Operational interpretation and estimation of quantum trace-norm asymmetry based on weak-value measurement and some bounds

The asymmetry of a quantum state relative to a translational group is a\ncentral concept in many areas of quantum science and technology. An important\nand geometrically intuitive measure of translational asymmetry of a state is\ngiven by the trace-norm asymmetry, which is defined as the trace norm of the\ncommutator between the state and the generator of the translation group. While\ntrace-norm asymmetry satisfies all the requirements for a bonafide measure of\ntranslational asymmetry of a state within the quantum resource theoretical\nframework, its meaning in terms of laboratory operations is still missing.\nHere, we first show that the trace-norm asymmetry is equal to the average\nabsolute imaginary part of the weak value of the generator of the translation\ngroup optimized over all possible orthonormal bases of the Hilbert space.\nHence, it can be estimated via the measurement of weak value combined with a\nclassical optimization in the fashion of quantum variational circuit which may\nbe implemented using the near-term quantum hardware. We then use the link\nbetween the trace-norm asymmetry and the nonreal weak value to derive the\nrelation between the trace-norm asymmetry with other basic concepts in quantum\nstatistics. We further obtain trade-off relations for the trace-norm asymmetry\nand quantum Fisher information, having analogous forms to the\nKennard-Weyl-Robertson uncertainty relation.\n