B Tech Project
Quantum Composite-Key Lock
By Chetan Gangwar 200101170
Dhirubhai Ambani Institute of Information & Communication Technology Gandhinagar, Gujarat April 2005
Quantum Composite Key Lock
Dhirubhai Ambani Institute of Information & Communication Technology Gandhinagar
CERTIFICATE This is to certify that the Project Report titled “Quantum Composite-Key Lock” submitted by Chetan Gangwar ID 200101170, for the partial fulfillment of the requirements of B Tech (ICT) degree, embodies the
work
done
by
him
off-campus
at
Physical
Research Laboratory, Ahmedabad under Prof.Prasanta K. Panigrahi along with guidance of the undersigned.
Date:________
Signature:_____________ (Prof. V.P.Sinha)
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Abstract
Quantum computation has in certain impressive ways exploded upon us during the last decade. This comes more than eight decades after the establishment by Max Planck in 1900 of Quantum Mechanics, the theory upon which quantum computation is based, something fundamentally different to classical Boolean logic. This difference leads to a greater efficiency of quantum computation over its classical counterpart. In this report, we explain the basic principles of quantum computation, including the construction of basic gates, and networks. We illustrate the power of quantum algorithms using the simple problem of Deutsch, and simple communication problems like quantum teleportation. We also present some basics of quantum error correction algorithms. Finally, we investigate a certain problem on Quantum Secret Multiparty-Sharing and propose a partial solution to it. We also explain the main obstacles in obtaining a complete solution to the same. This report aims to provide the necessary insights for an understanding of the field so that various non-experts can judge its fundamental and practical importance.
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Table of Contents 1. Introduction and overview to quantum computation............................................... 1.1.
5
From classical to quantum computers..........................................................
5
1.2. Fundamentals of quantum computation....................................................... 1.2.1. Single qubit and superposition.................................................................. 1.2.2. Bloch sphere representation...................................................................... 1.2.3. Multiple qubits and measurement............................................................. 1.2.4. Quantum gates and circuits....................................................................... 1.2.5. Bell states and entanglement.....................................................................
6 6 7 7 8 8
2. Illustrated algorithms............................................................................................... 2.1. Quantum teleportation.................................................................................. 2.2. Deutsch’s algorithm..................................................................................... 2.3. Deutsch-Jozsa algorithm............................................................................. 2.4. BB84 algorithm............................................................................................ 2.5. Quantum Fourier Transform........................................................................
9 9 10 11 12 13
3. Discussion................................................................................................................ 3.1. Basics of quantum cryptography.................................................................. 3.2. Quantum key distribution ............................................................................ 3.3. Quantum secret sharing basics..................................................................... 3.4. Secret locking problem and its variations. .................................................. 3.5. Proposed solutions using teleportation. ....................................................... 3.6. Quantum error correction............................................................................. 3.7. Persisting problems......................................................................................
13 14 15 15 17 17 19 20
4. Conclusion and Future directions............................................................................
21
5. Acknowledgements.................................................................................................
21
6. References and notes..............................................................................................
22
7. Appendix A1 : Quantum mechanics to quantum computation
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1. Introduction and overview to quantum computation Humans always try and discover new ways of exploiting various physical resources such as materials, forces and energies from nature. During previous century information was added to the list when the invention of computers allowed processing of information outside human brains. The evolutional phase of computer technology has involved a sequence of changes from one type of physical realisation of information to another i.e., from gears to relays to valves to transistors to integrated circuits and so on. Today's advanced lithographic techniques can squeeze fraction of micron wide logic gates and wires onto the surface of silicon chips. Soon they will yield even smaller parts and inevitably reach a point where logic gates are so small that they are made out of only a handful of atoms. On the atomic scale, matter obeys the rules of quantum mechanics, which are quite different from the classical rules that determine the properties of conventional logic gates. So if computers are to become smaller in the future, new quantum-computing technology must replace or supplement what we have now. The point is, however, that Quantum Computation can offer much more information processing power than its classical counterpart. It can bestow with qualitatively new algorithms based on quantum principles! For the readers of this report, we assume some basic knowledge about the way usual electronic digital computers process information represented by classical binary bits. Also some familiarity is assumed with Linear Algebra, and in particular, with real or complex vector spaces, their linear mappings between such spaces, the representation of such mappings by matrices, the eigenvectors and eigenvalues of such mappings or matrices, as well as the diagonalization of special classes of such mappings or matrices. Certain minimal knowledge on tensor products of vector spaces, as well as on complexity of computation will be required. This report has three major sections. The first section includes the basic fundamentals of quantum computation and its meaning from the point of view of quantum mechanics. Second section includes description of few basic and essential algorithms which are necessary for understanding of different phenomenon in quantum computation. Finally, the third section includes the work done on the new scheme of quantum composite-key lock and few quantum secret-sharing, error-correction basics to understand the scheme proposed.
1.1 From classical to quantum computers A classical computer uses bits 0 and 1 (usually implemented through voltages) to carry out all the computational processes. A number of problems are difficult to solve in a reasonable time frame in a classical computer, for example factorization of a big number is one such problem. The need to carry out different operations simultaneously so as to speed up the computational process has led to parallel processing. Quantum mechanics is based on superposition principle(see section 1.2.1), which intrinsically leads to enormous parallelization. In a significant development P.Shor showed in 1994 that this can be utilized to factorize a number in polynomial time. Since a number of cryptographic algorithms are based on the B.Tech Project Report Chetan Gangwar (200101170)
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Quantum Composite Key Lock difficulty of factorization into prime numbers on classical computers. Alarm bells started ringing in the meantime; secure transmission of information was shown to be possible through quantum algorithms. These developments have led to enormous activities in both theoretical and experimental fronts to realize a quantum computer. Quantum dots, Josephson junction, Bose-Einstein condensation of laser light can provide ingredients to build a quantum computer. Quantum computation is the level of abstraction above quantum mechanics.To get a brief overview of quantum mechanics and how it gives rise to the quantum computation see Appendix A1.
