Superpositions and mixed states
One of the more widely misunderstood aspects of quantum mechanics is the difference between superpositional states and mixed states. This is partly because we don’t do a good enough job defining our terms in physics (or rather agreeing to use the same consistent definition). This has particular relevance to my recent posts on decoherence and the Schrödinger cat paradox. As Hongwan Liu pointed out on Quora, while the latter paradox was originally formulated in relation to superpositional states, it has evolved to actually be about mixed states. So what are they?
A superpositional state is not actually as strange as it first appears (regardless of the interpretation of quantum mechanics that you happen to prefer). In theory, we could describe classical systems in terms of superpositional states just as easily if those systems were probabilistic. Consider a fair (unbiased) coin. Prior to flipping it (or, perhaps, after flipping it but prior to actually looking at it), we could describe its state as being
In some interpretations of quantum mechanics, quantum mechanical objects in such states are considered to be simultaneously in such states. This is partly why certain interpretations of QM take electrons in an atom to be in a ‘cloud’ (i.e. simultaneously in all energy eigenstates at once) prior to any type of measurement. But it is to be emphasized that this is merely an interpretation. It is based on results of interferometer experiments. In a purely statistical interpretation of QM as well as in certain epistemic interpretations, the view of the states is closer to the example of the coin that I just gave. But whatever your view, the fact remains that the above is a superpositional state.
So what’s a mixed state? A mixed state requires a density matrix representation. Using the above example, we can form a density matrix for the state of the coin as
Now we see that there do seem to be these strange situations in which the coin is both heads and tails simultaneously. These terms correspond to the off-diagonal elements of the density matrix and are truly quantum mechanical in that, while we can write such a matrix for a classical object like a coin, the meaning of the off-diagonal terms for non-quantum objects is not clear (in fact most people would simply say it has no meaning for classical objects, i.e. it makes no sense to even write down such a thing for a classical system). These off-diagonal terms are called coherences. The term ‘decoherence’ can be applied to this in two ways.
The interpretation-free way of defining decoherence is that it corresponds to any process that eliminates the coherences from a density matrix. Conversely, if we refer to decoherence as any process that irreversibly eliminates the coherences, we are, to some extent, biasing ourselves a bit towards interpretations that favor an environmental (or similar) explanation for the decay of such terms (assuming they decay to begin with). This is because many density operators are formed using the Schrödinger equation for density operators which naturally gives rise to exponential terms with time constants in them, i.e. unitary evolution produces terms with time-dependent exponentials. When these decay, the coherences go to zero and the diagonal terms become equal to classical probabilities (or, one might say the superpositional state turns into a mixed state). But I’m not entirely sure I agree that coherences are always time-dependent, exponentially decaying terms, in which case I don’t think the term ‘irreversible’ necessarily need apply (since there might be some other way to get rid of them). Or, if you were to insist that decoherence was an inherently irreversible process, then you would need a new term to describe a process by which the coherences vanish either reversibly or by a time independent process.
Either way, the superpositional state is a state that includes such coherences while a mixed state does not. Understanding the differences between them is more than merely a matter of learning QM. It gets at the heart of precisely what QM is and how it should (or can) be interpreted.