ATT & Verison Merger

Irish food 

If there is one country that none ever mentions when talking about international cuisine, that is England

A few pubs, an Italian place, just some standard restaurant place.

It was so bad after a week that I tried out some American chains like Planet Holywood. That was also terrible; although, it's not that good here either.

I got a nice cinnamon and sugar pancake from a cart outside the London Eye.

EDIT: I enjoy a toffee flavor in my beer, but not in my food. I also don't really eat desert.


Edited by Equality 7-2521 - January 12 2011 at 13:07

JJLehto wrote: Trollin? And if this is true, man I got no clue. All I know is I've been thinking a lot lately about John Bell's Inequality Theory As in the situation explored in the EPR paradox, Bell considered an experiment in which a source produces pairs of correlated particles. For example, a pair of particles with correlated spins is created; one particle is sent to Alice and the other to Bob. On each trial, each observer independently chooses between various detector settings and then performs an independent measurement on the particle. (Note: although the correlated property used here is the particles' spin, it could alternatively be any correlated "quantum state" that encodes exactly one quantum bit.) Same axis: pair 1 pair 2 pair 3 pair 4 ...n Alice, 0°: + - - + ... Bob, 180°: + - - + ... Correlation: ( +1 +1 +1 +1 ...)/n = +1 (100% identical) Orthogonal axes: pair 1 pair 2 pair 3 pair 4 ...n Alice, 0°: + - + - ... Bob, 90°: - - + + ... Correlation: ( -1 +1 +1 -1 ...)/n = 0.0 (50% identical) When Alice and Bob measure the spin of the particles along the same axis (but in opposite directions), they get identical results 100% of the time. But when Bob measures at orthogonal (right) angles to Alice's measurements, they get identical results only 50% of the time. In terms of mathematics, the two measurements have a correlation of 1, or perfect correlation when read the same way; when read at right angles, they have a correlation of 0; no correlation. (A correlation of -1 would indicate getting opposite results for each measurement.) So far, the results can be explained by positing local hidden variables — each pair of particles may have been sent out with instructions on how to behave when measured in the two axes (either '+' or '-' for each axis). Clearly, if the source only sends out particles whose instructions are identical for each axis, then when Alice and Bob measure on the same axis, they are bound to get identical results, either (+,+) or (-,-); but (if all four possible pairings of + and - instructions are generated equally) when they measure on perpendicular axes they will see zero correlation. Now, consider that Alice or Bob can rotate their apparatus relative to each other by any amount at any time before measuring the particles, even after the particles leave the source. If local hidden variables determine the outcome of such measurements, they must encode at the time of leaving the source a result for every possible eventual direction of measurement, not just for the results in one particular axis. Bob begins this experiment with his apparatus rotated by 45 degrees. We call Alice's axes a and a', and Bob's rotated axes b and b'. Alice and Bob then record the directions they measured the particles in, and the results they got. At the end, they will compare and tally up their results, scoring +1 for each time they got the same result and -1 for an opposite result - except that if Alice measured in a and Bob measured in b', they will score +1 for an opposite result and -1 for the same result. Using that scoring system, any possible combination of hidden variables would produce an expected average score of at most +0.5. (For example, see the table below, where the most correlated values of the hidden variables have an average correlation of +0.5, i.e. 75% identical. The unusual "scoring system" ensures that maximum average expected correlation is +0.5 for any possible system that relies on local hidden variables.) Classical model: highly correlated variables less correlated variables Hidden variable for 0° (a): + + + + - - - - + + + + - - - - Hidden variable for 45° (b): + + + - - - - + + - - - + + + - Hidden variable for 90° (a'): + + - - - - + + - + + - + - - + Hidden variable