Abstract
Quantum communications promises reliable transmission of quantum information, efficient distribution of entanglement and generation of completely secure keys. For all these tasks, we need to determine the optimal pointtopoint rates that are achievable by two remote parties at the ends of a quantum channel, without restrictions on their local operations and classical communication, which can be unlimited and twoway. These twoway assisted capacities represent the ultimate rates that are reachable without quantum repeaters. Here, by constructing an upper bound based on the relative entropy of entanglement and devising a dimensionindependent technique dubbed ‘teleportation stretching’, we establish these capacities for many fundamental channels, namely bosonic lossy channels, quantumlimited amplifiers, dephasing and erasure channels in arbitrary dimension. In particular, we exactly determine the fundamental rateloss tradeoff affecting any protocol of quantum key distribution. Our findings set the limits of pointtopoint quantum communications and provide precise and general benchmarks for quantum repeaters.
Introduction
Quantum information^{1,2,3} is evolving towards nextgeneration quantum technologies, such as the realization of completely secure quantum communications^{4,5,6} and the longterm construction of a quantum Internet^{7,8,9,10}. But quantum information is more fragile than its classical counterpart, so that the ideal performances of quantum protocols may rapidly degrade in realistic practical implementations. In particular, this is a basic limitation that affects any pointtopoint protocol of quantum communication over a quantum channel, where two remote parties transmit qubits, distribute entanglement or secret keys. In this communication context, it is a crucial open problem to determine the ultimate rates achievable by the remote parties, assuming that they may apply arbitrary local operations (LOs) assisted by unlimited twoway classical communication (CCs), that we may briefly call adaptive LOCCs. The maximum rates achievable by these adaptive protocols are known as twoway (assisted) capacities of a quantum channel and represent fundamental benchmarks for quantum repeaters^{11}.
Before our work, a single twoway capacity was known, discovered about 20 years ago^{12}. For the important case of bosonic channels^{2}, there were only partial results. Building on previous ideas^{13}, ref. 14 introduced the reverse coherent information. By exploiting this notion and other tools^{15,16}, the authors of ref. 17 established lower bounds for the twoway capacities of a Gaussian channel. This inspired a subsequent work^{18}, which exploited the notion of squashed entanglement^{19} to build upper bounds; unfortunately the latter were too large to close the gap with the bestknown lower bounds.
Our work addresses this basic problem. We devise a general methodology that completely simplifies the study of adaptive protocols and allows us to upperbound the twoway capacities of an arbitrary quantum channel with a computable singleletter quantity. In this way, we are able to establish exact formulas for the twoway capacities of several fundamental channels, such as bosonic lossy channels, quantumlimited amplifiers, dephasing and erasure channels in arbitrary dimension. For these channels, we determine the ultimate rates for transmitting quantum information (twoway quantum capacity Q_{2}), distributing entanglement (twoway entanglement distribution capacity D_{2}) and generating secret keys (secretkey capacity K). In particular, we establish the exact rateloss scaling that restricts any pointtopoint protocol of quantum key distribution (QKD) when implemented through a lossy communication line, such as an optical fibre or a freespace link.
Results
General overview of the results
As already mentioned in the Introduction, we establish the twoway capacities (Q_{2}, D_{2} and K) for a number of quantum channels at both finite and infinite dimension, that is, we consider channels defined on both discrete variable (DV) and continuous variable (CV) systems^{2}. Twoway capacities are benchmarks for quantum repeaters because they are derived by removing any technical limitation from the pointtopoint protocols between the remote parties, who may perform the most general strategies allowed by quantum mechanics in the absence of preshared entanglement. Clearly, these ultimate limits cannot be achieved by imposing restrictions on the number of channel uses or enforcing energy constraints at the input. The relaxation of such constraints has also practical reasons since it approximates the working regime of current QKD protocols, that exploit large data blocks and highenergy Gaussian modulations^{13,20}.
