Quantum Physics

It is well known that the derivation of the Bell Inequality rests on two major assumptions, usually called outcome independence and parameter independence. Parameter independence seems to have a straightforward motivation: it expresses a non-signalling requirement between space-like separated sites and is thus motivated by locality. The status of outcome independence is much les clear. Many authors have argued that this assumption too expresses a locality requirement, in the form of a \'screening off\' condition. I will argue that the assumption also admits of an entirely different interpretation, suggested by the concept of sufficiency in the general theory of statistical inference. In this view, the assumption of outcome independence can be explained as expressing the idea that the specification of the hidden variable is sufficient, i.e. it exhausts all the relevant statistical information about the measurement outcomes. In this view, the assumption has no roots in locality at all. Rather, I would claim, it stems from the assumption that there exists such an exhaustive state description in our putative hidden variable theories.

The focus of this talk is a particular feature of the statistical behavior of elementary particles, simple composite systems of them and the quantum probability theory to which this behavior gives rise. The standard interpretation of a generalized probability theory of the sort found in quantum mechanics is that its probabilities are probabilities of propositions belonging to particles, where a proposition belongs to a particle if its constituent dynamical property is a possible property of the particle. The feature of interest is the fact that there exist simple systems and finite combinations of propositions belonging to them for which no two-valued measures are possible. I will argue that quantum probabilities are not satisfactorily interpretable as probabilities of propositions belonging to particles, and that such an interpretation is possible only when the propositions to which probabilities are assigned form (an algebraic structure which is homomorphic to) a Boolean algebra. The idea I will develop is that the probabilities of quantum mechanics are probabilities of “effects,” probabilities of the traces of particleinteractions with objects and processes that are epistemically accessible to us. I hope to make it clear that such a view is not committed to any kind of anti-realism about the micro-world, that its mildly instrumentalist flavor is not a defect but a strength, and that it illuminates at least one otherwise paradoxical feature of quantum mechanics.

In an effort to better understand the class of operations on a bipartite system which preserve positivity of partial transpose (PPT operations), we have investigated the (non-asymptotic) transformation of pure states to pure states by operations in this class. Under local operations and classical communication (LOCC) Nielsen\'s majorization criterion provides a necessary and sufficient condition for such a transformation. This can be used to show that under LOCC a phenomenon called catalysis can occur, where an otherwise impossible transformation can be made possible by the provision of an entangled catalyst state, which must be recovered unchanged after the transformation (hence the name). I will present some recent work where we have found a necessary condition for obtaining a given pure state from a maximally entangled state via PPT operations. This condition is conjectured to be sufficient also, and we can prove this for the case where the goal state has Schmidt rank three. We have also shown that catalysis occurs under PPT operations, and have derived a necessary and sufficient condition for PPT pure state transformations where both the initial state and the catalyst are maximally entangled.

What does a fractional quantum Hall liquid and Kitaev\'s proposals for topological quantum computation have in common? It turns out that they are physical systems that exhibit degenerate ground states with properties seemingly different than ordinary (Landau-type) vacua, such as the ground states of a Heisenberg magnet. For example, those (topologically quantum ordered)states cannot be characterized by (local) order parameters such as magnetization. How does one characterize this new order? I will present a unifying framework which will allow us to engineer physical systems displaying topological quantum order. What are the physical properties of these new orders? How robust are they to temperature effects? What are they useful for? Topologically quantum ordered states of matter seem to be ideal physical systems to store and manipulate quantum information since they are believed to be robust against decoherence with an environment, and thus appropriate for building a quantum computer and quantum memories. I will discuss the role of temperature in the protection of quantum information. Have we finally found a technological application for quantum Hall liquids?

Constructing good quantum LDPC codes remains an important problem in quantum coding theory. We contribute to the ongoing discussion on this topic by proposing two approaches to constructing quantum LDPC codes. In the first, we rely on an algebraic method that uses a redundant description of the parity check matrix to overcome the problem of 4-cycles in the Tanner graph that degrade the performance of iterative decoding. In the second we use the fact that subsystem coding can simplify the decoding process. We show that if there exist classical LDPC codes with large error exponents, then we can construct degenerate subsystem LDPC codes with the stabilizer generators having low weight.

We investigate which families of quantum states can be used as resources for approximate and/or stochastic universal measurement-based quantum computation, in the sense that single-qubit operations and classical communication are sufficient to prepare (with some fixed precision and/or probability) any quantum state from the initial resource. We find entanglement-based criteria for non-universality in the approximate and/or stochastic case. By applying them, we are able to discard some families of states as not universal also in this weaker sense. Finally, we show that any family $Sigma$ of states that is \'close\' to an (approximate and/or stochastic) universal family $Gamma$ is approximate and stochastic universal, and we prove that if $Gamma$ was efficiently universal then also $Sigma$ is.

Quantum computers provide new resources to solve combinatorial optimization problems (COPs). Using techniques borrowed from quantum information theory, I will present a quantum algorithm that simulates classical annealing processes, where the (quantum) annealing rate greatly outperforms other classical methods like Markov chain Monte-Carlo based algorithms. Our quantum algorithm provides quadratic speedups to find both, the solution to particular instances of COPs, and the preparation of (quantum) Gibbs\' states.

I will present a new protocol that was developed entirely in the measurement-based model for quantum computation. Our protocol allows Alice to have Bob carry out a quantum computation for her such that Alice\'s inputs, outputs and computation remain perfectly private, and where Alice does not require any quantum computational power or memory. Alice only needs to be able to prepare single qubits from a finite set and send them to Bob, who has the balance of the required quantum computational resources. Our protocol is interactive: after the initial preparation of quantum states, Alice and Bob use two-way classical communication which enables Alice to drive the computation, giving single-qubit measurement instructions to Bob, depending on previous measurement outcomes. Our protocol is efficient and is presented for the special case of a classical-input, and classical-output; modifications allow the general case of quantum inputs and outputs. We also discuss the use of authentication in order for Alice to detect an uncooperative Bob. Based on joint work with Joseph Fitzsimons and Elham Kashefi