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DESCRIPTION:How to Obtain Quantum Advantage with Constant-Round Communication Jack Spilecki | UC Berkeley We consider the problem of approximate cloning of quantum states: given n copies of an unknown state ρ∈ℂd×d\, prepare an (n+k)-copy state with high fidelity to ρ⊗(n+k). Werner's pure state cloner is the optimal channel for the pure state case\, and shows that n=Θ(kd/ε) copies are necessary and sufficient to clone k additional copies of an unknown pure state to fidelity 1−ε. The random purification channel gives a straightforward extension of Werner's cloner to mixed state inputs: given n copies of a mixed state\, randomly purify your input\, apply Werner's channel in the larger Hilbert space\, and then trace out the auxiliary registers. This gives a mixed state cloner using n=O(krd/ε) copies to clone rank-r states. Can one do any better? We show that the answer is no: one must use n=Ω(krd/ε) copies. We prove our lower bound by studying the special case of projector cloning\, in which the input state ρ is promised to be of the form P/r\, where P is a rank-r orthogonal projector. As a further application of our techniques\, we consider the closely related problem of approximate transposition of quantum states\, where one seeks to convert ρ⊗n to a k-copy state with high fidelity to (ρT)⊗k. Here\, we again show n=Θ(krd/ε) copies are necessary and sufficient for this task.
X-ALT-DESC;FMTTYPE=text/html:How to Obtain Quantum Advantage with Constant-Round Communication Jack Spilecki | UC Berkeley We consider the problem of approximate cloning of quantum states: given n copies of an unknown state ρ∈ℂd×d, prepare an (n+k)-copy state with high fidelity to ρ⊗(n+k). Werner's pure state cloner is the optimal channel for the pure state case, and shows that n=Θ(kd/ε) copies are necessary and sufficient to clone k additional copies of an unknown pure state to fidelity 1−ε. The random purification channel gives a straightforward extension of Werner's cloner to mixed state inputs: given n copies of a mixed state, randomly purify your input, apply Werner's channel in the larger Hilbert space, and then trace out the auxiliary registers. This gives a mixed state cloner using n=O(krd/ε) copies to clone rank-r states. Can one do any better? We show that the answer is no: one must use n=Ω(krd/ε) copies. We prove our lower bound by studying the special case of projector cloning, in which the input state ρ is promised to be of the form P/r, where P is a rank-r orthogonal projector. As a further application of our techniques, we consider the closely related problem of approximate transposition of quantum states, where one seeks to convert ρ⊗n to a k-copy state with high fidelity to (ρT)⊗k. Here, we again show n=Θ(krd/ε) copies are necessary and sufficient for this task.
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SUMMARY:IQC Math and CS seminar featuring Jack Spilecki
DTSTART;TZID=America/New_York:20261001T130000
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