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.
Location
- QNC 1201
- Online on Zoom
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Meeting ID: 912 8146 6256
Passcode: 494237
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