1.2 Fundamentals of quantum computation 1.2.1 Single qubit and superposition A single bit is represented by 0 or 1, the quantum analog of it is a QUBIT. It is represented with |0 and |1, called the Dirac notation. A qubit can take the states even other than |0 and |1. It is also possible to form “ linear combination” of states, often called “a superposition state” |ψ = α | 0 + β | 1 , where α and β are complex. The state of a qubit is a vector in a 2-d complex vector space. |0 and |1 are known as the “computational basis” and form an orthonormal bases for this vector space. We cannot measure the qubit, we can only get the probability of |0 or |1 states as |α|2 or | β |2 respectively. Of course as the total probability is 1, |α|2 + | β |2 = 1. Qubit’s state is normalized to length 1; i.e. in general qubit’s state is a unit vector in a 2-dimension complex vector space. Qubit can exist in a continuum of states between | 0 and | 1 - until it is observed. A qubit can be realized by i. Polarization of photon, ii. Alignment of nuclear spin in uniform magnetic field, iii. 2 states of electron orbiting a single atom.
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Quantum Composite Key Lock 1.2.2
Bloch Sphere Representation- There are infinite points on the surface of the |0 z
θ φ
x
|ψ y
unit sphere, so in principle infinite memory is achieved by an infinite binomial expansion of θ. Any qubit that can be represented on the surface of the sphere is said to be in pure state, meaning that the net probability of finding 0 or 1 is 1. And a qubit is said to be in impure state if it suffers decoherence and such a state is can be represented by points inside the sphere only.
|1 Fig. 1.2.2.1. Bloch sphere
1.2.3 Multiple Qubits and Measurement If we have 2 bits, we have 4 possible states or precisely to say we have 4 computational basis states denoted by |00 , |01 , |10 and |11 . A pair of qubits can exist in superposition of these 4 states. | ψ = α00| 00 + α01| 01 + α10| 10 + α11| 11 The measurement result x ( = 00 , 01 , 10 , 11) occurs with probability |αx|2 ,with the state of qubit after the measurement collapsed to | x . By normalization condition, sum of the probabilities = 1, i.e. Σx є { 0,1 }2| αx|2 = 1 Measuring the first qubit alone gives 0 with the probability | α00|2 + | α01|2 and post measurement state would be, | ψ = (α00 | 00 + α01 | 01 ) / ( | α00|2 + | α01|2 )1/2 Observe here that post-measurement, the state is re-normalized to maintain the sum of probability equal to 1.
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1.2.4 Quantum Gates and Circuits Quantum gates implement quantum circuits. And quantum gates are bound by certain rules which we may mathematically sum up saying that they should be representable by Unitary matrices, i.e. U†U = I. Where U† is adjoint of U i.e. transpose of its complex conjugate. U† is read as “U dagger”. Pauli has defined 3 very important unitary matrices, they are: I=
1 0 0 1
0 1 ,X= 1 0
,Y =
0 -i i 0
,Z=
1 0 0 -1
A quantum circuit is to be read from left to right. The progress of line can be understood as passage of time. Another important constraint on the quantum circuit is that, we cant fan-in or fan-out or loop back. In other words we cannot copy a qubit. This follows from the no-cloning theorem, which states that: “It’s impossible to make a copy of an unknown state”. This is an important difference between classical and quantum circuits. Table 1.2.4.1 Examples of Quantum gates Quantum Gate
Matrix 01 10
NOT gate
Hadamard gate
½
Input
Output
α|0+β|1
β|0+α|1
|0
(| 0 + | 1 ) / 2
|1
(| 0 - | 1 ) / 2
11 -1 1
1.2.5 Bell States and Entanglement Bell states are important 2-qubit states. They are also called EPR pairs. They are ( |00 | 11 )/2, ( |01 | 10 )/2 . Consider ( |00 + | 11 )/2 These Bell state have property that upon measuring the first qubit, one obtains 2 possible results: 0 with probability ½ , leaving post measurement state | φ’ = | 00 and 1 with a probability ½ leaving, | φ’ = | 11 . As a result, a measurement of the second qubit always gives the same result as the measurement of the 1st qubit. That is, the measurement outcomes are correlated. This correlation property observed in the quantum domain is called the entanglement. It’s an essential, controversial and unique property of quantum mechanics that used as a pillar for most of quantum information and quantum cryptography.
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2.
Illustrative algorithms— In this section we describe few very simple,
basic
though essential algorithms that form standard base of many quantum processes.