To achieve our results we suitably combine the relative entropy of entanglement (REE)^{21,22,23} with teleportation^{9,24,25,26} to design a general reduction method, which remarkably simplifies the study of adaptive protocols and twoway capacities. The first step is to show that the twoway capacities of a quantum channel cannot exceed a general bound based on the REE. The second step is the application of a technique, dubbed ‘teleportation stretching’, which is valid for any channel at any dimension. This allows us to reduce any adaptive protocol into a block form, so that the general REE bound becomes a singleletter quantity. In this way, we easily upperbound the twoway capacities of any quantum channel, with closed formulas proven for bosonic Gaussian channels^{2}, Pauli channels, erasure channels and amplitude damping channels^{1}.
Most importantly, by showing coincidence with suitable lower bounds, we prove simple formulas for the twoway quantum capacity Q_{2} (=D_{2}) and the secretkey capacity K of several fundamental channels. In fact, for the erasure channel we show that K=1−p where p is the erasure probability (only its Q_{2} was previously known^{12}); for the dephasing channel we show that Q_{2}=K=1−H_{2}(p), where H_{2} is the binary Shannon entropy and p is the dephasing probability (these results for qubits are extended to any finite dimension). Then, for a quantumlimited amplifier, we show that where g is the gain. Finally, for the lossy channel, we prove that where η is the transmissivity. In particular, the secretkey capacity of the lossy channel is the maximum rate achievable by any optical implementation of QKD. At long distance, that is, high loss η≃0, we find the optimal rateloss scaling of K≃1.44 η secret bits per channel use, a fundamental bound that only quantum repeaters may surpass.
In the following, we start by giving the main definitions. Then we formulate our reduction method and we derive the analytical results for the various quantum channels.
Adaptive protocols and twoway capacities
Suppose that Alice and Bob are separated by a quantum channel and want to implement the most general protocol assisted by adaptive LOCCs. This protocol may be stated for an arbitrary quantum task and then specified for the transmission of quantum information, distribution of entanglement or secret correlations. Assume that Alice and Bob have countable sets of systems, a and b, respectively. These are local registers which are updated before and after each transmission. The steps of an arbitrary adaptive protocol are described in Fig. 1.
After n transmissions, Alice and Bob share an output state depending on the sequence of adaptive LOCCs . By definition, this adaptive protocol has a rate equal to R_{n} if the output is sufficiently close to a target state φ_{n} with nR_{n} bits, that is, we may write in trace norm. The rate of the protocol is an average quantity, which means that the sequence is assumed to be averaged over local measurements, so that it becomes tracepreserving. Thus, by taking the asymptotic limit in n and optimizing over , we define the generic twoway capacity of the channel as
In particular, if the aim of the protocol is entanglement distribution, then the target state φ_{n} is a maximally entangled state and . Because an ebit can teleport a qubit and a qubit can distribute an ebit, coincides with the twoway quantum capacity . If the goal is to implement QKD, then the target state φ_{n} is a private state^{27} and . Here the secretkey capacity satisfies , because ebits are specific types of secret bits and LOCCs are equivalent to LOs and public communication^{27}. Thus, the generic twoway capacity can be any of D_{2}, Q_{2} or K, and these capacities must satisfy D_{2}=Q_{2}K. Also, note that we may consider the twoway private capacity , which is the maximum rate at which classical messages can be securely transmitted^{15}. Because of the unlimited twoway CCs and the onetime pad, we have , so that this equivalence is implicitly assumed hereafter.
General bounds for twoway capacities
Let us design suitable bounds for . From below we know that we may use the coherent^{28,29} or reverse coherent^{14,17} information. Take a maximally entangled state of two systems A and B, that is, an Einstein–Podolsky–Rosen (EPR) state Φ_{AB}. Propagating the Bpart through the channel defines its Choi matrix . This allows us to introduce the coherent information of the channel and its reverse counterpart , defined as , where S(·) is the von Neumann entropy. These quantities represent lower bounds for the entanglement that is distillable from the Choi matrix via oneway CCs, denoted as . In other words, we can write the hashing inequality^{16}
For bosonic systems, the ideal EPR state has infiniteenergy, so that the Choi matrix of a bosonic channel is energyunbounded (see Methods for notions on bosonic systems). In this case we consider a sequence of twomode squeezed vacuum (TMSV) states^{2} Φ^{μ} with variance μ=+1/2, where is the mean number of thermal photons in each mode. This sequence defines the bosonic EPR state as . At the output of the channel, we have the sequence of quasiChoi matrices
defining the asymptotic Choi matrix . As a result, the coherent information quantities must be computed as limits on and the hashing inequality needs to be suitably extended (see Supplementary Note 2, which exploits the truncation tools of Supplementary Note 1).
