288 36 On the Interpretation of Quantum Mechanics Austria (see e.g., the paper in the journal “Spektrum der Wissenschaften” (1997, issue 1) which has already been mentioned, [32]). Starting with a laser beam propagating in the x-direction, the photons of this beam are linearly polarized in the vertical (±z)-direction (polariza- tion P 0 = ±0 ◦ ); a beam splitter then follows which produces two coherent beams propagating, e.g., in the x-direction and the y-direction, respectively (i.e., along two different paths W 1 and W 2 ) with complementary diagonal polarizations P 1 = ±45 ◦ and P 2 = ±135 ◦ . Thirdly, the two beams meet again at a crossing point “X”, where by inter- ference the original polarization P 0 = ±0 ◦ is restored. Next, a detector follows, which is inserted at the continuation of the path W 1 , but counts (per construction) only photons of polarization P D = ±90 ◦ . As a consequence, this detector will never count any photon of the beam, since all photons leaving X have the orthogonal polarization P 0 = ±0 ◦ . Thus, with such a detector one cannot state which way a photon emitted by the source reached the crossing point. Only when the interference is blocked, e.g., by interrupting the path W 2 , will the detector start counting photons: a photon emitted by the source reaches the point with probability 1 2 via the “non-blocked” path, i.e., with polarization P 1 = ±45 ◦ , and is counted by the detector P D = ±90 ◦ , again with probability 1 2 . In this way, i.e., if the detector records a count, one not only has a state- ment about, (i), which path the electron traveled (along W 1 ), but also one knows, (ii), that the alternative path W 2 is blocked (possibly by a container carrying a bomb; cf. the section on Schr¨odinger’s cat). Morever, this fact, i.e., the possible presence of a bomb in path W 2 , has been established here via an interaction-free quantum measurement, i.e., without making the bomb explode. Such experiments (and many similar ones) on the non-locality of quantum mechanics (of course without involving any bomb) have been realized by Zeilinger et al. 13 . It is obvious that this has potential applications. 13 In this connection, the recent book [32] should be mentioned. 36.6 Quantum Cryptography 289 36.6 Quantum Cryptography Quantum cryptography is another topical application of quantum mechanics which has been practicable for some years (long before the advent of quan- tum computing). Quantum cryptography should of course be contrasted to classical cryptography, which is largely based on the PGP concept that we described in some detail in a previous Sect. 36.3 on quantum computing. Whereas classical cryptography is intrinsically insecure,since a) the factorization of large integers, on which it is typically based, is always possible in principle; i.e., if one has enough computing power and patience (the computer may calculate for days, or weeks, or months) one can restore the private key of the receiver by “factorizing” his (or her) public key, and b) if this does not work in a reasonable time, a spy can find out the private key i.e., the way the receiver decodes the contents of a message encoded by his (or her) public key, simply by eavesdropping – in contrast, quantum cryptography is secure (which we stress again: although it is not based on quantum computers). Within quantum cryptography, a private key not only becomes nonessen- tial, rendering eavesdropping obsolete, but (almost incredibly) it is detri- mental, because, through the act of eavesdropping, a spy will automatically uncover his or her own presence. The reason is essentially that quantum me- chanical measurements usually perturb the measured state, and this can be discovered by cooperation between the sender and receiver. In the following a well known protocol used in quantum cryptography is described in some detail. The sender of the message (referred to as Alice) sends signals to the receiver (Bob) with linearly polarized photons with four different polarization directions P = ±0 ◦ ,P= ±90 ◦ P = ±45 ◦ ,P= ±135 ◦ (i.e., “horizontally/vertically/right-diagonally/left-diagonally”). The infor- mation assignment A of the signals shall be fixed, e.g., A ≡ 0 for either P = ±0 ◦ or P = ±135 ◦ , and A ≡ 1 for either P = ±90 ◦ or P = ±45 ◦ . 14 If one has a preference for spin systems, or if the equipment suggests doing so, instead of the optical polarization either P = ±90 ◦ or P = ±45 ◦ 14 In contrast to fixed assignments, random assignments are also possible, which the sender and the receiver of the message must agree about before the process, or during it. 290 36 On the Interpretation of Quantum Mechanics one can equivalently use the 2-spinors either 1 0 or 1 √ 2 1 1 for A ≡ 1; and instead of either P = ±0 ◦ or P = ±135 ◦ the spinors either 0 1 or 1 √ 2 1 −1 can be used for A ≡ 0. The two classes of spinors named by either and or yield the eigenvectors of ˆ S z and ˆ S x , with eigenvalues ± 2 . It is important here that these different eigenvectors with the same eigen- values of either ˆ S z or ˆ S x are not