Optical Communication with Invisible Photons

It has always been a self-evident and obvious feature of any kind of communication that there should be an exchange of objects like photons or electrons between the sender and the receiver to convey any information. In this chapter a protocol is presented in which information is transmitted between a sender and receiver with no particles in the transmission channel. The basic building block of this counterfactual communication protocol, the Mach–Zehnder interferometer, is discussed. The concept of interaction-free measurement is also introduced.

No-cloning Theorem and Quantum Copying

Heisenberg’s uncertainty relation and Bohr’s principle of complementarity form the foundations of quantum mechanics. If these are violated then the edifice of quantum mechanics can come crashing down. In this chapter, it is shown how cloning or perfect copying of a quantum state can potentially lead to a violation of these sacred principles. A no-cloning theorem is proven showing that the cloning of an arbitrary quantum state is not allowed. The foundation of quantum mechanics is therefore protected. It is also shown how quantum cloning can lead to superluminal communication. It is also discussed that, if making a perfect copy of a quantum state is forbidden, how best a copy of a state can be made.

The Schrödinger Equation

In this chapter, the Schrödinger equation is “derived” for particles that can be described by de Broglie waves. The Schrödinger equation is very different from the corresponding equation of motion in classical mechanics. In order to illustrate the fundamental differences between the two theories, one of the simplest problems of particle dynamics is solved in both Newtonian and quantum mechanics. This simple example also helps to show that quantum mechanics is the fundamental theory and classical mechanics is an approximation, a remarkably good approximation, when considering macroscopic objects. The solution of the Schrödinger equation is presented for a particle inside a box and the quantization condition is derived. The amazing possibility of quantum tunneling when a particle is incident on a barrier of height larger than the energy of the incident particle is also discussed. Finally the three-dimensional Schrödinger equation is solved for the hydrogen atom.

Quantum Computing II

This chapter deals with some of the most prominent successes of quantum computing. The most well-known quantum computing algorithm, Shor’s algorithm for factoring a number in its prime factors, is discussed in details. The key to Shor’s algorithm is the quantum Fourier transform that is explained with the help of simple examples. The role of quantum entanglement is also discussed. The next important quantum computing algorithm is Grover’s algorithm that helps in searching an item in an unsorted database. This algorithm is motivated by first discussing a quantum shell game in which a pea hidden under one of the four shells is found in one measurement with certainty each time. This amazing result is then generalized to an arbitrary number of objects and Grover’s algorithm.

Simplest Quantum Devices: Polarizers and Beam Splitters

Maxwell showed that light consists of electric and magnetic fields that oscillate in directions perpendicular to the direction of propagation. Associated with this picture of light as an electromagnetic wave is an important property—the polarization of light. The polarization of light is related to the direction of oscillation of the electric field in an electromagnetic wave. In this chapter, the basic principles of quantum mechanics are discussed by studying the polarization property of a single photon. First the properties of a polarizer are presented and Malus’ law for polarized light is derived. Next it is shown that the basic features of quantum mechanics can be understood via an analysis of a single photon passing through a polarizer. This simple system allows an introduction of Dirac’s ket and bra notations for a quantum state. Finally the transformation properties of the quantum beam splitter and the polarization beam splitters are discussed.

Fundamentals of Quantum Mechanics

The laws of quantum mechanics were formulated in the year 1925 through the work of Werner Heisenberg, followed by Max Born, Pascual Jordan, Paul Dirac, and Wolfgang Pauli. A separate but equivalent approach was independently developed by Erwin Schrödinger in early 1926. The laws governing quantum mechanics were highly mathematical and their aim was to explain many unresolved problems within the framework of a formal theory. The conceptual foundation emerged in the subsequent 2–3 years that indicated how radically different the new laws were from classical physics. In this chapter some of these salient features of quantum mechanics are discussed. The topics include the quantization of energy, wave–particle duality, the probabilistic nature of quantum mechanics, Heisenberg uncertainty relations, Bohr’s principle of complementarity, and quantum superposition and entanglement. This discussion should indicate how different and counterintuitive its fundamentals are from those of classical physics.

Quantum Computing I

A remarkable application of quantum mechanical concepts of coherent superposition and quantum entanglement is a quantum computer which can solve certain problems at speeds unbelievably faster than the conventional computer. In this chapter, the basic principles and the conditions for the implementation of the quantum computer are introduced and the limitations imposed by the probabilistic nature of quantum mechanics and the inevitable decoherence phenomenon are discussed. Next the basic building blocks, the quantum logic gates, are introduced. These include the Hadamard, the CNOT, and the quantum phase gates. After these preliminaries, the implementation of the Deutsch algorithm, quantum teleportation, and quantum dense coding in terms of the quantum logic gates are discussed. It is also shown how the Bell states can be produced and measured using a sequence of quantum logic gates.

Quantum Secure Communication

Cryptography is a method of secure communication between two or more parties. The crucial step is exchanging a key in a secure manner. There are, however, two problems with conventional cryptography. First the sender and the receiver should exchange the key through highly reliable and secure channels. The second problem is that a clever eavesdropper can, by a careful analysis of the sent information, reconstruct the key. In this chapter, schemes to overcome these problems are presented. First a scheme for exchanging a key over public channels, the so-called RSA algorithm, is discussed. Then the protocols for the quantum key distribution (QKD), the Bennett–Brassard-84 (BB-84) and Bennett-92(B-92) protocols, are then presented. The QKD protocols are exclusively derived using Bohr’s principle of complementarity. An application of these ideas to the design of secure quantum money is discussed.

EPR and Bell Theorem

The first round of the Einstein–Bohr debates took place when Einstein challenged Bohr’s principle of complementarity at the Solvay conference in 1927 and Bohr successfully defended it. The most serious challenge, however, came in 1935 when a paper by Einstein, Podolsky, and Rosen (EPR) argued that quantum mechanics was incomplete through a gedanken experiment motivating an approach based on hidden variables. In this chapter, EPR’s arguments about the incompleteness of quantum mechanics and Bohr’s reply to them are presented. The ultimate answer came almost 30 years later, almost ten years after Einstein’s death, and was nothing that Einstein would have expected. Bell’s inequality and the subsequent Bell-CHSH inequality, that are satisfied by all theories based on the “self-evident truths” of reality and locality are discussed. The startling results that quantum mechanics violates Bell’s inequality and the experimental results are in agreement with the prediction of quantum mechanics are presented.

Quantum Interference: Wave–Particle Duality

Young’s double-slit experiment played a crucial role in establishing the wave nature of light. In this chapter, the shocking result that incident electrons yield a similar interference pattern as that formed by light waves is described. It is shown that the only way the experimental results could be explained is via a wave function description of electrons. It is also shown that, in the same experiment, the interference fringes disappear if the which-path information becomes available. This is the essence of wave–particle duality. The first of the Einstein–Bohr debates on wave-particle duality and Bohr’s principle of complementarity in the double-slit experiment is also discussed. Also presented are the counterintuitive notions of delayed choice and quantum eraser effects showing how the availability or erasure of information generated in the future can affect how the data in the present can be interpreted.