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Scenario: ${\mathcal A}$ and ${\mathcal B}$ are two observables. Mathematically we model them by two Hermitian operators $A\colon H \to H$ and $B\colon H \to H$ on a separable Hilbert space. Physically they correspond to experiments $E_A$ and $E_B$, whose results are values in $Spec(A)$ and $Spec(B)$; repetitions produce value distributions on these spectra, expectation values, variances and higher momenta. The mathematical operator $A+B$ also is Hermitian. So let us look for an experiment which corresponds to this operator and let us study its expectation value in state $\varphi$.

Naive approach: Let us try pair-experiments. Assume we have a black box producing samples of state $\varphi$. Take a sample of the state, do experiment $E_A$ and get result $a$. Sample the state again, do experiment $E_B$ and get result $b$. Call the sum $a+b$ the result of the pair-experiment.

If $Spec(A) = \{ a_1, a_2 \}$ and $Spec(B) = \{b_1, b_2\}$ then the pair experiment has spectrum $\{ a_1 + b_1, a_1 + b_2, a_2 + b_1, a_2 + b_2 \}$. Obviously the pair-experiment has to be described in $H \otimes H$ and with a completely different observable. Details are straight forward, but we have no experiment for $A + B$. :-(

Second attempt: Let us assume that ${\mathcal A}$ and ${\mathcal B}$ are compatible and $A$ and $B$ commute. Then we can do the following: Sample the state once, on that sample do experiments $E_A$ and $E_B$ in whatever sequence, receive sequence independent values $a$ and $b$ and add them. Mathematically all is good. $A$ and $B$ share an eigenbasis, the spectrum of $A + B$ is the sum of the eigenvalues (belonging to the same shared eigenspace). Expectation values work out as expected. :-)

Now my question: $A + B$ still is a Hermitian operator, even if $A$ and $B$ do not commute. So I still am curious to which experiment this operator belongs to.

Note: In case of the product $A \cdot B$, the operator $A\cdot B$ is no longer Hermitian if the operators do not commute, and this makes it impossible for me to ask that question for the product. My question would break the preconditions of the formalism. But in $A + B$ the formalism allows to pose this question...

Update: In consequence of some comments I will try to specify my question more clearly: What is the physical meaning of the sum of two observables?

Obviously the "sum of two observables" is not the "sum of the values of the two observables". Assume that observable $A$ may have the values $2$ or $3$ and assume that observable $B$ may have the values $100$ or $200$ then the observable $A+B$ does not have the values $102$, $103$, $202$ or $203$ as a simple, naive approach might suggest or as an understanding of "sum of the values of the two observables" might suggest.

With this intuition failing, I would like to get an understanding of the physical meaning of $A + B$ starting from an understanding of $A$ and $B$.

Update 2: Adjusted the description of the pair experiment to a less misleading form.

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    $\begingroup$ This is a central part of QM; the core example is where $A$ is a particle's kinetic energy, and $B$ is its potential energy: both are trivial to diagonalize, but their sum rarely is. Basically all hard diagonalization problems boil down to sums of non-commuting operators where each individual term is easy or trivial to diagonalize. $\endgroup$ – Emilio Pisanty Aug 29 at 9:49
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    $\begingroup$ But that said, I really don't understand what the question really is here. $\endgroup$ – Emilio Pisanty Aug 29 at 9:54
  • $\begingroup$ SE posts are version controlled, so please do not make your post look like a revision table, instead just seamlessly integrate the new material into the post. There is an edit history button at the bottom of the post for those interested in seeing what changed. $\endgroup$ – Kyle Kanos Sep 11 at 11:31
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    $\begingroup$ the eigenvalues of $A+B$ can be written as convex combinations of the eigenvalues of $A$ and $B$, with the weights being dependent on the relations between their eigenvectors. So in some sense, $A+B$ is a measurement whose outcomes are something between those of $A$ and $B$, although to say more one would have to specify the relations between the eigenvectors of $A$ and $B$. Would this count as "physical meaning" of measuring $A+B$? $\endgroup$ – glS Sep 13 at 0:37
  • $\begingroup$ @gIS The hint on a convex combination is a useful mathematical remark, but it does not point us to an understanding of the physical aspect. Given an understanding of how $A$ and $B$ are measured in an experiment, how would I obtain an experiment for measuring $A + B$. So I really want to penetrate the physics aspect here, going beyond the mathematical side, and in particular for non-commuting $A$ and $B$. $\endgroup$ – Nobody-Knows-I-am-a-Dog Sep 19 at 21:05
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What you call "Naive approach" deal dit à two particules expérimentales (hence the espace $H\otimes H$) and what you describe is actually a measurement of $A+B$ as a shortcut for operator $A\otimes \mathbb{I}+\mathbb{I}\otimes B$, for the product state $\vert\phi\rangle\otimes \vert\phi\rangle$, where $\mathbb{I}$ is the identity of $H$. Its expectation value is $\langle \phi \vert A\vert\phi\rangle +\langle \phi \vert B\vert\phi\rangle$ ( using $\langle \phi \vert \mathbb{I}\vert\phi\rangle=1$) independently of the comparability or not of $A$ and $B$. As you have a product state, the measurement of $A$ on particle 1 can't influence the measurement of $B$ on particle 2.

