A Thermal State Conditioned on Pointer Bases and Its Properties

We introduce the conditional thermal state, which is a thermal state conditioned on the pointer bases, and discuss its resource-theoretic properties. We first demonstrate that a conditional thermal state is a resource state for the quantum thermometry, which could outperform the Gibbs state in the low temperature limit. Then, we show that its asymmetry plays a role as the informational contribution in upper bounding the quantum Fisher information for quantum thermometry. Focusing on the evolution governed by the time-dependent Hamiltonian, we show that there always exists a Gibbs-preserving map in the asymptotic limit with an arbitrarily small error in converting the final exact state to the conditional thermal state. Finally, we present the relation between the symmetric divergence, called quantum J-divergence, between the exact final state and the conditional thermal state, to the quantum work in the quantum system.