Quantum networking with short-range entanglement assistance

Siddhartha Santra and Vladimir S. MalinovskyUS Army Research Laboratory, Adelphi, Maryland 20783, USA

We propose an approach to distribute high-fidelity long-range entanglement in a quantum network assistedby the entanglement supplied by auxiliary short-range paths between the network nodes. Entanglement assis-tance in the form of shared catalyst states is utilized to maximize the efficiency of entanglement concentrationtransformations over the edges of the network. The catalyst states are recycled for use in adaptive operationsat the network nodes and replenished periodically using the auxiliary short-range paths. The rate of long-rangeentanglement distribution using such entanglement assistance is found to be significantly higher than possiblewithout using entanglement assistance.

Long-range entanglement in a quantum network [1] enablespromising applications ranging from unconditionally securecommunications [2] and loophole-free tests of quantum non-locality [3] to a network of quantum clocks [4] and quantum-enhanced interferometry [5]. The entanglement generated ini-tially in such networks is short-range, that is, in the form ofentangled states on edges between neighboring nodes. Sub-sequently, long-range entanglement may be obtained betweentwo remote nodes by identifying a path over the primary en-tangled edges and connecting them via entanglement swap-ping operations [6] at the intermediate nodes such as in thequantum repeater [7] approach.

A quantum network should be seen as more than an inter-laced collection of linear quantum repeaters. The latter utilizelocal quantum operations at the network nodes aided by clas-sical communication (LOCC), such as filtration [8] or entan-glement distillation [9], to distribute entanglement over longdistances. Yet, existing repeater protocols do not fully exploitthe entanglement along auxiliary short-range paths betweenthe nodes of the network that may be afforded, for example,by a dynamic topology [10], heterogenous links involving dif-ferent degrees of freedom of the flying qudits [11], or load-sharing in the quantum data plane of the network [12]. The in-efficient utilization of short-range entanglement in a quantumnetwork hinders high-fidelity long-range entanglement distri-bution where modest rates limit the quantum advantage of thepotential applications.

In this letter we propose a novel approach to efficientlyutilize short-range entanglement assistance for high-fidelitylong-range entanglement distribution in a quantum network.Here, auxiliary short-range paths supply catalyst states sharedbetween network nodes that enable a transformation of theentanglement assisted local operations and classical commu-nication (ELOCC) class [13]. This class of operations al-lows more general entanglement transformations in the net-work than LOCC. We utilize a catalytic ELOCC transforma-tion (catalysis) to concentrate the entanglement content of pri-mary low-fidelity states on network edges with a higher suc-cess probability compared to LOCC. Hence, a small numberof the primary states are required to obtain a state with a highfidelity to a Bell state that is necessary for long-range entan-glement distribution.

Catalysis is a one-shot procedure and if it succeeds the cat-

(a)

(b)

an+1 bn+1|C >

an bnρ

ELOCCan−1 bn−1

A B C D E

|C >an+1 bn+1

an bn

an−1 bn−1ρ

ρ

S S S S

a1 b1 a1 b1

A B

C

D E

FIG. 1. (color online) (a) Schematic of a quantum network. Yellow-colored edges comprise the long-range entanglement distributionpath connecting remote nodes A and E. Green-colored edges showavailable auxiliary short-range paths between neighboring nodes.(b) An ELOCC based long-range entanglement distribution path.Neighboring half-nodes share n-copies of a state ρ on qubit pairs(ai, bi)

i=ni=1 (Yellow spheres) and a catalyst state ∣C⟩ stored on the

qubit pair (an+1, bn+1) (Green spheres).

alyst state is recovered intact along with the desired Bell statecreation. The same catalyst state may be reused up to an ex-pected number of times determined by the catalysis successprobability. If the procedure fails no Bell state is obtainedand the catalyst state is lost as well. For sustained long-rangeentanglement distribution the primary states are continuouslygenerated over the edges along the path connecting the remotenodes in a quantum repeater-like manner. Whereas the stockof catalyst states needs to be replenished only periodically us-ing the auxiliary short-range paths between neighboring nodesallowed by the topology of the network, see Fig. 1.

