Ancilla dimensions needed to carry out positive-operator-valued measurement

Ping Xing Chen*, János A. Bergou, Shi Yao ZHU, Guang Can Guo

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

21 Citations (Scopus)

Abstract

To implement a positive-operator-valued measurement (POVM), which is defined on the dS -dimensional Hilbert space of a physical system, one has to extend the Hilbert space to include dA additional dimensions (called the ancilla). This is done via either the tensor product extension (TPE) or the direct sum extension (DSE). The implementation of a POVM utilizes the available resources more efficiently if it requires fewer additional dimensions. To determine how to implement a POVM with the least additional dimensions is, therefore, an important task in quantum information. We have determined the necessary and sufficient (hence minimal) number of the additional dimensions needed to implement the same POVM by the TPE and the DSE, respectively. If the POVM has n elements and ri is the rank of the i th element, then the dimension of the minimal ancilla is dA = i=1 n ri - dS for the DSE implementation, and this represents a lower bound for the added dimensions in the TPE implementation. In the proof, we explicitly construct the DSE implementation of a general POVM with elements of arbitrary rank. As an example, we determine dA for the unambiguous discrimination of N linearly independent states and provide the full DSE implementation of a state-discriminating POVM for N=2.

Original languageEnglish
Article number060303
JournalPhysical Review A - Atomic, Molecular, and Optical Physics
Volume76
Issue number6
DOIs
Publication statusPublished - 20 Dec 2007

Scopus Subject Areas

  • Atomic and Molecular Physics, and Optics

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