We provide an experimentally measurable local gauge $U(1)$ invariant Fubini-Study (FS) metric for mixed states. Like the FS metric for pure states, it also captures only the quantum part of the uncertainty in the evolution Hamiltonian. We show that this satisfies the quantum Cramer-Rao bound and thus arrive at a more general and measurable bound. Upon imposing the monotonicity condition, it reduces to the square-root derivative quantum Fisher Information. We show that on the Fisher information metric space dynamical phase is zero. A relation between square root derivative and logarithmic derivative is formulated such that both give the same Fisher information. We generalize the Fubini-Study metric for mixed states further and arrive at a set of Fubini-Study metric---called $α$ metric. This newly defined $α$ metric also satisfies the Cramer-Rao bound. Again by imposing the monotonicity condition on this metric, we derive the monotone $α$ metric. It reduces to the Fisher information metric for $α=1$.
Extremal properties of the variance and the quantum Fisher information
我们表明，方差是自己的凹屋顶。对于秩为2的密度矩阵和密度矩阵的本征基中对角元素为零的算子，我们解析地证明了量子费舍尔信息是方差凸屋顶的4倍。强有力的数值证据表明，这一说法是真实的，甚至对运营商与非零对角元素或密度矩阵的秩大于2。我们还发现，在[D. Petz，J.Phys.A：Math.Gen.35，929（2002）; P. Gibilisco，F. Hiai和D. Petz，IEEE Trans. Inf. Theory 55，439（2009）]，在适当的归一化之后，量子费舍尔信息是最大的。因此，我们推测，即使在一般情况下，量子费舍尔信息也是方差凸屋顶的四倍。
Versatile probe state preparation via generalized measurements for quantum sensing and thermometry
Faculdade de Ciências Exatas e Tecnologia, Universidade Federal da Grande Dourados（大杜拉多斯联邦大学精确科学与技术学院）；Xiamen University（厦门大学）；Universidade Federal de Lavras（拉夫拉斯联邦大学）
Quantum Fisher Information as the Speed Limit for Multipartite Entanglement
Argonne National Laboratory（阿贡国家实验室）；Stevens Institute of Technology（史蒂文斯理工学院）；Southern Illinois University（南伊利诺伊大学）；Center for Nanoscale Materials, Argonne National Laboratory（阿贡国家实验室纳米材料中心）；Indian Institute of Science Education and Research Thiruvananthapuram（印度科学教育研究 institute 特里凡得琅）
Enhancing Quantum Metrology with High-order Fisher Information and Experiments
School of Information Science and Technology, Southwest Jiaotong University（西南交通大学信息科学与技术学院）；School of Mathematical Sciences, Capital Normal University（首都师范大学数学科学学院）；Hefei National Laboratory（合肥国家实验室）
Generalized Exact Fractional Quantum Information Model with Memory Effects
Laboratory of Theoretical and Applied Physics, Echahid Cheikh Larbi Tebessi University（理论与应用物理实验室，埃沙希德谢赫拉里比特贝西大学）；Research Center of Astrophysics and Cosmology, Khazar University（天体物理学与宇宙学研究中心，哈萨尔大学）
