Quelques problèmes en analyse harmonique non commutative / Some problems on noncommutative harmonique analysis

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<p dir="ltr">In holographic quantum field theories, a bulk geometric semiclassical spacetime emerges from strongly coupled interacting conformal field theories in one less spatial dimension. This is the celebrated AdS/CFT correspondence. The entanglement entropy of a boundary spatial subregion can be calculated as the area of a codimension two bulk surface homologous to the boundary subregion known as the RT surface. The bulk region contained within the RT surface is known as the entanglement wedge and bulk reconstruction tells us that any operator in the entanglement wedge can be reconstructed as a non-local operator on the corresponding boundary subregion. This notion that entanglement creates geometry is dubbed "ER=EPR'' and has been the driving force behind recent progress in quantum gravity research. In this thesis, we put together two results that use Tomita-Takesaki modular theory and quantum ergodic theory to make progress on contemporary problems in quantum gravity.</p><p dir="ltr">A version of the black hole information loss paradox is the inconsistency between the decay of two-point functions of probe operators in large AdS black holes and the dual boundary CFT calculation where it is an almost periodic function of time. We show that any von Neumann algebra in a faithful normal state that is quantum strong mixing (two-point functions decay) with respect to its modular flow is a type III₁ factor and the state has a trivial centralizer. In particular, for Generalized Free Fields (GFF) in a thermofield double (KMS) state, we show that if the two-point functions are strong mixing, then the entire algebra is strong mixing and a type III₁ factor settling a recent conjecture of Liu and Leutheusser.</p><p dir="ltr">The semiclassical bulk geometry that emerges in the holographic description is a pseudo-Riemannian manifold and we expect a local approximate Poincaré algebra. Near a bifurcate Killing horizon, such a local two-dimensional Poincaré algebra is generated by the Killing flow and the outward null translations along the horizon. We show the emergence of such a Poincaré algebra in any quantum system with modular future and past subalgebras in a limit analogous to the near-horizon limit. These are known as quantum K-systems and they saturate the modular chaos bound. We also prove that the existence of (modular) future/past von Neumann subalgebras also implies a second law of (modular) thermodynamics.</p>

Derek Parfit är kanske en av vår tids mest kända moralfilosofer. Parfit inleder sin första bok Reasons and Persons med att ställa frågan: vad har vi mest skäl att göra? Hans fråga berör vad som egentligen har betydelse, en fråga som han fortsätter att beröra i sin andra bok On What Matters. Filosofen Toby Ord argumenterar i sin bok The Precipice för att den utmaning som definierar vår tid, och bör ha en central prioritering, är utmaningen att skydda mänskligheten emot så kallade existentiella risker. En existentiell risk är en typ av risk som hotar att förstöra, eller förhindra, mänsklighetens långsiktiga potential. Ord menar att vi idag befinner oss vid en kritisk tidpunkt i mänsklighetens historia som kan vara helt avgörande för om det ens kommer existera en framtid för mänskligheten. Men om vi bör skydda mänskligheten emot existentiella risker, så kan en lämplig följdfråga vara i vilken ordning vi bör prioritera olika existentiella risker. Den svenske filosofen Nick Bostrom har liksom Ord länge förespråkat att existentiella risker bör tas på allvar. Han menar att preventiva åtgärder bör vidtas. I sin bok Superintelligens argumenterar Bostrom, både omfattande och väl, för att den existentiella risk som kan te sig som mest brådskande, och kanske allvarligast, är artificiell intelligens. Bostrom menar att vi har goda skäl att tro att utveckling av artificiell intelligens kan eskalera till den grad att mänsklighetens öde kan hamna bortom vår egen kontroll. Det han syftar på är att människan just nu är den dominerande agenten på jorden och därför innehar en stor kontroll, men att så inte alltid behöver vara fallet. Bostroms tes kunde te sig som okonventionell då den presenterades, men kan även te sig så idag vid en första anblick. Han har dock fått explicit medhåll av personer som Bill Gates, Stephen Hawking, Elon Musk, Yuval Noah Harari och Max Tegmark, som antingen håller med eller resonerar i liknande banor. Även jag själv finner Bostroms antaganden välgrundade. Slutsatsen som många drar är därför att vi bör betrakta artificiell intelligens som en existentiell risk som ska prioriteras högt. Jag kommer dock i denna text att argumentera för tesen att vi inte bör betrakta artificiell intelligens som en existentiell risk. Tesen följer från en invändning som jag kommer att kalla för AI-paradoxen. Det tycks enligt invändningen som att artificiell intelligens inte kan leda till en existentiell katastrof givet vissa premisser som flera i debatten om artificiell intelligens tycks acceptera. Texten i uppsatsen är strukturerad på följande sätt. I avsnitt 2 kommer jag att återge det övergripande argumentet som cirkulerar i debatten om artificiell intelligens som ett hot. I avsnittet kommer jag också förklara några viktiga termer och begrepp. I avsnitt 3 börjar jag med att titta på den första premissen i argumentet, samt resonera om dess rimlighet. I avsnitt 4 går jag sedan vidare till den andra premissen i argumentet och gör samma sak med den. Väl i avsnitt 5 så väljer jag att presentera min egen idé som jag kallar för AI-paradoxen, vilket är en invändning mot argumentet. I avsnitt 6 diskuterar jag sedan AI-paradoxens implikationer. Avslutningsvis, i avsnitt 7, så ger jag en övergripande sammanfattning och en slutsats, samt några sista reflektioner. / Derek Parfit is perhaps one of the most famous moral philosophers of our time. Parfit begins his first book Reasons and Persons by asking the question: what do we have most reason to do? His question touches upon what really matters, a question he continues to touch upon in his second book On What Matters. The philosopher Toby Ord argues in his book The Precipice that the challenge that defines our time, and should have a central priority, is the challenge of safeguarding humanity from so-called existential risks. An existential risk is a type of risk that threatens to destroy, or prevent, humanity’s longterm potential. Ord means that today we are at a critical time in the history of humanity that can be absolutely decisive for whether there will even exist a future for humanity. But if we are to safeguard humanity from existential risks, then an appropriate question may be in what order we should prioritize different existential risks. The Swedish philosopher Nick Bostrom, like Ord, has long advocated that existential risks should be taken seriously. He believes that preventive measures should be taken. In his book Superintelligence Bostrom argues, both extensively and well, that the existential risk that may seem most urgent, and perhaps most severe, is artificial intelligence. Bostrom believes that we have good reason to believe that the development of artificial intelligence can escalate to the point that the fate of humanity can end up beyond our own control. What he is referring to is that humans are currently the dominant agent on earth and therefore has great control, but that this does not always have to be the case. Bostrom's thesis may have seemed unconventional when it was presented, but it can also seem so today at first glance. However, he has been explicitly supported by people like Bill Gates, Stephen Hawking, Elon Musk, Yuval Noah Harari and Max Tegmark, who either agree or reason similarly. I myself also find Bostrom's assumptions well-founded. The conclusion that many draw is therefore that we should regard artificial intelligence as an existential risk that should be given a high priority. However, in this text I will argue for the thesis that we should not regard artificial intelligence as an existential risk. The thesis follows from an objection of my own, which I call the AI ​​paradox. According to the objection, it seems that artificial intelligence cannot lead to an existential catastrophe given certain premises that many in the debate about artificial intelligence as a threat seem to accept. The text in the essay is structured as follows. In section 2 I will present the main argument circulating in the debate about artificial intelligence as a threat. In the section I will also explain some important terms and concepts. In section 3 I begin by looking at the first premise in the argument, and also reason about its plausibility. In section 4 I proceed to the second premise in the argument and examine it similarly. Once in section 5 I choose to present my own idea, which I call the AI ​​paradox, which is an objection to the argument. In section 6 I discuss the implications of the AI ​​paradox. Finally, in section 7, I give an overall summary and a conclusion, as well as some last reflections.

