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.

  • 1
    $\begingroup$ 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. $\endgroup$
    – Martin
    Oct 27, 2016 at 21:55
  • $\begingroup$ Holy trinities batman! $\endgroup$
    – lurscher
    Jun 6, 2020 at 19:42

1 Answer 1


The triorthogonal monomial basis for a Hilbert space decomposed into a triple tensor product is unique, the decomposition of the Hilbert space into such a triple product is not. So the "unique" basis into which a model decomposes is very much an artifact of the non-unique decomposition of the model into a system, an apparatus and an environment. A simple example of how non-unique that can be is the infamous "Heisenberg cut", moving which one can start the "observer" anywhere from the device's meter to the "consciousness" itself. Here is from Donald's Instability, Isolation, and the Tridecompositional Uniqueness Theorem

"Theorem 3.4 is concerned with variations in the global wave function. It is also important, for the question of the physical relevance of the tridecompositional uniqueness theorem, to recognize that the assumption that a physical Hilbert space has a fundamental tensor product structure may well be incorrect. This recognition can be supported by consideration of the derived and phenomenological nature of localized particles according to relativistic quantum field theory, but, even at a less sophisticated level, it is hard to justify the idea that there are the sort of natural boundaries which would allow the universe to be divided without ambiguity into a system, a measuring apparatus, and an environment. If the tensor product structure is not a fundamental aspect of reality, then the ultimate laws of nature cannot depend on it."

On the other hand, the usual hand-wringing over the non-objectivity and irreality of a preferred basis for decoherence in the bidecompositional case is itself a tempest in a teacup. Wallace in Everett and Structure criticizes Kent and Barret for what he calls the "fallacy of exactness". If the criterion for being real and objective is to be written into the axioms of a theory then, say, tigers would be neither real nor objective:

"In other words, a preferred basis must either be written into the quantum- mechanical axioms, or no such basis can exist — the idea of some approximate, emergent preferred basis is not acceptable... either there is some precise truth about transtemporal identity which must written into the basic formalism of quantum mechanics, or there are simply no facts at all about the past of a given world, or a given observer. (This seems to be what motivates Bell (1981) to say that in the Everett interpretation the past is an illusion.)..."

"To see why it is reasonable to reject the dichotomy of the previous section, consider that in science there are many examples of objects which are certainly real, but which are not directly represented in the axioms. A dramatic example of such an object is the tiger: tigers are unquestionably real in any reasonable sense of the word, but they are certainly not part of the basic ontology of any physical theory."

  • $\begingroup$ That makes a lot of sense. Ties together several threads nicely. I do like Wallace's philosophy (motivated by Dennett I believe) even if his multiverse ideas turn out to be wrong. $\endgroup$
    – isometry
    Oct 28, 2016 at 9:58

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.