CS 867/QIC 890 Semidefinite Programming in Quantum Information Winter Topic Suggestions

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1 CS 867/QIC 890 Semidefinite Programming in Quantum Information Winter 2017 Topic Suggestions Each student registered in this course is required to choose a topic connecting semidefinite programming with quantum information and computation, and to give one lecture and prepare a detailed set of lecture notes (or a paper) on the chosen topic. Individuals that wish to participate without registering for the course are also welcome to give a lecture if time permits. Below is a list of suggestions that may serve as a helpful starting point for choosing a topic. Please feel free to ask for clarifications, and keep in mind that these are just suggestions you are free to deviate from the suggestions, or choose a different topic altogether. 1. Quantum query complexity. Semidefinite programming has central importance in the study of quantum query complexity. In particular, the so-called general adversary bound, which can be represented by a semidefinite program, characterizes the quantum query complexity of function evaluation up to constant factors. This fact is established in these papers: Ben Reichardt. Reflections for quantum query complexity: The general adversary bound is tight for every boolean function. Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, pages , Peter Høyer, Troy Lee, and Robert Špalek. Negative weights make adversaries stronger. In Proceedings of the Thirty-Ninth Annual ACM Symposium on Theory of Computing, pages , An extension of this result, and some simplifications to the proof, can be found in this paper: Troy Lee, Rajat Mittal, Ben Reichardt, Robert Špalek, and Mario Szegedy. Quantum query complexity of state conversion. Proceedings of the 52nd Annual Symposium on Foundations of Computer Science, Distillable entanglement. Semidefinite programming can be used to obtain bounds on the distillable entanglement of bipartite states, as is described in the following papers: Eric Rains. A semidefinite program for distillable entanglement. IEEE Transactions on Information Theory 47(7): , 2001, Xin Wang and Runyao Duan. Improved semidefinite programming upper bound on distillable entanglement. Physical Review A 94: , A recent follow-up paper of the second group of authors is also relevant: Xin Wang and Runyao Duan. Rains bound is not additive. Available as arxiv.org e-print ,

2 3. Semidefinite programming hierarchies for nonlocal correlations. The set of input/output correlations that can be produced by two separated individuals who share an entangled state can be approximated by a hierarchy of semidefinite programs, as is described in these papers: Miguel Navascués, Stefano Pironio, Antonio Acín. Bounding the set of quantum correlations. Physical Review Letters 98: , Miguel Navascués, Stefano Pironio, Antonio Acín. A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations. New Journal of Physics 10: , Andrew Doherty, Yeong-Cherng Liang, Ben Toner, and Stephanie Wehner. The quantum moment problem and bounds on entangled multi-prover games. In Proceedings of the 23rd Annual IEEE Conference on Computational Complexity, pages , Extensions of this technique have also been considered, as in the following papers: Stefano Pironio, Miguel Navascués, Antonio Acín. Convergent relaxations of polynomial optimization problems with noncommuting variables. SIAM Journal on Optimization 20(5): , Mario Berta, Omar Fawzi, and Volkher Scholz. Quantum bilinear optimization. SIAM Journal on Optimization 26(3): , Semidefinite programming hierarchies for separable states. In a similar spirit to the previous topic, the set of separable density operators (for a given choice of local dimensions) can be approximated by a hierarchy of semidefinite programs, as was first discussed in this paper: Andrew Doherty, Pablo Parrilo, and Federico Spedalieri. Complete family of separability criteria. Physical Review A 69(2): , Further studies on this hierarchy (or variants of it) can be found in a number of papers, including these: Aram Harrow, Anand Natarajan, and Xiaodi Wu. An improved semidefinite programming hierarchy for testing entanglement. Available as arxiv.org e-print , Miguel Navascués, Masaki Owari, Martin Plenio. The power of symmetric extensions for entanglement detection. Physical Review A 80: , Quantum coin-flipping. Semidefinite programming has been used in multiple ways for the analysis of quantum coin-flipping. For example, Kitaev used semidefinite programming to prove a lower bound on the bias of any quantum strong coin-flipping protocol. Kitaev did not publish his proof, 2

