Approaches to Quantum Gravity

From quantum reference frames to deformed special relativity 515Relativity, it is not clear how QG could help to solve this problem. Different interpretationsto QM favor better understanding of the measurement procedure (thoughin general not solving it). In particular, treating QM as a theory about information(QIT) allows us to describe nicely what is a measurement in the presence of aquantum reference frame. This has been analyzed by [22] in the qubits universe.Let me recall the construction quickly in the case of the measurement of a qubitwith respect to another qubit. Since we have the tensor product of two spins 1 2 ,following the Schur lemma, 8 it is natural to decompose any measurement E λ withoutcome λ along the projectors 0,1 in the basis 0 ⊕ 1 ∼ 1 2 ⊗ 1 . The projectors2 0,1 are observable, that is invariant under global rotations:E λ = a λ,1 1 + a λ,0 0 ,where the coefficients a λ,i satisfy the necessary conditions to make an E λ a projectoroperator valued measurement (POVM) [23]. To be in the eigenspace of one ofthe projectors 12 ± 1 tells us if the spins are aligned or anti-aligned.2The idea is now to use the Bayes theorem, which from a prior distribution ofknowledge describes how to update it. One starts with a prior distribution p(α) onα. Upon obtaining the outcome λ, we can update our knowledge from the priordistribution p(α) to p λ (α) = p(α|λ):p(α|λ) = Tr(E λ ρ α ) p(α)p(λ) , (26.4)with ρ α a physical state that is rotationally invariant, and p(λ) = ∫ Tr (E λ ρ α )p(α)dα.♦Is our quantum reference frame robust?In the classical case, a reference frame can happen to be not a good referenceframe globally. This is related to the problem of invertibility of the partial observableas argued in the previous section. More physically a clock can for exampledecay, lose its precision, owing to various interactions with its environment. In thequantum case we can have some similar situations. For example, by making manyconsecutive measurements the QRF will get blurred since in general the QRF getsentangled with the system. Once again this has been analyzed in the context of QIT[24; 25]. For example after one measurement, forgetting about the outcome of theprevious measurement, one has the new QRF stateρ (1)RF = Tr S∑ a ρ RF ⊗ ρ S a .a=0,18 Since we want to make a physical measurement, that is RE λ R −1 = E λ , for any global rotation R.