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Entanglement Distribution and Quantum Discord Entanglement distribution and quantum discord 1, 2 2 Alexander Streltsov, ∗ Hermann Kampermann, and Dagmar Bruß 1Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, D-14195 Berlin, Germany 2Institut für Theoretische Physik III, Heinrich-Heine-Universität Düsseldorf, D-40225 Düsseldorf, Germany Establishing entanglement between distant parties is one of the most important problems of quan- tum technology, since long-distance entanglement is an essential part of such fundamental tasks as quantum cryptography or quantum teleportation. In this lecture we review basic properties of en- tanglement and quantum discord, and discuss recent results on entanglement distribution and the role of quantum discord therein. We also review entanglement distribution with separable states, and discuss important problems which still remain open. One such open problem is a possible advantage of indirect entanglement distribution, when compared to direct distribution protocols. P I. INTRODUCTION where pij are probabilities with i;j pij = 1. The set of all separable states will be denoted by . Any separable S This lecture presents an overview of the task of estab- state can be produced with local operations and classical lishing entanglement between two distant parties (Alice communication (LOCC). An entangled state cannot be and Bob) and its connection to quantum discord [1–5]. written as in Eq. (1). In order to produce an entangled Surprisingly, it is possible for the two parties to perform state, a non-local operation is needed. In SectionII we this task successfully by exchanging an ancilla which will review different ways to quantify the amount of has never been entangled with Alice and Bob. This puz- entanglement in a given state. zling quantum protocol was already suggested in [6], A state is called classically correlated (CC) if it can be but a thorough study [7–12] and experimental verifica- written as [18] tion [13–15] (see also [16]) had to wait for almost ten X AB A B years until recently, when interest in general quantum ρ = pij ei ei ej ej ; (2) cc j ih j ⊗ j ih j correlations arose and led to insights about their role in i;j the entanglement distribution protocol. A A composite quantum system does not need to be in a with ei ej = δij. Measuring ρcc in the local bases ei h jB i fj i g product state for the subsystems, but it can also occur as and ej does not change the state, i.e., a superposition of product states, or as a mixture of such fj i g superpositions. This feature does not exist in the classical A B AB AB Π Π [ρcc ] = ρcc ; (3) world, and a state exhibiting it is called entangled. In ⊗ general, a state is said to contain entanglement if it cannot where the von Neumann measurement Π is defined as be written as a mixture of projectors onto product states. X It is said to contain quantum correlations, if it cannot Π[σ] = ei ei σ ei ei : (4) j ih j j ih j be written as a mixture of projectors onto product states i with local orthogonality properties. And it is said to contain correlations (classical or quantum), if it cannot The set of all classically correlated states will be denoted by .A quantum correlated state cannot be written as be written as a product state. CC Let us formalize these notions. In the following def- in Eq. (2). The eigenbasis of a quantum correlated state initions we will consider for simplicity only bipartite is not a product basis with the property that the sets of quantum systems (with superscripts A and B for Alice local states are orthogonal ensembles. and Bob, respectively); the generalization to compos- It is also possible to combine the aforementioned ite quantum systems with more than two subsystems is frameworks of separability and classicality, thus arriv- arXiv:1610.05078v1 [quant-ph] 17 Oct 2016 ing at classical-quantum (CQ) states [18]: straightforward. Let us denote by ei a complete set of orthogonal basis states (which couldfj ig also be interpreted X ρAB = p e e A B : (5) as classical states), i.e. ei ej = δij, while Greek letters in- cq ij j ii h ij ⊗ j ji h jj dicate quantum statesh whichj i are not necessarily orthog- i;j onal, i.e., for the ensemble in general , δ i i j ij The set of all classical-quantum states will be denoted by holds. fj ig h j i . For any CQ state, there exists a local von Neumann A separable state ρ can be written as [17] sep measurementCQ on the subsystem A which leaves the state X ρAB = p A φ φ B ; (1) unchanged, i.e., sep ij j iih ij ⊗ j jih jj i;j ΠA 11B[ρAB] = ρAB; (6) ⊗ cq cq where the von