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Cambridge University Press eBooks

Letters to Michael Nielsen

Christopher A. Fuchs

Published Jan 6, 2011• 0 citations
Publisher

Abstract

February 1996, “Goo-Losh” Mmm, goulash…is that how you spell it? Thanks for the tip. I haven't heard the reaction of Bennett and company on this business yet, but when I do, I'll send a report your way. I've been thinking about Kochen–Specker theorems lately; they're very nice you know…especially the ones with small numbers of vectors. Implications for “quantum information”? There must be! In particular, why can't one go much further in this effort to prove the theorem with smaller and smaller numbers of vectors? Why is there no proof of this property (i.e., no noncontextual hidden variables for the outcomes of standard von Neumann measurements), the very second you admit a measurement that is nonorthogonal to the others? Just food for late night thoughts. March 1996, “The Commitments” It's a good movie; did you ever see it? Suppose you need some cash and so you go to your Automated Teller Machine (ATM). You insert your bank card and the machine asks you for your Personal Identification Number (PIN). It does this for the obvious reason that it wants to know that you are indeed who you say you are before it hands over the money. That's a good safeguard. However suppose, as really did happen a year or two ago, someone had set up a fake ATM in place of the real one.

DOI
10.1017/cbo9780511762789.025

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February 1996, “Goo-Losh” Mmm, goulash…is that how you spell it? Thanks for the tip. I haven't heard the reaction of Bennett and company on this business yet, but when I do, I'll send a report your way. I've been thinking about Kochen–Specker theorems lately; they're very nice you know…especially the ones with small numbers of vectors. Implications for “quantum information”? There must be! In particular, why can't one go much further in this effort to prove the theorem with smaller and smaller numbers of vectors? Why is there no proof of this property (i.e., no noncontextual hidden variables for the outcomes of standard von Neumann measurements), the very second you admit a measurement that is nonorthogonal to the others? Just food for late night thoughts. March 1996, “The Commitments” It's a good movie; did you ever see it? Suppose you need some cash and so you go to your Automated Teller Machine (ATM). You insert your bank card and the machine asks you for your Personal Identification Number (PIN). It does this for the obvious reason that it wants to know that you are indeed who you say you are before it hands over the money. That's a good safeguard. However suppose, as really did happen a year or two ago, someone had set up a fake ATM in place of the real one.
One of the statements one would hear pretty early in any quantum mechanics course is that for quantum particles there are properties which do not have definite values. For example, if we measure |+〉 in standard basis, the best we can do to describe the outcome is to say, that with probability 1/2 we will get 0 and with probability 1/2 we will get 1. Yet one could wonder, is this really the way things are or is it just because our description of this quantum particle is incomplete, but in fact there are some hidden variables inside every quantum system that contain enough information to tell the outcome of measurement (see Fig. 1). However, many people disdain the idea of hidden variables, so there has been rather long history of theorems that forbid theories involving hidden variables. Usually if one wants to prove some no hidden variables theorem he has to assume something about the hidden variables theories he is going to forbid. One of such assumptions is non-contextuality, which means that the outcome we get after measuring with an observable is the same regardless of what other observables we choose to measure at the same time. The first theorem forbidding hidden variables theories is due to von Neumann in 1932. Then for a third of a century everyone cited von Neumann’s work carelessly, till Bell in 1966 indicated that there was an unreasonable assumption in von Neumann’s proof and urged for further examination of the problem (see [6] and [1] for more historical details). Bell also constructed a hidden variables model for one qubit that could assign the outcome of any observable in a way consistent with quantum mechanics. He also proved two theorems that forbid non-contextual hidden variables theories using counterexamples with larger quantum systems. Yet surprisingly he himself thought that hidden variables do exists and considered these theorems only as characterizations of the hidden variables theory we should look for. In this essay we will discuss several versions of one of the two Bell’s theorems. The theorem we will consider was also independently proven by Kochen and Specker in 1967 (see [5]), hence it is called Bell-Kochen-Specker theorem. For shortness and in order not to mix this theorem with another Bell’s theorem from now on we will refer to it as Kochen-Specker theorem. We will be interested in proving several versions of this theorem and the following lemma will turn out to be of great use:
02 June 1999, “Please Do” You know what I'm really trying to get at here: I want the structure of quantum mechanics to be extremal for some kind of more abstract (i.e., QM-independent) information-disturbance problem. But the first thing that needs to be done is to pin down what quantum mechanics itself has to say. Another thing that keeps coming to mind is that the assumptions behind Gleason's theorem might have something to do with the allegory I presented you with at Charlie and Theo's house. God said, “Ok I will make information gathering invasive, but not so invasive that you will be forever science-less.” The connection is this. I think one can split the assumption behind Gleason up into at least two parts: (1) The questions that can be asked of a quantum system only correspond to orthonormal bases over some vector space, with there being no good notion of measuring two distinct questions simultaneously (there being no good notion of an AND operation). And (2) It is the task of physical theory to give probabilities for the outcomes of those questions, but we can say at least this much about the probabilities. They are context-independent in the sense that, for a given outcome, it does not matter which physical question (which basis) we've associated it with. I.e., the probabilities can be assumed to be of the form of a frame function.
An Automated Teller Machine (ATM) is an electronic device that allows people to conduct financial transactions without needing to interact directly with a bank. ATMs are very important in our daily lives because they let us withdraw cash, check account balances, and perform other banking tasks anytime. However, with the rise of cyber technology, ensuring the security of these machines is crucial. Traditional ATMs often face security issues like cash-trapping, skimming, and transaction reversal fraud, which is especially problematic in developing countries. Quantum Key Distribution (QKD) is a secured communication method which makes use of cryptographic protocols to overcome the issues faced by classical ATMs. This research aims to enhance ATM security using a QKD protocol known as BB84 which is based in quantum mechanics. Simulation results shown in this paper show that Quantum ATMs are more reliable and secure than traditional ones.