In this work the crucial tool is the upper bound. Recall that, for any bipartite state ρ, the REE is defined as , where σ_{s} is an arbitrary separable state and is the relative entropy^{23}. Hereafter we extend this definition to include asymptotic (energyunbounded) states. For an asymptotic state defined by a sequence of states σ^{μ}, we define its REE as
where is an arbitrary sequence of separable states such that →0 for some separable σ_{s}. In general, we also consider the regularized REE
where for an asymptotic state σ.
Thus, the REE of a Choi matrix is correctly defined for channels of any dimension, both finite and infinite. We may also define the channel’s REE as
where the supremum includes asymptotic states for bosonic channels. In the following, we prove that these singleletter quantities, and , bound the twoway capacity of basic channels. The first step is the following general result.
Theorem 1 (general weak converse): At any dimension, finite or infinite, the generic twoway capacity of a quantum channel is upper bounded by the REE bound
In Supplementary Note 3, we provide various equivalent proofs. The simplest one assumes an exponential growth of the shield system in the target private state^{27} as proven by ref. 30 and trivially adapted to CVs. Another proof is completely independent from the shield system. Once established the bound , our next step is to simplify it by applying the technique of teleportation stretching, which is in turn based on a suitable simulation of quantum channels.
Simulation of quantum channels
The idea of simulating channels by teleportation was first developed^{31,32} for Pauli channels^{33}, and further studied in finite dimension^{34,35,36} after the introduction of generalized teleportation protocols^{37}. Then, ref. 38 moved the first steps in the simulation of Gaussian channels via the CV teleportation protocol^{25,26}. Another type of simulation is a deterministic version^{39} of a programmable quantum gate array^{40}. Developed for DV systems, this is based on joint quantum operations, therefore failing to catch the LOCC structure of quantum communication. Here not only we fully extend the teleportationsimulation to CV systems, but we also design the most general channel simulation in a communication scenario; this is based on arbitrary LOCCs and may involve systems of any dimension, finite or infinite (see Supplementary Note 8 for comparisons and advances).
As explained in Fig. 2a, performing a teleportation LOCC (that is, Bell detection and unitary corrections) over a mixed state σ is a way to simulate a (certain type of) quantum channel from Alice to Bob. However, more generally, the channel simulation can be realized using an arbitrary tracepreserving LOCC and an arbitrary resource state σ (see Fig. 2b). Thus, at any dimension, we say that a channel is ‘σstretchable’ or ‘stretchable into σ’ if there is a tracepreserving LOCC such that
In general, we can simulate the same channel with different choices of and σ. In fact, any channel is stretchable into some state σ: A trivial choice is decomposing , inserting in Alice’s LO and simulating with teleportation over the ideal EPR state σ=Φ. Therefore, among all simulations, one needs to identify the best resource state that optimizes the functional under study. In our work, the best results are achieved when the state σ can be chosen as the Choi matrix of the channel. This is not a property of any channel but defines a class. Thus, we define ‘Choistretchable’ a channel that can be LOCCsimulated over its Choi matrix, so that we may write equation (8) with (see also Fig. 2c).
In infinite dimension, the LOCC simulation may involve limits and of sequences and σ^{μ}. For any finite μ, the simulation provides some teleportation channel . Now, suppose that an asymptotic channel is defined as a pointwise limit of the sequence , that is, we have for any bipartite state ρ. Then, we say that is stretchable with asymptotic simulation . This is important for bosonic channels, for which Choibased simulations can only be asymptotic and based on sequences .
Teleportation covariance
We now discuss a property which easily identifies Choistretchable channels. Call the random unitaries which are generated by the Bell detection in a teleportation process. For a qudit, is composed of generalized Pauli operators, that is, the generators of the Weyl–Heisenberg group. For a CV system, the set is composed of displacement operators^{9}, spanning the infinite dimensional version of the previous group. In arbitrary dimension (finite or infinite), we say that a quantum channel is ‘teleportationcovariant’ if, for any teleportation unitary , we may write
for some another unitary V (not necessarily in ).
