On Fri, Aug 9, 2019 at 10:20 AM Bruno Marchal <marc...@ulb.ac.be> wrote:
>
> On 9 Aug 2019, at 13:09, Jason Resch <jasonre...@gmail.com> wrote:
>
>
>
> On Fri, Aug 9, 2019 at 3:22 AM Bruno Marchal <marc...@ulb.ac.be> wrote:
>
>>
>> On 8 Aug 2019, at 17:41, Jason Resch <jasonre...@gmail.com> wrote:
>>
>>
>>
>> On Thu, Aug 8, 2019, 5:51 AM Bruno Marchal <marc...@ulb.ac.be> wrote:
>>
>>>
>>> On 8 Aug 2019, at 11:56, Bruce Kellett <bhkellet...@gmail.com> wrote:
>>>
>>> On Thu, Aug 8, 2019 at 7:21 PM Bruno Marchal <marc...@ulb.ac.be> wrote:
>>>
>>>> On 8 Aug 2019, at 02:23, Bruce Kellett <bhkellet...@gmail.com> wrote:
>>>>
>>>> On Wed, Aug 7, 2019 at 11:30 PM Bruno Marchal <marc...@ulb.ac.be>
>>>> wrote:
>>>>
>>>>> On 7 Aug 2019, at 14:41, Bruce Kellett <bhkellet...@gmail.com> wrote:
>>>>>
>>>>>
>>>>> Superpositions are fine. It is just that they do not consist of
>>>>> "parallel worlds”.
>>>>>
>>>>>
>>>>> But then by QM linearity, it is easy to prepare a superposition with
>>>>> orthogonal histories, like me seing a cat dead and me seeing a cat alive,
>>>>> when I look at the Schoredinger cat. Yes, decoherence makes hard for me to
>>>>> detect the superposition I am in, but it does not make it going away
>>>>> (unless you invoke some wave packet reduction of course)
>>>>>
>>>>>
>>>>>
>>>>>> “Parallel worlds/histories” are just a popular name to describe a
>>>>>> superposition.
>>>>>>
>>>>>
>>>>> In your dreams, maybe. There is a clear and precise definition of
>>>>> separate worlds: they are orthogonal states that do not interact. The
>>>>> absence of possible interaction means that they are not superpositions.
>>>>>
>>>>>
>>>>> That is weird.
>>>>> The branches of a superposition never interact. The point is that they
>>>>> can interfere statistically, if not there is no superposition, nor
>>>>> interference, only a mixture.
>>>>>
>>>>
>>>> There some to be some fluidity is the concepts of superposition and
>>>> basis vectors inherent in this discussion. Any vector space can be spanned
>>>> by a set of orthogonal basis vectors. There are an infinite number of such
>>>> bases, plus the possibility of non-orthogonal bases given by any set of
>>>> vectors that span the space. If the basis vectors are orthogonal, these
>>>> basis vectors do not interact. But any general vector can be expressed as a
>>>> superposition of these orthogonal basis vectors. (Orthonormal basis for a
>>>> normed Hilbert space.)
>>>>
>>>> So the question whether the branches of a superposition can interact
>>>> (interfere) or not is simply a matter of whether the branches are
>>>> orthogonal or not. If we have a superposition of orthogonal basis vectors,
>>>> then the branches do not interact. However, if we have a superposition of
>>>> non-orthogonal vector, then the branches can interact.
>>>>
>>>> For example, the wave packet for a free electron is a superposition of
>>>> momentum eigenstates (and position eigenstates). These momentum eigenstates
>>>> are orthogonal and do not interact. The overlap function <p|p'> = 0 for all
>>>> p not equal to p'. This is the definition of orthogonal states. But this
>>>> does not mean that the wave packet of the electron is a mixture: It is a
>>>> pure state since there is a basis of the corresponding Hilbert space for
>>>> which the actual state is one of the basis vectors. (We can construct an
>>>> orthonormal set of basis vectors around this vector.) On the other hand,
>>>> the two paths that can be taken by a particle traversing a two-slit
>>>> interference experiment are not orthogonal, so these paths can interact. So
>>>> when the quantum state is written as a superposition of such paths, there
>>>> is interference.
>>>>
>>>> Orthogonality is the key difference between things that can interfere
>>>> and those that cannot. So if separate worlds are orthogonal, there can be
>>>> no interference between them, and the absence of such interaction defines
>>>> the worlds as separate.
