# Knowledge by Direct Measurement versus Inference from Steering

## TL;DR

This paper compares direct measurement and inference from steering in quantum entanglement, showing that direct measurement provides more reliable information and challenging the assumption that steering reveals pre-existing states.

## Contribution

It demonstrates that information obtained through direct measurement surpasses inference from steering, clarifying misconceptions about quantum state revelation.

## Key findings

- Direct measurement yields more accurate state information.
- Steering-based inference can lead to incorrect conclusions.
- A technical bound on the overlap of steered states is established.

## Abstract

If Alice and Bob start out with an entangled state $|\Psi_{AB}\rangle$, Bob may update his state to $|\varphi_B\rangle$ either by performing a suitable measurement himself, or by receiving the information that a measurement by Alice has steered that state. While Bob's update on his state is identical, his update on Alice's state differs: if Bob has performed the measurement, he has steered the state $|\chi_{\leftarrow}(\varphi)\rangle_A$ of Alice; if Alice has made the measurement, to steer $|\varphi\rangle_B$ on Bob she must have found a different state $|\chi_{\rightarrow}(\varphi)\rangle_A$. Based on this observation, a consequence of the well-known `Hardy's ladder', 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 overlap between the steered and the steering state 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 $|\Psi\rangle_{AB}$.