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Positive Operator-Valued Measurement Essay

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How to obtain the state information of a given physical system by using a suitable measurement is a very important project in quantum mechanics. Basically, it is not easy to devise a measurement procedure which uniquely identifies the given quantum state from the statistical date produced by the measurements. For example, if the state of the quantum system is given by a K×K density matrix, the complete measurement statistics of one fixed von Neumann measurement is not sufficient to reconstruct the state, see, e.g., Ref. 1. However, it is possible to perform a somewhat general measurement procedure on a quantum system, namely, a positive operator-valued measurement (or POVM for short), see Ref. 2. Mathematically, a minimal informationally …show more content…

The extremal case in this sense arises when we are given a system of K2 normalized vectors {|ψi:i=1,…,K2} in CK for which|ψi|ψj|2=1K+1,1≤i≠j≤K2.(2)Such POVMs are called symmetric informationally complete POVMs, or simply, SIC-POVMs.SIC-POVMs constitute a basic ingredient in many applications of quantum information processing; see, for example, Refs. 6–13, etc., and references therein).For the existence of SIC-POVMs, we have the following facts: (I)Explicit analytical constructions of SIC-POVMs satisfying (2) have been given for small dimension K, •K = 2, 3, 4, 5, see Refs. 6 and 7;•K = 6, see Ref. 8;•K = 7, 19, see Ref. 9;•K = 8, 12, 28, see Refs. 10–12;•K = 9, 11, 13–15, 35, 48, see Ref. 12;•K = 16, see Ref. 13. (II)It has been conjectured that SIC-POVMs exist in all dimensions [see Ref. 6 (Sec. 3.4) or Ref. 7] and numerical evidence exists for dimensions up to 67 (see Ref. 12). The most recent development in this area can be found in Ref. 14.Note also that Appleby15 studied SIC-POVMs for operators with arbitrary rank.C.Our resultsGenerally speaking, it is hard to explicitly construct SIC-POVMs. In fact, there are no known infinite families of SIC-POVMs and it is not even clear whether there exist SIC-POVMs for infinitely many K. Based on this observation, Klappenecker et al.1 proposed to construct approximately symmetric informationally complete positive operator-valued measures (ASIC-POVM) for possible applications in

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