We develop a non-adiabatic generalization of holonomic quantum computation in which high-speed universal quantum gates can be realized using non-Abelian geometric phases. We show how a set of non-adiabatic holonomic one-and two-qubit gates can be implemented by utilizing optical transitions in a generic three-level configuration. Our scheme opens up the possibility of realizing universal holonomic quantum computation on qubits characterized by short coherence time.

Unpredictability, or randomness, of the outcomes of measurements made on an entangled state can be certified provided that the statistics violate a Bell inequality. In the standard Bell scenario where each party performs a single measurement on its share of the system, only a finite amount of randomness, of at most 4 log 2 d bits, can be certified from a pair of entangled particles of dimension d. Our work shows that this fundamental limitation can be overcome using sequences of (nonprojective) measurements on the same system. More precisely, we prove that one can certify any amount of random bits from a pair of qubits in a pure state as the resource, even if it is arbitrarily weakly entangled. In addition, this certification is achieved by near-maximal violation of a particular Bell inequality for each measurement in the sequence.Introduction.-Bell's theorem [1] has shown that the predictions of quantum mechanics demonstrate non-locality. That is, they cannot be described by a theory in which there are objective properties of a system prior to measurement that satisfy the no-signalling principle (sometimes referred to as "local realism"). Thus, if one requires the no-signalling principle to be satisfied at the operational level then the outcomes of measurements demonstrating non-locality must be unpredictable [1][2][3]. This unpredictability, or randomness, is not the result of ignorance about the system preparation but is intrinsic to the theory.Although the connection between quantum non-locality (via Bell's theorem) and the existence of intrinsic randomness is well known [1][2][3][4] it was analyzed in a quantitative way only recently [5,6]. It was shown how to use nonlocality (probability distributions that violate a Bell inequality) to certify the unpredictability of the outcomes of certain physical processes. This was termed device-independent randomness certification, because the certification only relies on the statistical properties of the outcomes and not on how they were produced. The development of information protocols exploiting this certified form of randomness, such as deviceindependent randomness expansion [5][6][7] and amplification protocols [8,9], followed.Entanglement is a necessary resource for quantum nonlocality, which in turn is required for randomness certification. It is thus crucial to understand qualitatively and quantitatively how these three fundamental quantities relate to one another. In our work, we focus on asking how much certifiable randomness can be obtained from a single entangled state as a resource. Progress has been made in this direction for entangled states shared between two parties, Alice (A) and Bob (B), in the standard scenario where each party makes a single measurement on his share of the system and then discards it. An argument adapted from Ref. [10] shows that either of the two parties, A or B can certify at most 2log 2 d bits of randomness [11], where d is the dimension of the local Hilbert space the state lives in, which in turn implies a bound of 4log 2 d b...

We investigate the relation between two approaches to the characterization of quantum Markovianity, divisibility and lack of information backflow. We show that a bijective dynamical map is completely positive divisible if and only if a monotonic nonincrease of distinguishability is observed for two equiprobable states of the evolving system and an ancilla. Moreover, our proof is constructive: given any such map that is not completely positive divisible, we give an explicit construction of two states that, when taken with the same a priori probability, exhibit information backflow. Finally, while an ancilla is necessary for the equivalence to hold in general, we show that it is always possible to witness the non-Markovianity of bijective maps without using any entanglement between the system and ancilla.