How Energy Conservation Limits Our Measurements

Observations in quantum mechanics are subject to complex restrictions arising from the principle of energy conservation. Determining such restrictions, however, has been so far an elusive task, and only partial results are known. In this Letter, we discuss how constraints on the energy spectrum of a measurement device translate into limitations on the measurements which we can effect on a target system with a nonconstant energy operator. We provide efficient algorithms to characterize such limitations and, in case the target is a two-level quantum system, we quantify them exactly. Our Letter, thus, identifies the boundaries between what is possible or impossible to measure, i.e., between what we can see or not, when energy conservation is at stake.

Figure 1

Bell experiment under energy-conserving interactions. Alice and Bob can violate the CHSH Bell inequality by means of beam splitters of varying transmittivity $T_A$, $T_B$, and shifters of phase ${\mathrm{\Phi }}_A$, ${\mathrm{\Phi }}_B$, if, besides the entangled state $|\phi ⟩$, they are distributed each a copy of the reference state $|+⟩$. The quantity $N_A^k$ ($N_B^k$) denotes the number of photons registered by Alice’s (Bob’s) $k$th detector.

Figure 2

Power states against coherent states. The plot shows the accuracy in reproducing $\mathcal{M}(2)$ for different values of the average energy $\overline{E}$ when our resource states are either the power states $|{\mathrm{\Psi }}_E^{\star }⟩$ or coherent states $|\alpha ⟩$.