for 135° (b'): + - - - - + + + + + - + - + - - Correlation score: If measured on a-b, score: +1 +1 +1 -1 +1 +1 +1 -1 +1 -1 -1 -1 -1 -1 -1 +1 If measured on a'-b, score: +1 +1 -1 +1 +1 +1 -1 +1 -1 -1 -1 +1 +1 -1 -1 -1 If measured on a'-b', score: +1 -1 +1 +1 +1 -1 +1 +1 -1 +1 -1 -1 -1 -1 +1 -1 If measured on a-b', score: -1 +1 +1 +1 -1 +1 +1 +1 -1 -1 +1 -1 -1 +1 -1 -1 Expected average score: +0.5 +0.5 +0.5 +0.5 +0.5 +0.5 +0.5 +0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 Bell's Theorem shows that if the particles behave as predicted by quantum mechanics, Alice and Bob can score higher than the classical hidden variable prediction of +0.5 correlation; if the apparatuses are rotated at 45° to each other, quantum mechanics predicts that the expected average score is 0.71. (Quantum prediction in detail: When observations at an angle of θ are made on two entangled particles, the predicted correlation is cosθ. The correlation is equal to the length of the projection of the particle's vector onto his measurement vector; by trigonometry, cosθ. θ is 45°, and cosθ is \tfrac{\sqrt{2}}{2}, for all pairs of axes except (a,b') – where they are 135° and -\tfrac{\sqrt{2}}{2} – but this last is taken in negative in the agreed scoring system, so the overall score is \tfrac{\sqrt{2}}{2}; 0.707. In one explanation, the particles behave as if when Alice or Bob makes a measurement, the other particle usually switches to take that direction instantaneously.) Multiple researchers have performed equivalent experiments using different methods. It appears most of these experiments produce results which agree with the predictions of quantum mechanics [1], leading to disproof of local-hidden-variable theories and proof of nonlocality. Still not everyone agrees with these findings [2]. There have been two loopholes found in the earlier of these experiments, the detection loophole and the communication loophole with associated experiments to close these loopholes. After all current experimentation it seems these experiments uphold prima facie support for quantum mechanics' predictions of nonlocality [3]. [edit] Importance of the theorem This theorem has even been called "the most profound in science."[1] Bell's seminal 1964 paper was entitled "On the Einstein Podolsky Rosen paradox."[2] The Einstein Podolsky Rosen paradox (EPR paradox) proves, on basis of the assumption of "locality" (physical effects have a finite propagation speed) and "reality" (physical states exist before they are measured) that particle attributes have definite values independent of the act of observation. Bell showed that local realism leads to a requirement for certain types of phenomena that are not present in quantum mechanics. This requirement is called Bell's inequality. After EPR (Einstein-Podolsky-Rosen), quantum mechanics was left in an unsatisfactory position: either it was incomplete, in the sense that it failed to account for some elements of physical reality, or it violated the principle of finite propagation speed of physical effects. In a modified version of the EPR thought experiment, two observers, now commonly referred to as Alice and Bob, perform independent measurements of spin on a pair of electrons, prepared at a source in a special state called a spin singlet state. It was equivalent to the conclusion of EPR that once Alice measured spin in one direction (e.g. on the x axis), Bob's measurement in that direction was determined with certainty, with opposite outcome to that of Alice, whereas immediately before Alice's measurement, Bob's outcome was only statistically determined. Thus, either the spin in each direction is an element of physical reality, or the effects travel from Alice to Bob instantly. In QM, predictions were formulated in terms of probabilities — for example, the probability that an electron might be detected in a particular region of space, or the probability that it would have spin up or down. The idea persisted, however, that the electron in fact has a definite position and spin, and that QM's weakness was its inability to predict those values precisely. The possibility remained that some yet unknown, but more powerful theory, such as a hidden variables theory, might be able to predict those quantities exactly, while at the same time also being in complete agreement with the probabilistic answers given by QM. If a hidden variables theory were correct, the hidden variables were not described by QM, and thus QM would be an incomplete theory. The desire for a local realist theory was based on two assumptions: 1. Objects have a definite state that determines the values of all other measurable properties, such as position and momentum. 