Quantum entanglement percolation on monolayer honeycomb lattice
QQuantum entanglement percolation on monolayer honeycomb lattice
Shashaank Khanna, , Saronath Halder, and Ujjwal Sen Discipline of Physics, Indian Institute of Technology Indore, Khandwa Road, Simrol, Indore
453 552 , India Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad
211 019 , India
The problem of establishing Bell and Greenberger-Horne-Zeilinger states between faraway placesor distant nodes of a circuit is a difficult and an extremely important one, and a strategy whichaddresses it is entanglement percolation. We provide a method for attaining the end through aquantum measurement strategy involving three-, two-, and single-qubit measurements on a single-layer honeycomb lattice of partially entangled bipartite entangled states.
I. INTRODUCTION
One of the major problems in quantum information[ ] is of distributing entangled states [ – ]. Entangle-ment is often a fragile resource and decoherence tendsto frequently make the problem of distributing entan-glement a difficult one. But distributing entanglement,be it between two remote positions on a lattice or be-tween two stations separated by a relatively large dis-tance, can have a multitude of uses, ranging from quan-tum computers [ ] to quantum key distribution [ , ]and quantum dense coding [ , ]. Quantum networkshave been employed as a solution to the problem of dis-tributing entanglement. They consist of nodes whereeach node can station many qubits. Different qubits ata particular node can be entangled with other qubits atother nodes. A network often has a well-defined geo-metric structure, and may form a lattice, for example,a triangular or a square lattice. Local quantum opera-tions at the nodes and classical communication betweenthe nodes are usually accessible in a realistic situation,and are thereby adopted in theoretical considerationsof manipulation of the structure and connectivity of aquantum network.An associated problem is that of establishing maxi-mally entangled states (also called Bell states) betweentwo faraway places. Using maximally entangled statesfor different protocols like quantum teleportation [ ]and quantum cryptography [ ] is extremely impor-tant, since they often provide the maximum advantageover the corresponding classical protocols. (See [ , ]however.)Nielsen’s majorization criterion answered the ques-tion whether a single copy of a bipartite pure state canbe converted to another, deterministically and underlocal operations and classical communication (LOCC)[ ]. Vidal derived the complete set of monotones forlocal pure state transformations and found the formulafor the maximum probability of successfully converting– under LOCC – a bipartite pure state to another [ ](see also [ – ]). The entanglement swapping schemewas earlier introduced by ˙Zukowski et al. in Refs. [ ](see also [ ]). Bose, Vedral, and Knight generalizedthis scheme to a multi-particle scenario and applied itto a communication network [ ].Acín, Cirac, and Lewenstein (ACL) [ ] developed the “classical entanglement percolation” (CEP) proto-col and used it to show how the concept of percola-tion [ ] in statistical mechanics can be applied in thecontext of sharing quantum entanglement. (See [ – ] for further studies.) Precisely, they showed howit can be utilized to achieve the task of distributingentanglement between faraway nodes on a networkand to establish a maximally entangled Bell state be-tween two distant nodes of an asymptotically large lat-tice. They further developed a protocol which theytermed as “quantum entanglement percolation” (QEP),and showed how QEP can succeed where CEP couldnot, in accomplishing the task of entanglement distri-bution.In this paper, we will mainly be concerned with QEP.In particular, a quantum measurement strategy is con-structed which helps to establish maximally entangledstates between two “antipodal” end-nodes of a lattice.The strategy utilizes a single-layer hexagonal lattice ofpartially entangled bipartite quantum states. Measure-ments involved are three-, two-, and single-qubit ones.The outcome is a Greenberger-Horne-Zeilinger state[ ] between an arbitrary number of nodes of the lattice,which can then be transformed to a maximally entan-gled two-qubit states between two faraway nodes. Wecompare our strategy with previous entanglement per-colation strategies in the literature with respect to theresources utilized.The rest of the paper is arranged in the followingway: In Sec. II, we provide a recapitulation of a fewtools that will be necessary for our analysis. Thereafter,in Sec. III, we present the main results, and finally, inSec. IV, we provide the concluding remarks. II. COLLECTING THE TOOLSA. Classical entanglement percolation