The key property of a teleportationcovariant channel is that the input teleportation unitaries can be pushed out of the channel, where they become other correctable unitaries. Because of this property, the transmission of a system through the channel can be simulated by a generalized teleportation protocol over its Choi matrix. This is the content of the following proposition.
Proposition 2 (telecovariance): At any dimension, a teleportationcovariant channel is Choistretchable. The simulation is a teleportation LOCC over its Choi matrix , which is asymptotic for a bosonic channel.
The simple proof is explained in Fig. 3. The class of teleportationcovariant channels is wide and includes bosonic Gaussian channels, Pauli and erasure channels at any dimension (see Methods for a more detailed classification). All these fundamental channels are therefore Choistretchable. There are channels that are not (or not known to be) Choistretchable but still have decompositions where the middle part is Choistretchable. In this case, and can be made part of Alice’s and Bob’s LOs, so that channel can be stretched into the state . An example is the amplitude damping channel as we will see afterwards.
Teleportation stretching of adaptive protocols
We are now ready to describe the reduction of arbitrary adaptive protocols. The procedure is schematically shown in Fig. 4. We start by considering the ith transmission through the channel , so that Alice and Bob’s register state is updated from to . By using a simulation , we show the input–output formula
for some ‘extended’ LOCC Δ_{i} (Fig. 4c). By iterating the previous formula n times, we may write the output state =Λ(⊗σ^{⊗n}) for (as in Fig. 4d). Because the initial state is separable, its preparation can be included in Λ and we may directly write =Λ(σ^{⊗n}). Finally, we average over all local measurements present in Λ, so that =(σ^{⊗n}) for a tracepreserving LOCC (Fig. 4e). More precisely, for any sequence of outcomes u with probability p(u), there is a conditional LOCC Λ_{u} with output (u)=p(u)^{−1}Λ_{u}(σ^{⊗n}). Thus, the mean output state is generated by =∑_{u}Λ_{u} (see Methods for more technical details on this LOCC averaging).
Note that the simulation of a bosonic channel is typically asymptotic, with infiniteenergy limits and . In this case, we repeat the procedure for some μ, with output , where _{μ} is derived assuming the finiteenergy LOCCs . Then, we take the limit for large μ, so that converges to in trace norm (see Methods for details on teleportation stretching with bosonic channels). Thus, at any dimension, we have proven the following result.
Lemma 3 (Stretching): Consider arbitrary n transmissions through a channel which is stretchable into a resource state σ. The output of an adaptive protocol can be decomposed into the block form
for some tracepreserving LOCC . If the channel is Choistretchable, then we may write
In particular, for an asymptotic channel simulation .
According to this Lemma, teleportation stretching reduces an adaptive protocol performing an arbitrary task (quantum communication, entanglement distribution or key generation) into an equivalent block protocol, whose output state is the same but suitably decomposed as in equation (11) for any number n of channel uses. In particular, for Choistretchable channels, the output is decomposed into a tensor product of Choi matrices. An essential feature that makes the technique applicable to many contexts is the fact that the adaptivetoblock reduction maintains task and output of the original protocol so that, for example, adaptive key generation is reduced to block key generation and not entanglement distillation.
Remark 4: Some aspects of our method might be traced back to a precursory but very specific argument discussed in Section V of ref. 31, where protocols of quantum communication (through Pauli channels) were transformed into protocols of entanglement distillation (the idea was developed for oneway CCs, with an implicit extension to twoway CCs). However, while this argument may be seen as precursory, it is certainly not developed at the level of generality of the present work where the adaptivetoblock reduction is explicitly proven for any type of protocol and any channel at any dimension (see Supplementary Notes 9 and 10 for remarks on previous literature).
REE as a singleletter converse bound
The combination of Theorem 1 and Lemma 3 provides the insight of our entire reduction method. In fact, let us compute the REE of the output state , decomposed as in equation (11). Using the monotonicity of the REE under tracepreserving LOCCs, we derive
where the complicated is fully discarded. Then, by replacing equation (13) into equation (7), we can ignore the supremum in the definition of and get the simple bound
Thus, we can state the following main result.
