Paradox-free classical non-causality and unambiguous non-locality without entanglement are equivalent

TL;DR

This work establishes a precise equivalence between paradox-free, deterministic classical process functions and unambiguous complete product bases, unifying two formalisms for studying nonstandard causal structures. It provides a complete recursive characterization of process functions and their (non-)causal nature, showing that non-causality corresponds exactly to quantum nonlocality without entanglement in the associated bases. The authors prove two key theorems: every unambiguous complete product basis yields a valid process function, and every process function can be encoded into such a basis, enabling systematic construction of non-causal PFs and QNLWE bases. The results also reveal a direct link between certain non-signaling and causal inequalities, with practical implications for constructing and certifying noncausal classical resources and their quantum-counterpart analogs. Overall, the work deepens the understanding of indefinite causal order in classical and quantum settings and offers tools for exploring new nonlocality-inspired resources without entanglement.

Abstract

Closed timelike curves (CTCs) challenge our conception of causality by allowing information to loop back into its own past. Any consistent description of such scenarios must avoid time-travel paradoxes while respecting the no-new-physics principle, which requires that the set of operations available within any local spacetime region remain unchanged, irrespective of whether CTCs exist elsewhere. Within an information-theoretic framework, this leads to process functions: deterministic classical communication structures that remain logically consistent under arbitrary local operations, yet can exhibit correlations incompatible with any definite causal order - a phenomenon known as non-causality. In this work, we provide the first complete recursive characterization of process functions and of (non-)causal process functions. We use it to establish a correspondence between process functions and unambiguous complete product bases, i.e., product bases in which every local state belongs to a unique local basis. This equivalence implies that non-causality of process functions is exactly mirrored by quantum nonlocality without entanglement (QNLWE) - the impossibility of perfectly distinguishing separable states using local operations and causal classical communication - for such bases. Our results generalize previous special cases to arbitrary local dimensions and any number of parties, enable systematic constructions of non-causal process functions and unambiguous QNLWE bases, and reveal an unexpected connection between certain non-signaling inequalities and causal inequalities.

Figures (4)

• Figure 1: In the process function scenario, $n$ isolated parties each can perform free local interventions $f_k$ while conducting single-round communication that is modeled by the process function $w$, which combines the classical channel $\text{\Large{\calligra w }}$ and retro-causal identity channels. Each party $k$ receives an external input $i_k$ and an incoming variable $x_k$ from $w$. For a given $f_k$, each party then produces the external output $o_k$ as well as an output $a_k$ which is fed back into $w$.
• Figure 2: Visualization of the reduced process function as defined in Eq. \ref{['eq:red_pf']} for the case of fixing $f_1$, a local operation of the first party.
• Figure 3: Circuit implementing the classical channel $\text{\calligra w}$ underlying the process function $w$ from the projective measurement on $\mathcal{S}$. For an unambiguous QNLWE basis $\mathcal{S}$, this circuit can be interpreted as a simulation of the non-causal process function, which would be genuinely implemented if each $x_k$ were respectively in the local pasts of $a_k$ (cf. Fig.\ref{['fig_pf']}).
• Figure 4: The LOPF scenario: $n$ parties perform local (projective) operations $(M_{a_k|x_k}^{A_k})_{a_k}$ for each $k$, on separated quantum systems $\mathcal{H}^{A_k}$ (orange wires). The process function $w$ (blue comb) maps the parties'outputs $\bm{a}$ into their respective inputs $\bm{x}$. This set-up generates an effective measurement $(E_{\bm{a}}^{\bm{A}})_{\bm{a}}$ (cf. Eq. \ref{['eq:dpvm']}). In Theorem \ref{['thm:pftob']}, we demonstrate that when for each $k$, $M_{a_k|x_k}^{A_k} := \mathinner{|{(a_k|x_k)}\rangle\!\langle{(a_k|x_k)}|}^{A_k}$, this measurement is precisely a projection on an unambiguous complete product basis $\mathcal{S}$.

Theorems & Definitions (28)

• Lemma 1
• Theorem 2
• proof
• Lemma 3
• proof
• Theorem 4
• proof
• Lemma 5: Non-self-signaling
• proof
• Theorem 6