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 quantum 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 entanglement 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 | ih | ⊗ | ih | 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 |B i {| i } 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 {| i } 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) | ih | | ih | 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 could{| i} also be interpreted X ρAB = p e e A ψ ψ B . (5) as classical states), i.e. ei ej = δij, while Greek letters in- cq ij | ii h i| ⊗ | ji h j| dicate quantum statesh which| i are not necessarily orthog- i,j onal, i.e., for the ensemble ψ in general ψ ψ , δ i i j ij The set of all classical-quantum states will be denoted by holds. {| i} h | 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 ψ ψ A φ φ B , (1) unchanged, i.e., sep ij | iih i| ⊗ | jih j| 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 general, 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 CCand 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) | d i √ | 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 )/ √2, which can be obtained from the state |φ+i −via | locali unitaries. In tum systems therefore form a nested structure [19] which | 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: | 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 →∞ (8) in the form of non-orthogonality. This makes them a where M = Tr √M M is the trace norm, φ+ is the potential resource for quantum information processing 1 † 2 projector|| onto|| the state φ+ [26], and the infimum is protocols. Counterintuitively, even though they do not 2 performed over all LOCC| 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 (9) bution and discuss the role of quantum discord therein. AB Conclusions in SectionV complete our lecture. For pure states ψ = ψ these two quantities coincide and are equal| i to| i the von Neumann entropy A of the reduced state [27]: d(ψ) = c(ψ) = 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. This phenomenon is also known as this lecture. More detailed reviews, also containing other bound entanglement [28]. 3

An important family of entanglement measures is ob- is related to the entanglement cost under quantum op- tained by taking the minimal distance to the set of sepa- erations preserving the positivity of the partial trans- rable states [23, 24]: pose [40]. S Several entanglement measures discussed above are (ρ) = inf D(ρ, σ). (10) subadditive, i.e., they fulfill the inequality E σ ∈S ρ σ ρ + (σ) (17) Here, D(ρ, σ) can be an arbitrary functional which is E ⊗ ≤ E E nonnegative and monotonic under quantum operations, for any two states ρ and σ. Examples for subadditive i.e., D(Λ[ρ], Λ[σ]) D(ρ, σ) for any quantum operation ≤ measures are entanglement cost, entanglement of for- Λ [29]. Examples for such distances are the trace dis- mation, and relative entropy of entanglement. The loga- tance ρ σ /2, the infidelity 1 F(ρ, σ) with fidelity || − ||1 − rithmic negativity is additive, i.e., it fulfills Eq. (17) with F(ρ, σ) = √ρ, √σ 2, and the quantum relative entropy || ||1 equality. It is conjectured [41] that the distillable entan- S(ρ σ) = Tr[ρ log2 ρ] Tr[ρ log2 σ]. In the latter case, the glement violates Eq. (17). corresponding|| measure− is known as the relative entropy of entanglement [23, 24]: III. QUANTUM DISCORD r(ρ) = min S(ρ σ). (11) E σ || ∈S Quantum discord was introduced in [1,2] as a quanti- The second important family of measures are convex fier for correlations different from entanglement. In the roof measures defined as [30] modern language of quantum information theory, quan- X tum discord of a state ρ = ρAB can be expressed in the (ρ) = inf p (ψ ), (12) E iE i following compact way [42, 43]: i δ(ρ) = I(ρ) sup I(Λ 11 ρ ). (18) where the infimum is taken over all pure state decom- − eb ⊗ P Λeb positions of ρ = i piψi. If for pure states entanglement is defined as the von Neumann entropy of the reduced Here, I(ρAB) = S(ρA) + S(ρB) S(ρAB) is the quantum mu- state (ψ) = S(ρA), the corresponding convex roof mea- tual information and the supremum− is performed over E sure is known as the entanglement of formation [31]: all entanglement breaking channels Λeb [44]. Quantum discord vanishes on CQ-states and is larger than zero X f(ρ) = min p S Tr ψ . (13) otherwise [45]. The quantity I(ρ) δ(ρ) was initially E i A i − i introduced in [2] as a measure of classical correlations. Interestingly, quantum discord is closely related to the In general, the relative entropy of entanglement is be- entanglement of formation via the Koashi-Winter rela- tween the distillable entanglement and the entanglement tion [46, 47]: of formation [32]: AB BC AB A δ(ρ ) = Ef(ρ ) S(ρ ) + S(ρ ), (19) d(ρ) r(ρ) f(ρ). (14) − E ≤ E ≤ E where the total state ρABC is pure [48]. Moreover, the regularized entanglement of forma- Similar as for entanglement, we can define distance- tion is equal to the entanglement cost [33]: c(ρ) = based measures of discord [49]: n E limn f(ρ⊗ )/n. We also mention that the geometric measure→∞ E of entanglement defined as (ρ) = inf D(ρ, Π 11[ρ]), (20) D Π ⊗ g(ρ) = 1 max F(ρ, σ) (15) where the infimum is performed over all local von Neu- E − σ ∈S mann measurements Π and D(ρ, σ) is a suitable distance is a distance-based and a convex roof measure simulta- between ρ and σ, such as the relative entropy. In the neously [34, 35]. latter case, the corresponding quantity is called relative Another important entanglement measure which will entropy of discord [50]: be used in this lecture is the logarithmic negativity. For a AB r(ρ) = min S(ρ Π 11[ρ]), (21) a bipartite state ρ = ρ it is defined as [36, 37] D Π || ⊗

