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

```> On 10 Nov 2018, at 01:27, agrayson2...@gmail.com wrote:
>
>
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> On Friday, November 9, 2018 at 12:26:52 PM UTC, Bruno Marchal wrote:
>
>> On 8 Nov 2018, at 18:25, agrays...@gmail.com <javascript:> wrote:
>>
>>
>>
>> On Thursday, November 8, 2018 at 11:04:20 AM UTC, Bruno Marchal wrote:
>>
>>> On 6 Nov 2018, at 12:22, agrays...@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 <> 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
>>
>> 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
>
>
> 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
>
> Then the up’ = 1/sqrt(2) (1 1), and down’ = 1/sqrt(2) (1 -1),
>
> So the operator, written in the base up down, will be
>
> 0 1
> 1 0
>
>  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```
```

If the measured spin state is Up or Dn along some axis, the measured spin state
will be Up + Dn or Up - Down along the axis obtained by rotating the measuring
apparatus adequately. That is physically realisable with spin (by just rotating
the Stern-Gerlach apparatus) of with light polarisation (rotating the polariser
or the CaCO3 crystal).

Bruno

>
> 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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