Our approach therefore goes beyond the linear repeaterparadigm and establishes a way of quantumly aggregatingthe entanglement resources provided by short-range paths ina quantum network. Together with its resource-efficient andone-shot feature, this can significantly enhance the long-rangeentanglement distribution rate between remote nodes in thenetwork compared to when no auxiliary short-range entangle-

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ment assistance is available.In catalysis, two parties at network nodes A and B uti-

lize a shared entangled pure state ∣C⟩ (the catalyst) to achievecertain entanglement transformations forbidden using onlyLOCC [14]. Rather than being consumed the catalyst stateis recovered after the transformation, i.e.,

σ /→LOCC

ρ

σ ⊗ ∣C⟩ ⟨C∣ →LOCC

ρ⊗ ∣C⟩ ⟨C∣ , (1)

where σ, ρ, ∣C⟩ are bipartite states shared by A and B. Moregenerally, catalysis can increase the success probability ofeven some probabilistic entanglement transformations com-patible with LOCC [15]. The set of possible catalysts for adesired transformation can be determined from the Schmidtcoefficients of the initial and final states [13]. For certain pairsof pure states, σ = ∣ψ⟩ ⟨ψ∣ and ρ = ∣φ⟩ ⟨φ∣, a catalyst enabling adeterministic transformation, i.e. with Pcat(I → F ) = 1, froman initial state ∣I⟩ = ∣ψ⟩ ∣C⟩ to a final state ∣F ⟩ = ∣φ⟩ ∣C⟩ can befound. This is possible iff the entanglement monotones [15],Ek(X) ∶= 1 −∑k−1

j=0 λXk , X = I,F , that are functions of the

ordered Schmidt coefficients, λX = (λX1 ≥ λX2 ≥ ... ≥ λXd ), ofthe initial and final states are non-increasing,

Ek(I) ≥ Ek(F )∀k ∈ [1, d], (2)

where d is the larger among the dimensions of the initial andfinal state.

For some pairs of ∣ψ⟩ , ∣φ⟩ no catalyst can make the trans-formation, ∣I⟩ → ∣F ⟩, deterministic, i.e. P (I → F ) ≠ 1, butcan still increase the success probability as compared to justLQCC, i.e. Pcat(I → F ) > P (ψ → φ) [16]. The successprobability in this case is given by,

Pcat(I → F ) = min1≤k≤d

Rk(I,F ), (3)

where Rk(I,F ) ∶= Ek(I)/Ek(F ), k ∈ [1, d], is the ratio ofthe entanglement monotones of the initial and final states. Ingeneral, catalysts fulfilling the inequality in Eq. (2) or maxi-mizing the R.H.S. of Eq. (3) can be easily obtained using lin-ear programming techniques [17]. Catalysis has been shownto be more powerful than LOCC also for the case when σ andρ are genuinely mixed states [18] although no straightforwardcriteria such as the inequality in Eq. (2) exists in this case.

Pure state probabilistic catalysis is already sufficient, how-ever, to boost the long-range entanglement distribution ratein a quantum network with assistance from the entanglementalong auxiliary short-range paths. We illustrate this in a casewhere both the primary and catalyst states are two-qubit en-tangled states although in general they may have different di-mensions. We first motivate the form of the primary statesgenerated on the network edges followed by a description ofthe catalysis process assuming availability of the catalyst. Wethen discuss how to obtain the said catalysts and demonstratethe enhancement that entanglement assistance obtained viaauxiliary short-range paths can provide to the long-range en-tanglement distribution rate.

