Partial independence suffices to rule out Real Quantum Theory experimentally

TL;DR

This work investigates whether the real-valued quantum framework can explain experimental correlations previously shown to contradict Real Quantum Theory under the assumption of fully independent sources. By relaxing independence to partial correlations, the authors derive a tradeoff between the amount of source correlation and the Bell value achievable in Real Quantum Theory, and they provide a constructive simulation showing that with one rebit per source, Real QM can reproduce Complex Quantum correlations in networks with independent sources. They establish that the ideal maximal quantum Bell score self-tests a specific entangled two-qubit state and quantify the minimal entanglement (in particular, $E_F=1$ for two rebits) required among sources to simulate complex correlations; they further develop a semidefinite programming hierarchy to bound the Bell value in non-ideal settings. The results imply that Real Quantum Theory cannot explain observed correlations unless a nontrivial amount of pre-shared entanglement is present among sources, thereby supporting Complex Quantum Theory in realistic scenarios, while also providing a universal Real-quantum simulation framework using delocalized reference frames. The work thus clarifies the boundary between Real and Complex quantum descriptions in networked experiments and offers both quantitative bounds and a practical simulation strategy for real-world tests of quantum theory.

Abstract

The role of complex quantities in quantum theory has been puzzling physicists since the beginnings. It is thus natural to ask whether, in order to describe our experiments, the mathematical structure of complex Hilbert spaces it is built on is really necessary. Recently, it was shown that this structure is inevitable in network scenarios with independent sources. More precisely, Real Quantum Theory cannot explain the predictions of (Complex) Quantum Theory [Renou et al., Nature 600, 2021]. Here, we revisit the independence assumption underlying this work. We show that assuming partial independence is sufficient for showing the inadequacy of Real Quantum Theory. We derive a tradeoff between source independence and the Bell value achievable in Real Quantum Theory, which also lower bounds the source correlations required to explain previous experiments by means of real quantum systems. We further show that 1 bit of entanglement is necessary and sufficient for recovering the complex quantum correlations by means of Real Quantum Theory in the scenario from [Renou et al., Nature 600, 2021]. Finally, building on [McKague et al., PRL 102, 2009], we provide a construction to simulate any complex quantum setup with m independent sources by means of Real Quantum Theory, by allowing the sources to share a m real-qubit entangled state in the first round of the experiment.

Figures (1)

• Figure 1: The setup. There are three parties Alice, Bob and Charlie. Alice and Charlie choose inputs $x, z$ and obtain outcomes $a,c$ respectively. Bob only obtains an outcome $b$. In the original bilocalilty scenario (black), according to (Real) Quantum Theory, two independent sources $S_1$ and $S_2$ distribute (real) quantum states $\rho_{AB_1}$ and $\rho_{B_2 C}$ to Alice with Bob and Bob with Charlie, respectively. The parties then perform (real) POVMs $\{ A_{a}^x \}_a$, $\{ B^{b} \}_b$ and $\{ C_z^{c} \}_c$ on their respective subsystems, leading to the distribution $P(a,b,c|x,z)= \hbox{tr}( (A_x^{a} \otimes B^{b} \otimes C_z^{c})( \rho_{AB_1} \otimes \rho_{B_2 C} ))$. In addition, we here allow the sources to initially share some (real) quantum correlations represented by the (real) quantum state $\varrho_{S_1S_2}$ (cyan), such that the state $\rho_{AB_1B_2 C}$ distributed by the source must no longer be product, nor separable, with respect to the bipartition $AB_1|B_2C$.

Theorems & Definitions (5)

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