Chapter 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 time toapply it to systems intended for information processing.The science and technology of quantum information is very new. The concept of the quantum bit (namedthe qubit) was first presented, in the form needed here, in 1995. There are still many unanswered questions(for example the quantum version of the channel capacity theorem is not known). As a result, the field is ina state of flux, and there are gaps in our knowledge which may become apparent in this chapter.13.1 Quantum Information StorageWe have used the bit as the mathematical model of the simplest classical system that can store information.Similarly, we need a quantum model, which will be called the “qubit.” At its simplest, a qubit can bethought of as a small physical object with two states, which can be placed in one of those states and whichcan subsequently be accessed by a measurement instrument that will reveal that state. However, quantummechanics both restricts the types of interactions that can be used to move information to or from thesystem, and permits additional modes of information storage that have no classical counterparts.An example of a qubit is the magnetic dipole which was used in Chapters9, 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 polarizations).Supp ose our system is a single magnetic dipole. The dip ole can be either “up” or “down,” and thesestates have different energies. The fact that the system consists of only a single dipole implies that thesystem is fragile. To preserve the state of the system, and therefore its information, the system must remainisolated. The slightest interaction with its environment is enough to change its state.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 thousands of 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).However, there are 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 same size or cost.The semiconductor industry is making rapid progress in this direction, and within a decade it should beAuthor: Paul Penfield, Jr.Version 1.1.0, May 5, 2004. Copyrightc 2004 Massachusetts Institute of TechnologyURL: http://www-mtl.mit.edu/Courses/6.050/notes/chapter13.pdfstart: http://www-mtl.mit.edu/Courses/6.050/notes/index.htmlback: http://www-mtl.mit.edu/Courses/6.050/notes/chapter12.pdfnext: http://www-mtl.mit.edu/Courses/6.050/notes/chapter13.pdf11813.2 Model 1: Tiny Classical Bits 119possible to make memory cells and gates that use so few atoms that statistical fluctuations in the numberof data-storing particles will be a serious problem. Second, sensitive information stored without redundancycould not be copied without altering it, so it would be possible to protect the information securely, or atleast know if its security had been compromised. And third, the properties of quantum mechanics couldpermit modes of c omputing and com munications 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 prob e ’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 b ec ause 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 instrument determines which state it is. If the system was already in one of thestationary states, then that one is the one selected. If, more generally, the system wave function is a linearcombination of stationary states, then one of those states is selected, with probability given by the squareof 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 enough toaccommodate many interesting properties of quantum information.This model is like the m agnetic dipole model, where only two state s (up and down) are possible. Everymeasurement restores the system to one of its two values, so small errors do not accumulate. Since measure-ments can be made without changing the system, it is possible to copy a bit. According to this model, thequantum bit behaves essentially like a classical bit except that the physical quantities associated with it arevery small.This model has proven useful for energy conversion systems. It was used in Chapter 12 of these notes.13.3 Model 2: Superposition of States (the Qubit)The second model makes use of the fact that the probabilities in quantum mechanics can be expressed interms of wave functions which obey Schr¨odinger’s equation. Since Schr¨odinger’s equation is linear, any linearcombination of wave functions that obey it also obeys it. Thus, if we associate the logical value 0 with thewave function ψ0and the logical value 1 with the wave function ψ1then any