Consider a source, S, of polarization entangled photon pairson an edge A −B along a path connecting two remote nodesA and E in a quantum network as shown in Fig. 1. S, pro-duces photon-pairs in the state, ∣ψ+⟩ = (∣HV ⟩ + ∣V H⟩)/

√2,

which travel through the connecting optical fibers of length,L0, to heralded quantum memories, e.g. [19], at A and B.However, due to polarization-dependent losses in the fiberand other connecting elements the state vector of a singlequbit in a superposition state undergoes the transformation,ζ0 ∣H⟩ + ζ1 ∣V ⟩ → ζ0(

√tH ∣H⟩ +

√1 − tH ∣0⟩) + ζ1 ∣V ⟩ [20].

That is, a horizontally polarized photon gets effectively mixedwith the vacuum mode at a beamsplitter with transmittivity0 < tH < 1 whereas a vertically polarized photon suffers norelative loss. The state of the two quantum memories condi-tioned on heralding, which we assume occurs with probabilityP0 is therefore ρ = ∣α⟩ ⟨α∣ with,

∣α⟩ =√α ∣HV ⟩ +

√1 − α ∣V H⟩ , (4)

where α = tLH/(tLH+tRH) and tL,RH are the transmitivities of the

left and the right channels relative to the source. In general,since α ≠ (1/2), the primary entanglement is in the form ofpartially entangled states of fidelity, FL0(ρ) = ∣ ⟨ψ+∣α⟩ ∣2 =12+√α(1 − α) < 1.

A direct utilization of the states ρ of the form in Eq. (4)to obtain a long-range entangled state, ρAE , via entanglementswapping at intermediate nodes leads to a rapid decrease ofthe expected fidelity of ρAE . Assuming primary entanglementgeneration over each edge as described above, the expectedfidelity of ρAE approaches the distillability threshold expo-nentially fast with the total length, i.e., ⟨ψ+∣ρAE ∣ψ+⟩ − 1

2∼

(α(1 − α))N/2, where N = Ltot/L0, is the number of edgesalong the path. Since much higher temporal resources arerequired for the entanglement manipulation of lower fidelitystates to obtain a high-fidelity state the rate of directly ob-taining the latter over long range is greatly suppressed. Thefidelity of the entangled states over each edge therefore needsto be enhanced before the entanglement connections are madein order to achieve a reasonable long-range entanglement dis-tribution rate.

An optimal catalytic transformation can efficiently provideBell states over the network edges utilizing a few copies ofthe state ρ and one copy of a non-maximally entangled cat-alyst, ∣C⟩ = √

c ∣HV ⟩ +√

1 − c ∣V H⟩ ,0.5 < c < 1. For this,at least (n + 1)-qubit quantum memories are required at eachhalf-node so that n-copies of the state ρ and one copy of thecatalyst shared with the neighboring node are available for im-plementing the transformation,

ρ⊗n ⊗ ∣C⟩ ⟨C∣→ ∣β⟩ ⟨β∣⊗ ∣00⟩ ⟨00∣⊗(n−1) ⊗ ∣C⟩ ⟨C∣ , (5)

with ∣β⟩ = (∣HV ⟩ + ∣V H⟩)/√

2. The transformation concen-trates the entanglement in n-copies of the partially entangledstate ρ into a single maximally entangled state ∣β⟩ in a two-step process using adaptive unitary operations and measure-ments dependent on the parameters α and c and one-round of

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two-way classical communication between the nodes [14, 15]and is probabilistic in general [16] (see supp. mat. for details).

In step one, a temporary shared state ∣γ(n,α, c)⟩ is deter-ministically obtained via a sequence of operations on the ini-tial joint state, ∣α⟩⊗n ∣C⟩, of qubits at A and B. The entries of∣γ(n,α, c)⟩ can be obtained algorithmically from the set of ra-tios,Rl(I,F ), l ∈ [1,2n+1], and the entanglement monotones,El(F ), for the initial and final states given by the left andright hand sides of the transformation in Eq. (5), respectively.Essentially, ∣γ(n,α, c)⟩ is a state that maximizes the successprobability in Eq. (5) achievable using only local measure-ments at either node. In step two, a generalized measurementwith two-outcomes is performed at one of the nodes on itsportion of ∣γ(n,α, c)⟩. Corresponding to one of the measure-ment outcomes the state, ∣β⟩⊗ ∣00⟩⊗(n−1)⊗ ∣C⟩, is successfullyobtained with probability, Pmax

