Download 7. Some Modern Applications of Quantum Mechanics

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
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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(N1/2). 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 1080 atoms in the universe. The parallesim of
quatum computing would require 10500 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)
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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.
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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.
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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
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original state. This is called the irreversibility of measurement . 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
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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.
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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.
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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,
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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.
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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
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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
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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:
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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.
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