We begin with a description of the protocol for CEP.Firstly, any node of a lattice can contain any number ofqubits and secondly, the qubits of two different nodescan be connected via partially entangled states. SeeFig. , where the nodes within a quantum network areshown. The geometry created due to these partially en-tangled qubits at different nodes, forms the structure a r X i v : . [ qu a n t - ph ] A ug FIG. . Schematic diagram of a quantum network. Thenetwork is formed by a collection of nodes, each of whichcontains a cluster of qubits. A qubit in one node is typicallyentangled with a qubit in a different node. The smaller cir-cles represent the qubits, while the larger ones represent thenodes. The lines represent the entangled states. of the lattice. The protocol begins by applying LOCCbetween the nodes to convert the partially entangledstates to maximally entangled states. After this, someof the previous links are broken and the probability thatan initial partially entangled state is converted to a max-imally entangled one, is governed by the singlet conver-sion probability (SCP) of the initial states.Now for every lattice, there exists a percolationthreshold which is the critical value of the occupationprobability in the lattice, such that infinite connectivity(percolation) occurs. In CEP, if the SCP is greater thanthe percolation threshold for the given lattice, then aninfinite cluster forms in the lattice. This infinite clusterconsists of nodes which are all linked with maximallyentangled states and thus, one finds many paths alongwhich one can do entanglement swapping (see Fig. )to create a maximally entangled state between two far-away nodes of the given lattice. This can be performed,provided the two such nodes lie in the same cluster, theprobability of which is θ ( p ) , which is strictly greaterthan zero if SCP is greater than the percolation thresh-old of the lattice. So, the task of creating a maximallyentangled state between two end nodes of the lattice hasbeen accomplished with a strictly non-vanishing prob-ability. The same would not have been possible usingonly entanglement swapping (without the singlet con-version step in CEP), since in that case, as was shownin Ref. [ ], if the initial states were partially entangled,then the probability of succeeding would have decayedexponentially, with increasing lattice distance betweenthe nodes. B. Quantum entanglement percolation
In QEP, the original lattice structure, using some par-ticular quantum measurements, is converted to someother lattice for which the percolation threshold is lower
FIG. . Entanglement swapping. Entanglement swapping,probably the earliest and the simplest quantum network, canbe seen as a method of entangling two quantum systems thathave never met, but are each entangled with two further quan-tum systems who have interacted in the past. In the schematicgiven, the quantum systems are represented by the blue cir-cles and the lines represent entangled states. The red ellipseenclosing the two blue circles indicate an interaction, possi-bly via a measurement, between the two blue circles. In thefigure, the two blue circles at the extremes get entangled bythe interaction. The arrow represents the flow of operationsin time. than that of the parent lattice. Thenceforth, CEP is ap-plied on the new lattice. To demonstrate the effective-ness of their protocol, ACL used a double-layered hon-eycomb lattice in which percolation is not possible asthe critical amount of entanglement (which is governedby the SCP here) is less than the percolation threshold[ ]. Carrying out measurements in the Bell basis atthe nodes, they converted their original lattice struc-ture to a triangular lattice which has a lower perco-lation threshold, and thus meets the criterion, of thecritical amount of entanglement being greater than thepercolation threshold, for entanglement percolation tosucceed in the new lattice. The Bell basis is given bythe set of four orthonormal states, ( √ )( | (cid:105) ± | (cid:105) ) , ( √ )( | (cid:105) ± | (cid:105) ) . As evident, this protocol of en-tanglement percolation uses the richness of the geome-try of two-dimensional lattices. Further, the particularlattice transformation used is one of the most impor-tant factors leading to the success of the QEP. It is im-portant to note here that a double-layered honeycomblattice was used, since using the measurement strategydescribed in Ref. [ ], it is not possible to convert thesingle-layered honeycomb lattice to a triangular lattice.In this paper, we