2. Effects of local actions, such as measurements, cannot travel faster than the speed of light (as a result of special relativity). If the observers are sufficiently far apart, a measurement taken by one has no effect on the measurement taken by the other. In the formalization of local realism used by Bell, the predictions of theory result from the application of classical probability theory to an underlying parameter space. By a simple (but clever) argument based on classical probability, he then showed that correlations between measurements are bounded in a way that is violated by QM. Bell's theorem seemed to put an end to local realist hopes for QM. Per Bell's theorem, either quantum mechanics or local realism is wrong. Experiments were needed to determine which is correct, but it took many years and many improvements in technology to perform them. Bell test experiments to date overwhelmingly show that Bell inequalities are violated. These results provide empirical evidence against local realism and in favor of QM. The no-communication theorem proves that the observers cannot use the inequality violations to communicate information to each other faster than the speed of light. John Bell's paper examines both John von Neumann's 1932 proof of the incompatibility of hidden variables with QM and Albert Einstein and his colleagues' seminal 1935 paper on the subject. [edit] Bell inequalities Bell inequalities concern measurements made by observers on pairs of particles that have interacted and then separated. According to quantum mechanics they are entangled while local realism limits the correlation of subsequent measurements of the particles. Different authors subsequently derived inequalities similar to Bell´s original inequality, collectively termed Bell inequalities. All Bell inequalities describe experiments in which the predicted result assuming entanglement differs from that following from local realism. The inequalities assume that each quantum-level object has a well defined state that accounts for all its measurable properties and that distant objects do not exchange information faster than the speed of light. These well defined states are often called hidden variables, the properties that Einstein posited when he stated his famous objection to quantum mechanics: "God does not play dice." Bell showed that under quantum mechanics, which lacks local hidden variables, the inequalities (the correlation limit) may be violated. Instead, properties of a particle are not clear to verify in quantum mechanics but may be correlated with those of another particle due to quantum entanglement, allowing their state to be well defined only after a measurement is made on either particle. That restriction agrees with the Heisenberg uncertainty principle, a fundamental and inescapable concept in quantum mechanics. In Bell's work: “ Theoretical physicists live in a classical world, looking out into a quantum-mechanical world. The latter we describe only subjectively, in terms of procedures and results in our classical domain. (...) Now nobody knows just where the boundary between the classical and the quantum domain is situated. (...) More plausible to me is that we will find that there is no boundary. The wave functions would prove to be a provisional or incomplete description of the quantum-mechanical part. It is this possibility, of a homogeneous account of the world, which is for me the chief motivation of the study of the so-called "hidden variable" possibility. (...) A second motivation is connected with the statistical character of quantum-mechanical predictions. Once the incompleteness of the wave function description is suspected, it can be conjectured that random statistical fluctuations are determined by the extra "hidden" variables -- "hidden" because at this stage we can only conjecture their existence and certainly cannot control them. (...) A third motivation is in the peculiar character of some quantum-mechanical predictions, which