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In order to formulate my question properly I need to explain a few things.

Cramer_Herbert

Zych_Brukner

Reference 1. - John Cramer, Nick Herbert, "An Inquiry into the Possibility of Nonlocal Quantum Communication", (article can be found at arXiv, see attached file).

Reference 2. - M. Zych, F. Costa, I. Pikovski. C. Brukner, "Quantum interferometric visibility as a witness of general relativistic proper time", Nature Communications, 18 Oct. 2011 (see attached file).

In reference 1 (Cramer_Herbert), Cramer and Herbert consider an experimental design with entangled photons in a path entangled dual interferometer. Their conclusion is that the intrinsic complementarity between two - photon interference and one - photon interference blocks any potential nonlocal signal. Without the coincidence circuits no nonlocal signal can be transmitted from Alice to Bob (in this particular Alice-Bob EPR setup). In terms of density matrix formalism, nothing that happens at Alice's end has any effect on Bob's density matrix, even when Bob and Alice's photons are maximally entangled (due to unitary evolution - conservation of energy).

In reference 2 (Zych_Brukner), the experimental design involves a Mach - Zehnder interferometer in a gravitational field. They consider interference of a "clock" particle with evolving degrees of freedom (for example an electron with the "clock" being the spin precession) that will not only display a phase shift, but also reduce visibility of the interference pattern. According to general relativity, proper time flows at different rates in different regions of space - time. Because of quantum complementarity the visibility of the interference pattern will drop as the which path information becomes available from reading out the proper time of the "clock" going through the interferometer (gravitationally induced decoherence).

The experiment that I propose. Let's consider a path entangled dual interferometer experiment involving entangled particles (electrons, for example), when one MZ - interferometer is in a gravitational field. When we consider the density matrix of the system composed of the entangled particles in the two MZ interferometers, and when we consider the partial trace over system A (Alice's subsystem situated in a gravitational field), then we see that the interference visibility will also be affected for system B (Bob's subsystem). This opens the door for nonlocal signalling since Alice can send binary messages to Bob by moving her MZ - interferometer in and out of the gravitational field, and Bob using statistical analysis can decode Alice's message based on high or low visibility of his interference pattern (and no need for coincidence circuits). In this case the evolution of the system represented by the entangled particles going through the dual MZ interferometers in the presence of a gravitational field (for Alice's subsystem) is not unitary.

The question then can be formulated as:

Could this experimental design open the door for nonlocal signalling, and based on the relation between Lorentz invariance and causality, how can macroscopic causality be preserved in this context? Is macroscopic causality an issue in the context of these quantum experiments?

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The short answer is no, this experiment would not lead to nonlocal signalling, as long as quantum mechanics holds.

The reduced density matrix for B, after tracing over A, is completely mixed if you start with a maximally entangled state, irrespective of what Alice is doing (and Alice's density matrix is mixed too, so she won't see interference using her data alone). In this case you have to take into account 4 degrees of freedom: the two paths and the two clocks. Gravity effectively induces a unitary interaction between Alice's clock and path degrees of freedom, which leaves unchanged the reduced density matrix of Bob's colck and path degrees of freedom. (Note that this remains true also if you start with a non-maximally entangled or any other state.) What this interaction does is to create entanglement between Alice's clock and path and this, because of monogamy of entanglement, results in a reduction of the entanglement beteen Alice's and Bob's path, preventing two-particle interference (and without restoring one-particle interference, as the path degrees of freedom remain mixed throughout).

To summarize: in the experiment considered there is never one-particle interference. The presence of gravity can reduce two-particle interference by creating entanglement with the clock degrees of freedom. 4-partite interference can be reconstructed by also measuring the clocks.

Note that a photon version of the setup you propose was considered in http://arxiv.org/abs/1206.0965, Sec. VB.

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