will present a quantum measure-ment strategy which helps to establish multiparticlegenuine entangled states between an arbitrarily largenumber of nodes and maximally entangled states be-tween two end nodes of the lattice by using a single-layered honeycomb lattice. C. Schmidt decomposition If | ψ (cid:105) is a pure state which belongs to a bipartitequantum system, described on the Hilbert space H = H A ⊗ H B , then there exist orthonormal bases {| i A (cid:105)} and {| i B (cid:105)} in H A and H B respectively, such that | ψ (cid:105) = ∑ i √ α i | i A (cid:105)| i B (cid:105) , ( )referred to as the Schmidt decomposition of | ψ (cid:105) , where √ α i are non-negative real numbers which satisfy thecondition ∑ i α i =
1. The √ α i are known as Schmidtcoefficients. D. Nielsen’s majorization criterion
Nielsen found the necessary and sufficient conditionthat a pure entangled state | ψ (cid:105) can be deterministicallyconverted into another pure entangled state | ψ (cid:105) underLOCC [ ]. Consider a pure entangled state | ψ (cid:105) ∈ C n ⊗ C n . Let the Schmidt decomposition of the state | ψ (cid:105) begiven by | ψ (cid:105) = n ∑ i = √ α i | i A (cid:105)| i B (cid:105) , ( )where ∑ ni = α i = 1 and α i ≥ α i + ≥
0. The problem is tofind whether it can be converted, exactly and determin-istically at the level of a single copy and under LOCC,to another pure state | φ (cid:105) ∈ C n ⊗ C n , whose Schmidtdecomposition is given by | φ (cid:105) = n ∑ i = (cid:112) β i | i A (cid:105)| i B (cid:105) , ( )where ∑ ni = β i = β i ≥ β i + ≥
0. Let us definethe Schmidt vectors, λ ψ = ( α , α , . . . , α n ) and λ φ =( β , β , . . . , β n ) . Then the Nielsen’s criterion tells us that | ψ (cid:105) can be converted to | φ (cid:105) under LOCC if and only if λ ψ is majorized by λ φ (written as λ ψ ≺ λ φ ), that is iff k ∑ i = α i ≤ k ∑ i = β i ( )for all k =
1, 2, . . . , n . E. Singlet conversion probability
Vidal [ ] showed that the pure state | ψ (cid:105) ∈ C n ⊗ C n can be locally converted to the pure state | φ (cid:105) of the sameHilbert space with a maximum probability given by P ( | ψ (cid:105) → | φ (cid:105) ) = min l ∈ [ n ] (cid:32) n ∑ i = l α i (cid:44) n ∑ i = l β i (cid:33) . ( ) For the conversion of a two-qubit pure partially en-tangled state with Schmidt coefficients √ φ and √ φ , φ > φ >
0, to a maximally entangled Bell state, theabove formula yields an SCP of φ . F. Locally converting generalized GHZ to GHZ state
We find here an LOCC-based strategy to convertan m -qubit partially entangled Greenberger-Horne-Zeilinger (GHZ) state [ ] to the m -qubit GHZ statewith maximal probability. The initial state for the mea-surement strategy is the partially entangled GHZ state(also called the generalized GHZ state), | ψ (cid:105) A A ... A m =cos θ |
00 . . . 0 (cid:105) + sin θ |
11 . . . 1 (cid:105) , | ii . . . i (cid:105) ≡ | i (cid:105) ⊗ m , ∀ i =
0, 1.Here we take 0 < θ < π and cos θ = √ φ > sin θ = √ φ . | (cid:105) and | (cid:105) are elements of the computational ba-sis, being eigenstates of the Pauli- z operator. We nowapply an LOCC-based measurement strategy to convertthe above state to the GHZ state, | ψ + (cid:105) = ( |
00 . . . 0 (cid:105) + |
11 . . . 1 (cid:105) ) / √
2. The strategy involves a measurement onjust any one of the m qubits. The corresponding mea-surement operators are given by M = (cid:32)(cid:113) φ φ
00 1 (cid:33) , M = (cid:32)(cid:113) − φ φ
00 0 (cid:33) , ( )where ∑ i = M † i M i = I , with I being the identity opera-tor acting on the qubit Hilbert space. The probability ofconversion is seen to be 2 φ , and is the same as that ofthe conversion of the same states in any bipartition, sothat the probability is optimal.In this paper, for m =
3, we will refer to the corre-sponding states as generalized GHZ and GHZ states.For larger m , we will refer to the GHZ state as the “cat”state [ , ]. III. MEASUREMENT STRATEGY LEADING TOENTANGLEMENT PERCOLATION IN ASINGLE-LAYERED HONEYCOMB LATTICE
