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Cambridge University Press eBooks

Other Letters

Christopher A. Fuchs

Published Jan 6, 2011• 0 citations
Publisher

Abstract

April 1996, to Michiel van Lambalgen, “Thesis” About myself (in case you are wondering) I am a physicist. My research specialty is “quantum information theory,” a field that concerns a hodgepodge of things including quantum cryptography, quantum computing, and statistical inference problems having to do with quantum mechanical systems. You may have heard of some of these things through Paul Vitanyi or André Berthiaume (Vitanyi's postdoc). I am presently a postdoc working for Gilles Brassard and Claude Crépeau in Montréal; starting October, I will have a three year position at Caltech. I was once interested in the mathematics of randomness because, though I am Bayesian through and through for all other uses of probability, I believed that probabilities for quantum mechanical measurement outcomes were something different…in fact something more akin to the frequentist conception. Thus I put a lot of effort into studying von Mises, Church, Kolmogorov, Martin-Löf, Chaitin, etc. (I didn't find your papers until I had pretty much abandoned this belief, though I'm not sure that I am completely over it!) I had hoped that there might be some mathematical connection between the structures found in quantum theory (vector spaces, positive-operator valued measures, etc.) and the structures required to formalize the notion of random sequences…at least that was my motivation.

DOI
10.1017/cbo9780511762789.037

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April 1996, to Michiel van Lambalgen, “Thesis” About myself (in case you are wondering) I am a physicist. My research specialty is “quantum information theory,” a field that concerns a hodgepodge of things including quantum cryptography, quantum computing, and statistical inference problems having to do with quantum mechanical systems. You may have heard of some of these things through Paul Vitanyi or André Berthiaume (Vitanyi's postdoc). I am presently a postdoc working for Gilles Brassard and Claude Crépeau in Montréal; starting October, I will have a three year position at Caltech. I was once interested in the mathematics of randomness because, though I am Bayesian through and through for all other uses of probability, I believed that probabilities for quantum mechanical measurement outcomes were something different…in fact something more akin to the frequentist conception. Thus I put a lot of effort into studying von Mises, Church, Kolmogorov, Martin-Löf, Chaitin, etc. (I didn't find your papers until I had pretty much abandoned this belief, though I'm not sure that I am completely over it!) I had hoped that there might be some mathematical connection between the structures found in quantum theory (vector spaces, positive-operator valued measures, etc.) and the structures required to formalize the notion of random sequences…at least that was my motivation.
La physique classique, par son caractère fondamentalement déterministe, ne permet pas la production d’aléa vrai. La physique quantique, quant à elle, fait des prévisions probabilistes, sur des processus qui paraissent fondamentalement aléatoires. De plus, les réponses apportées aux paradoxes de types EPR, comme la réponse formulée par les inégalités de Bell [16] et leurs expérimentations par Alain Aspect, montrent que la physique quantique ne peut être complétée en une théorie entièrement prédictive. Une question émane alors: Quelles sont les expériences quantiques fondamentalement aléatoires ? Cristian S.Calude et Al [4] montrent que presque toutes les mesures quantiques produisent de l’aléa (Théorème Strong Kochen Specker). Dans le cadre de cette thèse, notre but est ainsi de proposer de nouveaux schémas de génération quantique d’aléa (voir [10,13]) basés sur ce caractère aléatoire intrinsèque aux expériences quantiques. Ces protocoles explorent l’utilisation de produits d’observables incompatibles implémentées l’une à la suite de l’autre comme le font les auteurs de [36]. Pour un apport en sécurité à ces protocoles, nous dérivons, à partir d’inégalités de Bell déjà existantes notamment CHSH [28] et CHSH-3 (initialement CGLMP pour qutrits [29]), de nouvelles inégalités. Cette fois-ci, l’on considère des configurations constituées d’un unique Qutrit où la contrainte de commutativité des observables n’est pas imposée. La borne classique est cette fois-ci obtenue sous l’hypothèse du Réalisme macroscopique [51], comme celles des inégalités temporelles [36]. Ces bornes classiques sont égales à celles obtenues dans le cadre du réalisme locale. Cependant, les bornes quantiques, obtenues par SDP, peuvent être plus grandes que celles des expressions originelles permettant ainsi une meilleure résistance au bruit et donc à de potentielles attaques. Ces violations par rapport au cadre classique ne sont plus dues à la non localité mais plutôt au caractère indéfini des résultats de presque toutes les mesures quantiques. Avec les observables et états optimum de ces inégalités, nous donnons des arguments d'auto-test pour nos protocoles. Cette thèse a aussi été l’occasion de revisiter certains aspects de la cryptographie quantique comme la distribution quantique de clé.
