Quantum to classical transitions in causal relations

The landscape of causal relations that can hold among a set of systems in quantum theory is richer than in classical physics. In particular, a pair of time-ordered systems can be related as cause and effect or as the effects of a common cause, and each of these causal mechanisms can be coherent or not. Furthermore, one can combine these mechanisms in different ways: by probabilistically realizing either one or the other or by having both act simultaneously (termed a physical mixture). In the latter case, it is possible for the two mechanisms to be combined quantum coherently. Previous work has shown how to experimentally realize one example of each class of possible causal relations. Here, we make a theoretical and experimental study of the transitions between these classes. In particular, for each of the two distinct types of coherence that can exist in mixtures of common-cause and cause-effect relations—coherence in the individual causal pathways and coherence in the way the causal relations are combined—we determine how it degrades under noise and we confirm these expectations in a quantum-optical experiment.

Figure 1

Classes of causal relations. A causal relation that is either cause-effect, common-cause, or a combination of both can be classified according to two criteria. The more familiar question is whether the mechanisms that realize the common-cause and cause-effect relations are themselves classical (C) or quantum (Q)—that is, whether the common cause produces separable or entangled states and whether the cause-effect relation is a coherence-breaking or coherence-preserving channel. Another distinction is whether the two mechanisms are combined in a probabilistic mixture (Prob), a physical mixture (Phys), or in a way that is intrinsically quantum (Coh). (Formal definitions are provided in the text.) These distinctions give rise to five categories of interest, as shown above. (i)–(iii) Our experiment explores the transition from the most restrictive category, Coh, to ProbQ and PhysC (solid arrows), as well as the classical transition from PhysC to ProbC (dashed arrow).

Figure 2

General causal relation between two quantum systems. (a) When the causal relations are represented as a directed acyclic graph, with arrows representing causal influences, one can distinguish the cause-effect pathway and the common-cause pathway. (b) In order to fully characterize the general causal relation between A and B, the single system A is replaced by two distinct systems: C shares a common-cause relation with B, and D has a cause-effect relation to B. (c) The general causal relation between A and B is realized by the circuit fragment inside the dashed box, with C and D representing an output and input, respectively. The label E denotes the system that mediates the influence of the common cause on B.

Figure 3

Optical implementation for exploring causal relations. The initial preparation (yellow) targets the maximally entangled state $|{\mathrm{\Phi }}^{+}\rangle$ from Eq. (4). The gate (green) allows the implementation of different ways of combining a common-cause relation (between C and B) with a cause-effect relation (between D and B). Measurements of C and B and repreparations of D (blue) are used to characterize the causal relation. We explore the transitions (i) Coh to ProbQ, where we fix the dephasing probability ($p=0$) and phase ($\theta =\pi /2$) and vary the delay $\tau$ of the photon at E; (ii) Coh to PhysC, where we fix the dephasing directions along $(\widehat{x},\widehat{y},\widehat{z})$ and vary the phase $\theta$ (by tilting the glass plates in the interferometer) and the dephasing probability $p$ (by switching the LCRs on with probability $p$); (iii) PhysC to ProbC, where we fix the dephasing probability ($p=1$) and phase ($\theta =\pi /2$) while interpolating from $({\widehat{n}}_E,{\widehat{n}}_D,{\widehat{n}}_B)=(\widehat{z},\widehat{z},\widehat{z})$ to $(\widehat{x},\widehat{y},\widehat{z})$ using a single-parameter family of gates [see Eq. (10)]. Notation for optical elements: half-wave plate (HWP), quarter-wave plate (QWP), liquid-crystal retarder (LCR), polarizing beam splitter (PBS), nonpolarizing beam splitter (NPBS), avalanche photo diode (APD).

Figure 4

Transition from Coh to ProbQ. As the delay $\tau$ between photons is increased, quantum interference at the gate is gradually lost. We observe: (a) the witness of physical mixture ${\mathcal{C}}_{CD}$; (b) the negativity ${\mathcal{N}}_{BD}$ under postselection on eigenstates of ${\sigma }_X$ on C; (c) the negativity ${\mathcal{N}}_{CB}$ under preselection on eigenstates of ${\sigma }_Y$ on D; and (d) the negativity ${\mathcal{N}}_{CD}$ under postselection on eigenstates of ${\sigma }_Z$ on B. (b), (c) Nonzero values of ${\mathcal{N}}_{BD}$ and ${\mathcal{N}}_{CB}$ herald quantumness in the common-cause and cause-effect pathways, which is preserved throughout the transition, while (a), (d) the quantum Berkson effect is suppressed before the witness of physical mixture reaches zero. For intermediate values, between $\tau =0.27\mathrm{ps}$ ($q=0.23$) and $\tau =0.39\mathrm{ps}$ ($q=0.05$), we realize a causal map that belongs to PhysQ, but observe no evidence that it belongs to Coh. For larger delays, there is no evidence of physical mixture (${\mathcal{C}}_{CD}=0$), making the data consistent with ProbQ. Solid lines represent theoretical predictions in the ideal case (red) and taking into account the experimental implementation (blue), as described in the text.

Figure 5

Classifying a family of causal relations: transition from Coh to PhysC. (a) Witness of physical mixture ${\mathcal{C}}_{CD}$, (b) negativity ${\mathcal{N}}_{BD}$ for measurements of ${\sigma }_X$ eigenstates on C, (c) negativity ${\mathcal{N}}_{CB}$ for preparations of ${\sigma }_Y$ eigenstates on D, and (d) negativity ${\mathcal{N}}_{CD}$ for measurements of ${\sigma }_Z$ eigenstates on B. For each witness, we present: (left) theoretical predictions, which are identical for selection on either eigenstate; (middle) experimental data under selection on the $+1$ eigenstate; (right) a two-dimensional cross section through the contour plot at $\theta =92.8^{\circ }$, comparing experimental data for both selections (blue circles and triangles) with theoretical expectations assuming no experimental imperfections (red curve) and with theoretical expectations given an estimate of the experimental imperfections inferred from the tomographically reconstructed causal map realized at $p=0$ (blue curve). We observe: (a) The witness of physical mixture remains constant throughout the transition. (b)–(d) Quantumness in the common-cause and cause-effect pathways persists longer than quantumness in the way that the two mechanisms are combined. As we range from purely common-cause ($\theta =\pi$) to purely cause-effect ($\theta =0$), the signature of coherence in the cause-effect pathway increases, whereas the signature of coherence in the common-cause pathway diminishes.

Figure 6

Transition from PhysC to ProbC. A nonzero value of the witness ${\mathcal{C}}_{CD}$ heralds a physical mixture of causal structures. The witness is evaluated for different dephasing directions parametrized by Eq. (10), with the extremes, $\eta =0$ and $\eta =\pi /4$, corresponding to settings for our examples of PhysC and ProbC, respectively. For $\eta =\pi /4$, we measure ${\mathcal{C}}_{CD}=0.01\pm 0.02$, which is compatible with a probabilistic mixture, whereas $\eta =0$ produces ${\mathcal{C}}_{CD}=0.44\pm 0.02$, demonstrating with high confidence a nontrivial physical mixture of cause-effect and common-cause mechanisms. The solid lines indicate theoretical predictions assuming no experimental imperfections (red) and assuming the initial experimental implementation and dephasing operation inferred by comparing the causal maps with and without dephasing (blue).