7. Some Modern Applications of Quantum Mechanics

Transcription

1 7. Some Modern Applications of Quantum Mechanics What is particularly curious about quantum theory is that there can be actual physical effects arising from what philosophers refer to as counterfactuals-that is, things that might have happened, although they did not in fact happen [Penrose, 2005, p. 240]. 7.1 Introduction The rather strange nature of quantum mechanics is not only a question of ontological importance but can also be applied to various technological situations as well. To that end EPR s contributions cannot be overestimated. Quantum cryptography, quantum computing, quantum dense coding, quantum currency, Elitzur-Vaidman bomb detection and the like are some of potential application areas that are being fast developed into separate major fields. Quantum cryptography is truly a single particle application. Other applications make use of EPR type entanglements and Bell type state measurements. The phenomena like quantum Zeno effect, quantum teleportation, and quantum erasing, have at present only experimental consequences but may in future have potential applications. Some of these are discussed in this chapter. 7.2 Quantum Computing Certain computational capabilities that cannot be even imagined by conventional methods may be made possible in the not so far away future. For instance, Shor quantum algorithm [Shor,1997] would make an unparalleled and unheard of quantum leap as far as factoring is concerned. As an illustration, say one wants to 104

2 factor a very large integer (say, having 250 digits) what the existing fastest super computers is estimated to take a time of the order of the age of the Universe ( 13.6 billion years!). But it would only be a matter of few seconds or at the maximum a few minutes for a quantum computer equipped with quantum polynomial algorithm for factoring, like that of Shor [Julian Brown,2001]. Similarly the Grover s new efficient quantum search algorithms also would speed up searching phenomenally [Grover,1996]. If there are N number of the possible keys, then Grover s quantum search algorithm can speed up the time from O(N) to O(N 1/2 ). That means searching could be speeded up millions fold for very large value of N. What is impossible at present day might quite possible in future due to the developments in the field of quantum computing and computational algorithms. The technical difficulty in producing a quantum computer is the problem of decoherence, which is a big huddle to be surmounted. Yet the prototype quantum computers were developed with limited functions. These have lot of experimental value. According to David Deutsch the working of Quantum computer would prove the existence of many worlds of QM (chapter-6) and hence provide proof for this interpretation. David Deutsch, a great advocate of MWI says that the speed with which a quantum computer would work could be considered as a good demostration of the existence of the myriads of worlds. Deutsch arguments goes as follows. There are approximately atoms in the universe. The parallesim of quatum computing would require ways of computing in some cases. Therefore he argues that the existence of extra resources means more parallel ways of computing than can be afforded by the only resources of our universe proving the existence of parallel universes [Julian Brown, 2001]. Of course, we feel that recourse to consciousness might very well explain the above cited problem. 7.3 Quantum Cryptography Though cryptology is not really new, only in 1970 s the research in cryptology, crypto-analysis and related areas has gained observable impetus. Cryptography has precious applications in information communication. Cryptology includes a) 105

3 Cryptography, the art of encrypting /sending a message and b) crypto analysis, the art of decrypting. In order to do this, an algorithm (also called a cryptosystem or cipher) is utilized for jumbling up a message with what is known as the key to produce a cryptogram indecipherable and, therefore, unintelligible to any unauthorized party. In order to have a secure cryptosystem, it should be impossible to unlock the cryptogram without the key. The whole exercise is to safeguard the crucial secrecy of the original information of the message from any intrusion, technically called eavesdropping. Confidentiality was the traditional application of cryptogram. But, it has wider objectives now such as digital signatures, authentication, non-repudiation and so on [R-08]. Apart from several standard aspects of the field of number theory, there are its modern computational features and applications to cryptography. Any breakthrough in mathematics or in the field of computer science would, indeed, render the conventional cryptography not only useless and even having disastrous consequences. The potential power of quantum computing, as mentioned in the earlier section, would render the classical cryptography almost superfluous. For example, Shor algorithm of factorizing in quantum computing would render the classical RSA (Ronald Rivest, Adi Shamir and Leonard Adleman) type cryptography that relies on the computational complexity of factorization almost ineffective [Nicolas Gisin et al, 2009]. The speed with which a number can be factorized becomes phenomenal in quantum computing. Therefore any algorithm that relies on factorization like RSA type would become too inadequate. One of the main problems of cryptography is that of the key distribution (here after KD). Both public and private KD are employed. In conventional classical cryptography the private KD of RSA type algorithms are employed. Modern conventional classic cryptography makes use of a trap door algorithm for its public key. The assumption is on the complexity of computation, that it is difficult to factor a very large number by present computers. But any break-through in mathematics or leap in computation would invalidate such an assumption. Even private key or symmetric key distribution can be jeopardized by the eavesdropper. 106

