> 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:
>>> https://www.amazon.com/Conceptual-Foundations-Quantum-Mechanics-Advanced/dp/0738201049/ref=sr_1_1?s=books&ie=UTF8&qid=1540889778&sr=1-1&keywords=d%27espagnat+conceptual+foundation+of+quantum+mechanics&dpID=41NcluHD6fL&preST=_SY291_BO1,204,203,200_QL40_&dpSrc=srch
>>>  
>>> <https://www.amazon.com/Conceptual-Foundations-Quantum-Mechanics-Advanced/dp/0738201049/ref=sr_1_1?s=books&ie=UTF8&qid=1540889778&sr=1-1&keywords=d%27espagnat+conceptual+foundation+of+quantum+mechanics&dpID=41NcluHD6fL&preST=_SY291_BO1,204,203,200_QL40_&dpSrc=srch>
>>> 
>>> 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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