Our task is to establish maximally entangled statesbetween two distant nodes of an asymptotically largelattice. As shown in Ref. [ ], the average of the SCPsover all four possible outcomes at one node that may re-sult due to entanglement swapping between two iden-tical copies of a two-qubit state, is the same as that ofthe original states. Further, they used this result to con-vert two layers of partially entangled two-qubit purestates arranged on honeycomb lattices to a single layerof the same on a triangular lattice, via entanglementswapping measurements at the nodes of the bi-layeredhoneycomb lattice. This is done because the percolationthreshold of the honeycomb lattice is higher than that ofthe triangular one. If the partially entangled two-qubitpure states that acted as initial states of the bi-layeredhoneycomb lattice is such that the SCP is less than the amount needed to do entanglement percolation on ahoneycomb lattice, the ACL entanglement-swapping-based quantum measurement strategy enables one todo entanglement percolation via “moving” to the trian-gular lattice, provided the said SCP is higher than thecritical value needed for entanglement percolation onthe latter lattice. See also Ref. [ ].Beginning with a single layer of partially entangledpure states arranged on a honeycomb lattice, we pro-pose another quantum measurement strategy whichcan attain entanglement percolation. Since it is a singlelayer of the lattice that we use, the amount of entangle-ment used here is lower than that in the ACL strategywhich used two layers. However, while ACL requiredtwo-qubit measurements, we use three-qubit ones.The honeycomb lattice has three qubits at each node,and the three edges emerging from each node connectwith three other qubits of three neighbouring nodes,with each connection (edge) being a single copy of thepartially entangled state, | φ (cid:105) = √ φ | (cid:105) + √ φ | (cid:105) , φ > φ >
0, and φ + φ =
1. See Fig. . We now assumethat measurements are carried out, at a certain specifiedset of nodes of the lattice, in the three-qubit GHZ basiswhich is composed of the following eight states: ( | (cid:105) ± | (cid:105) ) / √ ( | (cid:105) ± | (cid:105) ) / √ ( | (cid:105) ± | (cid:105) ) / √ ( | (cid:105) ± | (cid:105) ) / √
2. ( )The honeycomb lattice is a “bipartite” lattice, whichmeans that its nodes can be colored by using two col-ors, say red and blue, such that all nearest neighbors ofany red node are blue, and vice versa. The GHZ-basismeasurements are carried out only at the nodes of aspecific color, say red. The measurements are carriedout on the three qubits at the nodes which are coloredred in Fig. , and which are circled in red in Fig. . Af-ter this measurement, we have successfully convertedour single-layered honeycomb lattice made up of par-tially entangled pure two-qubit states to a triangularlattice spanned by three-qubit generalized GHZ states.It is to be noted that the triangular lattice is such thatevery fundamental triangle that is filled with a general-ized GHZ state is surrounded by three empty trianglesthat are neighbors on its sides. And vis-à-vis, everyempty triangle is surrounded on its sides by filled tri-angles. The three-qubit generalized GHZ state betweenthe three relevant qubits of the three nodes (which forma triangle in Fig. ) neighboring the node at which theGHZ-basis measurement is carried out, can be obtainedby computing the following expression: ( I ⊗ |A i (cid:105)(cid:104)A i | ⊗ I ⊗ I ) | λ (cid:105) / √ p i . ( )The notations in the above state can be described as fol-lows. Consider four parties A , A , A , A which arefour neighboring nodes of a honeycomb lattice. Sup-pose that A is sharing three pure two-qubit partiallyentangled states √ φ | (cid:105) + √ φ | (cid:105) , φ > φ , φ + φ =
1, with each of the other three parties. So, A has three FIG. . Monolayer hexagonal lattice of bipartite states tomonolayer triangular lattice of tripartite states. The single-layer hexagonal lattice is formed by bipartite (possibly non-maximally) entangled states on each edge. Each node con-tains three qubits. Being a bipartite lattice, we can color thenodes of the hexagonal lattice with two colors, say, red andblue, such that each nearest neighbor of a red node is blueand vice versa. Measurement in the GHZ basis carried out onthe three qubits at the red nodes. For example, the GHZ-basismeasurement is performed at the node A , which has threequbits, each of which are connected to a qubit at a neigh-boring blue node via a bipartite entangled state. These bluenodes are denoted in the figure as A , A , and A . The rele-vant three qubits of these blue nodes transform into a gener-alized GHZ state, due to the GHZ-basis measurement at thered node, A . There are two more qubits at each of these bluenodes, which are in turn connected to neighboring red nodeson the other side with respect to A . Making such GHZ-basismeasurements at all the red nodes of the hexagonal latticeleads to a monolayer triangular lattice of generalzied GHZstates, a unit cell of which is depicted on the right-hand-sideof the figure. qubits, representing a node, on which the measurementin GHZ basis is carried out. The operators |A i (cid:105)(cid:104)A i | arethe projectors onto the elements of the three-qubit GHZbasis, which act on