A continuous-variable quantum repeater based on quantum scissors and mode multiplexing

Quantum repeaters are indispensable for high-rate, long-distance quantum communications. The vision of a future quantum internet strongly hinges on realizing quantum repeaters in practice. Numerous repeaters have been proposed for discrete-variable (DV) single-photon-based quantum communications. Continuous variable (CV) encodings over the quadrature degrees of freedom of the electromagnetic field mode offer an attractive alternative. For example, CV transmission systems are easier to integrate with existing optical telecom systems compared to their DV counterparts. Yet, repeaters for CV have remained elusive. We present a novel quantum repeater scheme for CV entanglement distribution over a lossy bosonic channel that beats the direct transmission exponential rate-loss tradeoff. The scheme involves repeater nodes consisting of a) two-mode squeezed vacuum (TMSV) CV entanglement sources, b) the quantum scissors operation to perform nondeterministic noiseless linear amplification of lossy TMSV states, c) a layer of switched mode multiplexing inspired by second-generation DV repeaters, which is the key ingredient apart from probabilistic entanglement purification that makes DV repeaters work, and d) a non-Gaussian entanglement swap operation. We report our exact results on the rate-loss envelope achieved by the scheme.

The quantum scissors operation lends itself as a tool for continuous variable entanglement distillation over lossy communication channels. We show that a quantum scissors can distill a near-perfect ebit of entanglement from a two-mode squeezed vacuum state whose one share has undergone arbitrary (pure) loss, with a success probability that scales proportional to the channel transmissivity. This is akin to entanglement distillation in single-photon discrete variable entanglement distribution protocols. Invoking a multiplexing-based design for a quantum repeater scheme that was proposed for discrete variable encodings, we show a repeater scheme for CV quantum communication using quantum scissors that beats the direct-transmission rate-loss tradeoff at large distances.

I. INTRODUCTION
A quantum internet [1] of shared entanglement, and shared secret keys distributed using quantum key distribution (QKD) over long distances exemplifies the goal of quantum communications. When in place, a quantum internet would enable, e.g., unconditionally secure multiparty classical communication [2], and distributed versions of quantum computation, sensing, and other applications of quantum information processing (QIP) [3][4][5][6][7][8]. The main hurdle in way of establishing such an infrastructure is photon loss and decoherence. Quantum states, in general, and entangled quantum states that form a resource for QIP, in particular, are fragile to these phenomena. The rate at which entanglement or shared secret keys can be distributed over lossy bosonic channels is known to suffer a tradeoff with loss, which is fundamental. For example, the capacity of a pure loss channel (where the environment injects vacuum noise) for these tasks when assisted by two-way classical communication is known to be C (η) = − log 2 (1 − η) [9, 10], which is ≈ 1.44η when η << 1, and drops exponentially with distance in fiber optic communication and inverse quadratically in free space optical communication [11].
Quantum repeaters [12,13] have been proposed to mitigate this rate-loss tradeoff in entanglement and secret key distribution. Repeater protocols of different types [14,15], based on matter memories [16] as well as optical memories [17,18], have been developed for discretevariable (DV) quantum resources such as single photon polarization or spatial dual rail qubits at rates beyond the direct transmission capacity. On the other hand, quantum continuous variables (CV), wherein the information is encoded in the continuous quadrature degrees of freedom of the electromagnetic field modes, are more naturally suited for integration with current telecommunications infrastructure, whereas the quest for quantum repeaters for CV quantum states of light remains to be conclusively settled.
It has been established that Gaussian operations alone cannot act as quantum repeaters for CV states [19,20]. Ralph proposed a scheme based on non-deterministic noiseless linear amplification (NLA) [21] and Gaussian CV teleportation that performs error correction [22] of CV states against the Gaussian noise arising from the action of a pure loss channel. Approximating the NLA operation using the quantum scissors [21,23], which is a non-Gaussian operation, Dias and Ralph [24,25] demonstrated higher logarithmic negativity and entanglement of formation of the entangled state distilled across a pure loss channel than with direct transmission. The present authors gave a lower bound on the reverse coherent information (RCI) [26][27][28][29] heralded by quantum scissors in a pure loss channel [30]. The use of NLA, both ideal [31] as well as approximate based on quantum scissors [32], was shown to increase the range of CV QKD in the presence of thermal noise. The CV error correction scheme of [22] was recently generalized to the thermal loss channel [33]. Furrer and Munro proposed a repeater scheme for continuous variable states that involves iterative use of entanglement distillation based on symmetric photon replacement and purifying distillation [34,35] and a non-Gaussian entanglement swapping scheme that beats the repeater-less capacity at large distances [36].
In this work, we investigate a CV quantum repeater scheme with NLA based on quantum scissors [21,23] and the non-Gaussian entanglement swap proposed in [36]. Our main contributions include the following: • We show that the optimal exact RCI heralded by a single quantum scissors on a pure loss channel is ≈ 1 independent of the channel transmissivity η, which corresponds to heralding of a near-perfect ebit of entanglement; while the heralding success probability scales proportional to η when η 1.
• Using these near-perfect ebits and the non-Gaussian entanglement swap operation, we show a mode multiplexing-based quantum repeater scheme that outperforms the direct transmission entanglement distribution rates at large distances. The multiplexed repeater differs from [36] in that it avoids iterative use of entanglement distillation, and is therefore simpler to implement.
In Sec. II, we review the multiplexed repeater scheme arXiv:1811.12393v1 [quant-ph] 29 Nov 2018 discussed in [15,16] for DV single photon sources. In Sec. III, we present an exact calculation of the RCI of the heralded state and the heralding success probability for any finite number of quantum scissors in a pure loss channel with CV entangled state input. In Sec. IV, we review the Bell state projection introduced in [36] for non-Gaussian entanglement swapping. Combining these different ingredients, in Sec. V we show a quantum repeater scheme based on quantum scissors.