seem almost to cry out for a hidden variable interpretation. This is the famous argument of Einstein, Podolsky and Rosen. (...) We will find, in fact, that no local deterministic hidden-variable theory can reproduce all the experimental predictions of quantum mechanics. This opens the possibility of bringing the question into the experimental domain, by trying to approximate as well as possible the idealized situations in which local hidden variables and quantum mechanics cannot agree ” In probability theory, repeated measurements of system properties can be regarded as repeated sampling of random variables. In Bell's experiment, Alice can choose a detector setting to measure either A(a) or A(a') and Bob can choose a detector setting to measure either B(b) or B(b'). Measurements of Alice and Bob may be somehow correlated with each other, but the Bell inequalities say that if the correlation stems from local random variables, there is a limit to the amount of correlation one might expect to see. [edit] Original Bell's inequality The original inequality that Bell derived was:[2] 1 + \operatorname{C}(b, c) \geq |\operatorname{C}(a, b) - \operatorname{C}(a, c)|, where C is the "correlation" of the particle pairs and a, b and c settings of the apparatus. This inequality is not used in practice. For one thing, it is true only for genuinely "two-outcome" systems, not for the "three-outcome" ones (with possible outcomes of zero as well as +1 and −1) encountered in real experiments. For another, it applies only to a very restricted set of hidden variable theories, namely those for which the outcomes on both sides of the experiment are always exactly anticorrelated when the analysers are parallel, in agreement with the quantum mechanical prediction. There is a simple limit of Bell's inequality which has the virtue of being completely intuitive. If the result of three different statistical coin-flips A,B,C have the property that: 1. A and B are the same (both heads or both tails) 99% of the time 2. B and C are the same 99% of the time then A and C are the same at least 98% of the time. The number of mismatches between A and B (1/100) plus the number of mismatches between B and C (1/100) are together the maximum possible number of mismatches between A and C. In quantum mechanics, by letting A,B,C be the values of the spin of two entangled particles measured relative to some axis at 0 degrees, θ degrees, and 2θ degrees respectively, the overlap of the wavefunction between the different angles is proportional to \scriptstyle \cos(S\theta)\approx 1-S^2\theta^2/2. The probability that A and B give the same answer is 1 − ε2, where ε is proportional to θ. This is also the probability that B and C give the same answer. But A and C are the same 1 − (2ε)2 of the time. Choosing the angle so that ε = .1, A and B are 99% correlated, B and C are 99% correlated and A and C are only 96% correlated. Imagine that two entangled particles in a spin singlet are shot out to two distant locations, and the spin of both are measured in the direction A. The spins are 100% correlated (actually, anti-correlated but for this argument that is equivalent). The same is true if both spins are measured in directions B or C. It is safe to conclude that any hidden variables which determine the A,B, and C measurements in the two particles are 100% correlated and can be used interchangeably. If A is measured on one particle and B on the other, the correlation between them is 99%. If B is measured on one and C on the other, the correlation is 99%. This allows us to conclude that the hidden variables determining A and B are 99% correlated and B and C are 99% correlated. But if A is measured in one particle and C in the other, the results are only 96% correlated, which is a contradiction. This intuitive formulation is due to David Mermin. [edit] CHSH inequality Main article: CHSH inequality In addition to Bell's original inequality,[2] the form given by John Clauser, Michael Horne, Abner Shimony and R. A. Holt,[3] (the CHSH form) is especially important[3], as it gives classical limits to the expected correlation for the above experiment conducted by Alice and Bob: \ (1) \quad \mathbf{C}[A(a), B(b)] + \mathbf{C}[A(a), B(b')] + \mathbf{C}[A(a'), B(b)] - \mathbf{C}[A(a'), B(b')]\leq 2, where C denotes correlation. Correlation of observables X, Y is defined as \mathbf{C}(X,Y) = \operatorname{E}(X Y). This is non-normalized form of the correlation coefficient considered in statistics (see Quantum correlation). In order to formulate Bell's theorem, we formalize local realism as follows: 1. There is a probability space Λ and the observed outcomes by both Alice and Bob result by random sampling of the parameter \lambda \in \Lambda. 