2.1 Quantum teleportation Quantum mechanics allows the phenomenon of quantum teleportation. An unknown quantum state (qubit) is destroyed at the transmiting place while a perfect replica appears at a receiving place.The problem statement is that there are two people A and B, who are far apart and can only communicate classically. A wants to transmit an unknown(some arbitrary state) qubit to B. It is given that initially they possess a one qubit of a shared entangled pair {00, 01, 10, 11}. Solution- A can do this by performing a simple quantum computation on his side, and communicating its result to B. B then performs the appropriate quantum computation on his side, after which his qubit assumes the state of the A’s qubit and the teleportation is achieved (for details see [2]). It should be noted that this experiment has been performed recently by a group in Innsbruck who achieved a successful teleportation of a singe qubit [3]. Again, since entangled states are nonexistent in classical physics this kind of protocol is impossible, leading to another advantage of quantum information transfer over its classical analogue. Following is circuit diagram of the quantum teleportation where, |
H
M M
00
X
|0
|1
|2
Z
| |3
Figure 2.1.1 Quantum Teleportation the unknown qubit | is represented by first line of the circuit. A’s has control of only first two particles (the top two lines in the figure) to which he does local unitary operations like C-Not and Hadamard (H) gate. Then A performs measurement (using blocks M) of his particles. Then A sends the classical result {00,01,10,11} through the classical channel to B, who accordingly applies unitary transformations on his particle (his half of the entangled pair), to retrieve the same unknown qubit |. The state of the system at various times (left to right) is given by | 0 = | (|00 + |11 ) = (1/√2) (α | 0 (| 00 + | 11 ) + β | 1 (| 00 + | 11 ) )
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Quantum Composite Key Lock
After the C-Not on first and second qubit (see figure 2.1.1) | 1 = (1/√2) (α | 0 (| 00 + | 11 ) + β | 1 (| 10 + | 01 ) ) After the Hadamard on first qubit (see figure 2.1.1) | 2 = (1/2) α (| 0 + | 1 ) (| 00 + | 11 ) + β (| 0 - | 1 ) (| 10 + | 01 ) By using the associativity of the tensor product, we further obtain | 2 = (1/2) [| 00 (α | 0 + β | 1 ) + | 01 (α | 1 + β | 0 ) + | 10 (α | 0 - β | 1 ) + | 11 (α | 1 - β | 0 ) ] The expression in the right hand side is quite useful. Its first term |00 (α |0 + β |1 ) has the two qubits of A in the state |00 and the single cubit of B in the state (α |0 + β |1) which is in fact | . Therefore, if A performs a measurement on her two qubits at obtains |00, then B will have obtained the desired |.Therefore, |3 = (1/2) (α | 0 + β | 1 ) = (1/2) | Finally, we should note that, during teleportation as performed above, both the original qubit | and the entangled EPR pair |β00, will in general be destroyed. In this way, teleportation has a price, and a nontrivial quantum one at that: One qubit teleported costs in general one entangled EPR pair! The experimental constraints and working shall be discussed in later sections.
2.2 Deutsch’s Algorithm As explained in the previous section (1.2) that superposition enables quantum parallelism, which can be understood very easily through Deutsch’s algorithm. To put it simply, quantum parallelism allows certain kind of simultaneous computations, thus saving computation time. Such a feature is used by Deutsch’s Algorithm to make simultaneous use (or comparison) of many results of a function in a single step, which is impossible in single classical computation. Lets start by a simple function f(x): {0,1} {0, 1}. It’s understood that the algorithm of this function is entirely independent of its inputs and outputs. Thus, f(0) can either be 0 or 1, and f(1) likewise can either be 0 or 1, giving altogether four possibilities. However, suppose that we are not interested in the particular values of the function at 0 and 1, but we need to know whether the function is: (i) constant, i.e. f(0) = f(1), or (ii) balanced, i.e. f(0) ≠ f(1). Now Deutsch poses the
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Quantum Composite Key Lock following task: by computing ‘f’ only once determine whether it is constant or balanced.[4]
Uf (x)
|
f()
Even though quantum parallelism produces multiple output values of any function in a single step, due to measurement constraints we are incapable of using any of the values individually for any purposes! For example, if | is a superposition of 8 states and after the unitary operator Uf(x), 8 output f() values are produced, any (1 or more) of which if measured will collapse into a single basis state. To make clever use of Quantum Parallelism is only to use some universal property of all the outputs like their {XOR, Product, and Sum etc.} could be measured. Deutsch’s Algorithm does exactly that by using modulo-2 (XOR) for binary function f(x), which we had defined previously, to determine whether its constant or balanced. Following is the circuit and expression of final state of the system for the function f(x) given the input state |0 |1
|0
H
x
x
H
Uf |1
H
y
|
yf(x)
The final state for this system using the binary function f(x) will be | = ± | f(0) f(1) (|0 - |1 ) / 2 As one can clearly see, that the first qubit contains an output of XOR (a universal property) of the two simultaneous output values. Measurement of the first qubit will decide the nature of the function (constant or balanced) in a single step. This kind of computation with classical computers would have taken not less than three instructions. The benefits in quantum computation using parallelism increase with the domain and range of the function used, as the Deutsch-Jozsa algorithm proves.
2.3 Deutsch-Jozsa algorithm The DJ algorithm is a massive extension of the quantum parallelism using the basic idea of Deutsch’s algorithm. For an arbitrary integer n ≥ 1, we shall define the nfold Walsh-Hadamard quantum gate Hn with n qubits input and n qubits output as B.Tech Project Report Chetan Gangwar (200101170)
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Quantum Composite Key Lock represented by this n-fold parallel device, and equivalent to n parallel Hadamard gates. n
Hn
n
Figure 2.3.1 Walsh Hadamard Gate This sort of n-qubit quantum gate when used (as done in Deutsch’s algorithm)for a binary function that gives either 0 or 1 as its output, the algorithm determines whether the function is constant(gives all same outputs,0/1) or balanced(equal number of 0s and 1s). The rest of the working of this algorithm is similar to the Deutsch’s algorithm. The detailed working of the algorithm is not as necessary as the basic idea (i.e., quantum parallelism)
2.4 BB84 Algorithm This algorithm is one of the fundamental ways to establish a secret key [5] between two parties, using the laws of quantum mechanics to protect it. In this protocol the sender (Alice) sends photons to the receiver (Bob) with random polarization from either the rectilinear basis: | H and | V , the former being a logic 0, the latter a logic 1, or from the diagonal basis: | R and | L , the first being a logic 0, the second a logic 1. Bob then measures, at random, in either of the bases. If he chooses the correct basis (which he does arbitrarily half the time) the results are perfectly correlated. If he chooses the wrong basis, then he gets the same bit as Alice with a 50 % chance. The raw key thus obtained has a 25 % error rate (without any assumed eavesdropping or external noise). Now Bob announces publicly (classically) for every bit in which basis he measured the polarization of the received photon (he does not tell the outcome) and Alice tells (classically) to Bob which measurements were done in the correct basis. In this way all the bits measured on a wrong basis are discarded, and Bob obtains a new ‘shorter’ correct key (called sifted key). In case of any eavesdropping by a third party, say Eve, who can intercept a photon traveling from Alice to Bob, and whenever she does that Bob does not receive any photon and the bit is discarded. This way, Eve does not gain any information; she only lowers the bit rate. What she could do to gain some information is measure the photon in either of the bases, like Bob does, and then sends a photon to Bob with the polarization she measured. Similar to Bob, she also arbitrarily chooses any basis, to measure the incoming bit. If she measures it using the right (same as Alice used to sent it) basis (which is 50% probable), she resends it to Bob and Bob does not realize any intervention of Eve. She also, necessarily, frequently chooses the wrong basis. Now, if Alice and Bob happen to use the same basis for a certain bit, say the rectilinear basis, and Eve uses the diagonal basis, then Alice and Bob would expect the same outcome. Since Eve has performed a measurement in the wrong basis projecting the polarization of the photon along one of the diagonal eigenstates, Alice and Bob could have different outcomes. Eve gains 50 % information on the key, but Alice and Bob obtained a sifted key, with an error rate of 25 %. In an ideal B.Tech Project Report Chetan Gangwar (200101170)
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Quantum Composite Key Lock situation, with no technical deficiencies, the key would contain no errors. A realistic error rate, using today’s technology, would be a few percent. Alice and Bob should compare a portion of their key to check whether or not they have been eavesdropped. If they decide they haven’t, they should discard only the portion used for comparison; otherwise, if they have noticed Eve’s intervention, they should discard the whole key and setup the distribution again.
