# Does uniqueness of the triorthogonal decomposition make quantum measurement objective?

Some books and articles on quantum measurement theory make use of a theorem (by Elby-Bub 1994) called the Triorthogonal Decomposition Theorem:

For three subsystems, a state vector $\lvert \Psi \rangle$ has a unique triorthogonal decomposition $$\lvert \Psi \rangle = \sum_j{c_j \lvert a_j \rangle \otimes \lvert b_j \rangle \otimes \lvert e_j \rangle}$$ even if some of the $\lvert c_j \rvert$ are equal.

Here is the basic idea as I understand it:

Suppose we model a quantum measurement as an interaction between a system (S) and measurement apparatus (A). Here because there are only two subsystems of the complete S+A system, it is possible to decompose into many different bases. We have basis degeneracy.

But now suppose we include the environment as a source of decoherence. Then apparently we now have three subsystems (S+A+E) so we can use this Triorthogonal Decomposition Theorem to argue that the measured system decomposes into a unique basis.

Auletta (Quantum Mechanics, 2009) describes this as follows:

The uniqueness of triorthogonal decomposition is a very important point. In fact while the tracing out [of the environment] is only relative to the system and the apparatus, the uniqueness of the triorthogonal decomposition introduces an objective character in the measurement theory that can account for irreversibility.

I find this intriguing yet a bit unbelievable. I appreciate that Auletta is giving a very idealised presentation - I don't object to that, but I wish to know if this idea really is such an important principle as claimed.

• Interesting - I didn't even know this existed and couldn't believe it at first. The problems with the case of two systems goes back to the non-uniqueness of the singular value decomposition (there is a unitary freedom even if the singular values are ordered decreasingly as long as the singular values are degenerate), but nothing of this sort exists for three or more systems. But as with you, I'm not completely sure about the consequences. It seems to primarily have importance in interpretations of measurement theory such as the many world theory. But I have to think more. Oct 27, 2016 at 21:55
• Holy trinities batman! Jun 6, 2020 at 19:42