II. A QUANTUM REPEATER SCHEME BASED ON MODE MULTIPLEXING
The direct transmission rate-loss tradeoff of a lossy communication channel can be circumvented by interspersing intermediate nodes that split the channel into smaller segments of manageable loss. These nodes, often called quantum repeaters, typically comprise quantum information processing elements such as a source of entanglement, entanglement distillation scheme and quantum memory. Consider a channel of transmissivity η, split into n repeater links by inserting n − 1 repeater nodes. The transmissivity of each repeater link is thus t = η 1/n . Among a few possible configurations, one possibility for a repeater link includes a single source of entanglement, followed by the loss segment, an entanglement distillation scheme and a quantum memory. In a DV single-photon-based quantum repeater scheme, the sources generate perfect dual rail Bell pairs (maximally entangled qubit pairs), one share of which are transmitted through the lossy channel segments. The transmitted photons survive the loss with a probability p ∝ t = cη 1/n . An entanglement distillation scheme in this case simply heralds the arrival of an entangled photon through a lossy channel segment at a repeater node. At the repeater node, one local photon and one photon received through the channel are combined on a Bell-basis entangling measurement. The measurement succeeds with a probability q, accomplishing entanglement swapping across the repeater node.
The success probability p can be boosted by multiplexing each repeater link, e.g., using a large number of spectral modes from the source. For M parallel repeater links, the probability that at least one of them succeeds in distilling an ebit of entanglement is given by For the n−repeater link channel, where each link is M −multiplexed, the rate at which an ebit of entanglement can be distributed between the end nodes equals the probability that at least one of the M modes succeeds in each of the n links and the entanglement swaps at each of the n − 1 repeater nodes succeeds. It is given (in ebits/mode) by From the first upper bound in (2), we have n = log (qR UB ) / log q. The two upper bounds intersect at η = 1/ (M c) n . From the intersection, we have n = − log η/ log (M c) . Combining the two, we have Clearly, for M > 1/(qc), s < 1, which beats the direct transmission rate. The R UB represents an upper bound on the envelope of achievable rates with the repeater scheme. The exact envelope of achievable rates with the repeater scheme was shown to be [15] where z is the unique solution of the transcendental equation

