Directed percolation effects emerging from superadditivity of quantum networks
aa r X i v : . [ qu a n t - ph ] J u l Directed percolation effects emerging from superadditivity of quantum networks
L. Czekaj,
1, 2
R. W. Chhajlany,
3, 1, 2 and P. Horodecki
1, 2 Faculty of Applied Physics and Mathematics, Gda´nsk University of Technology, 80-952 Gda´nsk, Poland National Quantum Information Center of Gda´nsk, 81-824 Sopot, Poland Faculty of Physics, Adam Mickiewicz University, Umultowska 85, 61-614 Pozna´n, Poland (Dated: August 1, 2018)Entanglement indcued non–additivity of classical communication capacity in networks consistingof quantum channels is considered. Communication lattices consisiting of butterfly-type entangle-ment breaking channels augmented, with some probability, by identity channels are analyzed. Thecapacity superadditivity in the network is manifested in directed correlated bond percolation whichwe consider in two flavours: simply directed and randomly oriented. The obtained percolation prop-erties show that high capacity information transfer sets in much faster in the regime of superadditivecommunication capacity than otherwise possible. As a byproduct, this sheds light on a new type ofentanglement based quantum capacity percolation phenomenon.
Introduction.
Percolation (see e.g. [1, 2]) is a naturalconcept that emerges in the description of spreading pro-cesses in the presence of medium imperfections. Percola-tion effects in quantum networks have recently been thesubject of increasing interest [3–7]. Most of the attentionhas been restricted to the generation of large scale net-works with maximally entangled states between elemen-tary nodes to allow for quantum communication applica-tions, starting from initial imperfect, i.e. non-maximallyentangled state networks. The interesting central newinsight introduced in [3, 4] is that local quantum opera-tions may be used not only to purify entanglement but,simultaneously, to change the topology of the lattice toa new one with lower percolation probability threshold.This idea has been developed for different states [5, 6]and lattice dimensions [7, 8]. In a different context, per-colation concepts also appeared in quantum informationtheory (QIT) in the study of cluster state generation [9].The capacity of a network determines its utility in thedomain of communication. The development of QIT hasled to the uncovering of interesting quantum effects onchannel capacities, e.g. superadditivity of quantum (Q-type) channel capacity [10]. This result followed the in-tuition developed in the bound entanglement activationeffect [11] (where two weak resources activate each otherbecoming collectively useful for some task) continued fur-ther in Refs. [12] and [13]. Independently, the first super-additivity effect of classical (C-type) capacity in quantummulti-access channels has been described [14]. Both Q-type and C-type superadditivities have been proven evenfor entanglement breaking channels [15].In this paper, we consider percolation effects in quan-tum networks from a channel perspective. In particular,we show that channel superadditivity can be used to en-hance percolation of information through networks. Theschemes are based on network models of classical infor-mation transfer through quantum multipartite channels(MACs), which can be mapped to certain types of di-rected bond percolation problems [16]. We first considera simple layered communication scheme (A) to demon- strate the basic idea of percolation assisted by superad-ditive capacities and then describe a more complicatedscheme of multidirectional communication (B). Interest-ingly, the percolation problem in case B does not seemto have been studied elsewhere in the literature.The basic ingredients of the quantum networks are:a passive fixed underlying network built up of elemen-tary entanglement breaking MACs, and an active aux-iliary incomplete network consisting of randomly gener-ated (open) bonds. The term passive means that no bondin the network allows a-priori high capacity communica-tion (HCC), whereas elements of the active network arehigh capacity channels. Importantly, the active channelscan serve to activate HCC through the passive channels.
Model A - Layered network communication.
