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Bulletin of the American Physical Society

Ordering relations for quantum states

Ian T. Durham

Published Mar 4, 2015• 0 citations
OpenAlex

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Abstract This paper studies the order relations among various quantum coherence measures in a single-qubit system. Ten representative coherence measures are selected from three categories: distance-based, fidelity-based, and relative entropy-based measures. The analytical expressions for the coherence values are derived in the Bloch sphere representation of the single-qubit state, and the order relations within the same category and across categories are systematically analyzed. The study shows that coherence measures within the same category generally maintain consistent order relations in the single-qubit system. However, there is typically no fixed order relation between fidelity-based coherence measures (represented by geometric coherence) and relative entropy-based coherence measures (represented by α -affinity coherence measures). Yet, in pure single-qubit states and certain restricted sets of mixed states, these two types of coherence measures exhibit the same order relation. The results indicate that the order relations between coherence measures depend not only on their construction but also on the quantum state space structure and the constraints on parameters.
It is found that two-mode output quantum electromagnetic field in two-mode squeezed states exhibits higher-order squeezing to all even orders. And the generalized uncertainty relations are also presented for the first time. The concept of higher-order squeezing of the single-mode quantum electromagnetic field was first introduced and applied to several processes by Hong and Mandel in 1985. Lately Li Xizeng and Shan Ying have calculated the higher-order squeezing in the process of degenerate four-wave mixing and presented the higher-order uncertainty relations of the fields in single-mode squeezed states. In this paper we generalize the above work to the higher-order squeezing in two-mode squeezed states. The generalized uncertainty relations are also presented for the first time.
It is found that the two-mode output quantum electromagnetic field in two-mode squeezed states exhibits higher-order squeezing to all even orders and the degree of higher-order squeezing is greater than that of the second-order. The higher-order squeezed parameter and squeezed limit due to the modulation frequency are investigated. The smaller the modulation frequency is, the stronger the degree of higher-order squeezing becomes. Furthermore, the higher-order uncertainty relations in two-mode squeezed states are presented for the first time. The product of higher-order noise moments is related to even order number N and the squeeze factor r .