orthogonal, whereas according to Einstein, Podolski and Rosen’s quasi-classical reasoning one would always tend to as- sociate orthogonal states to alternative polarizations ( ˆ S z alternatively to S x ) Thus far we have described the “public” part of the code. For the “secret” part, Alice and Bob, by some kind of “mutual preparation algorithm” (see the Appendix), come to an agreement, (i) as to which basis sequence b 1 , b 2 , b 3 , theywilluseinthecourseofthemessage,i.e.,forthen-th bit either the x-basis (diagonal polarization) or the z-basis (rectilinear polarization), and (ii) which bits n of the sequence of signals contain the message. This part of the agreement (ii) can even be published (e.g., on the internet), whereas the secret part of the agreement, ((i), detailing which basis is chosen for the n-th bit), remains unknown to the public. 15 The details of the “preparation algorithm” are complex, but straighfor- ward, and (as mentioned) are described in the Appendix. But the result (which has also just been mentioned) is simple: the public community, i.e., also a spy (“Adam” or “Eve”), only knows that Alice and Bob use the same basis (z or x) for the bits containing information, but they do not know which one. As a consequence, the spy does not know whether Alice and Bob inter- prete a signal as 0 or as 1. If for example the x-basis is used at the n-th digit (which the spy does not know), then he (or she) does not know whether the signal is in the first component only, or in both components. If he (or she) chooses erroneously (with 50% probability of error) the first alternative and received a signal in the first component (which we can assume to be positive), then the spy does not know whether this component must be completed by a second number. This means that he (or she) does not know whether this second component, which has not been measured, is positive (representing a “1”) or negative (representing a “0”); again there is 50% probability of error. Thus, if the spy continues his (or her) activity and sends instead of the 15 The public, including the “spy”, only know that sender and receiver use the same basis, but not which one. 36.6 Quantum Cryptography 291 original signals, which he (or she) has tried to analyse, “substitute signals” to Bob, these will be wrong with a high probability. Consequently, Bob, by cooperating with Alice, can uncover the activities of the spy. Thus the spy not only has no chance of obtaining information by eavesdropping, but instead of being successful will necessarily reveal his (or her) presence to Alice and Bob. For this concept at several places we have used essentially the nonclassical properties of the singlet state |ψ singlet ∝ |↑↓ − ↓↑ . A main property of this state is the Einstein-Podolski-Rosen property of “entanglement”. In fact, the above method can even be simplified by using entangled source states, e.g., where Bob and Alice receive complementary versions of the signal. In this case all digits can be used for the message, i.e., the mutual preparation becomes trivial with respect to the public part, (ii), but the secret part, (i), still applies. In this case too, the eavesdropping efforts of a spy would be unsuccessful and only reveal his own presence. One may compare the relevant paper in the journal “Physikalische Bl¨atter” 1999, issue 6. To enable quantum cryptography one must of course ensure coherence, i.e., the ability to interfere, along the whole length of propagation of the sig- nals. For distances which are larger than typically 10 to 100 km this condition is violated by the present glass fibre technology, since de-coherence by the necessary restoration of the original signal strength can only be avoided up to this sort of distance. With regard to the first commercial realization in the winter of 2003/2004 the reader is referred to a footnote in the section on quantum computing. 37 Quantum Mechanics: Retrospect and Prospect In retrospect, let us briefly look again at some of the main differences and similarities between classical and quantum mechanics. Classical mechanics of an N-particle system takes place in a 6N-dimen- sional phase space of coordinates plus momenta. Observables are arbitrary real functions of the 6N variables. The theory is deterministic and local, e.g., Newton’s equations apply with forces which act at the considered moment at the respective position. Measurements can in principle be performed arbi- trarily accurately, i.e., they only “state” the real properties of the system 1 . In contrast, in quantum mechanics the state of an N-particle system is described by an equivalence class of vectors ψ of a complex Hilbert space HR, where the equivalence relation is given by multiplication with a globally con- stant complex factor; i.e., one considers “rays” in HR. Already the description of the system is thus more complicated, the more so since the intrinsic angular momentum (spin) does not appear in classical physics and has unexpected non-classical properties, e.g., concerning the rotation behavior. The functions ψ must not only be (ia) square-integrable w.r.t. the position variables r j of N particles (j =1, ,N), but they must also be (ib) square-summable w.r.t. the respective spin variables (m s ) j , and (ii) for identical particles the Pauli principle, (iia) i.e. permutation-antisymmetry for fermions (half-integer spin quantum number, e.g., electrons or quarks), and (iib) permutation-symmetry for bosons (integer spin quantum number, e.g., pions or gluons), must be con- sidered. The Pauli principle has very important consequences in everyday life, e.g., the Mendeleev (or periodic) table of elements in chemistry. With regard to (i) one must in any case have m 1 =±1/2 m N =±1/2 r 1 d 3 r 1 r N d 3 r N |ψ(r 1 ,m 1 ; ; r N ,m N )| 2 ! =1. (37.1) The integrand in (37.1) has the meaning of a multidimensional probability density. 1 Newtonian mechanics is thus “realistic” in the sense of Einstein, Podolski and Rosen. 294 37 Quantum Mechanics: Retrospect and Prospect Furthermore, quantum mechanics is only semi-deterministic: a) On the one hand, between two measurements one has a deterministic equation of motion, e.g., in the Schr¨odinger representation the equation, i ˙ ψ = Hψ, and in the Heisenberg representation, which is equivalent, the equation of motion for the operators. These equations of motion are determined by the Hamilton operator, which corresponds largely, but not completely, to the classical Hamilton function H(r, p), where it is “almost only” necessary to replace the classical variables by operators, e.g., r by a multiplication operator and p by i ∇ ; “almost only”, but not completely, due to spin, which has no classical analogue. As a consequence of spin there are additions to the Hamilton operator, in which the spin (vector) operator appears, which, (i), has an anomalous behavior w.r.t. the g-factor (visible in the coupling to a magnetic field). Furthermore, (ii), spin plays a decisive role in the Pauli principle,whichis in itself most important. Finally, (iii), spin is coupled to the orbital angular momentum through the spin-orbit coupling, which is, e.g., responsible for the spectral fine structure of atoms, molecules and nuclei. b) On the other hand, for the results of a measurement there are only “prob- ability statements”. The reason is essentially that observables are now described by Hermitian operators, which once more correspond largely (but again not completely) to quantities appearing in classical mechanics. In general, operators are not commutable. As a consequence Heisenberg’s uncertainty principle restricts the product δ ˆ A 2 ψ · δ ˆ B 2 ψ of expectation values, by stating that this product of the variances of two series of measurements for non-commutable Hermitian operators ˆ A and ˆ B,with δ ˆ A 2 ψ := ψ| ˆ A 2 |ψ− ψ| ˆ A|ψ 2 , should be ≥ 1 4 ψ| ˆ A, ˆ B |ψ 2 , where ˆ A, ˆ B := ˆ A · ˆ B − ˆ B · ˆ A is the so-called commutator, i.e., [ˆp, ˆx]= i . Thus, by taking square roots, with appropriate definitions of the uncer- tainty or fuzziness of position and momentum in the state ψ,oneobtains 37 Quantum Mechanics: Retrospect and Prospect 295 the somewhat unsharp formulation: position and momentum cannot si- multaneously be precisely determined by a series of measurements, δˆx · δˆp ≥ 2 . With the definition δ ˆ k := δˆp/ for the fuzziness of the de-Broglie wavenumber one thus obtains δ ˆ k · δˆx ≥ 1/2 , i.e., a relation, in which the wave properties of matter dominate (k = 2π/λ). However, the wave aspect of matter is only one side of the coin; Heisen- berg’s uncertainty relation covers both sides. In fact, in quantum mechanics wave-particle duality with all its conse- quences applies: matter not only possesses particle properties, e.g., those de- scribed in Newtonian mechanics, but also wave properties, e.g., described by wave equations for probability amplitudes, with the essential property of co- herent superposition and interference of these amplitudes. At this place one should also mention the tunnel effect. Quantum mechanical measurements do not “state” properties of a system, but instead they “prepare” properties. In this respect quantum mechanics is not “realistic” but “preparing” in the sense of Einstein, Podolski and Rosen. In particular, typically (but not always) a measurement changes the state of a system. The totality of these statements is the so-called Copenhagen interpretation of quantum mechanics. After the objections of Einstein, Podolski and Rosen in their paper of 1935 were disproved by Bell experiments, it has now been accepted without controversy for several decades. The Copenhagen interpre- tation is also in agreement with non-local behavior, e.g., Aharonov-Bohm experiments. According to these experiments, the ψ-function, in contrast to the classical particle, does not “see” the local magnetic field and the Lorentz forces exerted by it, cf. Part II (so-called “local action”), but instead it “sees”, through the vector potential, its non-local flux (“remote action”). Other con- sequences of the interpretation based, e.g., on entanglement, coherence and interferences, have only recently been systematically exploited. After more than a century (nonrelativistic) quantum mechanics – which has been the main theme of this part of the text – is essentially a closed subject (as are Newtonian mechanics, Part I, and Maxwell’s electrodynamics, Part II), but with regard to future prospects, this does not necessarily imply that all consequences of these theories (particularly of quantum theory) have been fully understood or exploited. 