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Given two compatible observables $A$ and $B$ with a common eigenbasis, the completeness relation is: $\newcommand{\ket}[1]{|#1\rangle} \newcommand{\bra}[1]{\langle#1|}$ $$ \sum_{i,j}\ket{a^i,b^j}\bra{a^i,b^j} = 1 $$

Since $\ket{a^i,b^j}$ is not guaranteed to exist for all combinations of $i$ and $j$, does the sum imply we simply ignore the terms which don't exist?

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  • $\begingroup$ "Since $|a^i,b^j\rangle$ is not guaranteed to exist for all combinations of $i$ and $j$" - can you elaborate on this? Right now this seems to come from nowhere. $\endgroup$ Sep 15, 2020 at 0:51
  • $\begingroup$ Take a concrete example. Say the operators can be represented by 2x2 matrices and they have an identical eigenbasis set of size 2, each with their own eigenvalues (an example of which isn't too hard to find). That would mean that the total number of eigenkets given by combinations of indices is 4. Which is the confusing part since there are only 2 eigenkets. $\endgroup$ Sep 15, 2020 at 1:02
  • $\begingroup$ Ok, let's back up a bit. What exactly is the ket $|a^i,b^j\rangle$? What does this notation mean in this context? $\endgroup$ Sep 15, 2020 at 1:40
  • $\begingroup$ It is a notation that represents an eigenket of both $A$ and $B$ where the eigenvalue for $A$ is $a^i$ and for $B$ is $b^j$. $\endgroup$ Sep 15, 2020 at 1:45
  • $\begingroup$ Ok. What's the source for this relation? Where did you see it? $\endgroup$ Sep 15, 2020 at 1:48

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There is no stipulation that the set of $b$’s have same cardinality as the set of $a$’s, and the sum is limited to those sets where so there is no state for which $\vert a^i,b^j\rangle$ exist if $b^j$ is not in the set of $b$ and same for $a^i$. The indices $i$ and $j$ are independent so need not range over the same index set.

Thus, for instance, assuming that $b^i$ takes one of two values in $\{+,-\}$, and $a^i$ are energies of a harmonic oscillator, we would have $$ \sum_{i=0}^\infty\sum_{j=-,+}\vert E_i,j\rangle\langle E_i,j\vert = \sum_{i=0}^\infty\vert E_i,+\rangle\langle E_i,+\vert + \sum_{i=0}^\infty\vert E_i,-\rangle\langle E_i,-\vert=\mathbb{1} $$

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