4 So, any cryptography scheme based on the assumption of complexity of algorithm has a very high degree chance of letting down. Private KD runs the risk of being intercepted which cannot simply be wished away. More importantly, one is never sure if someone had eavesdropped or not, be it public or private KD scheme of the conventional classic cryptography [Reid et.al,2009]. Additionally as stated above, quantum crypto-analysis poses extremely high degree of potential threat to the present classical encryption systems. The necessity of the situation becomes acute and urgent when one considers the possibility of quantum retro-crypto analysis (or retroactive decryption) [Nicolas Gisin et al, 2009]. The evil eve might copy the existing public keys and information and create a bank of it, for potential quantum retro-analysis of the future. Hence, even the present state-of-the-art cryptology methods are neither fool proof nor future proof. For the above reasons, change over to the quantum cryptography would be compelling. Quantum cryptography is supposed to be tamper proof, fool proof and future proof. This is one of the applications where quantum weirdness has a positive, potential application. The EPR protocol and its connection to Bell-type measurement are important here. Quantum cryptography is of immense importance and value in the context of fast scientific developments, especially in the field of quantum computing as mentioned above. Ironically quantum cryptography provides a solution for the same. A future resistant way of crypto schemes is what is aimed at by Quantum cryptography. Quantum Cryptology is quite different from the classical one. It makes use of the features of quantum physics rather than depending on some kind of specific mathematical algorithm, as a key feature of its security to develop an undefeatable cryptosystem, one that is completely secure and safe against eavesdropping without the knowledge of the concerned sender or receiver of the messages. That is, quantum cryptography offers an ideally secure encryption of the data and transfer. 107

5 Quantum cryptology is based on fundamental quantum character of the individual elementary particle, like the photon. The following properties are basic and unique to quantum systems: a) Linearity, b) Superposition, c) Unitarity of evolution, d) No-cloning theorem [Wooters and Zurek, 1982]. Whereas it is possible to distinguish between two orthogonal quantum states any attempt to distinguish between two non-orthogonal states would irrevocably destroy the original state. This is called the irreversibility of measurement w. To see this from a closer stand point, consider the following facts. A state may be represented by a vector in a complex Hilbert space. Then measurement on a quantum system can be thought of as nothing but a projection operator acting on this vector in Hilbert space. Thus the projected state loses track of its original state giving rise to irreversibility. The memory of the past state is erased from the history of the universe itself. Thus any information on the past of the system is lost for good - a property when suitably manipulated gives rise to the phenomenon of quantum erasing. The last property mentioned above, namely the impossibility of cloning an unknown quantum state (no-cloning theorem) is mainly exploited in the quantum cryptography which in fact is only a corollary of the other three conditions. An infringement on no-cloning theorem would be a breach of linearity, causality and unitarity, or all put together. If cloning of an unknown quantum state were possible, it would amount to superluminal transmission of information. This property makes the passive listening by Eve impossible. Any elementary quantum particles can be used in cryptography but the best suited ones are of course photons, since they are reliable information carriers in the optical fiber cables. The polarizations of a set of the photons sent from Alice (sender) to Bob (receiver) form the coded information of the key. The actual encrypted message is transmitted via a public classic channel. Any attempt to measure the states of the w This fact is related with the infamous measurement problem. What is considered a bane in one context can possibly be a boon in another. Yet demonstration of these applications would confirm that the quantum weirdness can give more impetus to the study from a totally different perspective other than the usual ontological point of view. 108