The ele-mentary channel of the passive network is chosen in thispaper as a 2-sender 1-receiver noisy quantum MAC de-picted as the wedge shaped channels in Fig. 1(b): theslanted line is a two-qudit ( d –dimensional) input per-taining to one user; the vertical line is a single qudit inputof the second user. The single qudit system is modifiedby one of a chosen set of (orthogonal) d unitary oper-ations fired by the logical value of the two-qudit system(see Ref. [15] for details). This single qudit line is furthermodified by a depolarizing channel and is the sole outputsystem at the receiver’s end. For moderate to large de-polarization – the working regime considered here – thischannel has poor classical capacity C ′ for any user [15]: C ′ ≪ C ≤ C max ≡ log d, (1)where C max is the maximal attainable capacity for a sin-gle qudit output, corresponding to an identity channel.Communication can be improved by adding a high ca-pacity active channel, e.g. an ideal channel, along thevertical transmission line. Inputting a two-party maxi-mally entangled state to the ideal channel and verticalline of a wedge channel increases the capacity along theslanted line of the wedge channel to a much higher value C (see Eq.(1)) – a manifestation of superadditivity ofchannel capacities [21] (for high depolarization when the i,t N i,t+1 N i+1,t N i+1,t+1 a) b)c) d) BA BA
FIG. 1. Model A - layered communication network: a) passivenetwork; users are located at nodes of a square lattice, the lat-tice is filled with butterfly shaped primitives, information flowis directed from layer t → t +1; b) butterfly primitive consistsof two MACs (solid and dashed wedges) from Ref. [15]; Node(user) N i,t can communicate with N i − ,t +1 , N i,t +1 , N i +1 ,t +1 with maximal capacity C ′ ≪ C ; c) active network filled ran-domly by triples of ideal channels allowing HCC (no HCCfrom A to B); d) using entanglement changes the geometryof the HCC network leading to directed communication paths(black arrows) with capacities ≥ C from A to B. channel becomes entanglement breaking, C ≈ ( d + 1) C ′ [15]). We define HCC as transmission at a rate ≥ C .The structure of the wedge channel induces a sense ofdirection of communication. In the network context, wefirst consider the scenario where only forward communi-cation is allowed (see Fig.1). The passive channel inputsare shared by nearest neighbour pairs of sites in horizon-tal layers giving rise to a butterfly shaped fixed network.The active channels are only placed on vertical bonds.Suppose that these are available with probability p – onecan assume that identity channels are initially availableat all vertical bonds, yet due to fragility w.r.t. noise ei-ther remain useful (ideal) with probability p or becomeunuseful random operations with probability 1 − p effec-tively erasing information. In our scheme, we considera setup where a triple of identity channels is the basicactive channel [22]– this allows the possibility of simul-taneous HCC between a given sender and his verticallyplaced receiver as well as between the sender’s two hori-zontal nearest neighbours and that receiver (Fig. 1).The question of establishing long range HCC in such anetwork is a directed percolation problem. One asks, un-der what conditions, is it possible for any user to be ableto perform directed HCC, through intermediate nodes,with a user or users located at a distance scaling withthe length of the network. One may compare two sce-narios - (a) entanglement-free or classical, where no en-tanglement is allowed in the protocol used at any nodeand (b) entanglement-assisted (EA), which takes full ad- a)b)c) d) FIG. 2. Model B - Multidirectional communication networks:a) passive network, allowing communication in each direc-tion; b) structure of butterfly primitive; c) active channels;d) emerging network geometry induced by entanglement be-tween active and passive channels. vantage of the superadditive effect described earlier. Theformer case corresponds to communication only along 1–dimensional paths and the percolation threshold prob-ability is 1, rendering the network useless for HCC forfinite loss probability of active channels. The EA schemeinvolves changing the geometry of the HCC network andis mapped to a correlated directed bond percolation prob-lem where with probability p three directed bonds (form-ing an arrow shape) are placed on the lattice (Fig.1(e)).The threshold probability is significantly suppressed dueto entanglement induced increased connectivity. We per-formed a standard Monte Carlo simulation (see [23]) andfound the percolation threshold to be p c = 0 . . π/ p = 1) with axes along the verticallines passing through the starting nodes. Near the crit-ical point the clusters are very narrow and essentiallyquasi 1-dimensional, as the probability of obtaining aconnection with a site a large distance away from thesource and at an angle θ from the axis, Θ( p, θ ) > | θ | < δθ ( p ) ∼ ( p − p c ) b (see e.g. Dhar and Barma [17]).2 .0 0.1 0.2 0.3 0.4 0.5 0.6 . . . . . p + p − long range HCCachievable in EA model Blong range HCC achievable in EA and EF model B FIG. 3. (a) Comparison of the percolation critical lines( p c + , p c − ( p c + )) for percolation process on randomly orientedsquare lattice (classical scheme) and butterfly network (EAscheme). Dots are numerically obtained points. The EA scheme beats the classical scheme in that thereis a wide window in p for HCC, and that the horizon-tal extent of connections | θ | at a distance t → ∞ awaychanges from 0 to π/ p increases from p c to 1. Model B – Randomly oriented communication.
