User:Boris Tsirelson/Sandbox1

Nonlocality and entanglement

In general

The words "nonlocal" and "nonlocality" occur frequently in the literature on entanglement, which creates a lot of confusion: it seems that entanglement means nonlocality! This situation has two causes, pragmatical and philosophical.

Here is the pragmatical cause. The word "nonlocal" sounds good. The phrase "non-CFD" (where CFD denotes counterfactual definiteness) sounds much worse, but is also incorrect; the correct phrase, involving both CFD and locality (and no-conspiracy, see the lead) is very cumbersome. Thus, "nonlocal" is often used as a conventional substitute for "able to produce empirical entanglement".

The philosophical cause. Many people feel that CFD is more trustworthy than RLC (relativistic local causality), and NC (no-conspiracy) is even more trustworthy. Being forced to abandon one of them, these people are inclined to retain NC and CFD at the expence of abandoning RLC.

However, the quantum theory is compatible with RLC+NC. A violation of RLC+NC is called faster-than-light signaling (rather than entanglement); it was never observed, and never predicted by the quantum theory. Thus RLC and NC are corroborated, while CFD is not. In this sense CFD is less trustworthy than RLC and NC.

For quantum states

Quantitative measures for entanglement are scantily explored in general. However, for pure bipartite quantum states the amount of entanglement is usually measured by the so-called entropy of entanglement. On the other hand, several natural measures of nonlocality are invented (see above about the meaning of "nonlocality"). Strangely enough, non-maximally entangled states appear to be more nonlocal than maximally entangled states, which is known as "anomaly of nonlocality"; nonlocality and entanglement are not only different concepts, but are really quantitatively different resources. According to the asymptotic theory of Bell inequalities, even though entanglement is necessary to obtain violation of Bell inequalities, the entropy of entanglement is essentially irrelevant in obtaining large violation.

???

Further progress appears in the 17th century from the study of motion (Johannes Kepler, Galileo Galilei) and geometry (P. Fermat, R. Descartes). A formulation by Descartes (La Geometrie, 1637) appeals to graphic representation of a functional dependence and does not involve formulas (algebraic expressions):

If then we should take successively an infinite number of different

values for the line y, we should obtain an infinite number of values for the line x, and therefore an infinity of different points, such as C, by means of which the required curve could be

drawn.

The term function is adopted by Leibniz and Jean Bernoulli between 1694 and 1698, and disseminated by Bernoulli in 1718:

One calls here a function of a variable a quantity composed in any manner whatever of this variable and of constants.

This time a formula is required, which restricts the class of functions. However, what is a formula? Surely, y = 2x² - 3 is allowed; what about y = sin x? Is it "composed of x"? "In any manner whatever" is now interpreted much more widely than it was possible in 17th century.

... little by little, and often by very subtle detours, various

transcendental operations, the logarithm, the exponential, the trigonometric functions, quadratures, the solution of differential equations, passing to the limit, the summing of series, acquired the

right of being quoted. (Bourbaki, p. 193)

Surely, sin x is not a polynomial of x. However, it is the sum of a power series:

$\sin x=x-{\frac {x^{3}}{6}}+{\frac {x^{5}}{120}}-\dots$

which was found already by James Gregory in 1667. Many other functions were developed into power series by him, Isaac Barrow, Isaac Newton and others. Moreover, all these formulas appeared to be special cases of a much more general formula found by Brook Taylor in 1715.

But on the first stage the notion of an algebraic expression is quite restrictive. More general, possibly ill-behaving functions have to wait for the 19th century.

References

• Arnol'd, V.I. (1990), Huygens and Barrow, Newton and Hooke: pioneers in mathematical analysis and catastrophe theory from evolvents to quasicrystals, Birkhäuser.