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

|\psi\rangle=\frac{1}{\sqrt{2}}\left(|\textrm{heads}\rangle+|\textrm{tails}\rangle\right).

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

\begin{array}{lcl}\rho & = & |\psi\rangle\langle\psi| \\ & = & \frac{1}{2}|\textrm{heads}\rangle\langle\textrm{heads}|+\frac{1}{2}|\textrm{heads}\rangle\langle\textrm{tails}|\\ & & + \frac{1}{2}|\textrm{tails}\rangle\langle\textrm{heads}| + \frac{1}{2}|\textrm{tails}\rangle\langle\textrm{tails}| \\ & = & \left(\begin{array}{cc} \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} \end{array}\right)\end{array}.

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.

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15 Responses to “Superpositions and mixed states”

  1. Sergey Efremov Says:

    The density matrix was introduced in 1926 by Lev Landau when he was 18 years old. It has become one of the main concepts in quantum statistics.

    • quantummoxie Says:

      That’s amazing, though I am not surprised. I have heard he was quite an impressive guy. Do you have a reference to Landau’s work?

      • Sergey Efremov Says:

        3.^ Landau, L. D. (1927), “Das Dämpfungsproblem in der Wellenmechanik”, Zeitschrift für Physik 45 (5–6): 430–441, Bibcode 1927ZPhy…45..430L, doi:10.1007/BF01343064
        4.^ Landau, L. D., and Lifshitz, E. M. (1977), Quantum Mechanics, Non-Relativistic Theory: Volume 3, Oxford: Pergamon Press, pp. 41, ISBN 0-08-017801-4

      • quantummoxie Says:

        Thanks. That first one is the one I was looking for.

  2. I think you’ve made a couple of transposition errors:

    “(or, one might say the mixed state turns into a superpositional state)”

    Shouldn’t this be “the superpositional state turns into a mixed state”?

    “Either way, the mixed state is a state that includes such coherences while a superpositional state does not.”

    Again “superpositiona”l “mixed”

    One further comment is that I don’t agree that Schroedinger’s cat has evolved to become a problem about mixed states. It is still about pure states, it is just that those states are now states of systems that include the environment rather than just the cat on its own. You could say that it is a problem about proper vs. improper mixtures, but that amounts to the same thing because an improper mixture implies the existence of a pure state on a larger system.

    • quantummoxie Says:

      Oops! Yes, indeed, I did mix those up by accident. I have corrected them.

      As for Schrödinger’s cat, I did oversimplify the argument a bit. I think what I meant to say was that there’s a certain insight that the density matrix formalism can give that was a later development, i.e. the difference between a mixed state and a superpositional state. The point of QM is that the cat is in a superpositional state and not a mixed state, and I think, early on (’20s and ’30s) the difference between the two was not well understood.

  3. BlackGriffen Says:

    “(i.e. simultaneously in all energy eigenstates at once)”
    Um, I think that’s not right. It’s a cloud because the energy eigenstates are simultaneously in all position eigenstates.

    • quantummoxie Says:

      I disagree. It’s a cloud because the state is a superposition of all possible energy eigenstates until it gets measured, after which it is now in a definite energy eigenstate.

      • BlackGriffen Says:

        So if I’m holding a hydrogen atom known to be in its absolute ground state from measurement, you would not describe the electron orbital around the proton as a cloud?

  4. BlackGriffen Says:

    Adding to my last comment – in the instant I measure the exact position of an electron around a hydrogen atom it is in a position eigenstate, but in a superposition of all energy eigenstates. You would call that a cloud?

    • quantummoxie Says:

      Let me think about this a bit. Admittedly I was thinking in terms of one-dimensional wavefunctions. I’ll get back to you. 😉

  5. Ian, this is why my proposed experiment is so important – it would allow us to find that what is apparently a mixed state of photons coming out of either one side or another of a beamsplitter, actually retains phase information showing it is still a superposition. Yes, once the experiment has been explained, the results in retrospectcan be agreed as what quantum theory predicts – but what it is able to show, is still a surprise.

    Think – must someone proposing a quantum experiment predict results contradicting quantum theory? No, almost always the point is to use standard QT in creative ways, so that the results predicted by QT accomplish something surprising, something new and more than people thought possible or plausible. Think of interaction-free measurements, quantum Zeno, quantum eraser, and of course the Bell inequalities. I’m sharpening it up. They should read about it, let them decide what to think. Thanks!

    • quantummoxie Says:

      Actually I have been thinking about your proposed experiment in relation to this quite a bit (along with Chad Orzel’s reply). So I haven’t given up on it. I want to run through the calculation in density matrix terms to see what pops out.

      • Thanks for keeping up. I suspect what you’ll get, is that the DM just tells you the chances of hits, per se, for each output channel by itself – no consideration of any remaining. hidden relationships that could be exploited. I showed there is still something that can be recombined later, something that doesn’t show up in detector stats unless they are properly *combined* later on. Well, how will readers know what we’re talking about, unless they have something to read about? 😉

  6. Smiley didn’t make it at the end of the sentence FWIW. I’m just suggesting, the proposal deserves to be on “respectable” view in due course so people can mull it over. I have discussed the proposal at my blog but it doesn’t really support the best graphics etc and isn’t widely read – meanwhile anyone who is curious can ask for a copy of my article Ian is mulling over. You can write to paradoxer at tni dot net.

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