Inequalities for quantum Fisher information

Abstract

An inequality relating the Wigner-Yanase information and the SLD-quantum Fisher information was established by Luo (Proc. Amer. Math. Soc., 132, pp. 885-890, 2004). In this paper, we generalize Luo's inequality to any regular quantum Fisher information. Moreover, we show that this general inequality can be derived from the Kubo-Ando inequality, which states that any matrix mean is greater than the harmonic mean and smaller than the arithmetic mean

Characterising Two-Sided Quantum Correlations Beyond Entanglement via Metric-Adjusted f–Correlations

“…In this paper we instead introduce an infinite family of quantifiers of quantum correlations beyond entanglement which vanish on both classical-quantum and quantum-classical states and thus properly capture the quantum correlations with respect to both subsystems. More precisely, the 'quantum f −correlations' are here defined as the maximum metric-adjusted f −correlations between pairs of local observables with the same fixed equispaced spectrum and are in one-toone correspondence with the family of metric-adjusted skew informations [55][56][57][58][59][60][61]. While similar ideas were explored earlier in [62,63] to quantify entanglement, here we show that our quantifiers only reduce to entanglement monotones when restricted to pure states.…”

“…We have defined an infinite family of quantitative indicators of two-sided quantum correlations beyond entanglement, which vanish on both classical-quantum and quantum-classical states and thus properly capture quantumness with respect to both subsystems. These quantifiers, named 'quantum f −correlations', are in one-to-one correspondence with the metric-adjusted skew informations [55][56][57][58][59][60][61]. We have shown that the quantum f −correlations are entanglement monotones for pure states of qubit-qudit systems, having also provided closedform expressions for these quantifiers for two-qubit systems.…”

Quantum uncertainty based on metric adjusted skew information

“…Proof. It is well-known that among all the operator means, the harmonic one is the smallest one (see [5,12]), i.e.,…”

“…(c) Q f (ρ) has the spectrum representation (5), and has tight bounds (11). Which indicates that the maximally mixed state 1/n has no quantum uncertainty since it commutes with any observable, and all the pure states have the same maximal quantum uncertainty as we expect.…”

Information Geometry of Quantum Resources

“…Such properties are in fact met by all the regular quantum Fisher metrics, which are topologically equivalent to the SLDF [20]. Hence, they are legit measures of asymmetry.…”