In the last two posts (Quantum Measurement and Wavefunction Collapse), I’ve been exploring the notorious problem of measurement in quantum mechanics. This post picks up where I left off, so if you missed those first two, you should go read them now.
Here I’m going to venture into what we mean by quantum coherence and the Yin to its Yang, quantum decoherence. I’ll start by trying to explain what they are and then what the latter has to do with the measurement problem.
The punchline: Not very much. (But not exactly nothing, either.)
In order to understand decoherence it’s necessary to understand coherence, and to understand that we need to talk about phase. (Don’t be fazed, it’s not that bad.)
The starting point is the wave-like nature of matter. This is most apparent with light, where the wave/particle duality is ancient and relatively easily demonstrated. But ever since Louis deBroglie (pronounced “deh-broy”) we view all matter as having this duality. [For more, see What’s the Wavelength?].
As an aside, light, especially seen as photons, is matter. Light has energy but is a form of matter. Energy is a property all matter can have. All types of light, from radio waves to x-rays and gamma rays, are matter. What distinguishes them is their energy, which, in light, is expressed as frequency. Radio waves are low frequency (low energy) while x-rays and gamma rays are high frequency (high energy).
Frequency, casually speaking, is how often something happens. In physics, frequency refers to regularly occurring phenomena, such as swinging pendulums, vibrating springs, sine waves, or spinning objects. Here we’re interested in the last two, in particular the relationship between them:
There are three things to note in Figure 1.
Firstly, the direct relationship between the generated waves (red and blue) and the spinning wheel (green). This correlation is fundamental. Electrical power from the power companies waves in this manner because it’s generated by the spinning wheels of electrical generators.
Secondly, note that the peaks and valleys of the two waves don’t line up. This is because the red sine wave is a horizontal projection of the rotation while the blue cosine wave is a vertical projection. The vertical axis is 90° relative to the horizontal axis, so the blue wave is 90° out of phase from the red wave.
If the peaks and valleys lined up, we’d say the waves are exactly in phase or have a phase angle of 0° relative to each other. On the other hand, if the peaks of one matched the valleys of the other, we’d say their phase angle was 180°, which is as out of phase as it’s possible to get. They would be exactly out of phase, the opposite of being exactly in phase.
Obviously, projecting both waves onto the horizontal axis in the same direction generates identical in phase waves. Projecting one onto the horizontal axis in the opposite direction — 180° difference — generates a wave that is 180° out of phase. By varying the projection angles, we vary the phase angle.
As an aside, note that only the in 90° case can we say the blue wave is the cosine wave relative to the red sine wave. (In general, a wave generated by circular motion is called a sine wave.)
Thirdly, because both waves are generated by the same spinning circle, the distance between their respective peaks is the same for both. That distance is their wavelength and how frequently those peaks pass a fixed point is their frequency.
Phase is the angular distance between two waves relative to specific matching points on each, for instance their respective peaks or valleys or any other spot we identify.
Because of the relationship between sine waves and spinning wheels, phase is also the angular distance between two points on the wheel. If we think of a wheel as a clock, then, relative to 12 o’clock, 3 o’clock is 90° out of phase and 6 o’clock is 180° degrees out of phase. Above I said that 180° is as out of phase as possible — 6 being on the opposite side from 12. While 9 o’clock might seem like it’s 270° out of phase and, thus, more, it’s also -90° degrees, which is the same phase difference (in reverse) as 3 o’clock.
To segue into the notion of coherence, consider generating these waves with two spinning wheels, both projected left on the horizontal axis (like the red wave is). If the clocks were in phase — both striking 12 o’clock at the same moment — then the generated waves would also be in phase. To generate the above animation, the blue wave’s clock would have to strike 3 o’clock at the same moment the red wave’s clock struck 12 — the clocks would have to be 90° out of phase.
As an aside, we can calculate angular distance in degrees, which are arbitrary units (like inches or pounds), or we can use the natural units of radians where 180° degrees equals pi radians, and a full 360° is 2*pi radians. In physics we generally use radians. (One radian is about 57°. It’s the angle we get by moving along a circle the same distance as its radius.)
There is one more very important way we could control the phase between the two waves. Imagine generating two waves from one clock using the same projection. The waves would be exactly in phase but if one wave took a longer path to some comparison point, at that point they would be out of phase. How much out of phase depends on the relative lengths of the two paths.
Lastly, if the two clocks ran at different speeds, even if the generated waves were both from the same projection, then the phase angle constantly changes. How fast it changes depends on the clock speeds relative to each other. In the extreme case, think of how a stopped clock is right twice a day compared to a normally running clock.