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-08-01T21:11:37Z No. of bitstreams: 1 REX ANTONIO DA COSTA MEDEIROS - TESE PPGEE 2008..pdf: 1089371 bytes, checksum: ea0c95501b938e0d466779a06faaa4f6 (MD5) / Made available in DSpace on 2018-08-01T21:11:37Z (GMT). No. of bitstreams: 1 REX ANTONIO DA COSTA MEDEIROS - TESE PPGEE 2008..pdf: 1089371 bytes, checksum: ea0c95501b938e0d466779a06faaa4f6 (MD5) Previous issue date: 2008-05-09 / Nesta tese, a capacidade erro-zero de canais discretos sem memória é generalizada para canais quânticos. Uma nova capacidade para a transmissão de informação clássica através de canais quânticos é proposta. A capacidade erro-zero de canais quânticos (CEZQ) é definida como sendo a máxima quantidade de informação por uso do canal que pode ser enviada através de um canal quântico ruidoso, considerando uma probabilidade de erro igual a zero. O protocolo de comunicação restringe palavras-código a produtos tensoriais de estados quânticos de entrada, enquanto que medições coletivas entre várias saídas do canal são permitidas. Portanto, o protocolo empregado é similar ao protocolo de Holevo-Schumacher-Westmoreland. O problema de encontrar a CEZQ é reformulado usando elementos da teoria de grafos. Esta definição equivalente é usada para demonstrar propriedades de famílias de estados quânticos e medições que atingem a CEZQ. É mostrado que a capacidade de um canal quântico num espaço de Hilbert de dimensão d pode sempre ser alcançada usando famílias compostas de, no máximo,d estados puros. Com relação às medições, demonstra-se que medições coletivas de von Neumann são necessárias e suficientes para alcançar a capacidade. É discutido se a CEZQ é uma generalização não trivial da capacidade erro-zero clássica. O termo não trivial refere-se a existência de canais quânticos para os quais a CEZQ só pode ser alcançada através de famílias de estados quânticos não-ortogonais e usando códigos de comprimento maior ou igual a dois. É investigada a CEZQ de alguns canais quânticos. É mostrado que o problema de calcular a CEZQ de canais clássicos-quânticos é puramente clássico. Em particular, é exibido um canal quântico para o qual conjectura-se que a CEZQ só pode ser alcançada usando uma família de estados quânticos não-ortogonais. Se a conjectura é verdadeira, é possível calcular o valor exato da capacidade e construir um código de bloco quântico que alcança a capacidade. Finalmente, é demonstrado que a CEZQ é limitada superiormente pela capacidade de Holevo-Schumacher-Westmoreland.