cat (n,α, c). The success (or fail-ure) of the measurement is then relayed to the other node. Incase of success, agents at A and B can utilize the shared state∣β⟩ for entanglement swapping to extend its range (if the adja-cent edges on either side also report success) and the catalyststate for further local entanglement concentration operations.Whereas, in case of failure they restart the process of generat-ing n-copies of ∣α⟩ and reobtaining the catalyst, ∣C⟩.

The success probability, Pmaxcat (n,α, c), of the transforma-

tion in Eq. (5) increases with n for fixed α and c, that is, fora given initial state and choice of catalyst. However, generat-ing and storing more copies of ρ is temporally and spatiallyexpensive. Further, the depth of the quantum circuit to imple-ment the above steps at each network node scales as ∼ 2n+1.Therefore, in a quantum network catalytic transformationsthat require collective manipulations of a smaller number ofqubits are desirable. At least two-copies of ρ are required forany catalyst to boost, Pmax

cat (n,α, c), compared to the optimalLOCC success probability, Pmax

LOCC(n,α) = 2(1 − αn), of thetransformation, ρ⊗n → ∣β⟩ ⟨β∣ ⊗ ∣00⟩ ⟨00∣⊗(n−1). Whereas,for a large number of copies, n ≥ n∗(α), where n∗(α) ∶=⌈1/ log2(1/α)⌉, α ∈ [0.5,1), catalysis is not needed as inthat case the same transformation can be achieved with cer-tainty using LOCC without any catalyst. Accessing just afew copies of the primary entangled states in the range, n ∈[2, ..., (n∗(α) − 1)], and utilizing the optimal two-qubit cat-alyst, ∣Copt(n,α)⟩ =

√c0(n,α) ∣HV ⟩ +

√1 − c0(n,α) ∣V H⟩,

the probability, Pmaxcat (n,α, c0) = (1−αn)/(1−c0(n,α)), can

be significantly higher than PmaxLOCC(n,α). To quantify the in-

crease in efficiency due to catalysis we define, ηP (n,α, c) ∶=Pmax

cat (n,α, c)/PmaxLOCC(n,α). This ratio diverges when the op-

timal two-qubit catalyst is used in Eq. (5) for poor quality pri-mary entangled states, that is, ηP (n,α, c) ∼ 1/√n

√(1 − α),

for c = c0(n,α) and as α → 1.The catalyst state itself can be obtained from the resources

provided by the primary and auxiliary paths between the rel-evant nodes (A,B in our case). It may be obtained deter-ministically using a LOCC procedure if the paths providepure states that satisfy the criteria given by the inequality inEq. (2) else it may be obtained only probabilistically. For

example, in our case, it turns out that the optimal two-qubitcatalyst, ∣Copt(n,α)⟩, in Eq. (5) is completely determined byα and n with its Schmidt coefficient given by, c0(n,α) =(1+ 3αn − ((1+ 3αn)2 − 16α2n)1/2)/4αn, n ≥ 2. Therefore,it suffices to use ncat-copies of ∣α⟩ where, αncat ≤ c0(n,α),in order to implement, ∣α⟩⊗ncat → ∣Copt(n,α)⟩, with certaintyusing LOCC - if the only resources available are the primaryentangled states ρ. Note that the minimum number of copiesneeded to obtain the catalyst is determined by n and α since,ncat(n,α) ≥ ⌈log(c0(n,α))/ log(α)⌉.