Theorem 5 (oneshot REE bound): Let us stretch an arbitrary quantum channel into some resource state σ, according to equation (8). Then, we may write
In particular, if is Choistretchable, we have
See Methods for a detailed proof, with explicit derivations for bosonic channels. We have therefore reached our goal and found singleletter bounds. In particular, note that measures the entanglement distributed by a single EPR state, so that we may call it the ‘entanglement flux’ of the channel . Remarkably, there is a subclass of Choistretchable channels for which coincides with the lower bound in equation (2). We call these ‘distillable channels’. We establish all their twoway capacities as . They include lossy channels, quantumlimited amplifiers, dephasing and erasure channels. See also Fig. 5.
Immediate generalizations
Consider a fading channel, described by an ensemble , where channel occurs with probability p_{i}. Let us stretch into a resource state σ_{i}. For large n, we may decompose the output of an adaptive protocol as , so that the twoway capacity of this channel is bounded by
Then consider adaptive protocols of twoway quantum communication, where the parties have forward () and backward channels. The capacity maximizes the number of target bits per channel use. Stretching into a pair of states (σ, σ′), we find . For Choistretchable channels, this means , which reduces to if they are distillable. In the latter case, the optimal strategy is using the channel with the maximum capacity (see Methods).
Yet another scenario is the multiband channel , where Alice exploits m independent channels or ‘bands’ , so that the capacity maximizes the number of target bits per multiband transmission. By stretching the bands into resource states {σ_{i}}, we find . For Choistretchable bands, this means , giving the additive capacity if they are distillable (see Methods).
Ultimate limits of bosonic communications
We now apply our method to derive the ultimate rates for quantum and secure communication through bosonic Gaussian channels. These channels are Choistretchable with an asymptotic simulation involving . From equations (4) and (16), we may write
for a suitable converging sequence of separable states .
For Gaussian channels, the sequences in equation (18) involve Gaussian states, for which we easily compute the relative entropy. In fact, for any two Gaussian states, ρ_{1} and ρ_{2}, we prove the general formula S(ρ_{1}ρ_{2})=Σ(V_{1}, V_{2})−S(ρ_{1}), where Σ is a simple functional of their statistical moments (see Methods). After technical derivations (Supplementary Note 4), we then bound the twoway capacities of all Gaussian channels, starting from the most important, the lossy channel.
Fundamental rateloss scaling
Optical communications through freespace links or telecom fibres are inevitably lossy and the standard model to describe this scenario is the lossy channel. This is a bosonic Gaussian channel characterized by a transmissivity parameter η, which quantifies the fraction of input photons surviving the channel. It can be represented as a beam splitter mixing the signals with a zerotemperature environment (background thermal noise is negligible at optical and telecom frequencies).
For a lossy channel with arbitrary transmissivity η we apply our reduction method and compute the entanglement flux . This coincides with the reverse coherent information of this channel I_{RC}(η), first derived in ref. 17. Thus, we find that this channel is distillable and all its twoway capacities are given by
Interestingly, this capacity coincides with the maximum discord^{41} that can be distributed, since we may write^{42} I_{RC}(η)=D(BA), where the latter is the discord of the (asymptotic) Gaussian Choi matrix (ref. 43). We also prove the strict separation Q_{2}(η)>Q(η), where Q is the unassisted quantum capacity^{28,29}.
Expanding equation (19) at high loss , we find
or about η nats per channel use. This completely characterizes the fundamental rateloss scaling which rules longdistance quantum optical communications in the absence of quantum repeaters. It is important to remark that our work also proves the achievability of this scaling. This is a major advance with respect to existing literature, where previous studies with the squashed entanglement^{18} only identified a nonachievable upper bound. In Fig. 6, we compare the scaling of equation (20) with the maximum rates achievable by current QKD protocols.
The capacity in equation (19) is also valid for twoway quantum communication with lossy channels, assuming that η is the maximum transmissivity between the forward and feedback channels. It can also be extended to a multiband lossy channel, for which we write , where η_{i} are the transmissivities of the various bands or frequency components. For instance, for a multimode telecom fibre with constant transmissivity η and bandwidth W, we have
Finally, note that freespace satellite communications may be modelled as a fading lossy channel, that is, an ensemble of lossy channels with associated probabilities p_{i} (ref. 44). In particular, slow fading can be associated with variations of satelliteEarth radial distance^{45,46}. For a fading lossy channel , we may write
Quantum communications with Gaussian noise
The fundamental limit of the lossy channel bounds the twoway capacities of all channels decomposable as where is a lossy component while and are extra channels. A channel of this type is stretchable with resource state and we may write . For Gaussian channels, such decompositions are known but we achieve tighter bounds if we directly stretch them using their own Choi matrix.