III. REPEATER LINK BASED ON QUANTUM SCISSORS
In CV quantum communication, the source in each repeater link generates a two-mode squeezed vacuum (TMSV) state. For entanglement distillation across the lossy channel segment, in this work, we consider NLA based on quantum scissors. Consider a repeater link of transmissivity t with N −quantum scissors at the output of the channel that approximate NLA of gain g, as depicted in Fig. 1. For N > 1, the NLA operation consists of splitting the signal into N equal parts, where each part is acted on by a quantum scissors and recombined into one mode (c.f. [30, Fig. 1]) . The TMSV state can be expressed in the Fock basis as where χ = tanh sinh −1 √ µ , µ being the mean photon in each mode. By modeling the pure loss channel of transmissivity t as a beam splitter of the same transmissivity acting on the lossy mode and an environment mode that is in the vacuum state, we obtain a three-mode output state of the form When NLA is successfully applied on the mode Y using N −quantum scissors, the state heralded across Alice, Bob and the environment, and the heralding success probability are given by where being the NLA gain of the quantum scissors, κ = 1/ 1 + g 2 being an intrinsic parameter in the quantum scissors.
The final two-mode state heralded across the NLA is obtained by tracing over the loss mode E as ρ and the coefficients ζ m,u are given by The state ρ (N ) so that its entropy is given by (18) The state on system A is obtained by tracing over B as ρ where Thus, its entropy is given by Finally, the RCI of the heralded state follows from (18) and (22) as [26][27][28][29].
The RCI of a repeater link with a single quantum scissors, heralded upon successful operation of the scissors, is presented in Fig. 2 (a). When the heralded RCI is optimized over the mean photon number of the input TMSV state µ and the NLA gain g, the optimal value is found to be ≈ 1, independent of the link transmissivity t, which implies heralded distillation of a near perfect ebit of entanglement at all distances. The corresponding heralding success probability is seen to scale linearly with the transmissivity t for t 1, i.e., R = ct, with the proportionality constant c ≈ 5 × 10 −6 . This is akin to entanglement distillation in single photon-based DV quantum communication schemes. The heralded RCI × heralding success probability product optimized as a whole, though, is found to be still below the repeater-less rate-loss tradeoff, which implies that the quantum scissors on average does not outperform direct transmission. Nevertheless, the fact that the quantum scissors distills a perfect ebit of entanglement every time it succeeds, lends itself compatible with the repeater scheme presented in Sec. II. Figure 2 (b) shows a scatterplot of the heralded RCI and the heralding success probability for different values of the TMSV input mean photon number and NLA gain. The outer envelope of such a scatterplot represents the best pair of RCI and success probability achievable with a single quantum scissors at a given transmissivity, in this case t = 0.01. The optimal heralded RCI multiplied by the corresponding heralding success probability for N = 1, 2, 3, 4 quantum scissors. (The wiggles are due to numerical imprecision, but the trend is evident.) In Fig. 3 (a), the optimal heralded RCI is plotted as a function of the number of quantum scissors N for N = 1, 2, 3, 4. It is seen to be below the maximum possible value of log 2 (N + 1), where N + 1 is the output dimensionality of N quantum scissors. This indicates that the optimal heralded entangled states are not quite perfect "e-dits" other than for d = 2, i.e., with a single quantum scissors (N = 1). In Fig. 3 (b), the optimal heralded RCI (a constant) multiplied by the corresponding heralding success probability is plotted as a function of the transmission distance for different N. We see that this product scales proportional to t N for channel transmissivity t 1. Figure 4 shows that neither is the optimized product, heralded RCI × heralding success probability, improved with increasing number of quantum scissors. These observations question the utility of higher number of quantum scissors (other than N = 1) in a CV quantum  Figure 4. The optimal value of the product of heralded RCI and the heralding success probability for N = 1, 2, 3, 4 quantum scissors, as a function of transmission distance. The quantity is optimized over the input mean photon number µ and the NLA gain g. repeater scheme.