TA and has also been studied earlier in the context of ther- n(ρ) = log2 ρ (16) E 1 modynamics [51, 52]. If the distance is chosen to be 2 with the partial transposition TA. The logarithmic nega- the squared Hilbert-Schmidt distance Tr(ρ σ) , the tivity is zero for states which have positive partial trans- corresponding measure is known as the geometric− dis- pose, and thus there exist entangled states which have cord [53, 54]. Interestingly, the geometric discord can zero logarithmic negativity [38]. Nevertheless, these increase under local operations on any of the subsys- states cannot be distilled into singlets [28]. Interest- tems [55]. It was also shown to play a role for remote ingly, the logarithmic negativity is not convex [39], and state preparation [56]. 4

ABC Alice Bob tite state ρ = ρ , where Alice is in possession of the particles A and C, and the particle B is in Bob’s hands. preshared correlations Entanglement distribution is then achieved by send- A B ing the particle C from Alice to Bob, see Fig.2. The amount of distributed entanglement is then given by A BC AB AC B | (11 ΛC[ρ]) | (ρ). In the following, we will discussE recent⊗ results− onE these two types of entanglement distribution [11, 12].

quantum channel C A. Direct entanglement distribution

What is the maximal amount of entanglement that can Figure 2. Indirect entanglement distribution. Alice and Bob be directly distributed via a given quantum channel Λ? initially share the state ρ = ρABC, where Alice holds the par- For answering this question, we first introduce the cor- ticles A and C, and Bob holds the particle B. Entanglement responding figure of merit: distribution is achieved by sending the particle C from Alice to Bob via a (possibly noisy) quantum channel. The figure is direct(Λ) = sup (11 Λ[σ]) . (22) taken from [11]. E σ E ⊗ In general, the supremum is performed over all bipartite quantum states σ. However, if the entanglement quanti- The role of quantum discord in quantum informa- fier is convex, we can restrict ourselves to pure states. tion theory has been studied extensively in the last IfE the distribution channel is noiseless, i.e., Λ = 11, then years [3,4]. Several alternative quantifiers of discord Eq. (22) reduces to have been presented [5], and criteria for good discord measures have also been discussed [57]. As an impor- + direct(11) = (φd ), (23) tant example, we mention the interferometric power [58], E E which is a computable measure of discord and a figure of where d is the dimension of the carrier particle. It is merit in the task of phase estimation with bipartite states. tempting to believe that this also extends to noisy chan- Further results on the role of quantum discord in quan- nels, i.e., that for any noisy channel the optimal perfor- tum metrology have been presented in [59, 60]. The re- mance is achieved by sending one half of a maximally lation between quantum discord and entanglement cre- entangled state. Quite surprisingly, this procedure is not ation in the quantum measurement process has also been optimal in general [11, 71, 72]. In particular, for any convex entanglement measure there exists a noisy channel investigated, both theoretically [61, 62] and experimen- E tally [63]. Monogamy of quantum discord [64, 65] and Λ and a bipartite state ρ such that [71] its behavior under local noise [66] and non-Markovian (11 Λ[ρ]) > (11 Λ[φ+]). (24) dynamics [67] have also been studied. Experimentally E ⊗ E ⊗ d friendly measures of discord were presented in [68, 69], Even more, if entanglement is quantified via the log- and the possibility of local detection of discord has been arithmic negativity, then states with arbitrary little en- reported in [70]. As we will see in the next section, tanglement can outperform maximally entangled states quantum discord also plays an important role for entan- for some noisy channels [11]. Nevertheless, maximally glement distribution [7,8]. entangled states are still optimal in various scenarios, e.g. if the carrier particle is a qubit and entanglement quantifier is the entanglement of formation or the geo- IV. ENTANGLEMENT DISTRIBUTION metric entanglement [11]. If the distribution channel is a single-qubit Pauli channel, i.e.,