If you now have to measure the sum $A+B$ for one particle, the prediction will involve conditional probablities like $p(b|a)$ (assuming $b$ is one outcome of $B$ and $a$ the outcome is $A$ previously obtained) and these simplify only when $A$ and $B$ commute. This does not provide an expetimemt to measure $A+B$ but explain why the expected result is different.

Mathematically, the successive measurement of $A$ and $B$ has an expectation value $$ \langle \phi \vert A+B\vert\phi\rangle = \sum_a p(a) \Big(a+ \sum_b p(b|a) b\Big) $$ If $P_a$ denote the projector on eigenspace of $A$ with eigenvalue $a$, $p(a)= \langle \phi \vert P_a\vert\phi\rangle$ and $p(b|a)=\displaystyle\frac{\langle \phi \vert P_a P_b P_a\vert\phi\rangle}{\langle \phi \vert P_a\vert\phi\rangle}$

After expansion, the first term is simply $\langle \phi \vert A\vert\phi\rangle$ and the second, using $\sum_b b P_b =B$ is $$ \sum_{a,b}\langle \phi \vert P_a P_b P_a\vert\phi\rangle b = \langle\phi \vert \tilde B\vert\phi\rangle \quad\text{with}\quad \tilde B = \sum_a P_a B P_a $$ Finally : $$ \langle \phi \vert A+B\vert\phi\rangle = \langle\phi \vert A\vert\phi\rangle + \langle\phi \vert \tilde B\vert\phi\rangle $$ This reduces to the naive value in the special case where $\vert\phi\rangle$ is an eigen state of $A$, or more interesting for any $\vert\phi\rangle$ if $A$ and $B$ are compatible: then $P_a$ and $B$ commute, and $\tilde B = B$.

Nevertheless, in this scheme, the output state will be an eigen vector of $B$ and not of $A+B$...

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If $Spec(A) = \{ a_1, a_2 \}$ and $Spec(B) = \{b_1, b_2\}$ then the pair experiment has spectrum $\{ a_1 + b_1, a_1 + b_2, a_2 + b_1, a_2 + b_2 \}$.

I don't think this is even formally correct. Take $A = s_z = \frac\hbar2\left(\begin{array}{}1&0\\0&-1\end{array}\right)$ and $B = s_x = \frac\hbar2\left(\begin{array}{}0&1\\1& 0\end{array}\right)$. Compute the spectrum of $A+B$ and verify that it doesn't satisfy your condition.

To answer the question in the context of this example: the physical meaning of $A+B$ is the spin measurement along the $\vec{n}_x + \vec{n}_z$ axis (multiplied by $\sqrt{2}$, to be pedantic)

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  • $\begingroup$ My description of the pair experiment was misleading, so I adjusted it. The pair experiment does not correspond to the mathematical operator $A + B$. The physical meaning of the mathematical operator $A + B$ is just as you describe and it is not the sum of two values obtained from experiments $A$ and $B$ but rather a completely different experiment. How experiment $A+B$ is obtained in the general case from the individual experiments $A$ and $B$ is exactly my question. $\endgroup$ – Nobody-Knows-I-am-a-Dog Sep 9 at 15:39

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