2.5 Quantum Fourier Transform In this section we are going to explain the unitary transformation: the quantum Fourier transform. This is a discrete Fourier transform, not upon the data stored in the system state, but upon the state itself. Let’s look at the definition to make this a bit clearer. The discrete Fourier transform (DFM) of a discrete function f1, . . . , fN is given by N-1
Fk (1/N) e 2ijk / N fj j=0
and the inverse Fourier transform is N-1
fj (1/N) e -2ijk / N Fk k=0
In the quantum Fourier transform, we do a similar transformation, although the conventional notation for QFT is somewhat different. The QFT on an orthonormal basis |0,|1,........|N-1 is defined to be a linear operator with the following action on the basis states, N-1
| j ( 1/ N ) e
.
j=0
N-1
N-1
2ijk / N
|k
Equivalently, the action on an arbitrary state may be written
xj | j yk | k , where the amplitudes yk are the discrete Fourier transform of the amplitudes xj. It is not obvious from the definition, from this transformation is a unitary transformation, and thus can be implemented as the dynamics for a quantum computer.
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3. Discussion Till this section of the report, the general background necessary for quantum computation was explained. This section onwards, we will concentrate upon the precise direction of the work done. This might again require some more basics about quantum cryptography, quantum error correction, quantum secret-sharing schemes and their experimental possibilities. In the later section, we must also look into an idea of quantum secret sharing using quantum teleportation (using W-states) and the impediments in its realization.
3.1 Basics of quantum cryptography Cryptography is the science of designing systems that encrypt and decrypt data, one purpose of these systems is to protect (and successfully transmit) confidential data from unauthorized parties. Precisely, one wants to use a trapdoor like one-way function for encryption, which has the property that it is computationally hard to invert, unless some secret information, the trapdoor, is known. For example, factoring large integers and the discrete log problem are assumed to be computationally hard problems and form the basis of many cryptosystems. So if someone comes up with an algorithm that can factorize large numbers efficiently, a lot of cryptosystems would become worthless! Quantum paralellization can reduce these exponential problems into polynomial ones. Quantum also offers its own way of security. In quantum cryptography, the security of the cryptosystems is based on the laws of quantum mechanics, instead of mathematical assumptions. This enables us to design systems that are unconditionally secure, i.e. systems that are even secure in the presence of an eavesdropper with unlimited computational power. An essential and interesting consequence of quantum mechanics that makes quantum cryptography work is the no-cloning theorem[8], which states that one cannot make an exact copy of an unknown quantum state. This restricts an eavesdropper, who has managed to intercept the encrypted message, in his possibilities to attack, since the most obvious action, making a copy of the message and send it to the intended party, is not an option anymore. Furthermore, the fascinating phenomenon of quantum entanglement plays a key role in the field of quantum cryptography. As explained in previous sections, entanglement is the property of quantum mechanics, which states that the behavior of two quantum particles can be correlated, although they are spatially separated. Altogether, quantum mechanics could be able to destroy most known classical cryptosystems, but, on the other hand, also gives rise to a different, more powerful kind of systems, which makes quantum cryptography definitely worthwhile to consider. Nowadays the encrypting and decrypting algorithms are publicly available and secrecy depends only on the set of parameters used to encrypt or decrypt, otherwise known as the key. One such key is the Vernam key, after its inventor G. S. Vernam[9]. The simplest procedure is that owner possess a secret message m(like a B.Tech Project Report Chetan Gangwar (200101170)
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Quantum Composite Key Lock binary string) which he encrypts using the key k(another binary string), (say) does modulo-2 addition between them to obtain the cryptogram c = m k. On receiving the cryptogram, receiver has to decrypt by adding the shared key to the cryptogram, c k =m. The problem now lies in establishing a mutual key between sender and recipient. This introduces the new world of quantum key distribution, which C. H. Bennett and G. Brassard stepped into in 1984, by designing a way to establish a secret key between two parties, using the laws of quantum mechanics to protect it: the BB84 protocol [5], [See Section 2.4]. We shall be discussing details of quantum key distribution in [Section 3.2].
