Summary Gulls Theorem A Proof Using Fourier Theory arxiv.org
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The paper discusses the use of Gull's Theorem and Fourier theory to prove Bell's Theorem and the impossibility of classical distributed computer simulations of singlet correlations, as well as the history and development of Bell's inequalities and their relevance to quantum entanglement.
Key Points
- Gull's Theorem is a proof using Fourier theory that builds on perfect anti-correlation in the singlet correlations.
- Bell's Theorem states that conventional quantum mechanics is incompatible with local realism.
- The proof shows that classical computers cannot generate a full amplitude negative cosine correlation between outputs given inputs, which is a unique signature of quantum entanglement.
- The article discusses the history and development of Bell's inequalities, which were previously attributed to Itamar Pitowsky or Boole.
- The document emphasizes the significance of finding a simple and striking mathematical proof for quantum nonlocality.
Summaries
193 word summary
The document discusses the significance of finding a simple proof for quantum nonlocality and presents a list of references related to Bell's theorem and nonlocality proofs. Gull's Theorem uses Fourier theory to prove that classical distributed computer simulations of the singlet correlations are impossible. The paper proposes a classical Monte-Carlo computer simulation to generate empirical relative frequencies and predictions of statistics for hidden variables in quantum mechanics. Gull's Theorem, which states that it is impossible to write computer programs that can generate random numbers with identical outputs on two completely separate computers, can be proven using Fourier theory. Gull's Theorem uses Fourier theory to prove Bell's Theorem, which shows that classical computers cannot generate a full amplitude negative cosine correlation between outputs given inputs, a unique signature of quantum entanglement. The text discusses the history and development of Bell's inequalities, which were previously attributed to Itamar Pitowsky or Boole, as well as the EPR-B experiment and Malus' law in relation to spin and polarization measurements. The Bell-CHSH experiment involves two locations or labs with binary settings inputted by experimenters Alice and Bob. Bell's conclusions still hold if all correlations are only approximately true.
466 word summary
Gull's Theorem uses Fourier theory to prove Bell's Theorem, which shows that classical computers cannot generate a full amplitude negative cosine correlation between outputs given inputs, a unique signature of quantum entanglement. Bell's Theorem states that conventional quantum mechanics is incompatible with local realism. The text discusses the history and development of Bell's inequalities, which were previously attributed to Itamar Pitowsky or Boole. The article also mentions the EPR-B experiment and Malus' law in relation to spin and polarization measurements. The Bell-CHSH experiment involves two locations or labs with binary settings inputted by experimenters Alice and Bob. Bell's conclusions still hold if all correlations are only approximately true. The paper proposes a classical Monte-Carlo computer simulation to generate empirical relative frequencies and predictions of statistics for hidden variables in quantum mechanics. The author characterizes the theories as “local hidden variables theories” (LHV) and claims that the statistics predicted by quantum mechanics are merely the reflection of a more classical underlying theory. Gull's Theorem, which states that it is impossible to write computer programs that can generate random numbers with identical outputs on two completely separate computers, can be proven using Fourier theory. The proof is based on treating the physics implemented inside the computers as completely deterministic and using externally created streams of random binary setting choices to derive martingale properties of a suitable game score. A variant of the proof, designed specifically for deterministically operating computers, can also be proven using a Bell theorem. Gull's Theorem uses Fourier theory to prove that classical distributed computer simulations of the sin-glet correlations are impossible. The theorem focuses on CHSH-style experiments and martingale-based inequalities. The Gulls Theorem uses Fourier theory to prove that, under the assumption of local realism, the probability of outcomes for a classical distributed computer simulation performing a deterministic computation and communicating over a wired connection is stochastically smaller than a binomial distribution with parameters N and 0.75. The article discusses the proof of Bell's theorem using Fourier theory. The proof measures the statistical advantage in probing pairs of directions and shows that only four different test experiments have advantages over usual experiments. Marek Z?ukowski has introduced a concept of “functional Bell inequalities”. The document presents a list of references on Bell's theorem and its implications for physical reality. Inequalities are important in proving Bell's theorem, and experimental set-ups require a large amount of resources. The document emphasizes the significance of finding a simple mathematical proof for quantum nonlocality. The author acknowledges the contributions of former students and the use of Fourier theory in re-proving Bell's theorem. The project was inspired by a challenge to prove the CHSH inequality without certain mathematical tools. The work was stimulated by opposition to Bell's extension, and the author cites various sources related to Bell's theorem and nonlocality proofs.
