Zhi Li

We ask how quantum correlations can be distributed among many subsystems. To address this, we define entanglement sharing schemes (ESS) where certain pairs of subsystems allow entanglement to be recovered via local operations, while other pairs must not. ESS schemes come in two variants, one where the partner system with which entanglement should be prepared is known, and one where it is not. In the case of known partners, we fully characterize the access structures realizable for ESS when using stabilizer states, and construct efficient schemes for threshold access structures, and give a conjecture for the access structures realizable with general states. In the unknown partner case, we again give a complete characterization in the stabilizer setting, additionally give a complete characterization of the case where there are no restrictions on unauthorized pairs, and we prove a set of necessary conditions on general schemes which we conjecture are also sufficient. Finally, we give an application of the theory of entanglement sharing to resolve an open problem related to the distribution of entanglement in response to time sensitive requests in quantum networks.

Approximate quantum error correction (AQEC) provides a versatile framework for both quantum information processing and probing many-body entanglement. We reveal a fundamental tension between the error-correcting power of an AQEC and the hardness of code state preparation. More precisely, through a novel application of the Lov\'asz local lemma, we establish a fundamental trade-off between local indistinguishability and circuit complexity, showing that orthogonal short-range entangled states must be distinguishable via a local operator. These results offer a powerful tool for exploring quantum circuit complexity across diverse settings. As applications, we derive stronger constraints on the complexity of AQEC codes with transversal logical gates and establish strong complexity lower bounds for W state preparation. Our framework also provides a novel perspective for systems with Lieb-Schultz-Mattis type constraints.