We nowmove to a general scenario within the described frame-work of channels (Fig. 2). Now each node can com-municate in both directions with its nearest neighbours(main bonds) on a square lattice as well as all of its nextnearest neighbours (diagonal bonds) through the passivenetwork. The configuration leading to this possibility isshown in Fig.2 (a) and (b). The active network channelsare placed along the main bonds of the square lattice andare tagged for use in a particular direction.HCC between distant nodes is now a multidirectionalbond percolation problem. We consider the case whereupward, left-to-right oriented identities are present withprobability p + while downward, right-to-left orientedidentities are placed with probability p − . Note, that bidi-rectional bonds appear as the independent choice of twooppositely directed channels on a given bond.This setup is interesting already for the case when, say, p + = 0. Using an entanglement free protocol, capacity C ≥ C can already be obtained through the main bondsof the square lattice along the up or right directions, incontrast to Model A. This is the well known square latticedirected bond percolation problem for which the perco-lation threshold p c ≈ .
64 (see e.g. [17]). But, in the EAscheme, due to increased and correlated connectivity, the . . . . . .
15 0 . . . . . . . b p + t t class t EA b class b EA FIG. 4. Comparison of critical exponents τ , β for classicaland EA schemes of multidirectional oriented percolation. percolation threshold is almost halved w.r.t. the classi-cal scheme and has been calculated here to be p c ≈ . p ≈ .
6, the probability of a starting nodebelonging to an infinite or system spanning anisotropiccluster (in the thermodynamic limit) F ∞ > / F ∞ = 0 otherwise.The general multidirectional communication problemis the most interesting setup. In the classical scenario,this reduces to square lattice randomly oriented perco-lation. Problems of this category were studied in thecontext of random resistor diode networks (see e.g. [19])mainly using renormalization group calculations. How-ever not many results seem to be available in the litera-ture on the p ± percolation problem (see however [20] forsome analytical properties).Here, we map out the phase diagram in the ( p + , p − )plane in Fig.3 using Monte Carlo simulations (see Ap-pendix). Note that the diagram is symmetrical w.r.t.the line p + = p − due to system symmetry under the in-terchange p + ↔ p − . This result is to be compared withthe multidirectional correlated bond percolation phasediagram for the EA scheme (Fig. 2(d)). As can be seenin Fig.3, the EA scheme is drastically better than theclassical scheme in the whole parameter plane.An interesting feature of both schemes is that they al-low switching of universality classes of phase transitionsfrom the DP class to the Isotropic Percolation (IP) classto which standard bond/site percolation belongs. Thisis facilitated by the choice of parameters p ± , and as aresult one may accordingly change the properties of the3ong range clusters. To provide evidence for this, we cal-culate two universal critical exponents (Fig. 4) β and theFisher exponent τ (see [23]) as one moves along the criti-cal lines (Fig. 3) of the two models. Recall, that β deter-mines how the percolation probability F ∞ ∼ ( p − p c ) β in-creases above the percolation threshold, while the Fisherexponent determines how the probability of obtaining acluster larger than size n decreases with n at the criticalpoint p = p c : F n ∼ /n τ − . Note that the values of theseexponents in the DP class are β ≈ . , τ ≈ .