1468 word summary
The author of the paper acknowledges the contributions of former students and the use of Fourier theory in re-proving Bell's theorem. The project was inspired by a challenge to prove the CHSH inequality without using certain mathematical tools. The work was stimulated by opposition to Bell's extension. The author cites various sources related to Bell's theorem and nonlocality proofs. The document provides a list of references related to Bell's theorem and its implications for the completeness of physical reality. The references include comments on various papers, discussions of experimental proof, and mathematical models. The text highlights the importance of inequalities in proving Bell's theorem and notes that the amount of resources needed for an experimental set-up can be surprisingly large. The document emphasizes the significance of finding a simple and striking mathematical proof for quantum nonlocality. The article discusses the proof of Bell's theorem using Fourier theory. The proof measures the statistical advantage in probing pairs of directions and shows that only four different test experiments have advantages over usual experiments. The Grothendieck constants are a matter of conjecture in higher dimensions. Marek Z?ukowski has introduced a concept of “functional Bell inequalities”. A proof via Fourier analysis requires regularity assumptions and there is no use of memory or time. The proof requires two measurement functions and one probability distribution that must contain an implementation of a classical local hidden variables model. The computers must use the same seed and parameters on each computer. The proof shows that the correlation curve can be reduced to just four points, and the probabilities give an excellent basis for making bets. The Gulls Theorem uses Fourier theory to prove that, under the assumption of local realism, the probability of outcomes for a classical distributed computer simulation performing a deterministic computation and communicating over a wired connection is stochastically smaller than a binomial distribution with parameters N and 0.75. The experiment involves two classical PCs with labeled settings and measurement directions, and the success probability per independent trial is p. The theorem applies to a range of critical levels of x and is not affected by past settings or time trends. The experiment has been successfully tested in a "loophole-free" Bell type experiment by researchers in Delft, the Netherlands, and is presented in the supplementary material of their paper. The martingale structure of the experiment allows for easy exponential inequalities on the probabilities of large deviations upwards. Gull's Theorem uses Fourier theory to prove that classical distributed computer simulations of the sin-glet correlations are impossible. The theorem focuses on CHSH-style experiments and martingale-based inequalities. The proof involves averaging over hidden variables and setting pairs, and using the sample average of N independent, identically distributed random variables. The functions used in the proof have Fourier expansions, with only the k = -1 terms being significant. The functions square to +1 on average, but have nonzero coefficients for k ≠ -1. The text excerpt presents Gull's Theorem, which is a proof using Fourier theory. The theorem involves the expected sample correlation function for a sequence of N trials in which Alice and Bob each have 360 different settings at their disposal. The proof uses the Fourier series for the negative cosine and involves regularity assumptions for each function A n. The expected sample correlation function is defined by the correlation between two functions, and the proof shows that it converges almost surely as N approaches infinity. The theorem has implications for Bell-CHSH experiments, where the experimenter sets a digital dial to some whole number of degrees, and the analysis literature has numerous results giving stronger forms of convergence for the correlation between two functions. Gull's argument is presented without analytic niceties, but the authors plan to explore it further in the future. Gull's Theorem, which states that a convolution of functions maps to a product of Fourier series, is proven using Fourier theory. The periodic complex exponential functions on the circle form a complete orthonormal basis. The Fourier series transform converts a square integrable complex valued function to a square summable doubly infinite sequence of coefficients. Gull's theorem can be used to simulate Bell experiments on distributed computers. The proof is based on treating the physics implemented inside the computers as completely deterministic and using externally created streams of random binary setting choices to derive martingale properties of a suitable game score. A variant of the proof, designed specifically for deterministically