If auxiliary paths, i = 1, ...,Naux-path, each supplyingstates, ∣α(i)⟩, between the nodes are available then, n(i)cat ≥⌈log(c0(n,α))/ log(α(i))⌉, copies of the states are respec-tively enough to obtain a single copy of ∣Copt(n,α)⟩. Theresources provided by distinct paths can also be combinedto obtain the catalyst. For example, a subset of states withdifferent multiplicities among the ∣α(i)⟩ can provide the cata-lyst deterministically if their product, ∏i ∣α(i)⟩

⊗mi, mi ∈ N,

satisfies the inequality in Eq. (2) relative to ∣Copt(n,α)⟩. Ofcourse, more general catalysts (other than the optimal) [16]may also be used for entanglement manipulation over the net-work edges - the choice depends on the tradeoff between thetemporal resources expended to obtain the catalyst and the en-tanglement distribution rate.

The average temporal resource needed for every catalysisattempt over a network edge is the statistical average of thetemporal resources needed for success and failure events, i.e.,

⟨TL0⟩ = Pmaxcat ⟨Tpri⟩ + (1 − Pmax

cat ) ⟨Tpri+cat⟩ , (6)

where ⟨Tpri⟩ and ⟨Tpri+cat⟩ are the average times to obtainthe primary entangled states and the primary entangled statesalong with the catalyst respectively. We assume that the pri-mary entanglement generation takes much longer than the lo-cal operation time at the nodes which is typically the case.The terms on the right hand side of Eq. (6) also depend onseveral parameters as we now explain. The catalysis successprobability, Pmax

cat (n,α, c), clearly, depends on the number ofcopies, the primary entangled state and the choice of catalystvia n,α, c. Next, since the time-to-success of an entangle-ment generation attempt is geometrically distributed we ob-tain, ⟨Tpri⟩ = n(T0/P0), where T0 = 2L0/cf , with cf thespeed of light in the fiber, and P0 are the average time requiredand probability for primary entanglement generation.

The behavior of ⟨Tpri+cat⟩ depends on the availability ofauxiliary short-range paths between nodes A and B. If thecatalyst states are supplied exclusively by the auxiliary pathsthen, ⟨Tpri+cat⟩ = max{⟨Tpri⟩ , ⟨Tcat⟩}, is the larger among theprimary entanglement generation time and the catalyst gen-eration time. With, Naux-path auxiliary paths, for each ofwhich Pi, Ti, n

(i)cat are the entanglement generation probabil-

ity, entanglement generation time and number of states re-quired to obtain the catalyst, respectively, we obtain ⟨Tcat⟩ =1/∑i(n

(i)cat Pi/ ⟨Ti⟩) ≤ 1/Naux-pathmini(n(i)cat Pi/ ⟨Ti⟩). In the

limit of a large number of auxiliary paths, Naux-path >> 1, thetemporal resource needed to generate the catalyst is less than

4

the primary entanglement generation time, ⟨Tcat⟩ ≤ ⟨Tpri⟩,therefore ⟨Tpri+cat⟩ = ⟨Tpri⟩. This is effectively the case whenthe auxiliary paths can supply the catalyst faster than they areconsumed due to failure events which occur with an averagetime of ⟨Tpri⟩ /(1 − Pmax

cat ). In the other limit, when there areno auxiliary paths available, Naux-path = 0, in which case thetemporal resource needed to obtain the catalyts is budgetedfrom the primary entanglement generation process. Thus, inthis limit the overhead of temporal resource needed is addi-tive, ⟨Tpri+cat⟩ = ⟨Tpri⟩ + ⟨Tcat⟩ = (n + ncat(n,α))T0/P0.