Let us start from the thermalloss channel, which can be modelled as a beam splitter with transmissivity η in a thermal background with mean photons. Its action on input quadratures is given by → with E being a thermal mode. This channel is central for microwave communications^{47,48,49,50} but also important for CV QKD at optical and telecom frequencies, where Gaussian eavesdropping via entangling cloners results into a thermalloss channel^{2}.
For an arbitrary thermalloss channel we apply our reduction method and compute the entanglement flux
for <η/(1−η), while zero otherwise. Here we set
Combining this result with the lower bound given by the reverse coherent information^{17}, we write the following inequalities for the twoway capacity of this channel
As shown in Fig. 7a, the two bounds tend to coincide at sufficiently high transmissivity. We clearly retrieve the previous result of the lossy channel for =0.
Another important Gaussian channel is the quantum amplifier. This channel is described by →, where g>1 is the gain and E is the thermal environment with mean photons. We compute
for <(g−1)^{−1}, while zero otherwise. Combining this result with the coherent information^{51}, we get
whose behaviour is plotted in Fig. 7b.
In the absence of thermal noise (=0), the previous channel describes a quantumlimited amplifier , for which the bounds in equation (27) coincide. This channel is therefore distillable and its twoway capacities are
In particular, this proves that Q_{2}(g) coincides with the unassisted quantum capacity Q(g)^{51,52}. Note that a gain2 amplifier can transmit at most 1 qubit per use.
Finally, one of the simplest models of bosonic decoherence is the additivenoise Gaussian channel^{2}. This is the direct extension of the classical model of a Gaussian channel to the quantum regime. It can be seen as the action of a random Gaussian displacement over incoming states. In terms of input–output transformations, it is described by → where z is a classical Gaussian variable with zero mean and variance ξ≥0. For this channel we find the entanglement flux
for ξ<1, while zero otherwise. Including the lower bound given by the coherent information^{51}, we get
In Fig. 7c, see its behaviour and how the two bounds tend to rapidly coincide for small added noise.
Ultimate limits in qubit communications
We now study the ultimate rates for quantum communication, entanglement distribution and secretkey generation through qubit channels, with generalizations to any finite dimension. For any DV channel from dimension d_{A} to dimension d_{B}, we may write the dimensionality bound . This is because we may always decompose the channel into (or ), include in Alice’s (or Bob’s) LOs and stretch the identity map into a Bell state with dimension d_{B} (or d_{A}). For DV channels, we may also write the following simplified version of our Theorem 5 (see Methods for proof).
Proposition 6: For a Choistretchable channel in finite dimension, we may write the chain
where is the distillable key of .
In the following, we provide our results for DV channels, with technical details available in Supplementary Note 5.
Pauli channels
A general error model for the transmission of qubits is the Pauli channel
where X, Y, and Z are Pauli operators^{1} and p:={p_{k}} is a probability distribution. It is easy to check that this channel is Choistretchable and its Choi matrix is Belldiagonal. We compute its entanglement flux as
if p_{max}:=max{p_{k}}≥1/2, while zero otherwise. Since the channel is unital, we have that , where H is the Shannon entropy. Thus, the twoway capacity of a Pauli channel satisfies
This can be easily generalized to arbitrary finite dimension (see Supplementary Note 5).
Consider the depolarising channel, which is a Pauli channel shrinking the Bloch sphere. With probability p, an input state becomes the maximallymixed state
Setting κ(p):=1−H_{2} (3p/4), we may then write
for p2/3, while 0 otherwise (Fig. 8a). The result can be extended to any dimension d≥2. A qudit depolarising channel is defined as in equation (35) up to using the mixed state I/d. Setting f:=p(d^{2}−1)/d^{2} and , we find
for pd/(d+1), while zero otherwise.
