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

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On Tuesday, November 6, 2018 at 9:27:31 AM UTC, Bruno Marchal wrote:
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> On 4 Nov 2018, at 22:02, agrays...@gmail.com <javascript:> wrote:
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> On Sunday, November 4, 2018 at 8:33:10 PM UTC, jessem wrote:
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>> On Wed, Oct 31, 2018 at 7:30 AM Bruno Marchal <mar...@ulb.ac.be> wrote:
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>>>
>>> On 30 Oct 2018, at 14:21, agrays...@gmail.com wrote:
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>>>
>>>
>>> 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?
>>
>
> *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, *
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> 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.
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> *since they're not eigenstates of the spin operator, or any operator. *
>
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> This is were you are wrong. That are eigenstates of the spin operator when
> measured in some direction.
>```
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*If what you claim is true, then write down the operator for which up + dn
(or up - dn) is an eigenstate? AG *

>
> 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
>
>
>
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> *Do you understand Bruno's argument in a previous post on this topic? AG *
>
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