2. The values observed by Alice or Bob are functions of the local detector settings and the hidden parameter only. Thus * Value observed by Alice with detector setting a is A(a,λ) * Value observed by Bob with detector setting b is B(b,λ) Implicit in assumption 1) above, the hidden parameter space Λ has a probability measure ρ and the expectation of a random variable X on Λ with respect to ρ is written \operatorname{E}(X) = \int_\Lambda X(\lambda) \rho(\lambda) d \lambda where for accessibility of notation we assume that the probability measure has a density. Bell's inequality. The CHSH inequality (1) holds under the hidden variables assumptions above. For simplicity, let us first assume the observed values are +1 or −1; we remove this assumption in Remark 1 below. Let \lambda \in \Lambda. Then at least one of B(b, \lambda) + B(b', \lambda), \quad B(b, \lambda) - B(b', \lambda) is 0. Thus A(a, \lambda) \ B(b, \lambda) + A(a, \lambda) \ B(b', \lambda) +A(a', \lambda) \ B(b, \lambda) - A(a', \lambda) \ B(b', \lambda) = = A(a, \lambda) (B(b, \lambda) + B(b', \lambda))+ A(a', \lambda) (B(b, \lambda) - B(b', \lambda)) \quad \leq 2. and therefore \mathbf{C}(A(a), B(b)) + \mathbf{C}(A(a), B(b')) + \mathbf{C}(A(a'), B(b)) - \mathbf{C}(A(a'), B(b')) = = \int_\Lambda A(a, \lambda) \ B(b, \lambda) \rho(\lambda) d \lambda + \int_\Lambda A(a, \lambda) \ B(b', \lambda) \rho(\lambda) d \lambda + \int_\Lambda A(a', \lambda) \ B(b, \lambda) \rho(\lambda) d \lambda - \int_\Lambda A(a', \lambda) \ B(b', \lambda) \rho(\lambda) d \lambda = = \int_\Lambda \bigg\{A(a, \lambda) \ B(b, \lambda) + A(a, \lambda) \ B(b', \lambda) +A(a', \lambda) \ B(b, \lambda) - A(a', \lambda) \ B(b', \lambda)\bigg\} \rho(\lambda) d \lambda = = \int_\Lambda \bigg\{A(a, \lambda) (B(b, \lambda) + B(b', \lambda))+ A(a', \lambda) (B(b, \lambda) - B(b', \lambda)) \bigg\} \rho(\lambda) d \lambda \quad \leq 2. Remark 1. The correlation inequality (1) still holds if the variables A(a,λ), B(b,λ) are allowed to take on any real values between -1, +1. Indeed, the relevant idea is that each summand in the above average is bounded above by 2. This is easily seen to be true in the more general case: A(a, \lambda) \ B(b, \lambda) + A(a, \lambda) \ B(b', \lambda) + A(a', \lambda) \ B(b, \lambda) - A(a', \lambda) \ B(b', \lambda) = = A(a, \lambda) (B(b, \lambda) + B(b', \lambda)) +A(a', \lambda) (B(b, \lambda) - B(b', \lambda)) \quad \leq \bigg|A(a, \lambda) (B(b, \lambda) + B(b', \lambda)) +A(a', \lambda) (B(b, \lambda) - B(b', \lambda) ) \bigg| \quad \leq \bigg|A(a, \lambda) (B(b, \lambda) + B(b', \lambda))\bigg| +\bigg|A(a', \lambda) (B(b, \lambda) - B(b', \lambda))\bigg| \leq |B(b, \lambda) + B(b', \lambda)| +| B(b, \lambda) - B(b', \lambda)| \leq 2. To justify the upper bound 2 asserted in the last inequality, without loss of generality, we can assume that B(b, \lambda) \geq B(b', \lambda) \geq 0. In that case |B(b, \lambda) + B(b', \lambda)| +| B(b, \lambda) - B(b', \lambda)| =B(b, \lambda) + B(b', \lambda) + B(b, \lambda) - B(b', \lambda) = \quad = 2 B(b, \lambda) \leq 2. \quad . Remark 2. Though the important component of the hidden parameter λ in Bell's original proof is associated with the source and is shared by Alice and Bob, there may be others that are associated with the separate detectors, these others being independent. This argument was used by Bell in 1971, and again by Clauser and Horne in 1974,[4] to justify a generalisation of the theorem forced on them by the real experiments, in which detector were never 100% efficient. The derivations were given in terms of the averages of the outcomes over the local detector variables. The formalisation of local realism was thus effectively changed, replacing A and B by averages and retaining the symbol λ but with a slightly different meaning. It was henceforth restricted (in most theoretical work) to mean only those components that were associated with the source. However, with