Time delocalization and causality across temporal quantum reference frames

Abstract

In relational quantum dynamics, evolution emerges via the correlations between some system of interest and a clock system, which plays the role of a temporal reference frame. Their combined state satisfies a Wheeler-de Witt-like constraint equation, and therefore does not evolve, leading to a ``block universe'' picture. Here we investigate the interplay of two aspects, namely temporal localization and causal relations, when comparing emergent dynamics with respect to different choices of clock. We first explore the extent to which two clocks can agree on the temporal localization of events. Then, focussing on the operational notion of causality, we require a clearly defined notion of interventions, i.e. quantum operations, and consider two different approaches to modeling these operations within relational dynamics. The first considers their application via the choice of solutions to the constraint equation, i.e.~the choice of which ``history'' is considered. The second approach incorporates the operations into the constraint equation itself and thereby into its solutions, giving a dynamical picture of the interventions. From the perspective of a single clock, both approaches allow for a notion of operational causality in relational dynamics. However, for multiple clocks, only the second approach gives a consistent picture regarding causal relations, while necessarily manifesting some degree of temporal delocalization between frames. Moreover, this second approach, when considering certain cases of temporal delocalization, naturally describes scenarios with indefinite causal order, a well-known quantum feature of operational causality.

Figures (7)

• Figure 1: The relationship between the kinematical, physical, and two conditional Hilbert spaces corresponding to the reference frames of clocks $C_i$ and $C_j$. To transform from the reference frame of $C_i$ to that of $C_j$, one first applies the inverse of the reduction map $\mathcal{R}_i(\tau_i)$ which gives the conditional states of $C_i$. This takes us to the Hilbert space of solutions to the constraint equation, namely $\mathcal{H}_{\rm phys}$. One then applies the reduction map $\mathcal{R}_j(\tau_j)$, which is used to obtain the conditional states with respect to clock $C_j$. In total this gives the frame change map $\mathcal{S}_{i\rightarrow j}(\tau_i ,\tau_j )$ according to Eq. \ref{['eq:framechange']}.
• Figure 2: Operational causality. Let a collection of quantum systems $A,B,\dots$ evolve according to the CPTP map $\mathcal{E}$. We say that the input system $A$ signals to the output system $B$ if there exist a local map $\mathcal{M}_A$ on $A$ and an observable $\mathcal{O}_B$ on $B$ such that the application of $\mathcal{M}_A$ can be detected via $\mathcal{O}_B$. Signaling can occur if the reduced state $\rho_B$ of the output on $B$ differs depending on whether $\mathcal{M}_A$ is applied or not.
• Figure 3: Operational causality within the Page-Wootters formalism: a) The solution to the constraint equation determines the unitary evolution $\mathcal{U}_{|C}$ relative to clock C. By tracing out the ancilla system $X$ we obtain the CPTP map $\mathcal{E}$ we want to study. b) The operation $\mathcal{M}_A$ at some initial time can be absorbed into the choice of kinematical state used to construct a particular solution to the constraint equation $| {\Psi^A} \rangle\rangle=\mathcal{R}^{-1}_C(t_i)\mathcal{M}_A \langle t_i|\Psi\rangle\rangle$ giving conditional states $| \psi^A_{|C}(t) \rangle=\langle t| {\Psi^A} \rangle\rangle=\mathcal{U}_{|C}(t-t_i)\mathcal{M}_A\langle t_i| {\Psi} \rangle\rangle$. The probabilities for the outcomes of observable $\mathcal{O}_B$ at time $t_f$ can then be calculated from the respective conditional state $| \psi^A_{|C}(t_f) \rangle$. Investigating causal properties then, in general, means comparing said probabilities for different operations $\mathcal{M}_A$ and $\mathcal{M}_{A'}$ respectively.
• Figure 4: System-localization of operations for multiple clock systems: If a quantum operation is applied to a single subsystem in one temporal reference frame, here $C_1$, in the reference frame of another clock, $C_2$, this quantum operation acts on multiple subsystems. The change of temporal reference frame $\mathcal{M}_{|C_2}=\mathcal{S}_{1\rightarrow 2}\mathcal{M}_{|C_1}\mathcal{S}^{-1}_{1\rightarrow 2}$ does, in general, not preserve the subsystem localization of operations.
• Figure 5: Timed interventions as part of the constraint operator. The action of a quantum operation on a given subsystems, here $A$, is preserved when changing from the reference frame of clock $C_1$ to the frame of clock $C_2$. However, due to the necessary time delocalization between the two clocks the operations are not be well-localized in time according to the clock which is not coupled to $A$ via the interaction term, here $C_2$.