In the following discussion we will distinguish be- X3 tween direct and indirect entanglement distribution be- Λp[ρ] = piσiρσi, (25) tween two parties (Alice and Bob) [11, 12]. Direct entan- i=0 glement distribution is achieved if Alice prepares two particles in an entangled state ρ and sends one of them where σi are Pauli matrices with σ0 = 11, then maxi- to Bob. The amount of distributed entanglement is then mally entangled states are optimal for entanglement dis- given by (11 Λ[ρ]), where Λ describes the correspond- tribution, regardless of the particular entanglement mea- ing quantumE ⊗ channel, and is a suitable entanglement sure [11]: E measure. + direct(Λp) = (11 Λp[φ ]). (26) Indirect entanglement distribution is a more general E E ⊗ 2 scenario where Alice and Bob already share correlations This result also holds if entanglement distribution is per- initially. In this case the total initial state is a tripar- formed via a combination of (possibly different) Pauli 5 channels, also in this case sending one half of a max- system. In particular, there exist tripartite states ρ = ρABC imally entangled state is the best strategy. Finally, if such that entanglement is quantified via the logarithmic negativ- AC B AB C A BC | (ρ) = | (ρ) = 0, | (ρ) > 0. (30) ity, maximally entangled states are optimal for all unital E E E single-qubit channels [73]. The first example for a state fulfilling Eqs. (30) was pre- This completes our discussion on direct entanglement sented in [6], and can be written as distribution, and we will present the more general sce- 1 nario in the following. 1 X η = Ψ Ψ + β Π , (31) 3 | GHZih GHZ| ijk ijk i,j,k=0 B. Indirect entanglement distribution with Ψ = ( 000 + 111 )/ √2, Π = ijk ijk , and all | GHZi | i | i ijk | ih | Can Alice and Bob gain an advantage if they share βijk are zero apart from β001 = β010 = β101 = β110 = 1/6. some correlations initially? To answer this question, we These results were extended to Gaussian states in [75], first introduce a figure of merit for indirect entanglement and experiments verifying this phenomenon have also distribution: been reported [13–15]. n A BC AB AC B o Motivated by this result, Zuppardo et al.[12] proposed indirect(Λ) = sup | (11 ΛC[ρ]) | (ρ) , (27) a classification of entanglement distribution protocols. E ρ E ⊗ − E In particular, a noiseless distribution protocol is called where the supremum is taken over all tripartite states excessive if the amount of distributed entanglement is ρ = ρABC. In particular, we are interested in the question larger than the amount of entanglement between the if indirect is larger than direct for some noisy channel and carrier and the rest of the system, i.e., someE entanglement measure.E A BC AC B AB C Note that so far no general answer to this ques- | (ρ) | (ρ) > | (ρ). (32) E − E E tion is known, and partial results have been presented in [11, 12]. In particular, if the channel used for entangle- Otherwise, the protocol is called nonexcessive. As dis- ment distribution is a single-qubit Pauli channel given cussed above, the state η in Eq. (31) gives rise to an in Eq. (25) and entanglement is quantified via a subaddi- excessive distribution protocol. tive measure , then indirect entanglement distribution It is natural to ask if such entanglement distribution does not provideE any advantage [11]: with separable states can provide an advantage when compared to scenarios where the carrier particle is en- + indirect(Λp) = direct(Λp) = (11 Λp[φ ]). (28) tangled with the rest of the system. In particular, one E E E ⊗ 2 might ask if a separable state can show a better perfor- This means that in this case sending one half of a singlet mance for entanglement distribution when compared state is the optimal distribution strategy. This result can to maximally entangled states. This question could be be generalized to the case where entanglement is dis- especially relevant if the distribution channel is noisy. tributed via a combination of (possibly different) Pauli Despite attempts by several authors [9, 13], the question channels [11]. has not yet been settled. However, not all entanglement measures are subad- Finally, we mention that rank two separable states are ditive. An important example is the distillable entan- not useful for entanglement distribution if entanglement glement which was defined in Eq. (8) and is con- d is quantified via logarithmic negativity [10]. jectured [E41] to violate subadditivity. Interestingly, if this conjecture is true, then indirect entanglement distribution provides an advantage for the distribution of D. Role of quantum discord for entanglement distribution distillable entanglement [11]. Finally, we note that entanglement breaking channels cannot be used for entanglement distribution for any As was shown in [7,8], the amount of entanglement entanglement measure [12]: that can be distributed via a noiseless channel by using E a tripartite quantum state ρ = ρABC is bounded above by indirect(Λeb) = direct(Λeb) = 0 (29) the discord between the carrier particle C and the rest of E E the system: for any entanglement breaking channel Λeb. This can be A BC AC B C AB seen by noting that any entanglement breaking channel | (ρ) | (ρ) | (ρ). (33) is equivalent to an LOCC protocol [74]. E − E ≤ D This inequality is true for any distance-based measure of entanglement and discord given in Eqs. (10) and (20) C. Entanglement distribution with separable states if the corresponding distance does not increase under quantum operations and fulfills the triangle inequality. Entanglement can also be distributed by sending a Moreover, it is also true for the relative entropy of entan- carrier particle which is not entangled with the rest of the glement and discord [7,8]. 6