the three qubits present in a node(being in possession of A ). On the other hand, identityoperators act on the single qubits of A , A , and A . p i is the probability that the projector |A i (cid:105)(cid:104)A i | clicks. | λ (cid:105) denotes the total state of the six qubits in posses-sion of the four observers at the four nodes. Exploitingthe three-qubit measurement, we obtain the followingthree-qubit generalized GHZ states, which generate thetriangles spanning the new lattice: φ √ φ | (cid:105)± φ √ φ | (cid:105) √ φ + φ , φ √ φ | (cid:105)± φ √ φ | (cid:105) √ φ φ + φ φ , φ √ φ | (cid:105)± φ √ φ | (cid:105) √ φ φ + φ φ , φ √ φ | (cid:105)± φ √ φ | (cid:105) √ φ φ + φ φ . ( )These generalized GHZ states are created due to aGHZ-basis measurement, and they appear, respectively,with probabilities p i , given by p = p = φ + φ p j = φ φ + φ φ ∀ j =
3, . . . , 8. ( ) FIG. . Transforming a honeycomb lattice to a triangularone by measuring on the Greenberger-Horne-Zeilinger basison every other node. The nodes at which the measurementsare carried out are marked with a red circle on the hexagonallattice. These nodes do not appear any more on the triangularlattice. Further details appear in the text and in the caption ofFig. . The average SCP is calculated by averaging the SCPsover all the eight possible outcomes (all outcomes aregiven above) and is given as Avg. SCP = p = 2 φ ( φ + φ ) . Now we need to figure out the percolation thresh-old for our triangular lattice. It could be difficult to cal-culate the threshold for bond percolation in a triangularlattice spanned by GHZ states. However, we can mapour problem to a site percolation problem as shown inFig. .An essential point to note here is that now each sitein the mapped triangular lattice (the red dots on theright-hand-side lattice in Fig. ) denotes the presence of FIG. . Mapping bond percolation problem to site percola-tion one. The triangular lattice of generalized GHZ states thatwe obtained via the GHZ-basis measurements on every othernode of the hexagonal lattice is depicted on the left-hand-sideof the figure. The intent is to create a cat state (see text) be-tween qubits of an arbitrarily large number of nodes. On theleft-hand-side, this is a bond percolation problem, while wecan look at it as a site percolation problem by replacing ev-ery generalized GHZ states on the left panel by a dot on theright panel. Each of these dots has a three-qubit GHZ statewith probability p . The same intent as in the left panel isattained by a site percolation on the triangular lattice on theright panel. The change is only at the level of calculations,and does not require a physical transformation. FIG. . A specific scenario of percolation on triangular latticeof Greenberger-Horne-Zeilinger states. Suppose that we wishto create a cat state (see text) between the nodes marked as A ,... A . We are beginning with an initial situation where eachtriangle in the figure represents a GHZ state. For every suchtriangle, the GHZ state is present with probability p . Thecat state that we wished for, can be obtained by Bell measure-ments and single-qubit measurements along the boundary ofthe region formed by the nodes A , ... A . The Bell measure-ments are performed at the nodes marked by blue ellipses,on the two qubits inside those ellipses. The single-qubit mea-surements are marked by green circles. a GHZ state with probability p (and its absence withprobability 1 − p ). Percolation of GHZ states in theoriginal lattice is mapped to percolation of sites in themapped triangular lattice of sites in a one-to-one corre-spondence. It is to be noted that the mapping here isjust a mental picture that aids in the mathematics of theproblem, and does not represent a physical manoeuvre.The site percolation threshold for a triangular lat-tice is 1/2 [ ]. If the average SCP, p , of the origi-nal triangular lattice with generalized GHZ states (leftpanel in Fig. ) is larger than the percolation thresh-old p c ∆ ≡ ), arbitrarily large cat states, √ ( |
00 . . . 0 (cid:105) + |
11 . . . 1 (cid:105) ) , will be formed in the originaltriangular lattice. This will be effected in the followingway. There is a probability p for the the generalizedGHZ states in the original triangular lattice to be trans-formed locally, i.e., by local (with respect to the sites)quantum operations and classical communication (be-tween the sites), to a GHZ state. In cases when thetransformation is successful, we do two-qubit Bell-basismeasurements along with single-qubit measurementsat the sites of the original triangular lattice along theboundary of the region bounded by the sites formingthe cat state. See Fig. for an example. The condition,on the parameters of the bipartite non-maximally en-tangled states of the hexagonal lattice, for successfully creating an arbitrarily large GHZ state is given by2 φ ( φ + φ ) >
