Categorical quantum models and logics...Clearly this is a crude simpli cation, like other analogies...
Transcript of Categorical quantum models and logics...Clearly this is a crude simpli cation, like other analogies...
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Categorical quantum models and logics
Chris Heunen
7 januari 2010
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Welcome, thank you for coming.During the next hour I will defend my dissertation.
Categorical quantum models and logics
Chris Heunen
7 januari 2010
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Such an official ceremony is no place for experimental presentations,but nevertheless ...
Categorical quantum models and logics
Chris Heunen
7 januari 2010
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I will subtitle myself while introducing what the dissertation is about.
Categorical quantum models and logics
Chris Heunen
7 januari 2010
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Let’s start with ‘quantum’, perhaps the most intimidating word in the title.
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Quantum mechanics is the best description of nature on small scalesthat we have today.
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It is a very odd description: if one zooms in very far,nature’s behaviour is beyond our intuition.
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Why? Our intuition for what is odd, and what isn’t,is acquired on the much larger scale of everyday life.
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For example, we find it normal that one cannot walk on water,or carry it in one’s hands.
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But if one is small enough, like these merry flies,that is not so odd anymore at all.
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On the still smaller scale on which quantum mechanics reigns,there are still more fundamental oddities,
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but this example indicates that odd things can occur when one becomessmaller.
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This fundamental oddity of small scales has advantages.By using it a phone line can be made which can detect eavesdropping.
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Also, computers can be made that solve certain problemsessentially faster than current computers.
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Such socalled quantum computers use principles that fundamentallydiffer from those of current computers.
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Computer memory consists of bits; units that are either 0 or 1.Today, a single bit takes approximately 30 nanometers.
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Those dimensions change. Previous bits were bigger.Future bits will be smaller.
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Some day they will be so small that quantum mechanics comes into play.
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If one shrinks a bit enough, it becomes a qubit.When measured, its value is still either 0 or 1.
bit qubit
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But left unmeasured, its value can be something else entirely.In general, its state space is a sphere instead of two isolated points.
bit qubit
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This entails that strange things can happen.
bit qubit
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For example, a qubit in this state will give an outcome upon measurementof 0 half of the time, and 1 the other half.
qubit in superpositie
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In a certain sense the qubit is 0 and 1 at the same time,though that formulation is misleading.
qubit in superpositie
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A similar oddity happens when considering not one but two qubits.These can be entangled in such a state, that ...
entanglement
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if one is measured and gives outcome 0, then the other must give 1.This holds instantaneously, even if the two qubits are miles apart.
entanglement
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The water in the two ends of a bent pipe also has this property,but that is so because they form communicating vessels.
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The odd thing about entangled qubits is that this behaviour occurswithout a common cause.
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The third oddity I want to mention is (non)commutativity, which means itis not sensible to measure certain properties of qubits simultaneously.
(niet-)commutativiteit
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This is caused by the fact that the order of measurements matters.The effect of undressing and then taking a shower ...
(niet-)commutativiteit
=
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is completely different than vice versa!Thus there are also disadvantages to using quantum mechanics:
(niet-)commutativiteit
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one’s intuition will block attempts at understanding.It is better to reason purely mathematically.
(niet-)commutativiteit
=
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Traditionally one does so by making assumptions about the set of states.For a qubit, this is a sphere. The first half of the dissertation follows ...
categorie
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a different approach: let’s not be modest, and studyall possible state spaces at once, and all relationships between them.
categorie
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This is called a category, and is depicted on the coverand virtually every page of the thesis:
categorie
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such diagrams concern connections between objects,not so much the objects themselves.
categorie
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Instead of assuming some internal structure of a quantum system,we study how the system relates to others.
categorie
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After all, we don’t know what the internal structure is!One might say we elect a sociological approach over a neurological one.