IV. NON-GAUSSIAN ENTANGLEMENT SWAPPING
Since the entanglement distilled in each repeater link with a single quantum scissors is nearly a perfect ebit, the heralded state, to first approximation, is a pure state of the form At a repeater node, the entanglement in two such states |ψ A1B1 and |ψ A2B2 can be swapped by a Bell state projection of the formΠ = |φ φ| B1A2 , where |φ B1A2 = ξ |0 B1 |0 A2 + |1 B1 |1 A2 / 1 + ξ 2 . Such a Bell state projection can be implemented by Fock state filtering [34]  The success probability of this projection is found to be where sin 2 θ is the reflectivity of the beam splitter used for photon subtraction in Fig. 5.

V. IDEAL MULTIPLEXED REPEATER BASED ON QUANTUM SCISSORS
With c = 5×10 −6 and q as given in (24) and optimized over θ (the optimum value is found to be ≈ 0.00463), we can now apply the multiplexed repeater rate formula from (5) for the CV repeater scheme with quantum scissors. Figure 7 shows a schematic of the repeater scheme. With M ≈ 10 9 , the scheme can attain an entanglement distribution rate that scales as η τ , where τ = 0.655, which is found to beat the direct transmission rate at ≈ 1000 km transmission distance over an optical fiber of 0.2 dB/km loss. Likewise, with M ≈ 10 13 , the scheme can attain a rate exponent τ = 0.305, which beats the direct transmission rate at ≈ 800 kms. Figure 6 shows these rates in comparison to the direct transmission rate. The line corresponding to each value of τ represents the envelope of rates that can be attained with different number of repeater nodes along the channel with the corresponding M −multiplexing of each repeater link. For each distance there exists an optimal number of repeater nodes whose rate would match the rate η τ .
The simplicity of the above calculation of the repeater performance was a consequence of operating the quantum scissors at the optimal parameter values which yield a constant RCI of ≈ 1 at all transmissivities every time the scissors succeeds. The corresponding success probability scaled proportional to the transmissivity, which was boosted using multiplexing. However, this is not necessarily the optimal operation of the repeater scheme. The quantum scissors could be operated at a different set of parameter values, which, e.g., do not yield a perfect ebit of entanglement, but herald entanglement at a higher success probability. The rate supported by the multiplexed repeater scheme for such an operation is not as easy to calculate. It would involve determining an iterative sequence of states heralded after different number of entanglements swaps, the corresponding success probabilities and swapping success probability. Yet, it could lead to a higher repeater-enhanced entanglement distribution rates, which constitutes a task for future work.
In summary, we discussed a multiplexing-based repeater scheme using quantum scissors and a non-Gaussian entanglement swap operation for CV quantum communication over a pure loss bosonic channel. The scheme is similar in spirit to its DV counterpart [15]. We justified this proposal by showing that the quantum scissors can be optimized to distill nearly perfect ebits independent of the channel transmissivity, with a success probability that scales linearly with transmissivity, which is similar to the case of single-photon-based DV entanglement distillation schemes. We showed that the repeater scheme can beat the direct transmission entanglement distillation rate, e.g., at a distance ≈ 1000 km with a multiplexing M ≈ 10 9 . It's action on a Fock state input |n i ⊗ |0 j is given by U ij (θ) |n i |0 j = n k=0 n k cos 2 θ k/2 sin 2 θ (n−k)/2 |k i |n − k j . (A.4) Proposition 1. Consider the channel comprising of mixing an input mode i with a mode j in the Fock state |1 j on a beam splitter U ij (θ), followed by a projective measurement 1| i in mode i. Let us denote this non-trace-preserving map as N 11 i→j (θ). It can be written as where we use the notationn ij |n i = |n j to denote input to output transformation.
Proposition 2. Consider the channel comprising of mixing an input mode i with a mode j in the Fock state |0 j on a beam splitter U ij (θ), followed by a projective measurement 1| i in the mode i. Let us denote this non-trace-preserving map as N 01 i→j (θ). It can be written as N 01 i→j (θ) = cos θ (sin θ)n ijâ j (A.14) where we use the notationn ij |n i = |n j to denote the input to output transformation.
Proof. From (A.4), we have 1| i U ij (θ) |n i |0 j = √ n cos θ sin n−1 θ |n − 1 j =â j cos θ sin n−1 θ |n j = cos θ (sin θ)n ijâ j |n j which implies, for a general state |ψ i = n c n |n i , the action of the channel is as given in (A.14).