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

Sebastian Meznaric1, ∗ 1Clarendon Laboratory, , Oxford OX1 3PU, United Kingdom (Dated: August 9, 2021) We introduce a concept of the measurement space where the information that is not accessible using the particular type of measurements available is erased from the system. Each state from the Hilbert space is thus mapped to its counterpart in the measurement space. We then proceed to compute the entanglement of formation on this new space. We find that for local measurements this never exceeds the entanglement of formation computed on the original state. Finally we proceed to apply the concept to quantum communication protocols where we find that the success probability of the protocol using the state with erased information in combination with perfect measurements is the same as using the non-perfect measurements and the original state. We thus postulate that the so defined entanglement measure quantifies the amount of useful entanglement for quantum communication protocols. Keywords: measurement space, , entanglement

I. INTRODUCTION best possible measurements cannot extract more information than the chosen generalized measure- In an experimental setting it is likely never the ments in the original Hilbert space. case that we can achieve perfect rank-1 projective Physically, we can look at the first stage of the measurements. In such a setting it is impossible measurement process - the interaction between the for us to obtain all the possible information about measurement device and the measured system. We the state of the system. In this paper we construct add the measurement apparatus ancilla in some a theoretical framework that erases those proper- state |0i to the state |ψi to obtain |ψi |0i. The ties of the state that cannot affect the outcome of following unitary U then describes the interaction our (imperfect) measurements. (for more details see for instance [1, 2]): Mathematically, we define a new Hilbert space, X called the measurement space. The states in the U |ψi ⊗ |0i = Mm |ψi ⊗ |mi . (1) measurement space will contain only the informa- m tion that can be extracted using the available mea- The read out of the measurement is now conducted surements. We construct these states through a by conducting a measurement on the measure- procedure not unlike the Naimark’s dilation the- ment device. The measurement operators are of orem, whereby we change the state so that the the form 1 ⊗ |mi hm|. Notice that if you write measurements act just like rank-1 projectors in Mm |ψi ⊗ |mi as a single state vector, you get ex- the measurement space. Intuitively, the properties actly the measurement space state. The measur- that are preserved are those that are now accessi- able information in the state (1) is the same as ble only with the perfect measurements (rank-1 that in the measurement space. projectors). Next we explore how the quantum information Mathematically, the map can best be under- theoretic quantities behave on states where the stood as taking all states in the Hilbert space into non-measurable information has been removed. In P √ the form j pj kψji, where kψji are orthogonal particular we are interested in entanglement and measurement space states corresponding to differ- show that the state without any non-measurable ent measurement outcomes. The double bar in the information is just as effective in all quantum com- notation is there to remind us that these states are munication protocols as the original state. More arXiv:1004.4854v3 [quant-ph] 28 Aug 2010 the measurement space versions thereof. When we precisely, using a state with this amount of en- erased the non-measurable properties we changed tanglement and perfect measurement operators for the components of the state that correspond to the a quantum communicat