Symmetry and Randomness in Quantum Information Theory: Several Applications

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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 𝔰𝔲(2), and random symmetric invariant tensors under the action of 𝔰𝔲(𝘥) for any 𝘥, serving as an extension of the random invariant tensors under 𝔰𝔲(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 𝑂(ε⁻⁴ ln |𝒫|) copies of an unknown state to distinguish whether the unknown state is contained in or ε-far from a finite set 𝒫 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 𝑈 is equal to or ε-far from a known or unknown unitary 𝘝 in fixed dimension with 𝑂(ε⁻²) uses of the unitary, in which an ancilla system should be used. We give an algorithm that distinguishes the two cases with 𝑂(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 𝑛-fold parallel repetition of commuting nonsignaling extended nonlocal game 𝐺 decreases exponentially in 𝑛, provided that the game value of 𝐺 is strictly less than 1, following the approach used by Lancien and Winter based on de Finetti reduction.
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