Can we characterise how correlated the expectation values associated with a pair of observables are?

Consider a state $$\rho$$ and two observables $$P$$ and $$Q$$.

Is there a good way to characterise how correlated the associated expectation values are? Be it in terms of mutual information or something else.

As an example, if $$\rho$$ is a qubit and $$P_i=|i\rangle\!\langle i|$$ projectors onto the computational basis states, then we know that $$P_0$$ and $$P_1$$ are fully correlated, as $$\langle P_1\rangle = 1- \langle P_0\rangle$$. Knowing one tells the value of the other. On the other hand of the spectrum, if $$P_+\equiv |+\rangle\!\langle +|$$, then knowledge of $$\langle P_0\rangle$$ imposes the bound $$\frac12 - \sqrt{\langle P_0\rangle(1-\langle P_0\rangle)} \le \langle P_+\rangle \le \frac12 + \sqrt{\langle P_0\rangle(1-\langle P_0\rangle)}.$$ This can be seen starting from writing $$\rho=\frac12(I+\sum_k c_k \sigma_k)$$, observing that $$\langle P_0\rangle = (1+c_z)/2$$, $$\langle P_+\rangle = (1+c_x)/2$$, and $$c_z^2+c_x^2\le1$$, and thus concluding that $$|c_x|\le \sqrt{1-c_z^2}$$ with $$c_z=2\langle P_0\rangle-1$$. Thus, in the case of $$P_+$$, we have a continuous freedom compatible with an observed value of $$\langle P_0\rangle$$, albeit with constrained bounds.

Can similar characterisations be made in the general case of an arbitrary pair of observables $$P$$ and $$Q$$? To be more precise, I'm asking, given $$\alpha=\langle P\rangle$$, what is $$\{\langle Q\rangle_\rho\equiv \mathrm{Tr}(\rho Q) : \rho\in\mathrm D(\mathcal X)\text{ such that } \langle P\rangle_\rho=\alpha\},$$ where $$\mathrm D(\mathcal X)$$ is the set of all states in the relevant Hilbert space $$\mathcal X$$.

• Isn't this basically the commutator of the two observables? Also, have you looked at derivations of the Heisenberg Uncertainty Principle? Nov 25 '20 at 16:39
• You might also find entropic uncertainty relations interesting. Nov 25 '20 at 16:48
• @DaftWullie I mostly understand the HUP, in its more general form, as the statement $\sigma_A\sigma_B\ge \lvert \langle AB\rangle - \langle A\rangle\langle B \rangle\rvert$. Sure it relates to this problem, but how do you use this to find what you know about $\langle A\rangle$ given your knowledge of $\langle B\rangle$? I'm also not quite sure it holds in this form for general states. I mean I guess it does, but the derivation relies on the observation $\sigma_A^2=\| (A-\langle A\rangle I)|\psi\rangle\|^2$, so the proof might be a bit different for $\rho$ nonpure
– glS
Nov 25 '20 at 20:21
• @Rammus those are certainly interesting, but I still don't quite see the relation with what I'm asking. Those, as far as I understand, are different ways to formulate the HUP, which tells you, for a given state, how the variances/entropes of different observables relate to each other. Here I'm instead asking how observables' expvals are related to each other independently on the underlying state. In other words, I'm asking, given $\langle A\rangle$, what is $\{\langle B\rangle_\rho : \rho\}$ where $\rho$ ranges over all possible states compatible with the observation $\langle A\rangle$
– glS
Nov 26 '20 at 9:43