The contact temperature of arbitrary quantum states

An intuitive scheme to assign a temperature to an arbitrary state of a quantum system is to investigate the heat flow resulting from the coupling to a thermometer.

We introduce a simple model of a universal thermometer with the following property.

When it is prepared in a Gibbs equilibrium state at inverse temperature $\beta\in\mathbb R$ and brought into thermal contact with a system in any state, the heat flow between the system and thermometer vanishes for a unique value of $\beta$.

We call this value the contact temperature $\beta_{\rm op}\in\mathbb R$ of the system state.

The thermometer is universal in that it yields a unique contact temperature for arbitrary states of finite dimensional quantum systems.