Nies and Scholz defined quantum Martin-Löf randomness (q-MLR) for states (infinite qubitstrings). We define a notion of quantum Solovay randomness and show it to be equivalent to q-MLR using purely linear algebraic methods. Quantum Schnorr randomness is then introduced. A quantum analogue of the law of large numbers is shown to hold for quantum Schnorr random states. We introduce quantum-K, (QKQKQK) a measure of the descriptive complexity of density matrices using classical prefix-free Turing machines and show that the initial segments of weak Solovay random and quantum Schnorr random states are incompressible in the sense of QKQKQK. Several connections between Solovay randomness and KKK carry over to those between weak Solovay randomness and QKQKQK. We then define QKCQK_CQKC​, using computable measure machines and connect it to quantum Schnorr randomness. We then explore a notion of `measuring' a state. We formalize how `measurement' of a state induces a probability measure on the space of infinite bitstrings. A state is `measurement random' (mRmRmR) if the measure induced by it, under any computable basis, assigns probability one to the set of Martin-Löf randoms. I.e., measuring a mRmRmR state produces a Martin-Löf random bitstring almost surely. While quantum-Martin-Löf random states are mRmRmR, the converse fails: there is a mRmRmR state, ρρρ which is not quantum-Martin-Löf random. In fact, something stronger is true. While ρρρ is computable and can be easily constructed, measuring it in any computable basis yields an arithmetically random sequence with probability one. So, classical randomness can be generated from a computable state which is not quantum random. We conclude by studying the asymptotic von Neumann entropy of computable states.
12 June 1999, “Interest in Pauli” I have come across a reference to a paper of yours that I have not been able to obtain. May I ask of you to send me a reprint of it? The paper is: C. P. Enz, “Wolfgang Pauli and the Role of the Observer in Modern Physics,” in Philosophy of the Natural Sciences: Proceedings of the 13th International Wittgenstein Symposium , edited by P. Weingartner and G. Schurz (Verlag Holder–Pichler–Temsky, Vienna, 1989), pp. 110–119. I have read several of your other papers on Pauli's thought and have been uniformly pleased with each. I am especially interested in his ideas concerning the demise of the “detached-observer” notion within quantum theory. (I would also be interested in any of your other works on the subject that may have been written since your 1992 article, “Wolfgang Pauli between Quantum Reality and the Royal Path of Dreams.” This is the latest one that I am aware of.) About myself, I am a theoretical physicist at Caltech who specializes in quantum information theory and quantum computing. I am well acquainted with your colleague Nicolas Gisin at the University of Geneva. 21 August 1999, “One More Article?” Upon my return from Cambridge a couple of weeks ago, I found in my mailbox your package full of wonderful papers! Thank you so much.
The algebraic approach to physical theories provides a general framework encompassing both classical and quantum mechanics. Accordingly, by looking at the behaviour of the relevant algebras of observables one can investigate structural and conceptual differences between the theories. Interesting foundational questions can be formulated algebraically and their answers are then given in a mathematically compelling way. My dissertation focuses on some philosophical issues concerning entanglement and quantum information as they arise in the algebraic. These two concepts are connected in that one can exploit the non-local character of quantum theory to construct protocols of information theory which are not realized in the classical world. I first introduce the basic concepts of the algebraic formalism, by reviewing von Neumann's work on the mathematical foundations of quantum theories. After presenting the reasons why von Neumann abandoned the standard Hilbert space formulation in favour of the algebraic approach, I show how his axiomatic program remained a mathematical "utopia" in mathematical physics. The Bayesian interpretation of quantum mechanics is grounded in information-theoretical considerations. I take on some specific problems concerning the extension of Bayesian statistical inference in infinite dimensional Hilbert space. I demonstrate that the failure of a stability condition, formulated as a rationality constraint for classical Bayesian conditional probabilities, does not undermine the Bayesian interpretation of quantum probabilities. I then provide a solution to the problem of Bayesian noncommutative statistical inference in general probability spaces. Furthermore, I propose a derivation of the a priori probability state in quantum mechanics from symmetry considerations. Finally, Algebraic Quantum Field Theory offers a rigorous axiomatization of quantum field theory, namely the synthesis of quantum mechanics and special relativity. In such a framework one can raise the question of whether or not quantum correlations are made stronger by adding relativistic constraints. I argue that entanglement is more robust in the relativistic context than in ordinary quantum theory. In particular, I show how to generalize the claim that entanglement across space-like separated regions of Minkowski spacetime would persist, no matter how one acts locally.