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Instead of trying to find out how one individual’s brain works,we only consider how that person behaves in society.
categorie
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Thus we can qualitatively see what properties cause quantum behaviour.
categorie
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For example, an important such property is that every relation betweenquantum systems is invertible: if system A is somehow related to B, ...
axioma’s
Af // B
A Bf †oo
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then B is also connected to A. Thus one can make an assumption aboutthe category for every behaviour obtained from physical experiments.
axioma’s
Af // B
A Bf †oo
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Chapter 3 shows that if one assumes the three oddities we saw in this way,then the category always embeds into the traditional physical model.
inbedding
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In a sense this justifies the traditional model of quantum mechanics.
inbedding
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It turns out that (complex) numbers emerge in every model of quantummechanics, even if not explicitly assumed.
inbedding
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The second half of the thesis nonetheless tries to make sense of the oddities.That is, we try to set up a logic for quantum mechanics.
a2 + b2 = c2
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The formal mathematics that we replace naive intuition with,relies on the notion of proof.
a2 + b2 = c2
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I mean a different kind of proof than you might know, from, say, the law.I mean, for example, proof of the well-known Pythagorean theorem.
a2 + b2 = c2
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This is not something that has been tested on 100 right-angled triangles,and happened to hold for most.
a2 + b2 = c2
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No; from purely logical deductions, it is certain thatthe theorem holds for any right-angled triangle you will ever encounter.
a2 + b2 = c2
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Here is a very simple example of such a proof.It is raining. When it rains, one gets wet. Hence I get wet.
Het regent.Als het regent, word je nat.Dus ik word nat.
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In symbols: A and A⇒ B, hence B.Notice that logic is the grammar of a kind of language.
A en (A⇒ B), dus B.
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Another such grammar rule could be:(A or B) and C = (A and C ) or (B and C ).
(A of B) en C=
(A en C ) of (B en C )
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If offered tea or coffee with a biscuit,one expects either tea with a biscuit, or coffee with a biscuit.
(of
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(en
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But because of the oddities of quantum mechanics, this no longer holds!There are properties of qubits for which this equation doesn’t hold.
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Nevertheless this is traditionally called ‘quantum logic’. Chapter 4 showsthat this socalled logic holds in our categorical models unabated.
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But one cannot in good faith call something this unintuitive logic.That is why I try something else in chapter 5.
↓ ↓
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We play a technical trick there: by altering the grammar rules of one’s logic,one can pretend that a quantum system is intuitive.
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Imagine for the moment that our world is Smurf village,and hence that you are a Smurf.
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The world contains quantum systems, that we cannot intuitively understand.In the cartoon, a quantum system is a package, that looks mysterious.
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From the right perspective, that package would appear normal.If we were Snorks instead of Smurfs, and hence lived under water ...
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we would have seen through the water ripples,and the package had appeared perfectly normal.
↓
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Now, Smurfs are no Snorks. But a Smurf can live in Snorkland just fine.All he has to do is wear goggles and a snorkel, and forget he is a Smurf.
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Thus odd things can be made normal by “changing world”.That is the main trick of chapter 5.
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Given a single quantum system, we make a world we can live in just finewhich has fine logical laws, and in which the system looks perfectly normal.
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If we alter the grammar rules of our logic thus, oddities can be made normal.To ‘understand’ quantum mechanics, one must speak the right language.
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Clearly this is a crude simplification, like other analogies I used before.At some points what I said is not even entirely correct.
Categorical quantum models and logics
Chris Heunen
7 januari 2010
![Page 67: Categorical quantum models and logics...Clearly this is a crude simpli cation, like other analogies I used before. At some points what I said is not even entirely correct. Categorical](https://reader034.fdocuments.in/reader034/viewer/2022050409/5f862e1fa7fdc707b81a9755/html5/thumbnails/67.jpg)
But hopefully the topic of my dissertation is now somewhat clear.At least you can now interpret words of the title!
Categorical quantum models and logics
Chris Heunen
7 januari 2010
![Page 68: Categorical quantum models and logics...Clearly this is a crude simpli cation, like other analogies I used before. At some points what I said is not even entirely correct. Categorical](https://reader034.fdocuments.in/reader034/viewer/2022050409/5f862e1fa7fdc707b81a9755/html5/thumbnails/68.jpg)
Categorical quantum models and logics
Chris Heunen
7 januari 2010