Comment by dawnofdusk

Although I am not an expert in quantum information, I think the problem you pose is resolved by the fact that the no-signalling theorem is about measurements of a quantum state, which is a microscopic state, and heat transfer is a measurement of a thermodynamic quantity, which is macroscopic. In much the same way that measuring the temperature of a classical gas doesn't give information on the location or momenta of the constituent particles, a thermodynamic probe of entanglement doesn't necessarily furnish precise information on how a state is entangled (e.g., Eq. 2 in https://arxiv.org/pdf/quant-ph/0406040).