Quantum Nonlocal Boxes Exhibit Stronger Distillability

The hypothetical nonlocal box (\textsf{NLB}) proposed by Popescu and Rohrlich allows two spatially separated parties, Alice and Bob, to exhibit stronger than quantum correlations. If the generated correlations are weak, they can sometimes be distilled into a stronger correlation by repeated applications of the \textsf{NLB}. Motivated by the limited distillability of \textsf{NLB}s, we initiate here a study of the distillation of correlations for nonlocal boxes that output quantum states rather than classical bits (\textsf{qNLB}s). We propose a new protocol for distillation and show that it asymptotically distills a class of correlated quantum nonlocal boxes to the value $1/2 (3\sqrt{3}+1) \approx 3.098076$, whereas in contrast, the optimal non-adaptive parity protocol for classical nonlocal boxes asymptotically distills only to the value 3.0. We show that our protocol is an optimal non-adaptive protocol for 1, 2 and 3 \textsf{qNLB} copies by constructing a matching dual solution for the associated primal semidefinite program (SDP). We conclude that \textsf{qNLB}s are a stronger resource for nonlocality than \textsf{NLB}s. The main premise that develops from this conclusion is that the \textsf{NLB} model is not the strongest resource to investigate the fundamental principles that limit quantum nonlocality. As such, our work provides strong motivation to reconsider the status quo of the principles that are known to limit nonlocal correlations under the framework of \textsf{qNLB}s rather than \textsf{NLB}s.Comment: 25 pages, 7 figure

An Algorithmic Argument for Nonadaptive Query Complexity Lower Bounds on Advised Quantum Computation

This paper employs a powerful argument, called an algorithmic argument, to prove lower bounds of the quantum query complexity of a multiple-block ordered search problem in which, given a block number i, we are to find a location of a target keyword in an ordered list of the i-th block. Apart from much studied polynomial and adversary methods for quantum query complexity lower bounds, our argument shows that the multiple-block ordered search needs a large number of nonadaptive oracle queries on a black-box model of quantum computation that is also supplemented with advice. Our argument is also applied to the notions of computational complexity theory: quantum truth-table reducibility and quantum truth-table autoreducibility.Comment: 16 pages. An extended abstract will appear in the Proceedings of the 29th International Symposium on Mathematical Foundations of Computer Science, Lecture Notes in Computer Science, Springer-Verlag, Prague, August 22-27, 200

On the Minimum Degree up to Local Complementation: Bounds and Complexity

The local minimum degree of a graph is the minimum degree reached by means of a series of local complementations. In this paper, we investigate on this quantity which plays an important role in quantum computation and quantum error correcting codes. First, we show that the local minimum degree of the Paley graph of order p is greater than sqrt{p} - 3/2, which is, up to our knowledge, the highest known bound on an explicit family of graphs. Probabilistic methods allows us to derive the existence of an infinite number of graphs whose local minimum degree is linear in their order with constant 0.189 for graphs in general and 0.110 for bipartite graphs. As regards the computational complexity of the decision problem associated with the local minimum degree, we show that it is NP-complete and that there exists no k-approximation algorithm for this problem for any constant k unless P = NP.Comment: 11 page

Higher Order Decompositions of Ordered Operator Exponentials

We present a decomposition scheme based on Lie-Trotter-Suzuki product formulae to represent an ordered operator exponential as a product of ordinary operator exponentials. We provide a rigorous proof that does not use a time-displacement superoperator, and can be applied to non-analytic functions. Our proof provides explicit bounds on the error and includes cases where the functions are not infinitely differentiable. We show that Lie-Trotter-Suzuki product formulae can still be used for functions that are not infinitely differentiable, but that arbitrary order scaling may not be achieved.Comment: 16 pages, 1 figur

Multipartite Nonlocal Quantum Correlations Resistant to Imperfections

We use techniques for lower bounds on communication to derive necessary conditions in terms of detector efficiency or amount of super-luminal communication for being able to reproduce with classical local hidden-variable theories the quantum correlations occurring in EPR-type experiments in the presence of noise. We apply our method to an example involving n parties sharing a GHZ-type state on which they carry out measurements and show that for local-hidden variable theories, the amount of super-luminal classical communication c and the detector efficiency eta are constrained by eta 2^(-c/n) = O(n^(-1/6)) even for constant general error probability epsilon = O(1)

Onset of transcription of the aminopeptidase N (leukemia antigen CD 13) gene at the crypt/villus transition zone during rabbit enterocyte differentiation

AbstractThe sequence of a cDNA clone (2.82 kbp) of rabbit intestinal aminopeptidase N (CD 13) is reported. Using the corresponding anti-sense RNA probe, the distribution of aminopeptidase N mRNA along the crypt/villus axis of the rabbit small intestine was studied by in situ hybridization. The aminopeptidase N gene is expressed along the whole length of the villus with a maximum at its base. Expression was not detected in the crypt cells. The distribution of aminopeptidase N mRNA correlates with the presence of active enzyme as monitored by histochemical staining. The results are compatible with onset of transcription of the aminopeptidase N gene at the crypt/villus transition zone during the enterocyte differentiation

Necessary Condition for the Quantum Adiabatic Approximation

A gapped quantum system that is adiabatically perturbed remains approximately in its eigenstate after the evolution. We prove that, for constant gap, general quantum processes that approximately prepare the final eigenstate require a minimum time proportional to the ratio of the length of the eigenstate path to the gap. Thus, no rigorous adiabatic condition can yield a smaller cost. We also give a necessary condition for the adiabatic approximation that depends on local properties of the path, which is appropriate when the gap varies.Comment: 5 pages, 1 figur

Classical and quantum partition bound and detector inefficiency

We study randomized and quantum efficiency lower bounds in communication complexity. These arise from the study of zero-communication protocols in which players are allowed to abort. Our scenario is inspired by the physics setup of Bell experiments, where two players share a predefined entangled state but are not allowed to communicate. Each is given a measurement as input, which they perform on their share of the system. The outcomes of the measurements should follow a distribution predicted by quantum mechanics; however, in practice, the detectors may fail to produce an output in some of the runs. The efficiency of the experiment is the probability that the experiment succeeds (neither of the detectors fails). When the players share a quantum state, this gives rise to a new bound on quantum communication complexity (eff*) that subsumes the factorization norm. When players share randomness instead of a quantum state, the efficiency bound (eff), coincides with the partition bound of Jain and Klauck. This is one of the strongest lower bounds known for randomized communication complexity, which subsumes all the known combinatorial and algebraic methods including the rectangle (corruption) bound, the factorization norm, and discrepancy. The lower bound is formulated as a convex optimization problem. In practice, the dual form is more feasible to use, and we show that it amounts to constructing an explicit Bell inequality (for eff) or Tsirelson inequality (for eff*). We give an example of a quantum distribution where the violation can be exponentially bigger than the previously studied class of normalized Bell inequalities. For one-way communication, we show that the quantum one-way partition bound is tight for classical communication with shared entanglement up to arbitrarily small error.Comment: 21 pages, extended versio

A General SU(2) Formulation for Quantum Searching with Certainty

A general quantum search algorithm with arbitrary unitary transformations and an arbitrary initial state is considered in this work. To serach a marked state with certainty, we have derived, using an SU(2) representation: (1) the matching condition relating the phase rotations in the algorithm, (2) a concise formula for evaluating the required number of iterations for the search, and (3) the final state after the search, with a phase angle in its amplitude of unity modulus. Moreover, the optimal choices and modifications of the phase angles in the Grover kernel is also studied.Comment: 8 pages, 2 figure