Introduction to Measurement Space and Application to Operationally Useful Entanglement and Mode Entanglement

We introduce the idea that the knowable quantum reality depends not only on the state but also on measurements. Mathematically, we map the states from the ordinary Hilbert space into new states in what we call the measurement space. The state vectors in the measurement space contain only the information accessible with a particular set of measurements. The space can be used to find the operational equivalent of any information theoretic quantity. We use the formalism to define the operationally meaningful entanglement, explore its properties and apply it to mode entanglement. There we find that a mode-entangled state of a single particle has no operationally useful entanglement unless additional degrees of freedom are introduced to the system.