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

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On Friday, November 9, 2018 at 12:26:52 PM UTC, Bruno Marchal wrote:
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> On 8 Nov 2018, at 18:25, agrays...@gmail.com <javascript:> wrote:
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> On Thursday, November 8, 2018 at 11:04:20 AM UTC, Bruno Marchal wrote:
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>> On 6 Nov 2018, at 12:22, agrays...@gmail.com wrote:
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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 wrote:
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>>>
>>> On Sunday, November 4, 2018 at 8:33:10 PM UTC, jessem wrote:
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>>>>
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>>>> 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 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, *
>>>
>>>
>>> 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
>>
>
> *You don't have to do any calculation. Just write down the operator which,
> you allege, has up + dn or up - dn as an eigenstate. I don't think you can
> do it, because IMO it doesn't exist. AG *
>
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> If up and down are represented by the column (1 0) and (0 1) the
> corresponding observable is given by the diagonal matrix
>
> 1  0
> 0 -1
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> Then the up’ = 1/sqrt(2) (1 1), and down’ = 1/sqrt(2) (1 -1),
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> So the operator, written in the base up down, will be
>
> 0 1
> 1 0
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>  Here the eigenvalue +1 and -1 correspond to up (up’) or down (down’).
>
> I have no clue why you think that such operator would not exist.
>```
```
*Because the measured spin state is Up or Dn along some axis, never
anything in between. Up + Dn or Up - Dn is not physically realizable in
unprimed basis. AG*

All pure state can be seen as a superposition, in the rotated base, and you
> can always build an operator having them as eigenvalues.
>
> Bruno
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>>> 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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