3 but it may be found in this paper, which also considers quantum coin-flipping protocols involving more than two players: Andris Ambainis, Harry Buhrman, Yevgeniy Dodis, and Hein Röhrig. Multiparty quantum coin flipping. In Proceedings of the 19th Annual IEEE Conference on Computational Complexity, pages , Another example of the use of semidefinite programming in the analysis of quantum coin-flipping protocols can be found in this paper: Ashwin Nayak, Jamie Sikora, and Levent Tunçel. A search for quantum coinflipping protocols using optimization techniques. Mathematical Programming 156(1): , It should be mentioned that semidefinite programming plays an important role in Mochon s work on quantum weak coin-flipping, which is represented by these papers (which would be ambitious but not impossible for a course project): Carlos Mochon. Quantum weak coin flipping with arbitrarily small bias. Available as arxiv.org e-print , Dorit Aharonov, André Chailloux, Maor Ganz, Iordanis Kerenidis, and Loïck Magnin. A simpler proof of the existence of quantum weak coin flipping with arbitrarily small bias. SIAM Journal on Computing 45(3): , Quantum strategies. As is well-known, the Choi representation allows one to represent channels by positive semidefinite operators obeying certain linear constraints but in fact this representation can be adapted to allow for more complex interactions, in which quantum information is alternately input and output over the course of multiple steps. This framework is described in these papers, as well as several follow-up papers by the authors of the second paper: Gus Gutoski and John Watrous. Toward a general theory of quantum games. In Proceedings of the 39th Annual ACM Symposium on Theory of Computing, pages , Giulio Chiribella, Giacomo D Ariano, and Paolo Perinotti. Theoretical framework for quantum networks. Physical Review A 80(2): , Semidefinite programming is not explicitly used in these papers, but it does provide a convenient way to understand the general framework. The semidefinite programming connection is made explicit in Gus Gutoski s thesis: Gus Gutoski. Quantum Strategies and Local Operations. PhD Thesis, David R. Cheriton School of Computer Science, University of Waterloo,

4 Multiple projects involving this framework are possible. One possibility would be to describe the framework through semidefinite programming and present the alternative proof of Kitaev s bound for strong coin-flipping based on this framework. 7. Quantum variant of the Lovász ϑ-function. The (classical) Lovász ϑ-function is a function of a given graph, representable through semidefinite programming, that has various connections to the properties of the graph. A quantum analogue of this function was defined, and shown to provide an upper-bound on the zero-error entanglement assisted classical capacity of a given quantum channel, in the following paper: Runyao Duan, Simone Severini, and Andreas Winter. Zero-error communication via quantum channels, non-commutative graphs and a quantum Lovász theta function. IEEE Transactions on Information Theory 59(2): , An independent work proved that the ordinary Lovász ϑ-function provide an upperbound on the zero-error entanglement assisted classical capacity of quantum channels. Salman Beigi. Entanglement-assisted zero-error capacity is upper-bounded by the Lovász ϑ-function. Physical Review A 82: , A couple of variants of the quantum Lovász ϑ-function were studied in this paper: Toby Cubitt, Laura Mančinska, David Roberson, Simone Severini, Dan Stahlke, and Andreas Winter. Bounds on entanglement-assisted source-channel coding via the Lovász ϑ number and its variants. IEEE Transactions on Information Theory 60(11): Quantum steering. The notion of quantum steering, first suggested by Schrödinger, has been studied extensively in the past couple of years. Semidefinite programming is useful, for instance, for defining quantitative measures of steering, as is done in these (and other) papers: Paul Skrzypczyk, Miguel Navascués, and Daniel Cavalcanti. Quantifying Einstein Podolsky Rosen steering. Physical Review Letters 112: , Marco Piani and John Watrous. Necessary and sufficient quantum information characterization of Einstein Podolsky Rosen steering. Physical Review Letters 114: , The following article surveys work on quantum steering, with a focus on its connections to semidefinite programming: Daniel Cavalcanti and Paul Skrzypczyk. Quantum steering: a short review with focus on semidefinite programming. Available as arxiv.org e-print ,

5 9. Quantum communication complexity. Semidefinite programming is useful in the study of communication complexity. In the quantum setting, it was used to prove interesting relationships among the resources of shared randomness, shared entanglement, and quantum communication in this paper: Dmitry Gavinsky, Julia Kempe, Oded Regev, and Ronald de Wolf. Bounded-error quantum state identification and exponential separations in communication complexity. SIAM Journal of Computing 39(1): 1 24, Unique games with entangled provers. The unique games conjecture postulates that it is NP-hard to approximate the classical value for a class of games called unique games. The following paper used semidefinite programming together with a rounding technique to prove that this conjecture is false for entangled provers: Julia Kempe, Oded Regev, and Ben Toner. Unique games with entangled provers are easy. SIAM Journal on Computing 39(7): , Measurement compatibility. The relationship between commutativity and joint measurability of two measurements is simple in the case of projective measurements, but much more complicated for general measurements. Semidefinite programming is useful in the general case, as is demonstrated in these two papers: Michael Wolf, David Perez-Garcia, and Carlos Fernandez. Measurements incompatible in quantum theory cannot be measured jointly in any other local theory. Physical Review Letters 103: , Teiko Heinosaari and Michael Wolf. Nondisturbing quantum measurements. Journal of Mathematical Physics 51(9): , In particular, the first paper uses semidefinite programming to prove that any two incompatible general measurements allow for the violation of a Bell inequality. 12. Approximate degradable channels. The so-called degradable quantum channels are interesting for various reasons, particularly because their capacities for transmitting quantum information are relatively simple to analyze. The following paper formalizes an approximate notion of degradability, which can be computed through semidefinite programming, and uses this notion to compute new upper bounds on capacities for some interesting channels: David Sutter, Volkher Scholz, Andreas Winter, and Renato Renner. Approximate Degradable Quantum Channels. Available as arxiv.org e-print ,