s-)_z = sqrt(1/2) [(s+)_x - (s-)_x]
>>
>> (where _z, _x, are the z-component and the x-component of spin)
>>
>> so that psi = sqrt(1/2)[(s+)_z +(s-)_z] = (s+)_x.   (pure state, not
>> mixture state)..
>>
>> AGrayson2000 asked "If a system is in a superposition of states, whatever
>> value measured, will be repeated if the same system is repeatedly
>> measured.  But what happens if the system is in a mixed state?"
>>
>> Does Everett's "relative state interpretation" show how to interpret a
>> real superposition (like the above, in which the particle will always
>> emerge 'up') and how to interpret a mixture (in which the particle will
>> emerge 50% 'up' or 50% 'down')?
>>
>>
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