Entanglement witnesses from mutually unbiased measurements

Abstract

A new family of positive, trace-preserving maps is introduced. It is defined using the mutually unbiased measurements, which generalize the notion of mutual unbiasedness of orthonormal bases. This family allows one to define entanglement witnesses whose indecomposability depends on the characteristics of the associated measurement operators. We provide examples of indecomposable witnesses and compare their entanglement detection properties with the realignment criterion.

Indecomposable entanglement witnesses from symmetric measurements

“…Therefore, we show that x does not have to be optimal in order to produce indecomposable witnesses, which was inconclusive in ref. [41]. Also, note that the state ρ 1 is not detected if the witness in Example 3 is analogously constructed from the MUBs instead.…”

“…Therefore, it is possible to produce the same witnesses W ′ using different (N, M )-POVMs, provided that they arise from the same Hermitian orthonormal basis. In particular, for the Gell-Mann matrices, one recovers the reduction map W ′ = I d 2 − dP + [41,47].…”

“…The MUMs encompassing a full range of purity were shown to detect entanglement with as little as two measurement operators [40]. Recently, it has also been proven that the witnesses constructed from MUMs belong to the class based on the CCNR criterion [41].…”

Indecomposability of entanglement witnesses constructed from symmetric measurements

“…The corresponding entanglement witness has the form Therefore, it is possible to produce the same witnesses using different ( N , M )-POVMs, provided that they arise from the same Hermitian orthonormal basis. In particular, for the Gell-Mann matrices, one recovers the reduction map 7 , 40 .…”

“…The MUMs encompassing a full range of purity were shown to detect entanglement with as little as two measurement operators 36 . Recently, it has also been proven that the witnesses constructed from MUMs belong to the class based on the CCNR criterion 40 .…”