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

```> On 6 Nov 2018, at 12:22, agrayson2...@gmail.com wrote:
>
>
>
> On Tuesday, November 6, 2018 at 9:27:31 AM UTC, Bruno Marchal wrote:
>
>> On 4 Nov 2018, at 22:02, agrays...@gmail.com <javascript:> wrote:
>>
>>
>>
>> On Sunday, November 4, 2018 at 8:33:10 PM UTC, jessem wrote:
>>
>>
>> On Wed, Oct 31, 2018 at 7:30 AM Bruno Marchal <mar...@ulb.ac.be <>> wrote:
>>
>>> On 30 Oct 2018, at 14:21, agrays...@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 <> 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.
>>
>> I don't know if you're restricting the definition of phi1 and phi2 to some
>> particular type of eigenstate or not, but in general aren't there pure
>> states that are not eigenstates of any physically possible measurement
>> apparatus, so there is no way to directly measure that a system is in such a
>> state?
>>
>> Yes, such states exist IIUC. That's why I don't understand Bruno's claim
>> that Up + Dn and Up - Dn can be measured with any apparatus,
>
> Not *any*¨apparatus, but a precise one, which in this case is the same
> apparatus than for up and down, except that it has been rotated.
>
>
>
>
>> since they're not eigenstates of the spin operator, or any operator.
>
> This is were you are wrong. That are eigenstates of the spin operator when
> measured in some direction.
>
> If what you claim is true, then write down the operator for which up + dn (or
> up - dn) is an eigenstate? AG ```
```

It is the operator corresponding to the same device, just rotated from pi/2, or
pi (it is different for spin and photon). When I have more time, I might do the
calculation, but this is rather elementary quantum mechanics. (I am ultra-busy
up to the 15 November, sorry). It will have the same shape as the one for up
and down, in the base up’ and down’, so if you know a bit of linear algebra,
you should be able to do it by yourself.

Bruno

>
> Julian Swinger (and Townsend) showed that the formalism of (discrete, spin,
> qubit) quantum mechanics is derivable from 4 Stern-Gerlach experiments, using
> only real numbers, but for a last fifth one, you need the complex amplitudes,
> and you get the whole core of the formalism.
>
> Bruno
>
>
>
>
>> Do you understand Bruno's argument in a previous post on this topic? AG
>>
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