A Generalization of Schur–Weyl Duality with Applications in Quantum Estimation
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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.
KeywordsGauge Group Pure State Global Symmetry Density Operator Permutation Group
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- 2.Harrow A.: Applications of coherent classical communication and the Schur transform to quantum information theory, PhD thesis, MIT (2005). Arxiv preperint: quant-ph/0512255
- 4.Holevo, A.: Probabilistic and Statistical Aspects of Quantum Theory, Scuola Normale Superiore (Monographs) (2011)Google Scholar
- 5.Chiribella, G.: Optimal estimation of quantum signals in the presence of symmetry. PhD thesis, University of Pavia, Pavia, Italy (2006).Google Scholar
- 6.Marvian, I., Spekkens, R.W.: The theory of manipulations of pure state asymmetry: I. Basic tools, equivalence classes and single copy transformations. New J. Phys. 15, 033001 (2013). quant-ph/1104.0018
- 7.Marvian, I., Spekkens, R.W.: Under preparationGoogle Scholar
- 13.Zyczkowski, K., Sommers, H.J.: Induced measures in the space of mixed quantum states. J. Phys. A 34, 7111 (2001). quant-ph/0012101
- 14.Helstrom C.W.: Quantum Detection and Estimation Theory. Academic Press, New York (1976)Google Scholar