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MIT 6 050J - Quantum Information

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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 EntanglementChapter 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 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 infor-mation. Similarly, we need a quantum model, which will be called the “qubit.” At its simplest, a qubitcan be thought of as a small physical object with two states, which can be placed in one of those statesand which can subsequently be accessed by a measurement instrument that will reveal that state. However,quantum mechanics both restricts the types of interactions that can be used to move information to or fromthe system, 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 p olarizations).Supp ose our system is a single magnetic dipole. The dipole can be e ither “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 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).However, 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 same size orcost. The semiconductor industry is making rapid progress in this direction, and before 2015 it should beAuthor: Paul Penfield, Jr.This document: http://www.mtl.mit.edu/Courses/6.050/2006/notes/chapter13.pdfVersion 1.3, May 11, 2006. Copyrightc 2006 Massachusetts Institute of TechnologyStart of notes · back · next | 6.050J/2.110J home page | Site map | Search | About this document | Comments and inquiries13213.2 Model 1: Tiny Classical Bits 133possible 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 computing 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 many 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 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. This model of the quantum bitbehaves essentially like a classical bit except that the physical quantities associated with it are very 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 states in quantum mechanics can be expressed in termsof wave functions which obey the Schr¨odinger equation. Since the Schr¨odinger equation is linear, any linearcombination of wave


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