A generalization of Schur-Weyl duality with applications in quantum estimation

Abstract

Schur-Weyl duality is a powerful tool in representation theory which has many applications to quantum information theory. We provide a generalization of this duality and demonstrate some of its applications. In particular, we use it to develop a general framework for the study of a family of quantum estimation problems wherein one is given n copies of an unknown quantum state according to some prior and the goal is to estimate certain parameters of the given state. In particular, we are interested to know whether collective measurements are useful and if so to find an upper bound on the amount of entanglement which is required to achieve the optimal estimation. In the case of pure states, we show that commutativity of the set of observables that define the estimation problem implies the sufficiency of unentangled measurements.

The published version, Typos corrected, 40 pages, 2 figures

References in corpus (4)

• Induced measures in the space of mixed quantum states
• Classical and quantum communication without a shared reference frame
• Estimating the spectrum of a density operator
• Optimal estimation of a physical observable's expectation value for pure states

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• Clifford Group and Unitary Designs under Symmetry
• Linear programming with unitary-equivariant constraints
• Quantum superreplication of states and gates