Quantum Temporal Superposition: The Case of Quantum Field Theory

Quantum field theory is completely characterized by the field correlations between spacetime points. In turn, some of these can be accessed by locally coupling to the field simple quantum systems, also known as particle detectors. In this letter we consider what happens when a quantum-controlled superposition of detectors at different space-time points is used to probe the correlations of the field. We show that, due to quantum interference effects, two detectors can gain information on field correlations that would not be accessible, otherwise. This has relevant consequences for information theoretic quantities, like entanglement and mutual information harvested from the field. In particular, the quantum control allows for extraction of entanglement in scenarios where this is, otherwise, provably impossible.

Figure 1

Left: scenario (i) in which the two detectors are in a superposition of being active, either both in the past or both in the future, controlled by the state of the control qubit $|0⟩$ or $|1⟩$, respectively. In this case, the spacetime activation regions for the two detectors are spacelike in both branches of the quantum superposition. Right: scenario (ii) with indefinite causal order. In this case, the two detectors are active when timelike related. The quantum superposition between “$A$ before $B$” and “$B$ before $A$” is controlled by the state of the control qubit $|0⟩$ or $|1⟩$, respectively. Note that, in the main text, scenario (ii) also encompasses the case of spacelike separated detectors where $A$ and $B$ are placed at different slices of constant time.

Figure 2

Left: perturbative violation of the no-go theorem in $2+1$ dimensions for detectors with spatial compact support and Dirac-delta window function. The spacetime configuration of the two detectors—corresponding to a PF scenario with detectors $A$, $B$ active at times $t_i^A$ and $t_i^B$ with $i=0$, 1 determined by the control qubit—is shown on the lower right. The disks of radius $\sigma$ represent the spatial compact support. The violation happens whenever $|\mathcal{M}|-{\mathcal{P}}_D^{(+)}>0$. We plot this quantity as a function of spatial separation $s$ and temporal separation $T$ between activations in the different branches of the superposition. No violation occurs if ${\mathcal{P}}_D^{(+)}$ is replaced with ${\mathcal{P}}_D^{(\mathrm{tr})}$. Values used for the parameters are $\mathrm{\Omega }=3,\sigma =1,\delta ={10}^{-5}$. Note that, when $T<s-2\sigma -\delta \wedge s>2\sigma \wedge \delta <s-2\sigma$ the activation regions of detector one are spacelike related to the ones of detector two. In the figure, this situation corresponds to the lower triangular area of the plot. Center: nonperturbative violation of the no-go theorem for Gaussian spatial profiles in $3+1$ dimensions with CE superposition. Right: nonperturbative violation of the no-go theorem for Gaussian spatial profiles in $3+1$ dimensions with PF superposition. In the center and right panels, black dots represent the regions where no entanglement can be harvested. The detectors have a window function $\chi (t)=\eta \delta (t-t_i^D)$, a smearing function of $S(\mathbf{x})=\{\mathbf{(}\exp [-(\mathbf{x}/9{)}^2]\mathbf{)}/(9\sqrt{\pi }{)}^3\}$, an energy gap of $\mathrm{\Omega }\eta =1$, and couple to the field with a strength of $\lambda =1$. The elements of the reduced density matrix have been numerically calculated to 15 significant figures.

Figure 3

The concurrence of the perturbative reduced density matrix of two point-like detectors that couple to the field with a cosine window function in a (left) quantum controlled CE superposition and (right) a classical mixture of $A$ before $B$ and $B$ before $A$ plotted as a function of the spatial separation $s$ and the temporal separation $T$ of the detectors. The green lines mark the region of spacelike separation of the detectors, and the black dots represent the regions where no entanglement can be harvested. For the chosen parameters, spacelike entanglement harvesting is possible only when the detectors are in the CE superposition. The energy gap of the detectors is $\mathrm{\Omega }\eta =3$. The elements of the reduced density matrix have been numerically calculated to 15 significant figures.