Extended Nonlocal Games

The notions of entanglement and nonlocality are among the most striking ingredients found in quantum information theory. One tool to better understand these notions is the model of nonlocal games; a mathematical framework that abstractly models a physical system. The simplest instance of a nonlocal game involves two players, Alice and Bob, who are not allowed to communicate with each other once the game has started and who play cooperatively against an adversary referred to as the referee. The focus of this thesis is a class of games called extended nonlocal games, of which nonlocal games are a subset. In an extended nonlocal game, the players initially share a tripartite state with the referee. In such games, the winning conditions for Alice and Bob may depend on outcomes of measurements made by the referee, on its part of the shared quantum state, in addition to Alice and Bob's answers to the questions sent by the referee. We build up the framework for extended nonlocal games and study their properties and how they relate to nonlocal games.Comment: PhD thesis, Univ Waterloo, 2017. 151 pages, 11 figure

Quantum proof systems for iterated exponential time, and beyond

© 2019 Copyright held by the owner/author(s). Publication rights licensed to ACM. We show that any language solvable in nondeterministic time exp(exp(· · · exp(n))), where the number of iterated exponentials is an arbitrary function R(n), can be decided by a multiprover interactive proof system with a classical polynomial-time verifier and a constant number of quantum entangled provers, with completeness 1 and soundness 1 − exp(−C exp(· · · exp(n))), where the number of iterated exponentials is R(n) − 1 and C > 0 is a universal constant. The result was previously known for R = 1 and R = 2; we obtain it for any time-constructible function R. The result is based on a compression technique for interactive proof systems with entangled provers that significantly simplifies and strengthens a protocol compression result of Ji (STOC’17). As a separate consequence of this technique we obtain a different proof of Slofstra’s recent result on the uncomputability of the entangled value of multiprover games (Forum of Mathematics, Pi 2019). Finally, we show that even minor improvements to our compression result would yield remarkable consequences in computational complexity theory and the foundations of quantum mechanics: first, it would imply that the class MIP∗ contains all computable languages; second, it would provide a negative resolution to a multipartite version of Tsirelson’s problem on the relation between the commuting operator and tensor product models for quantum correlations

Symmetry and Randomness in Quantum Information Theory: Several Applications

University of Technology Sydney. Faculty of Engineering and Information Technology.This thesis studies four topics in quantum information theory using tools from representation theory and (high-dimensional) probability theory. First, we study the nonadditivity of minimum output von Neumann and Rényi entropy of quantum channels. A sketch of the proof by Aubrun, Szarek and Werner for nonadditivity of minimum output entropy is presented, and a slight simplification is given. We show that asymptotically the minimum output entropy of the random channel Ɛ ⊗ Ɛ ⊗ Ɛ* is achieved not by a tripartite genuinely entangled state, but by a tensor product of two states. We also study another model of random channel, and our estimation of the minimum output Rényi entropies fails to show the usefulness of genuine multipartite entanglement for the multiple nonadditivity. Second, we study the generic entanglement in the random near-invariant tensors under the action of (2), and random symmetric invariant tensors under the action of () for any , serving as an extension of the random invariant tensors under (2). We show that both the random tensors are asymptotically close to a maximally entangled state with respect to any bipartite cut. Third, we study efficient quantum certification for states and unitaries. We present an algorithm that uses (ε⁻⁴ ln ||) copies of an unknown state to distinguish whether the unknown state is contained in or ε-far from a finite set of known states with respect to the trace distance. This algorithm is more sample-efficient in some settings. The previous study showed that one can distinguish whether an unknown unitary is equal to or ε-far from a known or unknown unitary in fixed dimension with (ε⁻²) uses of the unitary, in which an ancilla system should be used. We give an algorithm that distinguishes the two cases with (ε⁻¹) uses of the unitary, without using ancilla system or using ancilla system of much smaller dimension. Finally, we study the parallel repetition of extended nonlocal game motivated by its connection with multipartite steering and entanglement detection. We show that the probability of winning an -fold parallel repetition of commuting nonsignaling extended nonlocal game decreases exponentially in , provided that the game value of is strictly less than 1, following the approach used by Lancien and Winter based on de Finetti reduction

Quantum no-signalling correlations and non-local games

We introduce and examine three subclasses of the family of quantum no-signalling (QNS) correlations introduced by Duan and Winter: quantum commuting, quantum and local. We formalise the notion of a universal TRO of a block operator isometry, define an operator system, universal for stochastic operator matrices, and realise it as a quotient of a matrix algebra. We describe the classes of QNS correlations in terms of states on the tensor products of two copies of the universal operator system, and specialise the correlation classes and their representations to classical-to-quantum correlations. We study various quantum versions of synchronous no-signalling correlations and show that they possess invariance properties for suitable sets of states. We introduce quantum non-local games as a generalisation of non-local games. We define the operation of quantum game composition and show that the perfect strategies belonging to a certain class are closed under channel composition. We specialise to the case of graph colourings, where we exhibit quantum versions of the orthogonal rank of a graph as the optimal output dimension for which perfect classical-to-quantum strategies of the graph colouring game exist, as well as to non-commutative graph homomorphisms, where we identify quantum versions of non-commutative graph homomorphisms introduced by Stahlke.Comment: 72 page

On the separation of correlation-assisted sum capacities of multiple access channels

The capacity of a channel characterizes the maximum rate at which information can be transmitted through the channel asymptotically faithfully. For a channel with multiple senders and a single receiver, computing its sum capacity is possible in theory, but challenging in practice because of the nonconvex optimization involved. In this work, we study the sum capacity of a family of multiple access channels (MACs) obtained from nonlocal games. For any MAC in this family, we obtain an upper bound on the sum rate that depends only on the properties of the game when allowing assistance from an arbitrary set of correlations between the senders. This approach can be used to prove separations between sum capacities when the senders are allowed to share different sets of correlations, such as classical, quantum or no-signalling correlations. We also construct a specific nonlocal game to show that the approach of bounding the sum capacity by relaxing the nonconvex optimization can give arbitrarily loose bounds. Towards a potential solution to this problem, we first prove a Lipschitz-like property for the mutual information. Using a modification of existing algorithms for optimizing Lipschitz-continuous functions, we then show that it is possible to compute the sum capacity of an arbitrary two-sender MAC to a fixed additive precision in quasi-polynomial time. We showcase our method by efficiently computing the sum capacity of a family of two-sender MACs for which one of the input alphabets has size two. Furthermore, we demonstrate with an example that our algorithm may compute the sum capacity to a higher precision than using the convex relaxation.Comment: v2: 64 pages, 2 figures; updated conclusion and acknowledgements sections, and added a referenc