>>>>
>>>>
>>>> What I use is the fact that when we have orthogonal states, like I0>
>>>> and I1>, I can prepare a state like (like I0> + I1>), and then I am myself
>>>> in the superposition state Ime>( I0> + I1>), Now, in that state, I have the
>>>> choice between measuring in the base {I0>, I1>} or in the base {I0> + I1>,
>>>> I0> - I1>). In the first case, the “parallel” history becomes indetectoble,
>>>> but not in the second case, so we have to take the superposition into
>>>> account to get the prediction right in all situations.
>>>>
>>>
>>> I don't think this is actually correct. Take a concrete example that we
>>> all understand. If we prepare a silver atom with spin 'up' in the
>>> x-direction, then a measurement in the x direction does not produce a
>>> superposition -- the answer is 'up' with 100% certainty. But is we measure
>>> this state in the transverse, y-direction, the result is either 'up-y' or
>>> 'down-y' with equal probabilities. This is because the initial state 'up-x'
>>> is already a superposition of 'up-y' and 'down-y'. When we measure this in
>>> the x-direction, there is no parallel history. When we measure in the
>>> y-direction, we get either 'up-y' or 'down-y'. MWI says that for either
>>> result, the alternative occurs in some other world. And that alternative
>>> result is just as undetectable as the 'down-x' result for the x-measurement.
>>>
>>>
>>>
>>> The pure state up-x is the same state as the superposition of up-y and
>>> down-y.
>>> Me in front of up-x and Me in front of up-y + down-y are only different
>>> description of the same state. When measuring that state in the
>>> x-direction, I don’t made that y-superposition disappears.
>>>
>>>
>>>
>>>
>>> The point being that whatever measurement we perform, we get only one
>>> result, and the alternative results that may or may not have been possible
>>> are undetectable.
>>>
>>>
>>> Yes, that is why we can exploit the parallel worlds (aka superposition
>>> of states relative to me) only by isolating the computer from from me, so
>>> that I don’t get entangled with it.
>>>
>>>
>>>
>>>
>>> However, it is interesting how this discussion has morphed. We started
>>> with the observation that a quantum computer does not demonstrate the
>>> existence of parallel worlds because its operation can be understood
>>> completely in terms of unitary rotations of the state vector in the one
>>> world of Hilbert space.
>>>
>>>
>>> Unitary rotations conserves the superposition (and the relative
>>> probabilities).
>>>
>>>
>>>
>>>
>>> Now we seem to have ended up with a discussion of the nature of
>>> superpositions, and the idea that unobserved outcomes from experiments have
>>> to be taken into account. How they are to be taken into account is never
>>> made clear.
>>>
>>>
>>>
>>> I don’t know why you say this. We need to take the superposition into
>>> account to get the probabilities right for arbitrary possible measurements.
>>>
>>>
>>>
>>> They are orthogonal, in fact, and cannot interact with the observed
>>> result. Parallel worlds, whether they "exist" or not, have no consequences
>>> for physics or experimental results. So Everett and MWI are otiose -- they
>>> have no conceivable effects, particularly in quantum computers, so they are
>>> irrelevant.
>>>
>>>
>>> If the superposition are not relevant, then I don’t have any minimal
>>> physical realist account of the two slit experience, or even the stability
>>> of the atoms.
>>>
>>> My goal is not in finding working theory, just to see if the current
>>> modern theory given by the physicists is consistent with digital mechanism,
>>> and indeed, its MWI aspect is the easiest prediction of mechanism. Then the
>>> math suggest we get also the negative interference and that QM confirms
>>> Digital Mechanism, unless we add the collapse postulate, which indeed is an
>>> option for the non-computationalist. But the collapse itself is not
>>> something that we can detect or observe in any way.
>>>
>>
>> Bruno,
>>
>> Forgive me if I have asked this before, but can you elaborate on the
>> how/why the math suggests negative interference?
>>
>> I currently have no intuition for why this should be.
>>
>> I recall reading something on continuous probability as being more
>> natural and leading to something much like the probability formulas in
>> quantum mechanics. Is that related?
>>
>>
>>
>> It is not intuitive at all. With the UDA, we can have have the intuition
>> coming from the first person indeterminacy on all all computational
>> continuation in arithmetic, but in the AUDA (the Arithmetical UDA), the
>> probabilities are constrained by the logic of self-reference G and G*. So
>> the reason why we can hope for negative amplitude of probability comes from
>> the fact that modal variant of the first person on the (halting)
>> computations, which is given by the arithmetical interpretation of:
>>
>> []p & p
>>
>> or
>>
>> []p & <>t
>>
>> or
>>
>> []p & <>t & p
>>
>> With, as usual, [] = Beweisbar, and p is an arbitrary sigma_1 sentences
>> (partial computable formula).