Coherence is a broad topic (see its Wiki disambiguation page), but as the Yang to the Yin of decoherence, we can think of it as the property of a wave-like system that allows consistent interference effects. A crucial point is that the phase of a coherent system remains controlled and stable. In a decoherent system the phase is unstable, usually due to the effects of some other uncontrolled system.
One example of a coherent system is laser light. It has amazing and useful properties because it is a coherent form of light. Its photons march in lockstep and have consistent phase.
In the quantum realm, phase and interference are aspects of the complex numbers used to describe quantum systems. These act like tiny clocks (with the caveat that, in some cases, the clocks are stopped, and we care about where they are stopped). This is why quantum math has so many constructs that look like this:
This complex exponential describes a “wheel” with frequency θ (theta). The x can stand for a rich variety of things depending on the system we’re describing. The t, if included, is the time component. In particular, note the i, the imaginary unit. Here again, complex numbers are important in quantum math. [See Circular Math and its many links (especially Beautiful Math) for more.]
For now, suffice to say constructs like this describe waves (which correspond to clocks or spinning wheels). We won’t delve further into the mathematics of it.
Quantum coherence is important in two regards: Firstly, it allows interference effects such as seen in two-slit and interferometry experiments. Secondly, as a “stopped clock”, it’s a critical part of quantum states (such as used in quantum computing).
Both of these are worthy of (multiple!) posts on their own. Here I’ll only summarize them so we can move on to decoherence and what (little) it has to do with measurement.
In two-slit experiments, a matter wave passes through two slits and interferes with itself to produce distinctive interference bands. While there is a classical wave description involving constructive and destructive interference, the quantum version is puzzling because, since we don’t really know what a matter wave is, we don’t really know what is interfering.
The presence of the bands tells us something is, but the phenomena isn’t like the classic version because what we see is based on probabilities, which remember, are the squares of the wavefunction at each point on the screen or detector. That wavefunction, by the way, contains complex exponentials similar to the equation above, and it’s these spinning wheels that interfere. The dark bands, unlike in the classical description, are not cancelations of wave energy but cancelations of probability.
Interferometry experiments, typically some form of the Mach–Zehnder interferometer, are more sophisticated and scientifically useful versions of the two-slit experiment. (The latter are mostly useful for just demonstrating the wave-like nature of matter.)
In quantum computing, the complex exponential describes the state of a qubit, and here we typically want a “stopped clock” — a fixed state. Recall that a (pure) quantum state is described by a vector with a length of one. The state is defined by where the vector points (see previous post) as well as by its phase.
In the Bloch sphere (image at top of this post), phase is represented as rotation around the sphere. (The basis vectors are the vertical axes labeled |0〉 and |1〉. See QM 101: Bloch Sphere for more.)
This brings us to decoherence. As the name implies, it’s the loss of a system’s coherence.
Two-slit and interferometry experiments depend on the phase of the waves being unperturbed. They require constant coherence. If environmental effects perturb the system enough, the interference effects are lost.
In quantum computing, if the phase of the qubit is perturbed, then the state of that qubit is no longer what it should be. This is similar to randomly flipping bits in classical computing — the computation becomes corrupted. Much of the engineering of quantum computers involves preserving the coherence of its qubits.
Note that, with regard to individual “particles” in a two-slit or interferometry experiment, altering the phase of their wave doesn’t mean it no longer interferes with itself. It means the interference shifts randomly due to the perturbation. If that happens to all the “particles”, the random nature of environmental influence smears the banding, destroying the overall effect.
So, quantum decoherence is the corruption of a quantum system’s phase by mixing it with the random phases from the environment. In turn, the quantum system’s phase is dispersed into the environment (imagine a tiny drop of ink in five gallons of water).
If coherence is a property a system can have, decoherence is a process it can experience.
Finally, what does decoherence have to do with measurement? Not much. At least, not much with the measurement itself. But I think it has a lot to do with the divide between a quantum system and the instrument that measures it.
In any large system, the phases of the individuals are dispersed, not just into the environment, but among the myriad individuals of the system. (Recall the 10²⁷ singers.) A large system has decohered and, thus, acts not as quantum system but as a classical one. In particular, quantum behaviors such as superposition and interference are no longer possible. (This is why cats are always either alive or not.)
Bottom line, when a classical system measures a quantum system, the effect is as if one more singer joined the crowd — that singer’s song is utterly and completely swamped by the “white noise” of the crowd.
Next time I’ll talk about measurement in some specific contexts, such as the infamous Schrödinger’s Cat experiment.
Stay coherent, my friends! Go forth and spread beauty and light.