3.2 Quantum key distribution— Aim of QKD is to facilitate provably secure protocol by which private keys between two parties be created over a public channel. The requirement is that qubits should be communicated over a public channel with error rate lower than a certain threshold. There are two basic ideas in quantum key distribution — 1. No cloning theorem [8], i.e., Eve can’t clone (exactly replicate) Alice’s qubit. 2. In any attempt to distinguish between two non-orthogonal states, information gain is only possible at the expense of disturbance to the signal. Example- |, | are two non-orthogonal states which Eve is trying to distinguish using an ancillary prepared in a standard state |u. | | u | | v | | u | | v1 Eve expects | v and | v1 to be different to be able to distinguish between | and |. However, inner products are preserved under unitary transformations, therefore v | v1 | = u | u | v | v1 = u | u = 1 which implies v and v1 must be identical. Thus, distinguishing between | and | must inevitably disturb at least one of the states. Hence, quantum key distribution is the first step of unconditional security, which is not due computational constraints, but because quantum mechanical principles guarantee ultimate secrecy.
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3.3 Quantum secret sharing schemes The problem that I have addressed in this report is closest to broad field of secret sharing, so this section will introduce the existing concepts and work in this direction. A simple problem for a classical secret sharing scheme [10-11] is like, suppose Alice and Bob wish to set up a joint checking account: neither of them can access it alone, but together, they can withdraw money. One way to do this is to require a secret password, for instance 011 100 001 010, to use the account. Neither Alice nor Bob alone has this password. Instead, Alice has a string of random numbers 101 010 010 001, and Bob has the bitwise XOR of the key and Alice's bit string: Key Alice Bob
011 100 001 010 101 010 010 001 110 110 011 011
Neither Alice nor Bob has any information about the key, but by putting their two codewords together, they can completely recover the key and access the bank account. Much more elaborate secret sharing schemes exist. For instance, it is possible to share a secret among 3 people so that any 2 of them can reconstruct it, but any single person has no information about it. In fact, it is possible to share a secret among n people so that any k of them can reconstruct it, but any k-1 of them have no information about the secret, for any n and k. This is known as a (k, n) threshold scheme[12]. In addition, secret sharing is a necessary component for performing secure distributed computations among a number of people who do not completely trust each other. Subsequently, extending the above idea to the quantum case, Cleve, Gottesman and Lo [16] proposed a quantum threshold scheme (QTS) as a method to split up an unknown secret quantum state | S into n pieces (shares) with the restriction that k > n/2 [17]. Quantum secret sharing protocols [18, 19, 20, 21] can accomplish distributing information securely where multi-photon entanglement (by using three-particle and four-particle entangled GHZ states) and procedure very similar to quantum teleportation [30] is employed. Recently, many kinds quantum secret sharing with entanglement have been proposed [15, 22, 23, 24]. Lance et al. have reported an experimental demonstration of a (2,3) threshold quantum secret sharing scheme [25]. The combination of quantum key distribution (QKD) and classical sharing protocol can realize secret sharing safely. Quantum secret sharing protocol provides for secure secret sharing by enabling one to determine whether an eavesdropper has been active during the secret sharing procedure. But it is not easy to implement such multi-party secret sharing tasks [18, 15], since the efficiency of preparing even tripartite or four-partite entangled states B.Tech Project Report Chetan Gangwar (200101170)
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Quantum Composite Key Lock is very low [26, 27], at the same time the efficiency of the existing quantum secret sharing protocols using quantum entanglement can only approach 50%. Recently, a protocol for quantum secret sharing without entanglement has been proposed by Guo and Guo [28]. They present an idea to directly encode the qubit of quantum key distribution and accomplish one splitting a message into many parts to achieve multiparty secret sharing only by product states. The theoretical efficiency is doubled to approach 100%. More recently, generalized quantum secret sharing schemes have been given by Singh and Srikanth[29] which shows inflation, compression and twin-thresholding in the shares of the players.
3.4 Secret locking problem and its variations. Sharing, locking, hiding and sealing some secret data are a series of cryptographical tasks that are very useful in today’s ordinary life, with many economic, industrial and even military purposes. This report addresses the following problem. Suppose that a boss Alice who has a quantum secret (a superposition state) in a locker, whose access she wishes to give to her two employees, Bob and Charlie, because she is usually away. Though she wishes that none of them individually could open the locker in her absence unless they act in concert. She also desires that whenever both Bob and Charlie, open the locker, she should get the secret qubit irrespective of her location. The above situation of composite key locking can be also be used to maintain a joint bank account, or access on a vault, or an industrial secret or to launch a missile with a nuclear warhead. This simply means that for security to be breached, two people must act in concert, thereby making it more difficult for any single person who wants to gain illegal access to the secret information.
3.5 Proposed solutions using teleportation and W-states. The basic idea of composite key locking is, two parties mutually own a secret, which is in a lock, controlled by an authority. The secret can be recovered if and only if, both parties act in concert. Here we will use entangled states to for secret hiding. Since the authority (like Alice, in above example) should be final destination of the secret, we will basically use a teleportation-like setup to transmit the secret. Recent works using controlled teleportation [31] for quantum secret sharing have been presented which use 3 particle GHZ states. In this report, a new quantum secret locking scheme is proposed, which partially accomplishes the pre-mentioned goal using 3-particle W-states [32]. Contrary to GHZ state, this kind of entanglement in the W-state has the highest degree of endurance against photon loss, which makes this scheme more robust for implementation. The core of the scheme could be visualized in the figure 3.5.1. This scheme uses an arbitrary binary string as a password (consisting of 0’s and 1’s) for each party and authority for their authentication. This length of the password will be a function of B.Tech Project Report Chetan Gangwar (200101170)
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Quantum Composite Key Lock number of people participating in the protocol. Suppose the Bob and Charlie, along with the authority, when in concert to reveal their password, an entangled state is prepared using Alice’s, Bob’s and Charlie’s password. For example, if the passwords of Alice, Bob and Charlie are x, y and z where x, y, z{0,1} then the entangled state formed (using method of spontaneous parametric downconversion) should be of the form 1/3 ( | xyz + | xyz + | xyz ) , where x means NOT(x). Evidently, for all the possible sets of values of x, y, and z the latter expression may not produce any entangled state, so any wrong password (from any one or more parties) may not work at all. Though there would be some particular constraints, on the passwords the parties can choose, which will become less strict when the scheme is generalized for more number of qubits. Once this entangled state is prepared, it is fed into the device having the (shown in Fig.3.5.1) quantum circuit, which contains the secret qubit locked inside. |system = 1/3 |secret ( | xyz + | xyz + | xyz ) where |secret is the secret superposition state like |secret = α | 0 + β | 1 .