operating computers, can also be proven using a Bell theorem. Gull's theorem is true and its mathematical proof is pretty and original. Gull's Theorem states that it is impossible to write computer programs that can generate random numbers with identical outputs on two completely separate computers. Gull's argument assumes the use of local hidden variables theory for the EPR-B "Gedanken Experiment". The theorem can be proven using Fourier theory. To prove Gull's Theorem, three networked computers are needed, two of which will represent the two measurement stations, and one will supply a stream of random numbers. The two computers need to be completely separated and running completely deterministic programs. If we could implement a local hidden variables theory, we could simulate the singlet correlations derived from one run of many trials on two completely separate computers. The two programs are started, and they both set up a dialogue, initially setting n to 1. Alice's computer waits for Alice to type an angle and then evaluates and outputs Xa(?n) = ?1, increments n by one. Bob's computer does the same thing. The paper discusses the use of pseudo-random number generators (RNG) in simulating hidden variables in quantum mechanics. The author proposes a classical Monte-Carlo computer simulation to generate empirical relative frequencies and predictions of statistics. The paper emphasizes that this is not about the properties of the real physical world but rather mathematical descriptions thereof. The author uses colorful language to describe what Alice would have observed if Bob's measurement setting had been different from what he actually chose. The paper addresses the mathematical existence of a classical probability space on which are defined a large collection of random variables Xa and Yb for all directions a and b. The author characterizes such theories as “local hidden variables theories” (LHV) and claims that the statistics predicted by quantum mechanics, and observed in experiments, are merely the reflection of a more classical underlying theory of an essentially deterministic and local nature. The author proposes a different way to prove Gull's theorem via the Bell-CHSH inequality. The article discusses Bell's Theorem and its relevance to quantum computing and experiments. The theorem involves joint probability distributions and correlations between quantum systems, particularly in the context of two-qubit or one-qubit quantum computers. The article also mentions the EPR-B experiment and Malus' law in relation to spin and polarization measurements. The Bell-CHSH experiment involves two locations or labs with binary settings inputted by experimenters Alice and Bob. The article clarifies terminology and attribution related to the topic. Additionally, the article notes that Bell's theorem elicits strong emotions and that many physicists and mathematicians lack knowledge of probability theory. The text discusses the history and development of Bell's inequalities, which were previously attributed to Itamar Pitowsky or Boole. Vorob'ev explored necessary and sufficient conditions for probability concerning indicator functions, similar to Bell's work. Boole derived six linear inequalities involving p, q, and r whose simultaneous validity is necessary for a probability space to exist. The abuse of notation is a problem in the philosophy of science concerning terminology. Fine's theorem is recognized in Boole's work, and the CHSH inequality is a simple result in elementary probability theory. Bell's conclusions still hold if all correlations are only approximately true. Bell's Theorem states that conventional quantum mechanics is incompatible with local realism. This theorem has many metaphysical ramifications. Gull's Theorem is a proof using Fourier theory that builds on perfect anti-correlation in the singlet correlations. The original proof involves four correlations used in two alternative proofs of the same theorem. Bell himself referred to his inequalities as "my theorem." The Born rule is argued to follow from the deterministic part of the theory. Gull and Bell were physicists, not in the business of writing out formal mathematical theorems. Gull's theorem clarifies the status of Bell's proof. This paper presents Gull's proof of Bell's theorem using Fourier theory, which is relevant to recent experimental progress and debates in quantum physics. The proof shows that classical computers cannot generate a full amplitude negative cosine correlation between outputs given inputs, which is a unique signature of quantum entanglement. The proof requires the assumption of shared generating pseudo-random numbers within each computer and a third computer supplying a stream of random numbers to the two computers. Gull's philosophy is that Bell's theorem is a no-go theorem for a project in distributed computing with classical, not quantum, computers.