Below is a brief summary of my main research areas over the past two years.

## Symmetric Extendability of Quantum States

For a given bipartite quantum state ρ^{AB} (describing, say, the quantum systems held by Alice and Bob), does there exist another quantum system B′ such that the joint tripartite state ρ^{ABB′} satisfies Tr_{B′}[ρ^{ABB′}] = Tr_{B}[ρ^{ABB′}] = ρ^{AB}? If such a tripartite state exists, we call ρ^{AB} *symmetrically extendable *and ρ^{ABB′} a *symmetric extension*. What we are essentially asking here is whether there exists a *copy *of Bob’s quantum system that is indistinguishable from Alice’s point of view.

Determining whether a quantum state is symmetrically extendable is not as easy as it might initially seem. In fact, we can currently only do it analytically for two very special types of states, states on two qubits and states on two qudits with several symmetries. Fortunately, using semi-definite programming, we can determine the symmetric extendability of *any *quantum state.

Relevant papers: arXiv:1310.3530v2, arXiv:0906.5255v1, arXiv:0812.3667v1

## Extreme Limits of Quantum Key Distribution and Bound Secrecy

Symmetrically extendable states turn out to be extremely important for quantum key distribution. In QKD, we are interested in how much information any eavesdropper (call her Eve) might have about the secret key Alice and Bob are trying to create between them. Since they use quantum systems to do this by the very nature of QKD, the states describing these quantum systems can be used to determine how much secret key, if any, they can create.

It turns out that if the state ρ^{AB} describing Alice and Bob’s quantum systems is symmetrically extendable, then since the system B′ is a copy of Bob’s system and is not in Alice or Bob’s possession, we cannot rule out the possibility that it belongs to Eve. This means that Bob and Eve could be indistinguishable from Alice’s point of view, which means that Eve will know just as much as Bob does about the information Alice is communication during the secret key creation process. This means that *no *secret key can be created if we just allow for communication from Alice to Bob.

They might still be able to create a secret key if we allow for *two-way *communication, meaning that we allow Bob to communicate some information as well, which might break the symmetry between himself and Eve. This has been the subject of my Master’s thesis research, in which we considered a particular kind of QKD protocol in the case when Alice and Bob are symmetrically extendable and no protocol has been found to date. We have numerical evidence to suggest that two-way protocols don’t exist. If this turns out to be true, then not only would we have a resolution to this long-standing QKD problem, but we would also have a proof for the existence of *bound secrecy*, which is a kind of correlation that contains secrecy but that secrecy cannot be extracted into a secret key.

Relevant work: Geir Ove Myhr’s PhD Thesis, arXiv:0812.3607v2, My Master’s Thesis, arXiv:1612.07734v2