α|0+β|1
H
Bob: x Charlie: y
M M M
Secure Qubit Retrieved
Alice: z Entangled State
|secret = α | 0 + β | 1
|1system
|2system
|3system
Figure 3.5.1 Quantum Composite Key Lock In the figure above, symbol “H” stands for Hadamard Gate, “M” for measurement of the channel on the basis 0 & 1(horizontal & vertical), “U”s stand for (different) unitary transforms necessary to project exact |secret on to the channel. Also, individual passwords are represented by x, y, z {0,1}. Double lines in the figure represent wires that can carry classical information. In the above setup, a secret state is contained inside and a 3-qubit entangled state is interacted with it. After the initial state of the system i.e., |1system, two C-Not gates and a Hadamard gates are applied, as shown in the figure. After that a measurement is performed on the top three channels. Just before and after the measurement states are given as |2system and |3system respectively. B.Tech Project Report Chetan Gangwar (200101170)
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Quantum Composite Key Lock |2system = [ |0x (yx) (|z + |z) + |0x(yx) (|z ) + |0x(yx) (|z + |z) + |1x (yx) (|z> - |z) + |1x(yx) (|z )+ |1x(yx) (|z> - |z ) - |1x (yx) ( |z ) ] This term is a superposition of many terms out of which only first term, contains the secret state(in bold terms), which could be retrieved if and only if the right values of x,y, and z were revealed by all the parties. ( denotes XOR function) Hence, after the measurement, top three channels are destroyed and state of system becomes the state of the fourth channel(which belongs to the authority, Alice) i.e.,
|3system = |secret = α | 0 + β | 1 . Generalization of this scheme for n parties (or 2 players having n/2-bit password) can be shown in a similar manner, which will thereby reduce the possibility of guessing other party’s password. In a 2-player scenario for n qubits, Alice can also give preferential treatment to Bob and Charlie by assigning unequal portion out of n qubits, which could make it easier for the preferred candidate to break the system in times of emergency. To add further complexity and thereby making the proposed scheme more secure against guessing or hacking threats, is using arbitrary orthogonal superposition states like | and | instead of basis states of | 0 and | 1 where | i = cosi | 0 + sini | 1 | i = - sin i | 0 + cosi | 1 where i would be some arbitrary angle ( [0,2) ) for the ith set of people. Assigning individual passwords from the set of superposition states { | i , | i }, the entangled state must be produced of the form 1/3 ( | i i i + | i ii + | ii i ) which is to be fed along with the secret qubit |secret = α | 0 + β | 1 into the composite-key lock circuit. Obviously, the C-Not gates for the superposition states will work similarly as these states are orthogonal to each other and are rotations of | 0 and | 1 . The fourth particle can be physically anywhere, irrespective of the first three (above it) which can give liberty for the boss (Alice) to be anywhere and utilize the quantum lock, thereby retrieving the quantum secret using just a classical channel.The setup also gives the freedom for the situation in which one of the parties could be authority themselves.
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Quantum Composite Key Lock This setup works only using W-states, which again is quite an interesting property of this quantum lock. Wootters, in his paper has worked length of entangled chains [36], i.e., linear arrangement of bell-states like links in a chain. These chains have links analogous to qubits pairs like (|00 - |11)/2. As we have already mentioned, the above solution works using only W-states, hence for n-qubit generalization of this scheme an entangled chain of W-states can also be prepared for using the quantum lock. This W-chain would be like .. (|001 + | 010 + | 001)(|001 + | 010 + | 001)(|001 + | 010 + | 001).. and could be fed into the circuit as a linearly entangled state. 3.6 Quantum error correction codes (QECC) This section aims at revealing the connection between secrets sharing schemes, to which we are already acquainted with in previous sections, and quantum error correction codes (because of their real life importance). Quantum computers are likely to be highly susceptible to errors from a variety of sources, much more so than classical computers. Therefore, the study of quantum error correction is vital not only to the task of quantum communications but also to building functional quantum computers. In addition, quantum error correction has many applications to quantum cryptography, like there is a strong connection between quantum errorcorrecting codes and secret sharing schemes [34]. Many quantum key distribution schemes also rely on ideas from quantum error-correction for their proofs of security. Classically, a (n, d)-secret sharing scheme splits a secret into n pieces so that no (d1) shares reveal any information about the secret, but any d shares allow one to reconstruct it. Such a scheme is already an error-correcting code, since it allows one to correct up to n - d erasures. Error-correcting codes need not be secret sharing schemes: a repetition code, for example, provides no secrecy at all. In the quantum world, the connection is much tighter. Cleve et al.[34] observed that any (perfect) QECC correcting t erasures is itself a secret sharing scheme, in that no t components of the code reveal any information about the message. This follows from the principle that information implies disturbance. Furthermore, most known (perfect) classical secret sharing schemes (and “ramp” schemes) can be directly transformed into (perfect) QECC’s with the related parameters [35], for example, any error detection/correction scheme can detect/correct changes by the environment or an eavesdropper in a similar fashion.