112 [17]and in IP: β = 5 / ≈ . , τ = 187 / ≈ . p c − ≈ .
4, since thisregion inherits the exponents at the DP point p c − = 0.At p c − = 1 / p c + ), which is an isotropic symmetrypoint, one obtains values of the exponents correspond-ing to the IP universality class. In between, there is acharacteristic crossover region between the two types ofbehaviour. In particular, this means that for a choice ofparameters approaching the isotropic point, one obtainsdifferent characteristic growth of clusters (determined bydifferent critical exponents) than when in the DP region.Secondly the cluster geometrical characteristics changefrom highly anisotropic to isotropic. The butterfly net-work percolation problem (EA scheme) basically followsthe same pattern (see Fig. 4) and can be considered arescaled version of the classical problem, wherein againlies the quantum advantage of the EA scheme. For thisnetwork, the isotropy point is found to be located at p c + = p c − ≈ . β inthe DP regime as compared to the square lattice prob-lem do not seem to be significant - and are a result of theextreme sensitivity of the calculated exponents on theaccuracy of critical probabilities. Concluding remarks.
We have described percolationeffects in a channel context showing how directed perco-lation effects (not considered before) emerge in the con-sideration of quantum networks. In the context of per-colation theory, to our knowledge, the multidirectionalcorrelated bond percolation problem has not been stud-ied before. Finally, quite remarkably, our results providea new entanglement percolation effect in the spirit of Ref.[3]. Keeping everything else unchanged, consider the Bellmeasurement (BM) channel of Fig. 1 of Ref. [15] insteadof the MAC used here, with AC (BC) playing the roleof diagonal (resp. vertical) bond of the square lattice.Then any randomly generated singlet between B and Ccan be switched, via entanglement swapping, to a singleton the diagonal AC. This leads to the same geometry asdiscussed here but now one asks about the possibility ofbuilding a long range network of singlets. Note that sin-glets are directionless. Without using the BM channel,we obtain a “classical” scheme related to square latticepercolation which has threshold p c = 0 .
5. Since direc- tionless percolation must certainly be at least as goodas directed percolation, an EA scheme making use ofthe switching mechanism of the BM channel will havethreshold at most equal to the calculated threshold atthe isotropic point of Model B p E Ac ≤ p c + = p c − ≈ . [1] D. Stauffer and A. Aharony, Introduction to percolationtheory, Taylor & Francis 2003.[2] G. Grimmett, Percolation, Springer-Verlag 1999.[3] A. Acin, J. I. Cirac, and M. Lewenstein, Nature Phys. ,256 (2007).[4] G. J. Lapeyre, Jr., J. Wehr, and M. Lewenstein, Phys.Rev. A , 042324 (2009).[5] S. Perseguers, L. Jiang, N. Schuch, F. Verstraete, M.Lukin, J. Cirac, and K. Vollbrecht, Phys. Rev. A ,062324 (2008); S. Perseguers, D. Cavalcanti, G. J.Lapeyre, Jr., M. Lewenstein, and A. Acin, Phys. Rev. A81 , 032327 (2010).[6] S. Broadfoot, U. Dorner, and D. Jaksch, EuroPhys. Lett.88, 50002 (2009).[7] S. Perseguers, Phys. Rev. A 81, 012310 (2010).[8] A. Grudka et al. , manuscript in preparation (2011).[9] K. Kieling, J. Eisert, ,,Percolation in quantum compu-tation and communication”, in:
