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

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Introduction to Measurement Space and Application to Operationally Useful Entanglement and Mode Entanglement Introduction to Measurement Space and Application to Operationally Useful Entanglement and Mode Entanglement Sebastian Meznaric1, ∗ 1Clarendon Laboratory, University of Oxford, 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, quantum information, 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 j0i to the state j i to obtain j i j0i. 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 j i ⊗ j0i = Mm j i ⊗ jmi : (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 ⊗ jmi hmj. Notice that if you write measurements act just like rank-1 projectors in Mm j i ⊗ jmi 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 p the form j pj k ji, where k ji 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 communication protocol results in the measurement outcomes - they were made orthogo- same fidelity as using the imperfect measurements nal (see figure 1). Since the perfect rank-1 projec- and the original state. tion measurements are best we can get, we effec- For some protocols the fidelity is a strictly tively transform the state into the form where the monotone increasing function of entanglement when perfect rank-1 projectors are used. In such cases we can claim that the entanglement in the measurement space state is the amount of entan- ∗ [email protected] glement that is useful for the particular protocol. Original Hilbert space Measurement space FIG. 1. When a state j i is mapped to the measurement space the components corresponding to the measure- ment outcomes are made orthogonal and their lengths adjusted to the square roots of the respective outcome probabilities. The operational entanglement measure thus ob- it as M. Every state j i 2 H has a correspond- tained encompasses those devised previously to ing measurement space state denoted as kφi. The deal with the indistinguishable particles (like for map from j i to k i is defined as example in [3, 4]) since the measurement operators X y 1=2 can always be made insensitive to the exchange of k i = mS(j i) = MmMm kmi ; particles. m We also prove that when the measurements are (2) local the map from the original state to the mea- surement space is LOCC. This implies that the where kmi are the orthonormal basis states of the entanglement in the measurement space is always measurement space M and Mm are the general- less than or equal to the standard entanglement. ized measurements. The dimension of M is there- Furthermore, we find an upper bound for the two- fore the same as the number of measurement out- dimensional systems and show when the upper comes in our quantum system S. As argued in the bound is attained. introduction, the measurement space state con- Finally, we apply the above formalism to the tains exactly the properties of the quantum real- mode entaglement. We find that the isolated ity measurable with the particular set of measure- single-particle mode entanglement cannot have an ments. operational meaning unless further degrees of free- The above formulation is a type of nonlinear dom are added to the system. We examine the Naimark dilation, where we pay the price of non- more general case of adding more particles and linearity in order to have simplicity in the calcula- more degrees of freedom and find the upper bound tion of the map. on the operational entanglement. Whenever the total number of measurement outcomes is not a III. ENTANGLEMENT IN prime number mode entanglement is found to have MEASUREMENT SPACE an operational meaning as well. Following this introduction, the paper is organ- ised into the definition of measurement space in In this section we shall examine the properties section II, the application to entanglement in sec- exhibited by the entanglement in the measurement tion III, followed by a look at the mode entan- space. In other words, we shall consider the fol- glement in section IV and concluding remarks in lowing quantity section V. Em(j i) = E(k i); (3) where E is an entanglement measure. II. DEFINITION OF MEASUREMENT We know that every protocol can be imple- SPACE mented by starting with a measurement by Al- ice, classical communication of the measurement Here we shall consider the quantum system S to result to Bob, followed by a unitary by Bob (for consist of a state j i and a set of generalized mea- proof of this see [1]). We shall label the measure- surement operators. We shall denote the original ment operators that Alice uses as M1; :::; Mn. In Hilbert space as H and we define a new Hilbert response to Alice obtaining the result k, Bob im- space, called the measurement space, and denote plements the unitary Uk. In order to measure the 2 fidelity of the protocol, Bob devises two measure- So as above, we obtain ments: My;k corresponding to the protocol suc- ceeding and Mn;k corresponds to protocol failing. pm;y As before, the index k enumerates Alice's measure- p(y j m) = : (11) pm;y + pm;n ment outcomes and is there for the cases where the success and failure of the protocol depends on Al- ice's measurement result. In many practical cases This completes the proof. this is so. For example, the quantum key distri- Next we shall show that as long as the mea- bution protocol succeeds if Bob obtains the same surement operators are local, entanglement in the measurement result as Alice when they measure in measurement space is always less than that in the the same basis. original Hilbert space. If the measurements are We are now ready to prove our theorem. not local, this can easily be shown not to be the case with an example. Theorem 1. Given a state j i to be used as a resource in a quantum communication protocol, Theorem 2. Given a set of local generalized mea- the success rate of the protocol is the same with surement operators M A ⊗ M B and a state j i, mA mB the original state and original imperfect measure- it is always true that E(k i) ≤ E(j i). ments as it is with the measurement space state and rank-1 projective measurements. Proof. The existence of an LOCC map taking j i y y Proof. Denote pk;y = h j Mk Mk ⊗ My;kMy;k j i to k i would immediately imply that E(k i) ≤ the probability of the outcome corresponding to E(j i) (see [1, 5, 6]).
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