Basic Quantum Theory

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6.2 Quantum Decoherence

Quantum decoherence is a very important topic and is, in fact, critical for quantum computing. Decoherence is directly related to the previous section on wave functions. Recall that a wave function is a mathematical representation of the state of a quantum system. As long as there exists a definite phase relation between the states, that system is coherent. Also, recall that interactions with the environment cause a wave function to collapse. If one could absolutely isolate a quantum system so that it had no interaction at all with any environment, it would maintain coherence indefinitely. However, only by interacting with the environment can it be measured; thus, data can be extracted.

What does it mean to have a definite phase relation between states? First, we must examine the concept of phase space, which is a concept from dynamical system theory. It is a space in which all the possible states of the system are represented. Each state corresponds to a unique point in the phase space. Each parameter of the system represents a degree of freedom. In turn, each degree of freedom is represented as an axis of a multidimensional space. If you have a one-dimensional system, it is a phase line. Two-dimensional systems are phase planes.

Two values, p and q, play an important role in phase space. In classical mechanics, the p is usually momentum and the q the position. Now, in quantum mechanics, this phase space is a Hilbert space. Thus, the p and q are Hermitian operators in that Hilbert space. While momentum and position are the most common observables, and are most often used to define phase space, there are other observables such as angular momentum and spin.

To refresh your memory, a Hermitian operator is also called a self-adjoint operator. Remember, we are dealing with matrices/vectors, so the operators are themselves matrices. Most operators in quantum mechanics are Hermitian. Hermitian operators have some specific properties. They always have real eigenvalues, but the eigenvectors or eigenfunctions might include complex numbers. A Hermitian operator can be “flipped” to the other side if it appears in an inner product—something like what you see here:

• f|Ag〉 = 〈Af|g〉

Hermitian operators’ eigenfunctions form a “complete set.” That term denotes that any function can be written as some linear combination of the eigenfunctions.

In general, if we are dealing with a non-relativistic model, the dimensionality of a system’s phase space is the number of degrees of freedom multiplied by the number of systems-free particles. Non-relativistic spacetime is conceptually rather simple. Relativistic spacetime uses n dimensional space and m dimensional time. Non-relativistic spacetime fuses that into a single continuum. Put another way, it is simply ignoring the effects of relativity. At the subatomic level that is perfectly reasonable, as relativistic effects are essentially irrelevant.

So, when the system interacts with the environment, each environmental degree of freedom contributes another dimension to the phase space of the system. Eventually, the system becomes decoupled. There is actually a formula for this called the Wigner quasi-probability distribution. This is sometimes called the Wigner-Ville distribution or just the Wigner function. The details may be a bit more than are needed in this book; however, the general outline is certainly something we can explore. Eugene Wigner first introduced this formula in 1932 to examine quantum modifications to classical mechanics. The purpose was to link the wave function we have studied in Schrödinger’s equation to a probability distribution in phase space.

Equation 6.9 shows the Wigner distribution.

EQUATION 6.9 Wigner Distribution

By this point, you should not be daunted by complex-looking equations, and much of this equation use symbols you already know. But let us briefly examine them. Obviously, the W is the Wigner distribution. X is usually position and p momentum, but they could be any pair (frequency and time of a signal, etc.). Of course, ψ is the wave function, and is the reduced Planck constant. We discussed the ∫ symbol earlier in the book; it denotes integration. For our purposes, you don’t have to have a detailed knowledge of the Wigner distribution, nor do you have to be able to “do the math.” Rather, you just need a general understanding of what is happening.

In classical mechanics, a harmonic oscillator’s motion could be completely described by a point in the phase space with the particle position x and momentum p. In quantum physics, this is not the case. Recall from Chapter 3 our discussion of Heisenberg’s uncertainty principle. You cannot know with precision the position and momentum simultaneously, but by measuring x, p, or their linear combination on a set of identical quantum states, you can realize a probability density associated with these observables (x and p). The Wigner function accomplishes this goal. Our goal is to understand decoherence. The Wigner distribution shows the decoupling process because it shows the probability of various states.

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