M C Jain Quantum Mechanics Pdf
Vannessa Rataj
This paper provides an overview of quantum key distribution targeted towards the computer science community. A brief description of the relevant principles from quantum mechanics is provided before surveying the most prominent quantum key distribution protocols present in the literature. In particular this paper describes the BB84 protocol and its many variants as well as Eckert's approach through quantum entanglement. A brief discussion of some of the issues arising in practical implementations are also presented including privacy amplification and the photon number splitting attack.
Classical cryptography can be divided into two major branches; secret or symmetric key cryptography and public key cryptography, which is also known as asymmetric cryptography. Secret key cryptography represents the most traditional form of cryptography in which two parties both encrypt and decrypt their messages using the same shared secret key. While some secret key schemes, such as one-time pads, are perfectly secure against an attacker with arbitrary computational power [Gisin02], they have the major practical disadvantage that before two parties can communicate securely they must somehow establish a secret key. In order to establish a secret key over an insecure channel, key distribution schemes basd on public key cryptography, such as Diffie-Hellman, are typically employed.
In contrast to secret key cryptography, a shared secret key does not need to be established prior to communication in public key cryptography. Instead each party has a private key, which remains secret, and a public key, which they may distribute freely. If one party, say Alice, wants to send a message to another party, Bob, she would encrypt her message with Bob's public key after which only Bob could decrypt the message using his private key. While there is no need for key exchange, the security of public key cryptography algorithms are currently all based on the unproven assumption of the difficulty of certain problems such as integer factorization or the discrete logarithm problem. This means that public key cryptography algorithms are potentially vulnerable to improvements in computational power or the discovery of efficient algorithms to solve their underlying problems. Indeed algorithms have already been proposed to perform both integer factorization and solve the discrete logarithm problem in polynomial time on a quantum computer [Shor97] [Bruss07].
While the advent of a feasible quantum computer would make current public key cryptosystems obsolete and threaten key distribution protocols such as Diffie-Hellman, some of the same principles that empower quantum computers also offer an unconditionally secure solution to the key distribution problem. Moreover, quantum mechanics also provides the ability to detect the presence of an eavesdropper who is attempting to learn the key, which is a new feature in the field of cryptography. Because the research community has been focused primarily on using quantum mechanics to enable secure key distribution, quantum cryptography and quantum key distribution (QKD) are generally synonymous in the literature.
The focus of this paper is to survey the most prominent quantum key distribution protocols and their security from the perspective a computer scientist and not that of a quantum physicist. In order to understand these protocols, however, we briefly describe the necessary principles from quantum mechanics. From these principles the protocols are divided into two categories; those based primarily on the Heisenberg Uncertainty Principle, and those utilizing quantum entanglement. While much of the recent research focus is on developing practical quantum cryptosystems [Bruss07], only a brief discussion of the practical security aspects of these protocols are included in an attempt to remain within the scope of the provided background on quantum mechanics.
The rest of the paper is structured as follows: Section 2 gives a lightweight background on the relevant principles of quantum mechanics. Section 3 describes the approach based on the Heisenberg Uncertainty Principle, in particular the BB84 protocol and its variants. Section 4 focuses on Eckert's approach using quantum entanglement for secure key distribution. Section 5 discusses the theoretical limitations of quantum cryptography and discusses security in light of practical issues. Finally, we conclude with a summary of the protocols and their security in section 6.
The basic model for QKD protocols involves two parties, referred to as Alice and Bob, wishing to exchange a key both with access to a classical public communication channel and a quantum communication channel. This is shown in figure 1. An eavesdropper, called Eve, is assumed to have access to both channels and no assumptions are made about the resources at her disposal. With this basic model established, we describe in layman's terms the necessary quantum principles needed to understand the QKD protocols.