The inequality (33) immediately implies that zero- this task. Despite substantial progress in recent years, discord states cannot be used for entanglement distri- several important questions in this research ﬁeld still re- bution. Moreover, this result can also be used to bound main open. In particular, it is still unclear if indirect the amount of entanglement in one cut of a tripartite entanglement distribution can provide an advantage in state ρ = ρABC in terms of entanglement and discord in comparison to direct distribution protocols. The ques- the other cuts [7,8]: tion also concerns entanglement distribution with separable states: also in this case it remains unclear if such AC B C AB A BC AC B C AB | (ρ) + | (ρ) | (ρ) | (ρ) | (ρ). scheme can be more useful than any direct distribution E D ≥ E ≥ E − D (34) procedure. For the relative entropy of entanglement and discord, We also mention that studying entanglement distribu- the inequality (33) is saturated for pure states of the form tion in relation to the resource theory of coherence [76– ψ AC φ B and also for the state η given in Eq. (31)[7]. | i ⊗ | i 78] and its extension to distributed scenarios [79–86] If the channel used for entanglement distribution is could potentially shed new light on these questions, and noisy, we get the following generalized inequality [11]: also lead to new independent results.

A BC AC B n C AB C AB o | (ρ0) | (ρ) min | (ρ), | (ρ0) . (35) E − E ≤ D D AB Here, we used the notation ρ0 = 11 ΛC[ρ], and and are any measures of entanglement⊗ and discord whichE ACKNOWLEDGMENTS fulﬁllD Eq. (33).

We thank Remigiusz Augusiak, Maciej Demianow- V. CONCLUSIONS icz, Jens Eisert, and Maciej Lewenstein for discussion. This work was supported by the Alexander von In this lecture we discussed recent results on entan- Humboldt-Foundation, Bundesministerium für Bildung glement distribution and the role of quantum discord in und Forschung, and Deutsche Forschungsgemeinschaft.

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