Majorization for superpositions of nonorthogonal states and the existence of golden states

Abstract

In the resource theory of superposition relative to linearly independent, nonorthogonal reference states, golden states are known for two reference states and for special overlap matrices, and all of them are maximally coherent in the Löwdin basis. We clarify what decides their existence and their Löwdin structure. The robustness and the geometric measure of superposition are maximized by the same states, which overlap equally with every reference state; their maximal values are set by the ground-state energy of a classical $XY$ model on the overlaps of the dual basis. For every overlap matrix, these states, and only these, are golden under the maximal superposition-free operations, but they need not be maximally coherent in the Löwdin basis. For the Kraus-type free operations, we prove a necessary and sufficient criterion, Gram majorization, for converting pure states into targets with nonzero coefficients; it reduces to majorization for orthonormal reference states. It shows that golden states of these operations must be maximally coherent in the Löwdin basis, but that this does not ensure their existence: three reference states that are not orthonormal admit a golden state if and only if their overlaps have a common modulus $s$ and their Bargmann phase $Φ$ obeys $s\geq2\cos[(π+|Φ|)/3]$. Equal-overlap states also yield a complete family of superposition witnesses in the Löwdin basis, which show that white noise on the span of the reference states is a resource unless these states are orthonormal.