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

```> On 30 Oct 2018, at 14:21, agrayson2...@gmail.com wrote:
>
>
>
> On Tuesday, October 30, 2018 at 8:58:30 AM UTC, Bruno Marchal wrote:
>
>> On 29 Oct 2018, at 13:55, agrays...@gmail.com <javascript:> wrote:
>>
>>
>>
>> On Monday, October 29, 2018 at 10:22:02 AM UTC, Bruno Marchal wrote:
>>
>>> On 28 Oct 2018, at 13:21, agrays...@gmail.com <> wrote:
>>>
>>>
>>>
>>> On Sunday, October 28, 2018 at 9:27:56 AM UTC, Bruno Marchal wrote:
>>>
>>>> On 25 Oct 2018, at 17:12, agrays...@gmail.com <> wrote:
>>>>
>>>>
>>>>
>>>> On Tuesday, October 23, 2018 at 10:39:11 PM UTC, agrays...@gmail.com
>>>> <http://gmail.com/> wrote:
>>>> 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? TIA, AG
>>>>
>>>> If you think about it, whatever value you get on a single trial for a
>>>> mixed state, repeated on the same system, will result in the same value
>>>> measured repeatedly. If this is true, how does measurement distinguish
>>>> superposition of states, with mixed states? AG
>>>
>>> That is not correct. You can distinguish a mixture of particles in the up
>>> or down states with a set of 1/sqrt(2)(up+down) by measuring them with the
>>> {1/sqrt(2)(up+down), 1/sqrt(2)(up-down}) discriminating apparatus. With the
>>> mixture, half the particles will be defected in one direction, with the
>>> pure state, they will all pass in the same direction. Superposition would
>>> not have been discovered if that was not the case.
>>>
>>> And someone will supply the apparatus measuring (up + down), and (up -
>>> down)? No such apparatuses are possible since those states are inherently
>>> contradictory. We can only measure up / down. AG
>>
>> You can do the experience by yourself using a simple crystal of calcium
>> (CaCO3, Island Spath), or with polarising glass. Or with Stern-Gerlach
>> devices and electron spin. Just rotating (90° or 180°) an app/down
>> apparatus, gives you an (up + down)/(up - down) apparatus.
>>
>> I don't understand. With SG one can change the up/down axis by rotation, but
>> that doesn't result in an (up + down), or (up - down) measurement. If that
>> were the case, what is the operator for which those states are eigenstates?
>> Which book by Albert? AG
>
> David Z Albert, Quantum Mechanics and Experience, Harvard University Press,
> 1992.
> https://www.amazon.com/Quantum-Mechanics-Experience-David-Albert/dp/0674741137
>
> <https://www.amazon.com/Quantum-Mechanics-Experience-David-Albert/dp/0674741137>
>
> Another very good books is
>
> D’Espagnat B. Conceptual foundations of Quantum mechanics,  I see there is a
> new edition here:
>
>
> It explains very well the difference between mixtures and pure states.
>
> Bruno
>
> Thanks for the references. I think I have a reasonable decent understanding
> of mixed states. Say a system is in a mixed state of phi1 and phi2 with some
> probability for each. IIUC, a measurement will always result in an eigenstate
> of either phi1 or phi2 (with relative probabilities applying).```
```
If the measurement is done with a phi1/phi2 discriminating apparatus. Keep in
mind that any state can be seen as a superposition of other oblique or
orthogonal states.

> So if we're measuring spin, the result will always be an eigenstate of the
> spin vector for phi1 or phi2. But (up + down) and (up - down) are NOT
> eigenstates of either. How then can you justify your claim of measuring (up +
> down) or (up - down)? AG

By rotating the polariser filter: (abstracting from some normalising factor
which is supposed to be 1/sqrt(2))

up’ = up + down
down’ = up - down

up = up’ + down’
down = up’ - down’

If you measure a set of photons all in the state up’= up+down in the base {up’,
down’}, they all get the same result up'.
If you measure a set of photons from a mixture of up and down, in the same base
{up’, down’}, half of them will be up’, and half of them will be down’.

Bruno

>
>>
>> Bruno
>>
>>
>>
>>
>>
>>>
>>> Bruno
>>>
>>>
>>>
>>>
>>>
>>>>
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