3.7 Persisting problems The existing problems/disadvantages in the proposed scheme are the following. In the experimental realization, each time this scheme is used, it necessitates production of an entangled state and the secret qubit (within the lock). It also needs measurement of an entangled state. These two are the most challenging tasks for any experimental realization. There exist no experimentally realizable procedures to B.Tech Project Report Chetan Gangwar (200101170)
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Quantum Composite Key Lock identify even two qubit-entangled states (i.e., all the 4 Bell states). The fidelity of experimentally determined teleportation for an EPR pair has been achieved as F=0.58. (Note that the fidelity is an average over all input states and so measures the ability to transfer an arbitrary, unknown superposition from A to B). Besides the experimental problems, even the theoretical scheme does not scale up exponentially with increase in number of players, which makes the proposed solution partial enough. Probably, in this scheme with some different kinds of entanglement, one could achieve reliability exponentially close to one.
4. Conclusion and Future directions This report has aimed at providing a basic overall understanding of Quantum Computation and Information, hence covered the basics, few essential algorithms and in the later sections quantum cryptography and secret sharing scheme concepts. Using this description, a new scheme of Quantum Lock was proposed and its merits-demerits followed. This report also tried to reveal the vital connection between quantum cryptography and quantum error correction, which helps in physical realization of quantum computers. The persisting problems with the proposed scheme could be the future directions on which the work could be improved upon.
5. Acknowledgments This entire work had been carried out under the joint supervision of Prof. P.K. Panigrahi and Prof. V.P.Sinha, to whom I would like to cordially thank for their helpful and stimulating discussions. Also thanks to Prof. Pashupati for his enlightening directions in my work. It was a nice research experience on the whole. Furthermore, thanks to my colleagues Pallavi, Shailaja and colleagues from PRL, for the nice lunch breaks and discussions during them. I would also like to thank PRL, Ahmedabad and DA-IICT for their support and cooperation during the process.
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Quantum Composite Key Lock
6. References and Notes: 1. A quantum state(like | ), describing all we know about a system, can be thought of as a vector in some abstract space called Hilbert space. A vector space S with a valid inner product defined on it is called an inner product space, which is also a normed linear space. A Hilbert space is an inner product space that is complete with respect to the norm defined using the inner product. The dimensionality of the Hilbert space of K spin-1/2 particle-system grows exponentially with k. In some very real sense quantum computations make use of this enormous size latent in even the smallest systems. 2. C. H. Benett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres and W. K. Wootters, Phys. Rev. Lett. 70, 1895 (1993). 3. D. Bouwmeester, J. W. Pan, K. Mattle, M. Eibl, H. Weinfurter, A. Zeilinger, Nature 390, 575-579 (1997). 4. D. Deutsch and R. Jozsa, Proc. R. Soc. Lond. A 439, 553 (1992). D.S. Simon, Proceedings of the 35th Annual Symposium on the Foundations of Computer Science, edited by S. Goldwasser (IEEE Computer Society Press, Los Alamitos, CA), 16 (1994). These kinds of problems are known as promise algorithms because one property out of a certain number of properties is initially promised to hold, and our task is to determine computationally which one holds. 5. Bennett C. H. and G. Brassard, in Proceedings of the International Conference on Computers, Systems & Signal Processing, Bangalore, India, p. 175(1984) Many more protocols for quantum key distribution exist, many are based on the concepts of the BB84 protocol. Some key distribution schemes do not use the photon’s polarization, but its phase. Another variation uses entangled photons[6]. 6. Ekert, A. K., Quantum cryptography based on Bell’s theorem Phys. Rev. Lett. 67, 661-663 (1991) 7. Michael A. Nielsen and Isaac L. Chuang, Quantum Computation and Quantum Information (Cambridge, UK, 2000) 8. W.K. Wootters and W.H. Zurek: A single quantum cannot be cloned, Nature 299 (1982) 802 9. Vernam, G. S., Cipher printing telegraph systems for secret wire and radio telegraphic communications J. Am. Inst. Electr. Eng. 45, 109-115 (1926) 10. G.R. Blakeley, in Proc. AFIPS 1979 NCC, pp.317-317; A.Shamir, Commun. ACM 22, 612(1979). 11. B. Schneier, Applied Cryptography, Wiley, New York, 1996. See also J. Gruska, Foundations of Computing Thomson Computer Press, London, 1997. 12. A particularly symmetric variety of secret splitting (sharing) is called a threshold scheme: in a (k, n) classical threshold scheme (CTS), the secret is split up into n pieces (shares), of which any k shares form a set authorized to reconstruct the secret, while any set of k-1 shares or fewer shares has no information about the secret. Blakely[13] and Shamir[10] showed that CTS’s exist for all values of k and n with n k. Hillery et al. [14] and Karlsson et al. [15] proposed methods for implementing CTSs that use quantum information to transmit shares securely in the presence of eavesdroppers. B.Tech Project Report Chetan Gangwar (200101170)
