Abstract
When transforming pairs of independent quantum operations according to the fundamental rules of quantum theory, an intriguing phenomenon emerges: some such higher-order operations may act on the input operations in an indefinite causal order. Recently, the formalism of process matrices has been developed to investigate these noncausal properties of higher-order operations. This formalism predicts, in principle, statistics that ensure indefinite causal order even in a device-independent scenario, where the involved operations are not characterised. Nevertheless, all physical implementations of process matrices proposed so far require full characterisation of the involved operations in order to certify such phenomena. Here we consider a semi-device-independent scenario, which does not require all operations to be characterised. We introduce a framework for certifying noncausal properties of process matrices in this intermediate regime and use it to analyse the quantum switch, a well-known higher-order operation, to show that, although it can only lead to causal statistics in a device-independent scenario, it can exhibit noncausal properties in semi-device-independent scenarios. This proves that the quantum switch generates stronger noncausal correlations than it was previously known.
Highlights
A common quantum information task consists in certifying that some uncharacterised source is preparing a system with some features
We focus on quantum theory, indefinite causal order could in principle be certified even without relying on the laws of quantum mechanics
All three semidefinite programming (SDP) formulations we presented in eqs. (18), (19) and (34) are feasibility problems which can be turned into optimisation problems that allow for a robust certification of indefinite causal order
Summary
We will deal with statistical data in the form of behaviours. Following the same reasoning as the one in the definition of a general process matrix, in order to associate causal properties to process matrices we define causally ordered process matrices as the most general operator that takes pairs of local instruments to causally ordered behaviours, that is, pA≺B(ab|xy) = Tr (Aa|x ⊗ Bb|y) W A≺B , (12). [20, 18], which states that a bipartite process matrix W A≺B ∈ L(HAI ⊗ HAO ⊗ HBI ⊗HBO ) is causally ordered from Alice to Bob if it satisfies. In line with the definition of a causal behaviour, a causally separable process matrix W sep ∈ L(HAI ⊗ HAO ⊗ HBI ⊗ HBO ) is a process matrix that can be expressed as a convex combination of causally ordered process matrices, i.e., Wsep := qW A≺B + (1 − q)W B≺A,. Process matrices that do not satisfy eq (14) are called causally nonseparable process matrices
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