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

`On Wed, Oct 31, 2018 at 7:30 AM Bruno Marchal <marc...@ulb.ac.be> wrote:`
```
>
> 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 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
>>>> 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
>>
>> 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?

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