13 Quantum Information13.1 Quantum Information Storage13.2 Model 1: Tiny Classical Bits13.3 Model 2: Superposition of States (the Qubit)13.4 Model 3: Multiple Qubits with Entanglement13.5 Detail: Qubit and Applications13.6 Bracket Notation for Qubits13.6.1 Kets, Bras, Brackets, and Operators13.6.2 Tensor Product---Composite Systems13.6.3 Entangled qubits13.7 No Cloning Theorem13.8 Representation of Qubits13.8.1 Qubits in the Bloch sphere13.8.2 Qubits and symmetries13.8.3 Quantum Gates13.9 Quantum Communication13.9.1 Teleportation - Alice and Bob's story13.9.2 Quantum Cryptography13.10 Quantum Algorithms13.10.1 Deutsch Josza13.10.2 Grover13.11 Quantum Information ScienceChapter 13Quantum InformationIn Chapter 10 of these notes the multi-state model for quantum systems was presented. This model wasthen applied to systems intended for energy conversion in Chapter 11 and Chapter 12. Now it is applied tosystems intended for information processing.The science and technology of quantum information is relatively new. The concept of the quantum bit(named the qubit) was first presented, in the form needed here, in 1995. There are still many unansweredquestions about quantum information (for example the quantum version of the channel capacity theorem isnot known precisely). As a result, the field is in a state of flux. There are gaps in our knowledge.13.1 Quantum Information StorageWe have used the bit as the mathematical model of the simplest classical system that can store informa-tion. Similarly, we need a model for the simplest quantum system that can store information. It is calledthe “qubit.” At its simplest, a qubit can be thought of as a small physical object with two states, whichcan be placed in one of those states and which can subsequently be accessed by a measurement instrumentthat will reveal that state. However, quantum mechanics both restricts the types of interactions that can beused to move information to or from the system, and permits additional modes of information storage andprocessing that have no classical counterparts.An example of a qubit is the magnetic dipole which was used in Chapters 9, 11, and 12 of these notes.Other examples of potential technological importance are quantum dots (three-dimensional wells for trappingelectrons) and photons (particles of light with various p olarizations).Qubits are difficult to deal with physically. That’s why quantum computers are not yet available. Whileit may not be hard to create qubits, it is often hard to measure them, and usually very hard to keep themfrom interacting with the rest of the universe and thereby changing their state unpredictably.Supp ose our system is a single magnetic dipole. The dipole can be either “up” or “down,” and thesestates have different energies. The fact that the system consists of only a single dipole makes the systemfragile.The reason that classical bits are not as fragile is that they use more physical material. For example,a semiconductor memory may represent a bit by the presence or absence of a thousand electrons. If oneis missing, the rest are still present and a measurement can still work. In other words, there is massiveredundancy in the mechanism that stores the data. Redundancy is effective in correcting errors. For asimilar reason, it is possible to read a classical bit without changing its state, and it is possible for one bitto control the input of two or more gates (in other words, the bit can be copied).Author: Paul Penfield, Jr.This document: http://www.mtl.mit.ed u/Co urse s/6. 050/ 200 8/no tes/ chap ter1 3.p dfVersion 1.5, May 12, 2008. Copyrightc 2008 Massachusetts Institu te of TechnologyStart of notes · back · next | 6.050J/2.110J home page | Site map | Search | About this document | Comments and inquiries13713.2 Model 1: Tiny Classical Bits 138However, there are at least three reasons why we may want to store bits without such massive redundancy.First, it would be more efficient. More bits could be stored or processed in a structure of the s ame size orcost. The semiconductor industry is making rapid progress in this direction, and before 2015 it should bepossible to make memory cells and gates that use so few atoms that statistical fluctuations in the number ofdata-storing particles will be a problem. Second, sensitive information stored without redundancy could notbe copied without altering it, so it would be possible to protect the information securely, or at least know ifits s ec urity had been compromised. And third, the properties of quantum mechanics could permit modes ofcomputing and communications that cannot be done classically.A model for reading and writing the quantum bit is needed. Our model for writing (sometimes called“preparing” the bit) is that a “probe” with known state (either “up” or “down”) is brought into contactwith the single dipole of the system. The system and the probe then exchange their states. The system endsup with the probe’s previous value, and the probe ends up with the system’s previous value. If the previoussystem state was known, then the state of the probe after writing is known and the probe can be used again.If not, then the probe cannot be reused because of uncertainty about its state. Thus writing to a systemthat has unknown data increases the uncertainty about the environment. The general principle here is thatdiscarding unknown data increases entropy.The model for reading the quantum bit is not as simple. We assume that the measuring instrumentinteracts with the bit in some way to determine its state. This interaction forces the system into one of itsstationary states, and the state of the instrument changes in a way determined by which state the systemends up in. If the system was already in one of the stationary states, then that one is the one selected.If, more ge nerally, the system wave function is a linear combination of stationary states, then one of thosestates is selected, with probability given by the square of the magnitude of the expansion coefficient.We now present three models of quantum bits, with increasingly complicated behavior.13.2 Model 1: Tiny Classical BitsThe simplest model of a quantum bit is one which we will consider only briefly. It is not general enoughto accommodate the most interesting properties of quantum information.This model is like the magnetic dipole model, where only two states (up and down) are possible. Everymeasurement restores the system to one