>>
>> They all give a quantum logic enough close to Dalla Chiara’s presentation
>> of them, to have the quantum features like complimentary observable, and
>> what I have called a sort of abstract linear evolution build on a highly
>> symmetrical core (than to LASE: the little Schroeder equation: p -> []<>p,
>> which provides a quantisation of the sigma_1 arithmetical reality.
>>
>> It is mainly the presence of this quantisation which justify that the
>> probabilities behave in a quantum non boolean way, but this is hard to
>> verify because the nesting of boxes in the G* translation makes those
>> formula … well, probably in need of a quantum computer to be evaluated. But
>> normally, if mechanism (and QM) are correct this should work.
>>
>> This is explained with more detail in “Conscience et Mécanisme”.
>>
>> Bruno
>>
>>
> Thank you Bruno for your explanation and references.
>
>
> Y’re welcome.
>
>
> Regarding “Conscience et Mécanisme”, is there a web/html or English
> version available? Unfortunately my browser cannot do translations of PDFs
> but can translate web pages. If not don't worry, I can copy and paste into
> a translator.
>
>
> Yes, There is no HTML page for the long text. But you can consult also my
> paper:
>
> Marchal B. The Universal Numbers. From Biology to Physics, Progress in
> Biophysics and Molecular Biology, 2015, Vol. 119, Issue 3, 368-381.
> https://www.ncbi.nlm.nih.gov/pubmed/26140993
>
> You will still need some background in quantum logic, like the paper by
> Goldblatt which makes the link between minimal quantum logic and the B
> modal logic.
>
> There is also a paper by Rawling and Selesnick which shows how to build a
> quantum NOT gate, from the Kripke semantics of the B logic. It is not
> entirely clear if this can be used in arithmetic, because we loss the
> necessitation rule in “our” B logic. Open problem. A positive solution on
> this would be a great step toward an explanation that the universal machine
> has necessarily a quantum structure and can exploit the “parallel
> computations in arithmetic” in the limit of the 1p indeterminacy..
>
> Rawling JP and Selesnick SA, 2000, Orthologic and Quantum Logic: Models
> and Computational Elements, Journal of the ACM, Vol. 47, n° 4, pp. 721-T51.
>
> Ask question, online or here. It *is* rather technical at some point.
>
> Bruno
>
>
>
I've been reading those references, and have found a few more which might
be related and of interest. Effectively, they provide arguments for the
quantum probability theory based on the requirement for continuous
reversible operations, or the juxtaposition between finite
information-carry capacity and smoothness.
Lucien Hardy's "Quantum Theory From Five Reasonable Axioms"
https://arxiv.org/abs/quant-ph/0101012
The usual formulation of quantum theory is based on rather obscure axioms
(employing complex Hilbert spaces, Hermitean operators, and the trace rule
for calculating probabilities). In this paper it is shown that quantum
theory can be derived from five very reasonable axioms. The first four of
these are obviously consistent with both quantum theory and classical
probability theory. Axiom 5 (which requires that there exists continuous
reversible transformations between pure states) rules out classical
probability theory. If Axiom 5 (or even just the word "continuous" from
Axiom 5) is dropped then we obtain classical probability theory instead.
This work provides some insight into the reasons quantum theory is the way
it is. For example, it explains the need for complex numbers and where the
trace formula comes from. We also gain insight into the relationship
between quantum theory and classical probability theory.
and Jochen Rau's "On quantum vs. classical probability"
https://arxiv.org/abs/0710.2119v2
The key (and novel) technical result, on the other hand, will pertain to
the second objective: I will show that the single distinguishing property
of quantum theory is the juxtaposition of finite information-carrying
capacity and smoothness, where the concept of smoothness will be carefully
defined and motivated. The mathematical derivation of this result will
involve close inspection of the symmetry group, with successive constraints
leading unequivocally to the unitary group of transformations in complex
Hilbert space. As for the final objective, I will provide arguments why
there is likely no further probabilistic theory that satisfies basic
physical desiderata.
Would you say these properties are inherent in the computations of the UD?
In so far as any computational thread representing an observer or a system
the observer interacts with is finite in its information carrying capacity,
but all the threads of similar indistinguishable computations for a
continuum? Is there a reason to suppose operations are reversible (could
this be due to some conservation of information principal in non-halting
programs?).
Is the appearance of complex numbers in the quantum probability sufficient
to get interference?
Jason
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