Correlations and quantum circuits with dynamical causal order

TL;DR

The paper introduces a refined framework to classify correlations and quantum processes by their causal order, identifying a new dynamical yet non-influenceable class that emerges only for $N\ge 4$ parties. It defines four polytopes of causal correlations (convFO, NIO, NIO', causal) and proves strict inclusions among them, alongside an analogous hierarchy for quantum circuits using classical and quantum control of causal order (QC-convFO, QC-NICC, QC-CC, QC-supFO, QC-NIQC, QC-QC). By employing the process-matrix formalism, they provide explicit SDP-characterisations of various QC-CCs and QC-QCs, construct concrete examples (including NICC and NIQC processes), and analyze how dynamical and indefinite causal orders interplay to produce or constrain correlations. They show that while some quantum-circuit classes saturate the convFO or causal bounds on I4, others can violate convFO bounds, yet fully reach the larger NIO/NIO' bounds remains an open challenge. The work thereby clarifies when dynamicality and indefiniteness coincide or diverge and opens pathways to explore higher-party scenarios, alternative graph-based representations, and potential information-processing advantages of indefinite and dynamical causal orders.

Abstract

Requiring that the causal structure between different parties is well-defined imposes constraints on the correlations they can establish, which define so-called causal correlations. Some of these are known to have a "dynamical" causal order in the sense that their causal structure is not fixed a priori but is instead established on the fly, with for instance the causal order between future parties depending on some choice of action of parties in the past. Here we identify a new way that the causal order between the parties can be dynamical: with at least four parties, there can be some dynamical order which can nevertheless not be influenced by the actions of past parties. This leads us to introduce an intermediate class of correlations with what we call non-influenceable causal order, in between the set of correlations with static (non-dynamical) causal order and the set of general causal correlations. We then define analogous classes of quantum processes, considering recently introduced classes of quantum circuits with classical or quantum control of causal order -- the latter being the largest class within the process matrix formalism known to have a clear interpretation in terms of coherent superpositions of causal orders. This allows us to formalise precisely in which sense certain quantum processes can have both indefinite and dynamical causal order.

Figures (16)

• Figure 1: Illustration of the inclusion relations between the polytopes: ${\cal P}_\textup{convFO} \subset {\cal P}_\textup{NIO} \subset {\cal P}_{\textup{NIO}'} \subset {\cal P}_\textup{causal}$. In general (for $N\ge 4$) the inclusions are strict; for $N\le 3$ however, ${\cal P}_\textup{NIO}$ and ${\cal P}_{\textup{NIO}'}$ reduce to ${\cal P}_\textup{convFO}$; and for $N \le 2$, they all coincide. The polytopes are also all (in general strictly) included in the polytope of all correlations (all valid probability distributions), illustrated by the dashed lines.
• Figure 2: Graphical representation of a quantum circuit with classical control (QC-CC) of causal order (a simplified version of Fig. 9 from Ref. wechs21, also reproduced with more details as Fig. \ref{['fig:QCCC_graph_full']} in Appendix \ref{['app:subsec_QCCC_classes']}). The QC-CC is defined as the composition of the "internal" operations $\tilde{\mathcal{M}}_1,\ldots,\tilde{\mathcal{M}}_{N+1}$ (shown in the red shaded area), leaving "open slots" for the parties' "external" operations $\mathcal{A}_k$---or the CP maps $\mathcal{M}^{A_k}_{a_k|x_k}$, as we denote them in the main text, for a choice of setting $x_k$ and associated with a classical output $a_k$ (not shown on the figure). Each internal operation $\tilde{\mathcal{M}}_n$ (for $n\le N$) determines which party $A_{k_{n}}$ should act next; it transmits a target system from the previous party $A_{k_{n-1}}$ to the next party $A_{k_{n}}$ (or from the global past space $\mathcal{H}^P$ to $A_{k_1}$ for $\tilde{\mathcal{M}}_1$), potentially along with an auxiliary system (mid-height line on the figure), and updates the state of the classical control system (bottom double-stroke line) into the state $[\![(k_1,\ldots,k_{n-1},k_{n})]\!]\coloneqq {|(k_1,\ldots,k_{n-1},k_n) \rangle \!\langle (k_1,\ldots,k_{n-1},k_n)|}$, which keeps track of the ordered list of all external operations used so far. The last internal operation $\tilde{\mathcal{M}}_{N+1}$ just outputs a state in the global future space $\mathcal{H}^F$ (when the latter is nontrivial). According to this description, the causal order between the external operations is thus established on the fly or dynamically, as each internal operation is applied.
• Figure 3: The upper part of the figure depicts the inclusion relations between the different (sub)classes of QC-CCs considered in this paper. For $N\ge 4$ (and nontrivial input and output spaces for the different parties) the inclusions are strict; for $N\le3$, QC-NICC reduces to QC-convFO. The arrows directed to the lower part of the figure indicate the kind of causal correlations (defined through the generalised Born rule, Eq. \ref{['eq:Born_rule']}) that process matrices in each class can produce: process matrices in QC-convFO only give correlations in ${\cal P}_\textup{convFO}$; those in QC-NICC only give correlations in ${\cal P}_\textup{NIO}$; those in QC-CC give causal correlations in ${\cal P}_\textup{causal}$.
• Figure 4: A circuit-like representation of $W_{\textup{NICC}}$. $A$ and $B$ are each initially given half of a maximally entangled pair of qubits in the state ${|\Phi^+ \rangle}~=~\frac{1}{\sqrt{2}}({|0,0 \rangle} + {|1,1 \rangle})$. The global output state is then measured by the projective measurement $\{2W_{+\alpha},2W_{-\alpha}\}$. A new system initialised in the state ${|\psi \rangle}$ is then sent to $C$ and subsequently forwarded to $D$, or sent to $D$ and forwarded to $C$, with the order depending on the outcome of the previous measurement. The two branches of the circuit containing $C$ and $D$ are exclusive: on each run only one "happens", so that $C$ and $D$ (like $A$ and $B$) are each applied once and only once.
• Figure 5: Graphical representation of a quantum circuit with quantum control (QC-QC) of causal order (a simplified version of Fig. 10 from Ref. wechs21, also reproduced with more details as Fig. \ref{['fig:QCQC_graph_full']} in Appendix \ref{['app:subsec_QCQC_classes']}). The QC-QC is defined as the composition of the "internal" operations $\tilde{V}_1,\ldots,\tilde{V}_{N+1}$ (in the red shaded area), leaving "open slots" for the parties' "external" operations $A_k$. The construction is similar to that of QC-CCs (see Fig. \ref{['fig:QCCC_graph']}), except that here the control system (the now single-stroke bottom line) acts in a quantum manner, and can coherently control different causal orders, in a superposition. Furthermore, the state of the control system, in the form ${|\mathcal{K}_{n-1},k_{n} \rangle}$, only keeps track of the unordered set $\mathcal{K}_{n-1}$ of parties that have already acted, together with the next party $A_{k_n}$ to come. The causal order between the external operations can again be seen as being established on the fly or dynamically, as each internal operation is applied.