Knowledge by direct measurement versus inference from steering

Inspired by recent debates on Wigner’s friend extensions to bipartite scenarios, we analyze the following situation: Alice and Bob start out with an entangled state $$|\varPsi \rangle _{AB}$$; following a measurement by one of the parties, Bob’s state is updated to $$|\varphi \rangle _B$$. If the measurement was performed by Bob himself, Bob knows that Alice’s state has been steered to $$|\chi _{\leftarrow }(\varphi )\rangle _A$$. If the measurement was performed by Alice, Bob knows that Alice must have found a $$|\chi _{\rightarrow }(\varphi )\rangle _A$$ which is generally different, as a consequence of the well-known “Hardy’s ladder”. Based on this observation, we show that information from direct measurement must trump inference from steering. The erroneous belief that both paths should lead to identical conclusions can be traced to the usual prejudice that measurements should reveal a pre-existing state of affairs. We also prove a technical result on Hardy’s ladder: the minimum value of $$|\langle {\chi _{\leftarrow }(\varphi )}|{\chi _{\rightarrow }(\varphi )}\rangle |$$ is $$2\sqrt{p_{0}p_{n-1}}/(p_0+p_{n-1})$$, where $$p_0$$ and $$p_{n-1}$$ are the smallest (non-zero) and the largest Schmidt coefficients of $$|\varPsi \rangle _{AB}$$.