Neumann measurement Π is given in ∗ [email protected] Eq. (4). If a state cannot be written as in Eq. (5), we say 2 entanglement measures, can be found in [20–22]. In gen- eral, we require that a measure of entanglement fulfills the following two properties [23, 24]: E CC Nonnegativity: (ρ) 0 for all states ρ with equal- • ity for all separableE states≥ [25], S Monotonicity: (Λ[ρ]) (ρ) for any LOCC oper- • ation Λ. E ≤ E Many entanglement measures also have additional properties such as strong monotonicity in the sense that entanglement does not increase on average under selec- P Figure 1. State space for composite quantum systems [19]: tive LOCC operations [23, 24]: i pi (σi) (ρ), where classically correlated states form a connected set of measure E ≤ E the states σi are obtained from the state ρ by the means zero; separable states form a convex set (green,CC containing S of LOCC with the corresponding probabilities pi. More- ). Quantum correlated states are all states outside of , over, many entanglement measures are also convex in andCC entangled states (blue) are all states outside of . CC P P S the state, i.e., ( i piρi) i pi (ρi)[23, 24]. From now onE we will≤ focusE on the bipartite scenario with two parties A and B of the same dimension d. In this that the state has nonzero quantum discord with respect case, any entanglement measure is maximal on states of to the subsystem A. Measuring a state with nonzero the form discord in any orthogonal basis on the subsystem A nec- d 1 essarily changes the state. In Section III we will present 1 X− different ways to quantify the amount of discord in a φ+ = ii ; (7) j d i p j i given state. d i=0 From the above definitions it is clear that classically since from this state any quantum state can be created correlated states are a subset of separable states, and via LOCC operations [22]. Of particular importance is that entangled states are a subset of quantum correlated states. These different types of states for composite quan- the two-qubit singlet state ( 01 10 )= p2, which can be obtained from the state jφ+i −via j locali unitaries. In tum systems therefore form a nested structure [19] which j 2 i is sketched in Fig.1. Note that separable states form a entanglement theory local unitaries do not change the convex set, due to their definition in Eq. (1). However, properties of a state, and thus we will refer to the state φ+ as a singlet. classically correlated states do not form a convex set: j 2 i one can produce a quantum correlated state by mixing Operational measures of entanglement are distillable two classically correlated states. entanglement and entanglement cost. Distillable entangle- Those states which are not entangled, but nevertheless ment quantifies the maximal rate for extracting singlets possess quantum correlations a la discord, may exhibit from a state via LOCC operations [21, 22]: puzzling features. They can be produced via LOCC, but nR n + ⊗ nevertheless they carry quantum properties. Namely, in d(ρ) = sup R : lim inf Λ ρ⊗ φ2 = 0 ; E n Λ − 1 order to produce them one has to create quantumness !1 (8) in the form of non-orthogonality. This makes them a where M = Tr pM M is the trace norm, φ+ is the potential resource for quantum information processing 1 y 2 projectorjj ontojj the state φ+ [26], and the infimum is protocols. Counterintuitively, even though they do not 2 performed over all LOCCj operationsi Λ. Entanglement carry entanglement, they may be used for the distribu- cost on the other hand quantifies the minimal singlet rate tion of entanglement, as we will see below. required for creating a state via LOCC operations [21, 22]: The structure of this lecture is as follows: in SectionII ( ! ) we review different measures of quantum entanglement, nR + ⊗ n discord quantifiers are reviewed in Section III. In Sec- c(ρ) = inf R : lim inf Λ φ2 ρ⊗ = 0 : E n Λ − tionIV we review recent results on entanglement distri- !1 1 (9) bution and discuss the role of quantum discord therein. AB Conclusions in SectionV complete our lecture. For pure states = these two quantities co- incide and are equalj i toj i the von Neumann entropy A of the reduced state [27]: d( ) = c( ) = S(ρ ) = A A E E Tr[ρ log2 ρ ]. This implies that the resource theory II. QUANTUM ENTANGLEMENT of− entanglement is reversible for pure states [21, 22]. In general, it holds that d(ρ) c(ρ), and there exist states Here, we will review different entanglement mea- which have zero distillableE ≤ entanglement E but nonzero sures, mainly focusing on measures which are used in entanglement cost.
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