Causal-Order Identification of Memoryless Sequential Quantum Processes from Restricted Projective Data

TL;DR

This paper establishes necessary and sufficient conditions for determining the causal order of memoryless sequential quantum processes from limited measurement data, especially in the two-qubit Pauli setting.

Contribution

It introduces a complete criterion combining directional conditional independence, positivity, and algebraic consistency for identifying quantum causal order from restricted data.

Findings

Directional conditional independence alone is insufficient for causal identification.

Positivity based on pseudo-density matrices is not enough without algebraic consistency.

The criteria are explicitly characterized in the two-qubit Pauli setting, distinguishing memoryless from memoryful strategies.

Abstract

Identifying causal order from restricted projective data is generally nontrivial. When two quantum players interact only through an unobserved environment, the available local measurement statistics are typically not tomographically complete, so the underlying process cannot in general be reconstructed exactly from the observed distribution. As a result, causal direction can be statistically identifiable in some cases but fundamentally indistinguishable in others. In this work, we determine necessary and sufficient conditions for deciding when an observed distribution is compatible with a memoryless sequential quantum process in a fixed direction. We show that directional conditional-independence structure and the positivity criterion based on the pseudo-density matrix, as developed in recent work by Liu, Qiu, Dahlsten, and Vedral, are not sufficient by themselves. The missing ingredient is an additional algebraic consistency requirement, and together these conditions yield a complete criterion for membership in the memoryless sequential class. We then specialize to the two-qubit Pauli setting, where the problem remains non-tomographic but becomes explicitly tractable. In this regime, we characterize when the two sequential directions are statistically indistinguishable, and we show by example that positivity alone does not exclude more general memoryful strategies, whereas the additional algebraic consistency requirement does.