the extension proved in Remark 1, CHSH inequality still holds even if the instruments themselves contain hidden variables. In that case, averaging over the instrument hidden variables gives new variables: \overline{A}(a, \lambda), \quad \overline{B}(b, \lambda) on Λ which still have values in the range [-1, +1] to which we can apply the previous result. [edit] Bell inequalities are violated by quantum mechanical predictions In the usual quantum mechanical formalism, the observables X and Y are represented as self-adjoint operators on a Hilbert space. To compute the correlation, assume that X and Y are represented by matrices in a finite dimensional space and that X and Y commute; this special case suffices for our purposes below. The von Neumann measurement postulate states: a series of measurements of an observable X on a series of identical systems in state φ produces a distribution of real values. By the assumption that observables are finite matrices, this distribution is discrete. The probability of observing λ is non-zero if and only if λ is an eigenvalue of the matrix X and moreover the probability is \|\operatorname{E}_X(\lambda) \phi\|^2 where EX (λ) is the projector corresponding to the eigenvalue λ. The system state immediately after the measurement is \|\operatorname{E}_X(\lambda) \phi\|^{-1} \operatorname{E}_X(\lambda) \phi. From this, we can show that the correlation of commuting observables X and Y in a pure state ψ is \langle X Y \rangle = \langle X Y \psi \mid \psi \rangle. We apply this fact in the context of the EPR paradox. The measurements performed by Alice and Bob are spin measurements on electrons. Alice can choose between two detector settings labelled a and a′; these settings correspond to measurement of spin along the z or the x axis. Bob can choose between two detector settings labelled b and b′; these correspond to measurement of spin along the z′ or x′ axis, where the x′ – z′ coordinate system is rotated 45° relative to the x – z coordinate system. The spin observables are represented by the 2 × 2 self-adjoint matrices: S_x = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} , S_z = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}. These are the Pauli spin matrices normalized so that the corresponding eigenvalues are +1, −1. As is customary, we denote the eigenvectors of Sx by \left|+x\right\rang, \quad \left|-x\right\rang. Let φ be the spin singlet state for a pair of electrons discussed in the EPR paradox. This is a specially constructed state described by the following vector in the tensor product \left|\phi\right\rang = \frac{1}{\sqrt{2}} \bigg(\left|+x\right\rang \otimes \left|-x\right\rang - \left|-x\right\rang \otimes \left|+x\right\rang \bigg). Now let us apply the CHSH formalism to the measurements that can be performed by Alice and Bob. Illustration of Bell test for spin 1/2 particles. Source produces spin singlet pairs, one particle of each pair is sent to Alice and the other to Bob. Each performs one of the two spin measurements. Illustration of Bell test for spin 1/2 particles. Source produces spin singlet pairs, one particle of each pair is sent to Alice and the other to Bob. Each performs one of the two spin measurements. A(a) = S_z \otimes I A(a') = S_x \otimes I B(b) = - \frac{1}{\sqrt{2}} \ I \otimes (S_z + S_x) B(b') = \frac{1}{\sqrt{2}} \ I \otimes (S_z - S_x). The operators B(b'), B(b) correspond to Bob's spin measurements along x′ and z′. Note that the A operators commute with the B operators, so we can apply our calculation for the correlation. In this case, we can show that the CHSH inequality fails. In fact, a straightforward calculation shows that \langle A(a) B(b) \rangle = \langle A(a') B(b) \rangle =\langle A(a') B(b') \rangle = \frac{1}{\sqrt{2}}, and \langle A(a) B(b') \rangle = - \frac{1}{\sqrt{2}}. so that \langle A(a) B(b) \rangle + \langle A(a') B(b') \rangle + \langle A(a') B(b) \rangle - \langle A(a) B(b') \rangle = \frac{4}{\sqrt{2}} = 2 \sqrt{2} > 2. Bell's Theorem: If the quantum mechanical formalism is correct, then the system consisting of a pair of entangled electrons cannot satisfy the principle of local realism. Note that 2 \sqrt{2} is indeed the upper bound for quantum mechanics called Tsirelson's bound. The operators giving this maximal value are always isomorphic to the Pauli matrices.