As mentioned, the security of quantum cryptography rests on several principles from quantum physics. The most fundamental of these principles is the Heisenberg Uncertainty Principle (HUP) which states that in a quantum system only one property of a pair of conjugate properties can be known with certainty. Heisenberg, who was initially referring to the position and momentum of a particle, described how any conceivable measurement of a particle's position would disturbs its conjugate property, the momentum. It is therefore impossible to simultaneously know both properties with certainty. Quantum cryptography can leverage this principle but generally uses the polarization of photons on different bases as the conjugate properties in question. This is because photons can be exchanged over fiber optic links and are perhaps the most practical quantum systems for transmission between two parties wishing to perform key exchange.
One principle of quantum mechanics, the no cloning theorem, intuitively follows from Heisenberg's Uncertainty Principle. The no cloning theorem, published by Wooters, Zurek, and Dieks in 1982 stated that it is impossible to create identical copies of an arbitrary unknown quantum state [Bruss07] [Wooters82]. One could see that without the no cloning theorem, it would be possible to circumvent Heisenberg's uncertainty principle by creating multiple copies of a quantum state and measuring a different conjugate property on each copy. This would allow one to simultaneously know with certainty both conjugate properties of the original quantum particle which would violate HUP.
The other important principle on which QKD can be based is the principle of quantum entanglement. It is possible for two particles to become entangled such that when a particular property is measured in one particle, the opposite state will be observed on the entangled particle instantaneously. This is true regardless of the distance between the entangled particles. It is impossible, however, to predict prior to measurement what state will be observed thus it is not possible to communicate via entangled particles without discussing the observations over a classical channel. The process of communicating using entangled states, aided by a classical information channel, is known as quantum teleportation and is the basis of Eckert's protocol as will be described in Section 4 [Eckert91] .
This section covered the basic key distribution model employed in quantum cryptography. A short overview of the necessary principles from quantum mechanics were also included with an emphasis on the Heisenberg Uncertainty Principle and the principle of quantum entanglement. With this necessary background, the next section describes the QKD protocols based on the first of these key principles.
In 1984 Charles Bennett and Gilles Brassard published the first QKD protocol [BB84]. It was based on Heisenberg's Uncertainty Principle and is simply known as the BB84 protocol after the authors names and the year in which it was published. It is still one of the most prominent protocols and one could argue that all of the other HUP based protocols are essentially variants of the BB84 idea. The basic idea for all of these protocols then is that Alice can transmit a random secret key to Bob by sending a string of photons where the secret key's bits are encoded in the polarization of the photons. Heisenberg's Uncertainty Principle can be used to guarantee that an Eavesdropper cannot measure these photons and transmit them on to Bob without disturbing the photon's state in a detectable way thus revealing her presence.
Figure 2 shows how a bit can be encoded in the polarization state of a photon in BB84. We define a binary 0 as a polarization of 0 degrees in the rectilinear bases or 45 degrees in the diagonal bases [CKI-BB84] [Gisin02]. Similarly a binary 1 can be 90 degrees in the rectilinear bases or 135 in diagonal bases. Thus a bit can be represented by polarizing the photon in either one of two bases.
In the first phase, Alice will communicate to Bob over a quantum channel. Alice begins by choosing a random string of bits and for each bit, Alice will randomly choose a basis, rectilinear or diagonal, by which to encode the bit. She will transmit a photon for each bit with the corresponding polarization, as just described, to Bob. For every photon Bob receives, he will measure the photon's polarization by a randomly chosen basis. If, for a particular photon, Bob chose the same basis as Alice, then in principle, Bob should measure the same polarization and thus he can correctly infer the bit that Alice intended to send. If he chose the wrong basis, his result, and thus the bit he reads, will be random. In the second phase, Bob will notify Alice over any insecure channel what basis he used to measure each photon. Alice will report back to Bob whether he chose the correct basis for each photon. At this point Alice and Bob will discard the bits corresponding to the photons which Bob measured with a different basis. Provided no errors occurred or no one manipulated the photons, Bob and Alice should now both have an identical string of bits which is called a sifted key. The example below shows the bits Alice chose, the bases she encoded them in, the bases Bob used for measurement, and the resulting sifted key after Bob and Alice discarded their bits as just mentioned [Wiki-SIFT].
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