Abstract
This chapter is exceptional insofar as the formalism is not developed further. Rather, it serves to collect our previously acquired knowledge, to compare and to check where there are open questions of form or content. In the second part of the chapter, we take up quantum cryptography.
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Notes
- 1.
A comment on the notation: although the Hamiltonians of the two approaches in (10.1) are completely different mathematical objects, it is customary to denote them with the same symbol H. The same holds for the eigenfunctions and vectors.
- 2.
In other words, if the initial and final states are not the same.
- 3.
Quantum mechanics is very well-behaved in this sense.
- 4.
In this case one speaks of ‘state preparation’.
- 5.
This makes it perhaps understandable that Einstein dismissed it as ‘spooky action at a distance’. It can be shown that the effect is not suitable for the superluminal transmission of information—the validity of the theory of relativity thus remains unquestioned.
- 6.
In an extension of Bell’s one-liner, those theories which, on the one hand, one cannot really (or does not want to) justify, but which, on the other hand, agree well with experimental results and are very useful for all practical purposes, are called fapp theories. Quantum mechanics may be such a theory, if one regards it only as a tool (or judges it primarily by its usefulness) and is not willing (or able) to reflect upon its meaning.
- 7.
However, the movie title ‘Quantum of Solace’ promises not a ‘quantum jump’, but rather a minimum in terms of comfort for James Bond—quantum solace, so to speak.
- 8.
We have already seen that this is not always true, e.g. in the algebraic approach, where the basic ideas can be formulated using simple vector algebra.
- 9.
This term is short and to the point, but also a bit misleading. As we shall see shortly, quantum mechanics does not help to encrypt a message, but rather ensures that the key cannot be discovered by a spy.
- 10.
For this reason, the topic is also very well suited for discussion at the school level.
- 11.
Another method, called the E91 protocol (the ‘E’ designates Artur Ekert), works with entangled photons (for this concept see Chap. 20, Vol. 2).
- 12.
Unfortunately, one must not be too far ahead of one’s time. Depicting blue horses in the 15th century probably caused (at most) some head-shaking. That applies also in science.
- 13.
The \(\boxtimes \) plane is of course the \(\boxplus \) plane, rotated by \( 45^{\circ }\). Moreover, the \(\boxplus \) states are the eigenvectors of \( \sigma _{z}\), and the \(\boxtimes \) states, up to a sign, are the eigenvectors of \(\sigma _{x}\); cf. Chap. 4.
- 14.
For example, the mapping \(0\,\hat{=}\,\left| h\right\rangle \) and \(1\,\hat{=}\,\left| v\right\rangle \) would be just as good.
- 15.
By a bit, one denotes a quantity that can take on only two values, here 0 and 1.
- 16.
We remark that Bob, in his measurements with a ‘wrong’ basis, may of course also obtain other values, and these with equal probability. The last row in the table above is a concrete example of a total of 16. Other possibilities for Bob’s actual measurements are e.g. \(\begin{array}{|c|c|c|c|c|c|c|c|c|c|} 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 \\ \end{array} \) or \(\begin{array}{|c|c|c|c|c|c|c|c|c|c|} 1 &{} 0 &{} 0 &{} 1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ \end{array}\).
- 17.
C. Kurtsiefer et al., ‘A step towards global key distribution’, Nature 419 (2002), p. 450.
- 18.
See the webpage ‘Experimental Quantum Physics’, http://xqp.physik.uni-muenchen.de/.
- 19.
R. Ursin et al., ‘Entanglement-based quantum communication over 144 km’, Nature Physics 3 (2007), p. 481.
- 20.
See e.g. S. Liao et al., Satellite-to-ground quantum key distribution, Nature 549, 43–47, https://doi.org/10.1038/nature23655 (Sep 2017), where a quantum key distribution over a distance of up to 1,200 km is reported. Quantum keys may also be distributed in optical fibers over remarkable distances of up to 100 km; see e.g. K.A. Patel et al., Coexistence of high-bit-rate quantum key distribution and data on optical fiber, Phys. Rev. X 2, 041010 (2012)), or Paul Jouguet et al., Experimental demonstration of long-distance continuous-variable quantum key distribution, Nature Photonics (2013), https://doi.org/10.1038/nphoton.2013.63. In addition, the feasibility of BB84 quantum key distribution between an aircraft moving at 290 km/h at a distance of 20 km was recently proven for the first time; see: Sebastian Nauert et al., Air-to-ground quantum communication, Nature Photonics (2013), https://doi.org/10.1038/nphoton.2013.46.
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Pade, J. (2018). Stopover; Then on to Quantum Cryptography. In: Quantum Mechanics for Pedestrians 1. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-00464-4_10
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