Consider now the dephasing channel. This is a Pauli channel, which deteriorates quantum information without energy decay, as it occurs in spinspin relaxation or photonic scattering through waveguides. It is defined as
where p is the probability of a phase flip. We can easily check that the two bounds of equation (34) coincide, so that this channel is distillable and its twoway capacities are
Note that this also proves , where the latter was derived in ref. 53.
For an arbitrary qudit with computational basis {j〉}, the generalized dephasing channel is defined as
where P_{i} is the probability of i phase flips, with a single flip being . This channel is distillable and its twoway capacities are functionals of P={P_{i}}
Quantum erasure channel
A simple decoherence model is the erasure channel. This is described by
where p is the probability of getting an orthogonal erasure state . We already know that (ref. 12). Therefore, we compute the secretkey capacity.
Following ref. 12, one shows that . In fact, suppose that Alice sends halves of EPR states to Bob. A fraction 1−p will be perfectly distributed. These good cases can be identified by Bob applying the measurement on each output system, and communicating the results back to Alice in a single and final CC. Therefore, they distill at least 1−p ebits per copy. It is then easy to check that this channel is Choistretchable and we compute . Thus, the erasure channel is distillable and we may write
In arbitrary dimension d, the generalized erasure channel is defined as in equation (42), where ρ is now the state of a qudit and the erasure state lives in the extra d+1 dimension. We can easily generalize the previous derivations to find that this channel is distillable and
Note that the latter result can also be obtained by computing the squashed entanglement of the erasure channel, as shown by the independent derivation of ref. 54.
Amplitude damping channel
Finally, an important model of decoherence in spins or optical cavities is energy dissipation or amplitude damping^{55,56}. The action of this channel on a qubit is
where , , and p is the damping probability. Note that is not teleportationcovariant. However, it is decomposable as
where teleports the original qubit into a singlerail bosonic qubit^{9}; then, is a lossy channel with transmissivity η(p):=1−p; and _{CV→DV} teleports the singlerail qubit back to the original qubit. Thus, is stretchable into the asymptotic Choi matrix of the lossy channel . This shows that we need a dimensionindependent theory even for stretching DV channels.
From Theorem 5 we get , implying
while the reverse coherent information implies^{14}
The bound in equation (47) is simple but only good for strong damping (p>0.9). A shown in Fig. 8b, we find a tighter bound using the squashed entanglement, that is,
Discussion
In this work, we have established the ultimate rates for pointtopoint quantum communication, entanglement distribution and secretkey generation at any dimension, from qubits to bosonic systems. These limits provide the fundamental benchmarks that only quantum repeaters may surpass. To achieve our results we have designed a general reduction method for adaptive protocols, based on teleportation stretching and the relative entropy of entanglement, suitably extended to quantum channels. This method has allowed us to bound the twoway capacities (Q_{2}, D_{2} and K) with singleletter quantities, establishing exact formulas for bosonic lossy channels, quantumlimited amplifiers, dephasing and erasure channels, after about 20 years since the first studies^{12,31}.
In particular, we have characterized the fundamental rateloss scaling which affects any quantum optical communication, setting the ultimate achievable rate for repeaterless QKD at bits per channel use, that is, about 1.44η bits per use at high loss. There are two remarkable aspects to stress about this bound. First, it remains sufficiently tight even when we consider input energy constraints (down to ≃1 mean photon). Second, it can be reached by using oneway CCs with a maximum cost of just classical bits per channel use; this means that our bound directly provides the throughput in terms of bits per second, once a clock is specified (see Supplementary Note 7 for more details).
Our reduction method is very general and goes well beyond the scope of this work. It has been already used to extend the results to quantum repeaters.^{57} has shown how to simplify the most general adaptive protocols of quantum and private communication between two endpoints of a repeater chain and, more generally, of an arbitrary multihop quantum network, where systems may be routed though single or multiple paths. Depending on the type of routing, the endtoend capacities are determined by quantum versions of the widest path problem and the maxflow mincut theorem. More recently, teleportation stretching has been also used to completely simplify adaptive protocols of quantum parameter estimation and quantum channel discrimination^{58}. See Supplementary Discussion for a summary of our findings, other followup works and further remarks.