12 , ( )which, solving the cubic, provides the range, φ > φ ∈ (cid:18) − sin π
18 , 12 (cid:19) , ( )that is, φ > φ ∈ (cid:18) (cid:19) , ( )with the left end of the last interval being correct to fivesignificant figures. This threshold is exactly the same asfor entanglement percolation using CEP on the hexag-onal lattice [ ]. Each site of the original triangular lat-tice contains three qubits. However, the sites that willform the cat state will of course have one “active” qubit(i.e., the qubit used in the construction of the cat state),and the remaining two qubits will remain “passive”. Bymeasuring in the σ x -basis on the active qubits at all buttwo of these sites, we can create a maximally entangledbipartite state between two sites that have an arbitrarilylarge distance between them on the lattice.Comparing our result with that in Ref. [ ], we seethat there the authors converted the bilayer honeycomblattice of partially entangled two-qubit states to a tri-angular lattice, and they succeeded in attaining bipar-tite entanglement percolation. All measurements per-formed were two-qubit ones. Our measurement strat-egy uses only a single layer of their honeycomb lat-tice, but uses three-, two-, and single-qubit measure-ments, to attain multipartite (and hence also bipartite)entanglement percolation. The percolation thresholdobtained in Ref. [ ] was better than the one obtainedhere.In Ref. [ ], the authors have used a different mea-surement strategy and have attained multipartite en-tanglement percolation. For the success of their strat-egy, the authors needed to do between four and fivemeasurements per unit cell of their honeycomb lattice,whereas we need to do three measurements per unitcell of our honeycomb lattice. Due to the reduced num-ber of measurements (per unit cell), it is plausible that noise effects on a realization of our strategy will belower than the same on the one in Ref. [ ].Finally, we compare our result of using QEP on themonolayer hexagonal lattice with that of using CEP onthe same lattice [ ]. The thresholds obtained are ex-actly the same. However, the number of measurementsare different. While CEP uses lower-qubit measure-ments, we use less measurements. Precisely, for an l × l square box encompassing a part of the hexagonal lat-tice, to create a cat state between nodes on the bound-ary of the square, CEP requires 6 l + O ( l ) single-qubitand O ( l ) two-qubit measurements, while the QEP pro-posed here requires 2 l + O ( l ) single-qubit, O ( l ) two-qubit, and 2 l + O ( l ) three-qubit measurements. It is tobe noted that the length and breadth of the square boxare counted such that the hexagonal lattice in Fig. isof breadth two. IV. CONCLUSION
Entanglement percolation is an interesting techniqueto achieve entangled states between two or more nodesof a lattice that can be arbitrarily distant. We haveused a single layer of a honeycomb lattice made ofnon-maximally entangled bipartite quantum states, andprovided a quantum measurement strategy involvingthree-, two-, and single-qubit measurements, to ob-tain Greenberger-Horne-Zeilinger (cat) states sharedbetween an arbitrarily large number of lattice nodes.The cat states can then be reduced to a two-qubit Bellstate shared between faraway nodes. A feature of ourstrategy is that after our entanglement swapping mea-surement on the initial hexagonal lattice, every othernode with all their qubits is totally removed from theprotocol and does not play any further role in the strat-egy, which is significantly different from the measure-ment strategy used in Ref. [ ].The resources used in an entanglement percolationstrategy are the entangled states of the original lat-tice and the measurements performed in between. Thenumber of measurements performed is potentially animportant parameter for estimating the noise effects ona realization of the strategy. We have compared our en-tanglement percolation strategy with existing ones inthe literature with respect to both these resources. [ ] M. A. Nielsen and I. J. Chuang, Quantum Computation andQuantum Information (Cambridge University Press, Cam-bridge, ).[ ] R. Horodecki, P. Horodecki, M. Horodecki, and K.Horodecki, Rev. Mod. Phys. , ( );[ ] O. Gühne and G. Tóth, Physics Reports , ( ).[ ] S. Das, T. Chanda, M. Lewenstein, A. Sanpera, A.Sen(De), and U. Sen, The separability versus entangle-ment problem , in
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