A fundamental problem in quantum computation and quantum information is finding the minimum quantum dimension needed for a task. For tasks involving state preparation and measurements, this problem can be addressed using only the input–output correlations. This has been applied to Bell, prepare-and-measure, and Kochen–Specker contextuality scenarios. Here, we introduce a novel approach to quantum dimension witnessing for scenarios with one preparation and several measurements, which uses the graphs of mutual exclusivity between sets of measurement events. We present the concepts and tools needed for graph-theoretic quantum dimension witnessing and illustrate their use by identifying novel quantum dimension witnesses, including a family that can certify arbitrarily high quantum dimensions with few events.

In a recent breakthrough result, Chattopadhyay, Mande and Sherif showed an exponential separation between the log approximate rank and randomized communication complexity of a total function f , hence refuting the log approximate rank conjecture of Lee and Shraibman [2009]. We provide an alternate proof of their randomized communication complexity lower bound using the information complexity approach. Using the intuition developed there, we derive a polynomially-related quantum communication complexity lower bound using the quantum information complexity approach, thus providing an exponential separation between the log approximate rank and quantum communication complexity of f . Previously, the best known separation between these two measures was (almost) quadratic, due to Anshu, Ben-David, Garg, Jain, Kothari and Lee [CCC, 2017]. This settles one of the main question left open by Chattopadhyay, Mande and Sherif, and refutes the quantum log approximate rank conjecture of Lee and Shraibman [2009]. Along the way, we develop a Shearer-type protocol embedding for product input distributions that might be of independent interest.

Multi-source-extractors are functions that extract uniform randomness from multiple (weak) sources of randomness. Quantum multi-source-extractors were considered by Kasher and Kempe [27] (for the quantum-independent-adversary and the quantum-bounded-storage-adversary), Chung, Li and Wu [13] (for the general-entangled-adversary) and Arnon-Friedman, Portmann and Scholz [2] (for the quantum-Markov-adversary).One of the main objectives of this work is to unify all the existing quantum multi-source adversary models. We propose two new models of adversaries: 1) the quantum-measurementadversary (qm-adv), which generates side-information using entanglement and on post-measurement and 2) the quantum-communication-adversary (qc-adv), which generates side-information using entanglement and communication between multiple sources. We show that, 1. qm-adv is the strongest adversary among all the known adversaries, in the sense that the side-information of all other adversaries can be generated by qm-adv.

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