6 photons would invariably disturb the state and hence will be noticed by the parties concerned. The sender and the receiver would compare their measurements of the polarizations of the photons to check for any potential eavesdropping. The quantum key distribution cannot prevent eavesdropping but can surely be detected with 100% efficiency. The key can then be discarded. Quantum cryptology works according to different models, a famous one being due to Bennett and Brassard [Bennett, and Brassard,1984]. Detection of intrusion by eavesdropper is vital to quantum cryptography. It is a way to combine the relative convenience and ease of key exchange in public key cryptography (PKC) with the ultimate security of a one time pad. Heisenberg s Uncertainty Principle of QM plays a dominant role in actual practice here. Historically Quantum Cryptology was first proposed by Stephen Wiesner in early 1970 s introducing the concept of quantum conjugate coding. In 1990 Arthur Ekert developed a different approach to quantum cryptology that was based on the quantum correlation due to quantum entanglement [Ekert,1991]. Quantum cryptology has the ability to detect any interception of the key, whereas with classical ones the key security cannot be proved. Quantum Key Distribution (QKD) would mean secure communication though not guaranteed against jamming. It enables two parties to produce a shared random bit string known only to them, which can be used as a key to encrypt and decrypt messages. QKD systems are automatic with greater reliability factor. The Bennett and Brassard scheme of 1984 for quantum key distribution is known as BB84 scheme. It is a procedure utilizing a set of individual photons. BB84 solves the QKD problem but safe storage of the key is still a problem. Therefore Ekert proposed EPR-based protocol for key distribution [Bennett, Brassard and Ekert,1992]. The photons of EPR pair have undefined polarizations (chapter-5). These two properties together are mysterious yet it is the basis of many modern quantum applications. 109

7 Quantum cryptography is one of the useful modern applications of EPR and Bell states which assume now greater significance in several other technological contexts as well. What was started as pure philosophical debate between Einstein and Bohr several decades ago is likely to give efficient technological spin-offs in the modern times. Some noted physicists had considered the interpretational aspects only to be of academic interest with very little experimental content. Bell s theorem did change the whole situation so considerably. The bottom line is not to neglect the true understanding of science [R-05]. It will definitely pay off well, even as technological spin offs, quantum cryptology being a glaring example for it, not to talk about various other applications. 7.4 Quantum Teleportation Quantum teleportation is one of the modern applications that exploit the concept of Quantum entanglement. The experimental confirmation of teleportation [Zeilinger,2005a] is the proof of the quantum nature of entanglement and hence it becomes very important from the point of understanding of QM. To see how exactly it is done, let us suppose that Alice and Bob share a pair of photons prepared in an entangled Bell state of polarization. That is, each of them possesses one of the entangled photons. Also suppose that Alice has an additional photon in an unknown state of polarization, u. Alice now performs measuremts on both the photons in her possession. There are four possible random outcomes. Bob's photon will be transformed into any one of the four states, depending on the four possible outcomes of Alice's measurement: either the state u, or a state that is related to u in a definite way. This operation of Alice entangles the two photons of her and disentangles that of Bob, forcing it into a state u*. Once Alice communicates the outcome of her operation to Bob, Bob knows either that u* = u, or how to transform u* to u by a local operation. Now the photon in Alice s possession can be considered as teleported to Bob as the quantum state of a photon is transferred from Alice to Bob. This phenomenon clearly demonstrates the non-local quantum correlation. Here again, 110

8 the no-cloning prevents superluminal transmission as in quantum cryptography which is why it does not defy SRT. 7.5 Elitzur Vaidman Bomb-Tester Avshalom Elitzur and Lev Vaidman in 1993 proposed [Elitzur and Vaidman,1993] the following bomb testing problem which was constructed and tested with success by Anton Zeilinger, et al. in 1994 using Mach-Zehnder interferometer. This is referred to as interaction free measurement. The bomb-testing problem can be described as follows. Say, in a collection of bombs some are duds. The bombs can be detonated by a single photon. Dud bombs will not absorb the photon but good ones will absorb and explode. We can use the counterfactual phenomenon of QM to separate the usable bombs from the duds. If we try to test by detonating the usable ones, then it will destroy all the usable bombs. (The alternate version of the problem would be to detect a bomb without detonating at least some of them). A very sensitive mirror attached to the plunger activates the detonator when a photon that impinges on it is pushing the plunger. The plungers of the duds are stuck, so that they do not get pushed and therefore no detonation occurs. It means a dud one effectively reflects the photons. The fact that the photon did not actually hit the bomb's mirror is enough to know that the photon went through the other path (a "null" measurement). The experimental set-up consisting of Mach-Zehnder interferometer is arranged as shown in the figure 7.1. The light source is of very low intensity that it emits only single photon at a time. A photon reaching the beam splitter BS1 has equal chances of passing through or of getting reflected by it. Say, on path 1, a bomb is placed with the triggering mechanism by photon as described above. If the bomb is usable, then the photon is absorbed triggering the bomb. If the bomb is dud one, the photon will pass through unaffected. Let us consider the two cases separately in what follows. 111