296 37 Quantum Mechanics: Retrospect and Prospect For “John Q. Public” some aspects of quantum mechanics, e.g., spin and the Pauli principle, are (unwittingly) of crucial importance in everyday life, e.g., for the simple reason that the periodic table of elements, and therefore the whole of chemistry and essential parts of biology, depend strongly on these properties, which are not at all “classical”. Simply for this reason, at school or university, one should thus tolerate a partial lack of understanding of these topics; however, one should be careful about introducing potentially wrong explanations (e.g., spinning-top models for spin, which are popular, but essentially inadequate). Instead it may be better to admit that the reasons for the behavior are truly complicated. In any case, quantum mechanics is a discipline which contains considerable scope for intellectual and philosophical discussion about the nature of matter, radiation, interaction, and beyond. Finally, in the sections on quantum computation and quantum crypto- graphy we have tried to make clear that quantum mechanics is a field with a promising future and that the next few decades might bring surprises, perhaps not only with regard to novel applications. 38 Appendix: “Mutual Preparation Algorithm” for Quantum Cryptography Firstly, Alice (the sender of the message) with her private key 1 generates a sequence consisting of (+)- or (-)-symbols, which correspond to the in- stantaneous alignment of her “transmitter”, e.g., an optical polarizer. The instantaneous polarization can be horizontally/vertically (0 ◦ /90 ◦ ,ˆ= ˆ S z )or left/right diagonally (135 ◦ /45 ◦ ,ˆ= ˆ S x ). Next she publicly sends to Bob (the receiver) – for their mutual prepara- tion of the encoding of the message to follow – a data set consisting of a long random sequence of (0)- or (1)-bits, a test message, where the bits of the (0/1)-sequence are closely correlated to the (+/-)-sequence according to the following rules. The test message contains at the position n either a “1”, e.g., for vertical polarization, P =90 ◦ , and for right-diagonal polarization, P =45 ◦ ;ora“0” for horizontal polarization and for left-diagonal polarization, P =0 ◦ and P = 135 ◦ . Thirdly: Bob, using the same rules, receives the test message with another (+/-) analyzer sequence, taken from his own private key (which is not known by any other person, not even by Alice!). Fourthly: Thereafter Bob informs Alice publicly which sequence he actu- ally received, i.e. he sends to Alice the message “?” (e.g., an empty bit), if his analyzer was in the wrong polarization (e.g., if Alice transmitted a 0 ◦ -signal at the n-th place, whereas Bob’s analyzer was (i) positioned to 90 ◦ ,sothat no signal was received; or (ii) positioned to 135 ◦ or 45 ◦ , so that the meaning was not clear, since the incoming signal had equally strong components for 1 and 0). In the remaining cases he sends to Alice either a “1” or a “0”, according to what he received. Fifthly: Alice finally compares this message from Bob with her own test message and fixes the numbers n 1 , n 2 , n 3 , etc., where both agree. She informs Bob publicly about these numbers, and that her message, which she will send next, will be completely contained in these “sensible bits”andshouldbe interpreted according to the known rules, whereas the remaining bits can be skipped. Thus it is publicly known, which are the “sensible bits”, and that at these bits Bob’s analyzer and Alice’s polarizer have the same polarization, but 1 if existent; otherwise the sequence is generated ad hoc. 298 38 Appendix: “Mutual Preparation Algorithm” for Quantum Cryptography whether this is the horizontal/vertical polarization or, instead, the left/right- diagonal polarization is not known to anyone, except to Alice and Bob, inspite of the fact that the communications, including the rules, were completely public. Although this procedure is somewhat laborious, the final result is simple, as stated above (see Sect. 36.6). . between classical and quantum mechanics. Classical mechanics of an N-particle system takes place in a 6N-dimen- sional phase space of coordinates plus momenta. Observables are arbitrary real functions. Einstein, Podolski and Rosen’s quasi-classical reasoning one would always tend to as- sociate orthogonal states to alternative polarizations ( ˆ S z alternatively to S x ) Thus far we have described. are popular, but essentially inadequate). Instead it may be better to admit that the reasons for the behavior are truly complicated. In any case, quantum mechanics is a discipline which contains