13.1: Quantum Information Storage

We have used the bit as the mathematical model of the simplest classical system that can store information. Similarly, we need a model for the simplest quantum system that can store information. It is called the “qubit.” At its simplest, a qubit can be thought of as a small physical object with two states, which can be placed in one of those states and 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 from the system, and permits additional modes of information storage and processing 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 trapping electrons) and photons (particles of light with various polarizations).

Qubits are difficult to deal with physically. That’s why quantum computers are not yet available. While it may not be hard to create qubits, it is often hard to measure them, and usually very hard to keep them from interacting with the rest of the universe and thereby changing their state unpredictably.

Suppose our system is a single magnetic dipole. The dipole can be either “up” or “down,” and these states have different energies. The fact that the system consists of only a single dipole makes the system fragile.

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 one is missing, the rest are still present and a measurement can still work. In other words, there is massive redundancy in the mechanism that stores the data. Redundancy is effective in correcting errors. For a similar reason, it is possible to read a classical bit without changing its state, and it is possible for one bit to 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 or cost. The semiconductor industry is making rapid progress in this direction, and before 2015 it should be possible to make memory cells and gates that use so few atoms that statistical fluctuations in the number of data-storing particles will be a problem. Second, sensitive information stored without redundancy could not be copied without altering it, so it would be possible to protect the information securely, or at least know if its security had been compromised. And third, the properties of quantum mechanics could permit 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 contact with the single dipole of the system. The system and the probe then exchange their states. The system ends up with the probe’s previous value, and the probe ends up with the system’s previous value. If the previous system 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 system that has unknown data increases the uncertainty about the environment. The general principle here is that discarding unknown data increases entropy.

The model for reading the quantum bit is not as simple. We assume that the measuring instrument interacts with the bit in some way to determine its state. This interaction forces the system into one of its stationary states, and the state of the instrument changes in a way determined by which state the system ends up in. If the system was already in one of the stationary states, then that one is the one selected. If, more generally, the system wave function is a linear combination of stationary states, then one of those states 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.