Simulating Noncausality with Quantum Control of Causal Orders

TL;DR

This work investigates the physical realizability of noncausal classical processes by connecting the SHIFT measurement to the Lugano process. It demonstrates that SHIFT can be implemented using local operations with quantum control of classical communications, effectively simulating the Lugano noncausal structure with a quantum switch (QC-QC). The authors show that the SHIFT measurement certifies causal nonseparability rather than true noncausality in standard tripartite LOCC scenarios, while introducing the broader LOSupCC framework and a class of SupCC processes that realize similar measurements. They also discuss time-delocalized-subsystem pictures, purification routes, and prospects for photonic and network-DI certifications, clarifying how indefinite causal order can manifest in physically realizable devices and what it implies for interpretations of noncausal correlations.

Abstract

Logical consistency with free local operations is compatible with non-trivial classical communications, where all parties can be both in each other's past and future-a phenomenon known as noncausality. Noncausal processes, such as the "Lugano (AF/BW) process", violate causal inequalities, yet their physical realizability remains an open question. In contrast, the quantum switch-a physically realizable process with indefinite causal order-can only generate causal correlations. Building on a recently established correspondence [Kunjwal & Baumeler, PRL 131, 120201 (2023)] between the SHIFT measurement, which exhibits nonlocality without entanglement, and the Lugano process, we demonstrate that the SHIFT measurement can be implemented using a quantum switch of classical communications in a scenario with quantum inputs. This shows that the structure of the Lugano process can be simulated by a quantum switch and that successful SHIFT discrimination witnesses causal nonseparability rather than noncausality. Finally, we identify a broad class of "superposition of classical communications" derived from classical processes without global past capable of realizing similar causally indefinite measurements. We examine these results in relation to the ongoing debate on implementations of indefinite causal orders.

Figures (11)

• Figure 1: The LOPF scenario: three parties perform local (projective) operations $(M_{a|x}^A)_a,(M_{b|y}^B)_b,(M_{c|z}^C)_c$ on separated quantum systems (orange wires). The Boolean process function (blue comb) maps the parties' outputs $(a,b,c)$ into their respective inputs $(x,y,z)$, generating an effective projective measurement $(E_{a,b,c}^{ABC})_{a,b,c}$.
• Figure 2: The quantum switch realization of the SHIFT measurement: Alice and Bob perform classical measure and prepare operations on the process spaces that respectively define and are defined by their projective measurements $(M_{a|x}^A=\mathinner{|{a|x}\rangle\!\langle{a|x}|}^A)_a$ and $(M_{b|y}^B=\mathinner{|{b|y}\rangle\!\langle{b|y}|}^B)_b$ on their respective quantum systems (orange wires). The order of their operations is controlled by the quantum system in $\mathcal{H}^{PF}$ via a quantum switch (blue comb) $W_{QS}\in\mathcal{L}(\mathcal{H}^{PA_{IO}B_{IO}F})$ and Fiona's final measurement $(M_{f|a,b}^F=\mathinner{|{f|b(a\oplus 1)}\rangle\!\langle{f|b(a\oplus 1)}|}^F)_f$, determined by the communication of $a$ and $b$ through side classical channels. This set-up generates an effective set of projectors $(E_{f,a,b}^{PAB})_{f,a,b}$, the SHIFT measurement.
• Figure 3: Circuit implementing the classical channel underlying the Lugano process from the SHIFT measurement. (Fig. 2 from kunjwal23a). This circuit can be interpreted as a simulation of the noncausal Lugano process, which would be genuinely implemented if $x,y,z$ were respectively in the local pasts of $a,b,c$.
• Figure 4: From the Lugano process to the QC-QC: All the global past spaces of the purified Lugano process $W_L$ are initialized in state $\mathinner{|{0}\rangle}^{P_X}$, and the systems in $\mathcal{H}^{F_A}$ and $\mathcal{H}^{F_B}$ are traced out. Alice and Bob operations are unchanged: $M_{a|x}^A=\mathinner{|{a|x}\rangle\!\langle{a|x}|}^A$ with $\mathinner{|{a|x}\rangle}=H^x\mathinner{|{a}\rangle}$ and $M_{b|y}^B=\mathinner{|{b|y}\rangle\!\langle{b|y}|}^B$ with $\mathinner{|{b|y}\rangle}=H^y\mathinner{|{b}\rangle}$. The global future from the purification of Charlie's systems is left open, i.e. is identified as the global future of the process, $\mathcal{H}^{F_C}\equiv\mathcal{H}^{F}$. Charlie performs a $SWAP$ between his process systems and his auxiliary systems. His auxiliary input system, i.e. the SHIFT substate he receives, can be identified as the global past of the process, $\mathcal{H}^{C_O}\equiv \mathcal{H}^{P}$. The input received from the process is sent to Fiona $\mathcal{H}^{C_I}\equiv \mathcal{H}^{F_t}$, who measures it in the computational basis, giving outcome $z$. Fiona then measures the process global future system following $M_{f|z}^F=\mathinner{|{f|z}\rangle\!\langle{f|z}|}^F$ with $\mathinner{|{f|z}\rangle}=H^z\mathinner{|{f}\rangle}$.
• Figure 5: QC-QC realization of the SHIFT measurement

Theorems & Definitions (2)

• Definition 1
• Definition 2