9 Case 1 - The bomb is a dud one: The photon either gets reflected by the first beam splitter, BS1 and takes path-2 or after passing through BS1, is reflected by the mirror on the trigger of the bomb along path-1. The plunger will not get pushed as it is dud. The system is now like the basic Mach-Zehnder apparatus in which constructive interference occurs along the horizontal path along D1 and destructive along the vertical path towards D2. Therefore, the detector D1 will click, and the detector at D2 will not. Figure 7.1 Elitzur-Vaidman Bomb-Tester Case 2-The bomb is usable: As in the above case the two different possibilities are that the photon can take either path-1 or path-2 after encountering the beam splitter BS1. If it takes path-1 it surely gets reflected and by the plunger mirror and the bomb will definitely explode. Since the bomb acts like a detector, the wave function collapses and therefore cannot be in superposition. If the photon takes the upper route path-2 the bomb will not explode yet there will be no interference effect. The photon now either passes through the BS2 or is reflected. The photon must be in either of the detectors D1 or D2. Hence, on the whole, there are only three outcomes: a) The bomb explodes, b) The bomb does not explode and only detector D2 detects the 112

10 photon. In this case we are sure that the bomb is live though it has not exploded and the photon has not interacted with it, c) The bomb does not explode and only detector D1 detects photon. It is possible that the bomb is usable or that it is a dud. In the last case, the test has to be repeated to see if the bomb will explode or if D2 will click. Usually this is sufficient to recognize all of the dud ones. The tests will identify the one third of the usable bombs without detonating but detonate two thirds of the remaining good ones. Kwiat et al in 1996 devised a technique, using a series of polarizing devices to yield a rate arbitrarily close to one. Here the answer to the query what would happen is determined without the bomb going off. This provides an example of an experimental method to answer a counterfactual question. 7.6 Discussions While there are several potential applications that can directly exploit the quantum entanglement, quantum cryptography is the most potential one developed so far. The quantum teleportation proves that the quantum state identifies a particle (which has no other identity) as the particles cannot be associated with their history. Conveying the polarization state of photon is amounting to transmitting infinite amount of classical information as to specify an angle to arbitary acurracy. The outcome of Alice's operation can be communicated to Bob by two classical bits of information since it has only four possible outcomes with equal a priori probabilities, which is remarkable. Bob can now reconstruct the state u on the basis of just two bits of classical information. The entangled state is exploited as a quantum communication channel for the transfer of rest of the information. The no-cloning theorem is also sort of verified indirectly by these phenomena. The uniqueness of a quantum state and the almost infinite amount of classical information that it contains qualify these to be connected with the mental state. For example, when we look at an ordinary scene in life, it contains infinite classical bits of information. In a man s life such infinite amount of information can be stored in the memory. The instant pattern 113

11 recognition capability of the brain is the demostration of tremendous (parrallel) computation capability. Otherwise the resources of the brain in terms of number of nerve synapses or even the number of states that the atoms in it have would be pretty very much limited. There were attempts to explain this phenomenon by holographic theory of memory by the Magellan of brain science Karl Pribram. Some people may think that the proposal to explian the brain using basic quantum phenomena, though in the right direction, is far-fetched at the moment as is reflected in the following quote, The recent proposal by Penrose and Hameroff exceeds the domain of present-day quantum theory by far and is the most speculative example [Atmanspacher, Harald,2008]. But we do feel that the powerful quantum phenomena and the emerging revolutionanary ideas can really revolutionise this new field in due course of time. Elitzur-Vaidman bomb tester is another example that demonstrates a further strange aspect of QM. The laboratory version of it has already been successfully demonstrated. The interaction free measurement (IFM) emphasizes the counterfactual phenomena are important in QM. That is, in a quantum interaction the possible events and situations that could have happened but not really taken place are important. This is an important non-classical property that has direct bearing on the quantum reality. The omniscience of the elementary particles has now tangible effects and potential applications. A variation of this quantum property is made use of in quantum computing where parallelism is achieved by the counterfactual way in what is called computation without computation. When a photon interacts with the beam splitter its state is non-deterministically altered only to go into quantum superposition state until an observer interacts with it causing the state vector reduction forcing the photon to a deterministic state. We relate this phenomenon of counterfactual phenomenon and IFM with the Feynman s formalism mentioned in chapter-2. Equations (2.30) and (2.33) depict this feature (see also fig 2.1). The many histories take into account not the classical paths but the non-classical ones as well. The quantum particles can sniff the surroundings paths. The above facts demonstrate three facets of QM: 114

12 1) Non-locality, 2) Counter factuality and 3) Strange quantum reality. This formulation even considers amplitudes for speeds greater than the velocity of light as well as that for particle traversing backward in time as well. The triumphs of QED and Feynman graphs justify these counter intuitive, non-classical histories. 115