# What does QM say about the past rather than the future?

In QM, the wave function (in the Copenhagen interpretation) is not an actual physical wave but a device to derive probabilities about the outcomes of experiments. The wave function encodes all the information about the system we want to derive predictions for. Predictions are about future measurements. Once the measurement has been performed and the result is known, we adjust accordingly our expectation: the so-called collapse of the wavefunction just took place (let me add, in our minds). This subjective knowledge about the predictions of QM is crucial to avoid problem with causality in relativity when studying entangled systems. Fine.

What I am a bit confused about is what QM says about the past, rather than the future. What is the analog picture that QM gives about the state of a system in the past? What does QM say about the conditional probabilities of events? What does QM tell about, say, cosmology and the far past of the universe when e.g. string theory becomes relevant? I hope it is not a trivial, naive, question.

• I think, your question is not clear. You should ask more explicitly. – Self-Made Man Jun 2 '14 at 16:14
• (Not an answer to your question) I am personally unsatisfied with the Copenhagen interpretation. Thus the thing follows Copenhagen interpretation is suspicious. The non-deterministic wavefunction collapse $$| \psi \rangle \rightarrow |n \rangle$$ seems in contradiction with deterministic time-dependent Schrodinger equation. To me, the right theory is to say the time evolution of the entire universe is unitary, Schrodinger equation. The non-unitary, collapse, is a subsystem phenomenon. The non deterministic is just like statistical mechanics. – user26143 Jun 2 '14 at 16:23
• "The future's uncertain and the end is always near." <-- see, no mention of the past. :-) . And keep in mind we can only claim a 'good guess' at the past based on our measurements and our interpretation of the results. – Carl Witthoft Jun 2 '14 at 16:56

Quantum mechanics can be used to answer questions about the past in a fairly straightforward way as any question of that type can be phrased as a question about expectation value of operators (or as transition amplitudes). As a simple example consider a two state system (e.g. spin 1/2). Suppose someone else prepares the state in either spin up or spin down but doesn't tell you. Also suppose that the dynamics are unitary and known ($U$). Then you can use quantum mechanics to ask, for example, what is the probability that the state was prepared in the 'up' state if I measure it in the up state now?

$$p = | \langle \mbox{up}|U|\mbox{up} \rangle|^2$$

So really there is nothing new, you just apply quantum mechanics to whatever question you mean to ask about the past. You might have to be a bit careful in phrasing the question however.

For the general case of reconstructing the past state given present measurements, see for example the wikipedia article on Quantum tomography (http://en.wikipedia.org/wiki/Quantum_tomography)

• If I understand correctly your example, you are evolving back in time with $U^\dagger$ the $|up\rangle$ state and then projecting it on $|up\rangle$: $p=|\langle up| U |up\rangle|^2$. Because $U$ is unitary, this is the same probability to find today $|up\rangle$ for a system that was prepared in the state $|up\rangle$ in the past, and has been evolved it with $U$ until now. Is it that right? If so, you are dealing with the past and the future in an exactly symmetric way. Not sure this is the right answer though. – TwoBs Jun 3 '14 at 8:00
• You are correct! I was going to mention that you can think of it as evolving the final state back in time, but thought it might clutter the answer. – DrEntropy Jun 3 '14 at 14:55
• What I can't clearly see is why such a process, that is evolving back in time my wave function that I have at hand after I measured the outcome (and hence adjusted the wavefunction accordingly), should give me the correct probability that the state was originally prepared in the up state. I mean, why should I treat the probability about a past event on the same foot of probability for future events? In other words, what tells me that the intrinsic randomness of QM applies to the dynamics reversed in time? I guess it is a silly question – TwoBs Jun 3 '14 at 16:37
• Let me elaborate: the uncertainty about the past state could seem, at least superficially, different in nature than the uncertainty about the future measurement outcomes. But I think you are right, there is no a fundamental difference instead with the randomness about what the state was in the past after I measure it now. After all, preparing a system in a certain definite state requires a measurement and it makes no difference whether I am looking at it backward in time – TwoBs Jun 3 '14 at 16:53
• I think you might find it fruitful to think about these same questions in the context of classical probability theory, as perhaps your difficulty really has nothing to do with the quantum mechanical aspects. – DrEntropy Jun 3 '14 at 17:42

On the fundamental level, QM is time-symmetric, hence it says the same about the past as about the future. The dynamics of the state is deterministic and given by the quantum Liouville equation (or, if you consider an isolated system in a pure state, by the Schroedinger equation). This state determines the probability distribution of measurable events at any time, the past as well as the future.

Accounting for new information about the past (or the future), as it becomes used by the observer to improve predictions about the past (or the future), is accounted for by projecting the state to the invariant subspace determined by the new information. This is the quantum analogue of taking conditional expectations in a classical stochastic model when new information about the past (or the future) becomes available.