Quantum and Semi-classical Percolation and Breakdown in Disordered Solids ,pages 287-319 (Springer, Berlin, 2009).[10] G. Smith, J. Yard, Science , 1812 - 1815 (2008).[11] P. Horodecki, M. Horodecki, R. Horodecki, Phys. Rev.Lett. , 1056 (1999).[12] P. Shor, J. A. Smolin, and A. V. Thapliyal, Phys. Rev.Lett. , 107901 (2003).[13] W. Duer, J. I. Cirac, and P. Horodecki,Phys. Rev. Lett. , 020503 (2004).[14] L. Czekaj and P. Horodecki, Phys. Rev. Lett. , 110505(2009).[15] A. Grudka and P. Horodecki, Phys. Rev. A , 060305(R)(2010).[16] S. R. Broadbent and J. M. Hammersley, Proc. Camb.Phil. Soc. , 629 (1957).[17] D. Dhar and M. Barma, J. Phys. C: Solid State Phys., , L1 (1981).[18] H. Hinrichsen, Adv.Phys. , 815 (2000).[19] S. Redner, J. Phys. A : Math. Gen. , L349 (1981); ibid , L685 (1982); Phys. Rev. B 25 , 3242 (1982).[20] Geoffrey R. Grimmett, Random Structures & Algorithms
257 (2001)[21] This is a superadditive effect as the identity is unavailabeto the user of the slanted input line of the wedge channel.[22] For qubit channels, e.g. this can be physically realizedas a process of ideal transmission with probability p of asingle photon having three frequency states and two po-larization degrees of freedom for each frequency, where thedominating noise process is photon loss.
23] See supplemental material for methods. upplemental Material Directed percolation effects emerging from superadditivity of quantum networks
Lukasz Czekaj, Ravindra W. Chhajlany and Pawe l Horodecki
Methods
The critical percolation properties of the two models were studied using direct Monte Carlo simulations in thespirit of Dhar and Barma [1]. We studied the change in behaviour of the probability F n , of appearance of clusters ofsize greater than n , with bond probabilities p and looked for characteristic scaling behaviour expected in the criticalregion to identify percolation thresholds plotted. Below the probability threshold, F n ∝ exp( − n ) for large n while F n → constant in the supercritical phase [1]. The critical region is characterized by scaling laws with F n ∝ n − τ described by a power law dependence at the critical value p c , which we localized by sweeping through super– andsub–critical probabilities using an interval bisection method. For model A, cluster size distribution data was obtainedby performing 10 realizations (per value of probability p ) of cluster growth starting from a single node. Model B simulations were performed on a fixed 2 × by 2 × square lattice also with 10 realizations for each p , wherecluster connectivity was identified using a breadth–first search algorithm.The qualitative values of the critical exponents τ, β presented in the paper for model B are defined as follows: F n ( p ) ∼ n − ( τ − at p = p c and F ∞ ( p ) ∼ ( p − p c ) β A simple method used to obtain these was to directly calculate the slope of the plots of these two functions in log-logscale. Since the problem contains two parameters p + , p − , we chose one of them p + do be the independent parameterwith p − ( p + ) determined so as to be on the critical line of the model. For the determination of β , we consideredclusters of size n greater than 10 to be “infinite” or system spanning clusters on the finite lattice.Alternatively, we also determined β from τ and an auxiliary exponent γ , which describes the critical behaviour ofmean cluster size h n i also readily available from the simulation: h n i ∼ ( p c − p ) − γ for p − p c → + The following universal equation, derived from scaling relations, is known to hold for directed (uncorrelated) bondpercolation [1] and isotropic percolation [2]: β = (cid:16) τ − − τ (cid:17) γ We assumed the equation to be true for the entire critical plane and obtained results for β in agreement with thoseobtained using the first method. Results obtained in the latter manner are those presented in the paper. [1] D. Dhar and M. Barma, J. Phys. C: Solid State Phys., , L1 (1981).[2] D. Stauffer and A. Aharony, Introduction to percolation theory, Taylor & Francis 2003., L1 (1981).[2] D. Stauffer and A. Aharony, Introduction to percolation theory, Taylor & Francis 2003.