Decoherence of spin-entangled triplet-pair states in the solid state: local vs delocalized vibrations

Question

The context: We are in the solid state. After a photon absortion by a system with a singlet ground state, the system undergoes the spin-conserving fission of one spin singlet exciton into two spin triplet excitons (for context, see The entangled triplet pair state in acene and heteroacene materials). These spin triplet pair propagates in the solid, still entangled. The quantum-computing-related goal of all this operation would be to transfer the entanglement of the two flying qubits to two positions that are fixed in space and are also well protected from decoherence (low-energy excitations of nuclear spins in a paramagnetic ion, for example).

The problem at hand (2), and the question: Eventually, the entanglement between the two triplets is lost, and moreover inevitably the triplets find a way to relax back to the singlet ground state, emitting energy in form of photons. I would like to calculate how these processes are affected by vibrations. I assume the independent relaxation of each of the two triplets can be calculated mostly considering local vibrations, e.g. following a procedure similar to the one we employed here (Determining key local vibrations in the relaxation of molecular spin qubits and single-molecule magnets). Would the calculation of the loss of entanglement be necessarily related to delocalized vibrational modes that simultaneously involve the local environment of both triplets?

Answer 1

Let me go for a self-learner experience. After some reading, my short answer to my own question

Would the calculation of the loss of entanglement be necessarily related to delocalized vibrational modes that simultaneously involve the local environment of both triplets?

is: probably yes, but not necessarily/primarily.

A longer answer follows. With a previous familiarity with decoherence but non-familiarity with disentanglement, this paper was extremely helpful: Entanglement loss in molecular quantum-dot qubits due to interaction with the environment (Enrique P Blair et al, 2018, J. Phys.: Condens. Matter, 30, 195602). The physical scenario is not identical, but allows for a few key insights:

• Like coherence, entanglement is maintained by default, not by a process, which is to say: we only need to look for processes explicitly destroying it. This is whay one gets better numbers for entangled photons compared with solid state qubits, see What is the maximum separation between two entangled qubits that has been achieved experimentally?
• From the above point (and from the paper above), let us first consider the case where two qubits are far enough to avoid interaction among each other, and also to avoid interaction with a common environment. For thus isolated qubits just by accounting for decoherence we will fully account for disentanglement.
• Entanglement is exclusivity: entanglement between two parties is gradually lost as these parties entangle more and more with other parties. So, with entanglement between two qubits -as with coherence of one qubit- the primary focus of our attention should be how does the qubit interact with its environment. In the case under consideration: with the spin bath and the phonon bath. The same processes that destroy coherence will destroy entanglement, essentially at the same rate. For details, calculate fidelities and/or entanglement witnesses.
• If the two qubits are not perfectly isolated from each other, there is an interaction between them, which can be direct or via a common environment. In that case, the two qubits can experience a collective evolution which, beyond affecting their individual coherence, also alters their entanglement. This is what the question is asking, and here the answer would be a conditional yes. Collective vibrational modes affecting both qubits need to be considered, since they promote a collective evolution that can either create or destroy entanglement.