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Quantum Composite Key Lock 13. G. R.Blakely, Proc. AFIPS 48, 313(1979). 14. M. Hillery, V. Bu¡zek and A. Berthiaume, Phys. Rev. A59, 1829 (1999). 15. A. Karlsson, M. Koashi and N. Imoto, Phys. Rev. A59, 162 (1999). 16. R. Cleve, D. Gottesman and H.-K. Lo, Phys. Rev. Lett. 83, 648 (1999). 17. For if this inequality were violated, two disjoint sets of players can reconstruct the secret, in the violation of no-cloning theorem[8]. 18. M. Hillery, V. Bu¡zek, and A. Berthiaume, Phys. Rev. A 59, 1829 (1999). 19. W. Tittel, H. Zbinden, and N. Gisin, Phys. Rev. A 63,042301 (2001). 20. D. Gottesman, Phys. Rev. A 61, 042311 (2000). 21. A. C. A. Nascimento, J. M. Quade, and H. Imai, Phys.Rev. A 64, 042311 (2001). 22. R. Cleve, D. Gottesman, and H. K. Lo, Phys. Rev. Lett.83, 648 (1999). 23. V. Karimipour, A. Bahraminasab, and S. Bagherinezhad, Phys. Rev. A 65, 042320. 24. S. Bagherinezhad and V. Karimipour, arXiv: quant-ph/0204124 25. A.M.Lance, T.Symul, W. P. Bowen, B. C. Sanders, and P. K. Lam, Phy. Rev. Lett. 92, 177903 (2004). 26. D. Bouwmeester, J.W. Pan, M. Daniell, H. Weinfurter,and A. Zeilinger, Phys. Rev. Lett. 82, 1345 (1999). 27. J. W. Pan,S. Gasparoni, G. Weihs, and A.Zeilinger, P.R Lett. 86,4435 (2001). 28. G. P. Guo and G. C. Guo, Phys. Lett. A 310, 247 (2003). 29. S.K. Singh and R. Srikanth, arXiv: quant-ph/0307200. 30. C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, Phys. Rev. Lett. 70, (1895-1993). For an experimental realization of quantum teleportation see D. Boumeester, J. W. Pan, K. Mattle, M. Eisl, H. Weinfurter, and A. Zeilinger, Nature, London, 360, 575 (1997); D. Boschi, S. Branca, F. De Martini, L. Hardy, and S. Popescu, Phys. Rev.Lett. 80, 1121 ~1998!. 31. Fu-Guo Deng, Chun-Yan Li, Yan-Song Li, Hong-Yu Zhou, and Yan Wang. arXiv:quant-ph/0501129 32. Three Photon W-State, Nikolai Kiesel, Mohamed Bourennane, Christian Kurtsiefer, Harald Weinfurter, D. Kaszlikowski, W. Laskowski, and Marek Zukowski 33. Li-Yi Hsu, Che-Ming Li, Phy.Rev. A 71,022321 (2005) 34. Richard Cleve, Daniel Gottesman, Hoi-Kwong Lo. How to share a quantum secret. Phys.Rev.Lett. 83, p.648–651, 1999. 1,3 35. A. Smith. Quantum secret sharing for general access structures. E-print quantph/0001087, 2000. 36. Entangled Chains, William K. Wootters, quant-ph/0001114.
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Quantum Composite Key Lock APPENDIX A1- Quantum “mechanics” to quantum “computation”. Quantum computation is the level of abstraction above quantum mechanics. Hence, before delving quantum computation we will slightly try and understand the reality below it. Quantum mechanics deals with both continuous variables like coordinate and momentum x and p, which can take values ranging from - to as different variable like spin of electron or polarization of light which can take finite number of discrete values. The wave function ψ representing a particle or a system contains all the information about the same, like ψ(r) as a function of coordinate space, ψ(p) in momentum space or the coefficients ci in a complete set expansion, can be thought of as the components of the vector along some basis vectors. The wave functions are vectors on the Hilbert Space[1], satisfying additional theorems like ψ = a ψ1 + b ψ2 (a and b are complex), which is nothing but the superposition principle. Superposition, counter-intuitively, means existence of a particle(like a photon etc.) simultaneously in two states. Though it can also be interpreted in a probabilistic sense where the particle could be in either of the two states with certain probability. Measurement of a physically observable quantity is done through an operator. For example, the energy of a system can be measured through the hamiltonian operator Ĥop: Ĥopψ = Eψ
(1)
The measurable quantities are like, momentum p, coordinates x, projection of spin along zdirection etc. These are hermitian operations giving real eigen values. If two operators don’t commute we cannot measure them simultaneously. To visualize this we represent the operators by matrices and any observation is mathematically diagonalization of the matrix. The eigen values are the possible results of measurement. The eigen are the possible result of the measurement. The eigen states are the corresponding wave functions. It is known that hermitian matrices have real eigen values and if two matrices do not commute we cannot diagonalize them simultaneously. As an illustration consider the electron spin, which can have two values (½ or - ½) along x, y or z direction, in which it can be measured. The corresponding operators(C2 C2) for measurement of electron spin are:
0 1 Sx = ½ 1 0
0 -i Sy = ½ i 0
1 0 Sz = ½ 0 -1
Since Sz is diagonal (see above), it is clear that, we should measured the spin along the zdirection. Its respective eigen vectors are
1 0
and
0 -1
Since from equation (1)
Sz . 1 0
= ½
1 0
Sz . 0 1
B.Tech Project Report Chetan Gangwar (200101170)
= -½
0 1
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Quantum Composite Key Lock Since Sx and S y don’t commute with Sz , so we cannot measure x and y comprising of the spin simultaneously with the z. Note that, S2 = Sx2 + Sy2 + Sz2 and it commutes with S x, S y, and Sz and hence can be measured along with the spin in one direction. It should also be noted that the Sx or Sy, both have two eigen values ½ and - ½ . The corresponding eigen states are
½
1 1
, ½
1 -1
1 1
½
and
, ½
1 -1
We observe that ,
½
1 1
= ½
1 0
+
0 1
,
implying that the eigen state along the x-direction is a linear superposition with equal weight of the eigen states of Sz . Suppose, we are provided with the above state and we decide to measure the spin values along z-direction — over large number of measurements, we may find ½ or – ½ , we will find exactly half the time spin up, and half the time as spin down (along z-direction). This probabilistic scenario is completely different from the classical intuition. In the above example, prior to the measurement, the state was a linear superposition and at the time of measurement, the wave function collapses, either to |0 or |1 state (the eigen basis vectors are here represented in dirac notation for the ease of notation).
|0 =
1 0
, |1 =
0 1
This is called collapse of the wave function. Subsequently, any measurements on spin along z-direction will produce only |0 or |1 state. To illustrate the counter intuitive nature of quantum mechanics, we observe that |0 itself can be written as a superposition of two states: |0 = ½
½
1 -1 +½
1 1
If one now measures the spin observable along x-direction again, one finds that, the spin projection along x-direction comes with equal probability. Namely, a large number of measurements will produce a ½ or – ½ result along the x-direction.
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