Entanglement breaking in self-referential quantum feedback: a record bound and an exactly solvable loop

Abstract

A feedback loop that returns a message into the process it describes defines a quantum channel on the message. We ask when such a loop preserves entanglement with a reference, that is, when its channel is not entanglement breaking. For any qubit loop, the negativity of the Choi state is at most half the fidelity between the states that the two values of the message leave in the discarded registers, in any basis. This channel analogue of the duality between fringe visibility and which-way information is attained by pure dephasing, and a perfect record of the message in the discarded registers makes the loop entanglement breaking. We then solve a three-qubit loop in which the message steers a future event, a probe reads the event, and a partial swap feeds the probe back. Every partial feedback is a strict contraction, and the loop is entanglement breaking exactly when an explicit trigonometric polynomial in its coupling angles is not positive. A partial record of the message in the future event can restore entanglement that the feedback alone destroys. Stability, a coherent fixed point, information about the future event and entanglement preservation coexist on about three quarters of the parameter space.