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Finite quantum de Finetti theorems

by Michael Nielsen on April 8, 2005

Attention conservation notice: This post needs knowledge of some elementary quantum mechanics to make sense.

David Poulin gave a seminar today, in which he described some beautiful results in quant-ph/0410229, by Koenig and Renner.

The main result is, I think, quite incredible.

Suppose we have N quantum systems, all with identical state spaces. Suppose rho is some fixed but completely arbitrary quantum state of those N systems.

Now suppose we pick out M of those systems. M is fixed, but we choose the subset of systems completely at random. Let’s call the resulting quantum state when we disregard the remaining systems rho’.

What can we say about rho’? Konig and Renner prove that, to a very good approximation, the state can be written in the form:

(*) int P(tau) tau^{\otimes M} d tau,

where the integral is over single-copy density matrices tau, and P(tau) is a normalized probability distribution.

How good an approximation? They prove that the trace distance between rho’ and the form (*) is at most c M / sqrt{N-M}, where c is some constant. As N goes to infinity, this goes to zero. (This gives rise to the result known as the quantum de Finetti theorem.)

Why is this representation theorem incredible?

From a mathematical point of view, the reason is that states of the form (*) are very special. The way we constructed rho’, it’s not difficult to see that the resulting state ought to be symmetric, but that is only enough to ensure that rho’ has a representation like (*), but possibly with P(tau) taking negative values, or with the tau operators not being density matrices. States of the form (*) are far more special.

There is also a reason that is more physical, almost philosophical, in nature. Imagine we have some very large number of systems, all with identical state spaces, but otherwise arbitrary. For example, consider the set of all electrons in the Universe.

Now pick out a small number of those systems, at random. The representation (*) tells us that to a very good approximation we can imagine that the state of the system was prepared by first sampling from the distribution P(tau), and then prepared M identical copies of tau. Furthermore, this is true, no matter what the original state was! The only way the original state enters the picture is that it controls what the distribution P(.) is.

In the N -> infinity limit all of this has been known for years; in recent years it’s received a lot of attention, due to the work of Caves, Fuchs, Schack and collaborators. I must admit, though, that until David’s talk I hadn’t appreciated how remarkable the results were, perhaps because I am always suspicious of any result involving infinite tensor products.

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