We do need quantum theory to describe many physical phenomena, but, at the same time, there are many other phenomena where the classical theory of probability works pretty well. Indeed, we hardly see quantum interference on a daily basis. Why? The answer is decoherence. The addition of probability amplitudes, rather than probabilities, applies to physical systems which are completely isolated. However, it is almost impossible to isolate a complex quantum system, such as a quantum computer, from the rest of the world: there will always be spurious interactions with the environment (such as heat transfer), and when we add amplitudes, we have to take into account not only different configurations of the physical system at hand, but also different configurations of the environment.
For example, consider an isolated system composed of a quantum computer and its environment.
The computer is prepared in some input state
- The computer is isolated and quantum computation does not affect the environment.
The computer and the environment evolve independently from each other and, as a result, the environment does not hold any physical record of how the computer reached output
O. In this case we add the amplitudes for each of the two alternative computational paths.
- Quantum computation affects the environment.
The environment now holds a physical record of how the computer reached output
O, which results in two final states of the composed system (computer and environment) which we denote O_1and O_2. We add the probabilities for each of the two alternative computational paths.
When quantum computation affects the environment (or vice versa), we have to include the environment in our analysis, since it is now involved in the computation.
Depending on which computational path was taken, the environment may end up in two distinct states.
The computer itself may show output
We shall derive this formula later on, and you will see that
Decoherence suppresses quantum interference.
Decoherence is chiefly responsible for our classical description of the world: without interference terms we may as well add probabilities instead of amplitudes (thus recovering the additivity axiom). While decoherence is a serious impediment to building quantum computers, depriving us of the power of quantum interference, it is not all doom and gloom: there are clever ways around decoherence, such as quantum error correction and fault-tolerant methods, both of which we will meet later.