Expected value in usual quantum mechanics vs quantum information

Question

In standard Quantum Mechanics, one computes the expected value of an operator $A$ (arbitrary state $|\Psi\rangle$) as

$$ \langle\Psi|A|\Psi\rangle. $$

This has the virtue that we can compute for instance the average energy of the system $E=\langle \Psi|H|\Psi\rangle$ and also we can compute, if we want, the expected value of non-Hermitian operators like $\alpha = \langle \Psi|a|\Psi\rangle$ in the Harmonic oscillator. Now, in Quantum Information or Quantum Open Systems, one usually works with a complete set of measurement (Kraus) operators $\{ K_m\}$ that fulfill

$$ \sum_m K^\dagger_m K_m=\mathbb{I}. $$

Then, for instance, the majority of quantities that we can measure (if not all) will be Hermitian, since for instance

$$ \langle\Psi|a^\dagger a|\Psi\rangle. $$

On the other hand, as it is true that e.g. $a$ is not a Hermitian operator, in phase space techniques one manipulates such quantities theoretically and they are very useful for studying the system dynamics.

My questions are: can the set of measurement operators $\lbrace K_m\rbrace$ can at some point accommodate such non-Hermitian quantities? Why have a formalism that in principle "loses" a big amount of (theoretically) useful information?

Answer 1

You can measure the expectation value of any operator $O$ if you can measure hermitian operators, by decomposing it as $$ O=H+iA\ , $$ where $H=(O+O^\dagger)/2$ and A=$(O-O^\dagger)/2i$ are both hermitian, and measuring the expectation value of $H$ and $A$ individually.

Thus, any measurement formalism which allows you to measure hermitian observables allows you to measure any observable.

In particular, for the POVM measurement formalism you describe above (what you term "Kraus operators"), given the eigenvalue decompositions of $H$ and $A$, $$ H=\sum_{i=1}^n h_i|H_i\rangle\langle H_i|\,,\ A=\sum_{i=1}^n a_i|A_i\rangle\langle A_i|\,, $$ you can define a POVM measurement with operators $$\{K_k\}=\{\tfrac{1}{\sqrt{2}}|H_1\rangle\langle H_1|,\ldots,\tfrac{1}{\sqrt{2}}|H_n\rangle\langle H_n|, \tfrac{1}{\sqrt{2}}|A_1\rangle\langle A_1|,\ldots,\tfrac{1}{\sqrt{2}}|A_n\rangle\langle A_n|\}\ . $$ Given the outcome probabilities $p_k$, you can then compute $\langle O\rangle$ as $$ \langle O \rangle = \sum_{k=1}^n (p_k h_k + i p_{k+n} a_k)\ . $$

Finally, note that whenever $O$ is unitarily diagonalizable (but not necessarily hermitian - namely, this holds if and only if $O$ is normal), you can compute $\langle O\rangle$ by defining POVM operators $K_k$ as the projectors onto the eigenvectors of $O$, as above.