The rate of long-range entanglement distribution in a quan-tum network along a path where the edges use optimal catal-ysis for entanglement concentration can be calculated as,

R(N)cat (n,α, c) = 1

⟨TL0⟩Z(N)(Pmaxcat (n,α, c)) (7)

where Z(N)(P ) = ∑Nj=1 (Nj )

(−1)j+11−(1−P )j , [21] is the waiting-

time [22] to obtain successful events in all of the N -edgesthat comprise the path of length, NL0. On the other hand,in a quantum network where the edges utilize the optimalLOCC procedure to obtain Bell states the rate is given byR(N)LOCC(n,α) = 1/ ⟨Tpri⟩Z(N)(Pmax

LOCC(n,α)). Note that boththe rates, R(N)cat (n,α, c) and R(N)LOCC(n,α), are independent ofthe final fidelity (which is 1) but depend on the fidelity of theprimary entangled states through α.

The ratio of the two rates, η(N)R (n,α, c) ∶=

R(N)cat (n,α, c)/R(N)LOCC(n,α), estimates the enhancement

due to short-range entanglement assistance in the network.It is especially amplified when the primary entangledstates are of poor quality and the optimal catalyst is usedin the transformation in Eq. (5), that is, as α → 1 andc = c0(n,α), see Fig. 2. In this limit, when no aux-iliary paths are available the ratio of rates approaches,η(N)R (n,α, c0) → (⟨Tpri⟩ / ⟨Tcat⟩)ηP (n,α, c0) ≈ 1. This

is because the temporal overhead to generate the catalystsdiminishes the advantage in rate due to the increase in theentanglement concentration success probability throughcatalysis. Whereas, in the same limit, α → 1, when a largenumber of auxiliary paths are available the ratio diverges,η(N)R (n,α, c0) → ηP (n,α, c0) ∼ 1/√n

√(1 − α) - for paths

of arbitrary length. In Fig. 2, the marginal enhancementobserved in the, Naux-path = 0 and α /→ 1 regime (Blue line),is due to the fact that catalytic entanglement concentrationinvolves the collective manipulation of a slightly largernumber of qubits in each attempt - making it a bit moreefficient.

Catalysis based long-range entanglement distribution in aquantum network works particularly well under resource lim-ited conditions - when only a small number of copies of poorquality primary entangled states are available. Its advantageis contingent upon many auxiliary short-range paths provid-ing the catalysts. In this approach the spatial and temporal re-sources needed at the network nodes are constant with respectto the total length of the path and the long-range entanglement

R

0.70 0.75 0.80 0.85 0.90 0.95 1.00

2

4

6

8

10

12

α

0.955 0.960 0.965 0.970 0.975 0.980

1.10

1.15

1.20

1.25

η(N)

R(n,

α,c0)

FIG. 2. (color online) Ratio of long-range entanglement distributionrate using optimal catalysis to the rate using optimal LOCC for thetransformation in Eq. (5) with n = 2 on N = 25 edges along a net-work path. Orange (Brown) curve shows the ratio using the optimal2 × 2 (4 × 4) catalyst when, Naux-path >> 1. The discontinuous Bluecurve shows the ratio with the optimal 2×2 catalyst for,Naux-path = 0.Inset shows the nature of discontinuity - it occurs whenever the num-ber of copies of the primary entangled state needed to obtain thecatalyst jumps by 1.

is obtained in the form of high-fidelity maximally entangledBell states. Finally, note that the identification of suitable cat-alysts and design of the adaptive quantum circuit required forthe catalytic transformation can be fully automated using clas-sical computational resources at the nodes given an estimateof the primary entangled states.

In a quantum network, catalysis can enhance or enablemore general entanglement transformations beyond the purestate case since mixed state entanglement transformations canalso be catalysed. Although, it does not improve the purifi-cation efficiency of Werner states [23], what advantage cancatalysis provide for output states of quantum channels rele-vant for long-range entanglement distribution, e.g., the depo-larizing and dephasing channels [24]? Can catalysis be usefulwhen only partial state information is available? Finally, cancatalysis be used in networks to efficiently distribute multi-party entangled states, such as GHZ or W states [25]?

We believe the results obtained here will stimulate researchon the above mentioned questions and can facilitate develop-ments in long-range entanglement distribution necessary forthe future quantum internet.

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