arXiv:quant-ph/0106070 · IEEE Transactions on Information Theory

Optimal tight frames and quantum measurement

Yonina C. Eldar, G. David Forney

Published Mar 1, 2002• 149 citations

Abstract

Tight frames and rank-one quantum measurements are shown to be intimately related. In fact, the family of normalized tight frames for the space in which a quantum-mechanical system lies is precisely the family of rank-one generalized quantum measurements on that space. Using this relationship, frame-theoretical analogs of various quantum-mechanical concepts and results are developed. The analog of a least-squares quantum measurement is a tight frame that is closest in a least-squares sense to a given set of vectors. The least-squares tight frame is found for both the case in which the scaling of the frame is specified (constrained least-squares frame (CLSF)) and the case in which the scaling is chosen to minimize the least-squares error (unconstrained least-squares frame (ULSF)). The well-known canonical frame is shown to be proportional to the ULSF and to coincide with the CLSF with a certain scaling.

DOI: 10.1109/18.985949

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