Methods
Basics of bosonic systems and Gaussian states
Consider n bosonic modes with quadrature operators . The latter satisfy the canonical commutation relations^{59}
with I_{n} being the n × n identity matrix. An arbitrary multimode Gaussian state ρ(u, V), with mean value u and covariance matrix (CM) V, can be written as^{60}
where the Gibbs matrix G is specified by
Using symplectic transformations^{2}, the CM V can be decomposed into the Williamson’s form where the generic symplectic eigenvalue ν_{k} satisfies the uncertainty principle ν_{k}≥1/2. Similarly, we may write ν_{k}=_{k}+1/2 where _{k} are thermal numbers, that is, mean number of photons in each mode. The von Neumann entropy of a Gaussian state can be easily computed as
where h(x) is given in equation (24).
A twomode squeezed vacuum (TMSV) state Φ^{μ} is a zeromean pure Gaussian state with CM
where and μ=+1/2. Here is the mean photon number of the reduced thermal state associated with each mode A and B. The Wigner function of a TMSV state Φ^{μ} is the Gaussian
where x :=(q_{A}, q_{B}, p_{A}, p_{B})^{T}. For large μ, this function assumes the deltalike expression^{25}
where N is a normalization factor, function of the antisqueezed quadratures q_{+}:=q_{A}+q_{B} and p_{−}:=p_{A}−p_{B}, such that . Thus, the infiniteenergy limit of TMSV states defines the asymptotic CV EPR state Φ, realizing the ideal EPR conditions for position and for momentum.
Finally, recall that singlemode Gaussian channels can be put in canonical form^{2}, so that their action on input quadratures is
where T and N are diagonal matrices, E is an environmental mode with _{E} mean photons, and z is a classical Gaussian variable, with zero mean and CM ξI ≥0.
Relative entropy between Gaussian states
We now provide a simple formula for the relative entropy between two arbitrary Gaussian states ρ_{1}(u_{1}, V_{1}) and ρ_{2}(u_{2}, V_{2}) directly in terms of their statistical moments. Because of this feature, our formula supersedes previous expressions^{61,62}. We have the following.
Theorem 7: For two arbitrary multimode Gaussian states, ρ_{1}(u_{1}, V_{1}) and ρ_{2}(u_{2}, V_{2}), the entropic functional
is given by
where δ:=u_{1}−u_{2} and G:=g(V) as given in equation (52). As a consequence, the von Neumann entropy of a Gaussian state ρ(u, V) is equal to
and the relative entropy of two Gaussian states ρ_{1}(u_{1}, V_{1}) and ρ_{2}(u_{2}, V_{2}) is given by
Proof: The starting point is the use of the Gibbsexponential form for Gaussian states^{60} given in equation (51). Start with zeromean Gaussian states, which can be written as , where G_{i}=g(V_{i})is the Gibbsmatrix and Z_{i}=det(V_{i}+iΩ/2)^{1/2} is the normalization factor (with i=1, 2). Then, replacing into the definition of Σ given in equation (58), we find
Using the commutator and the anticommutator , we derive
where we also exploit the fact that Tr(ΩG)=0, because Ω is antisymmetric and G is symmetric (as V).
Let us now extend the formula to nonzero mean values (with difference δ=u_{1}−u_{2}). This means to perform the replacement →, so that
By replacing this expression in equation (63), we get
Thus, by combining equations (62) and (65), we achieve equation (59). The other equations (60) and (61) are immediate consequences. This completes the proof of Theorem 7.
As discussed in ref. 60, the Gibbsmatrix G becomes singular for a pure state or, more generally, for a mixed state containing vacuum contributions (that is, with some of the symplectic eigenvalues equal to 1/2). In this case the Gibbsexponential form must be used carefully by making a suitable limit. Since Σ is basis independent, we can perform the calculations in the basis in which V_{2}, and therefore G_{2}, is diagonal. In this basis
where {v_{2k}} is the symplectic spectrum of V_{2}, and
Now, if v_{2k}=1/2 for some k, then its contribution to the sum in equation (66) is either zero or infinity.
Basics of quantum teleportation
Ideal teleportation exploits an ideal EPR state of systems A (for Alice) and B (for Bob). In finite dimension d, this is the maximally entangled Bell state
In particular, it is the usual Bell state for a qubit. To teleport, we need to apply a Bell detection on the input system a and the EPR system A (that is, Alice’s part of the EPR state). This detection corresponds to projecting onto a basis of Bell states where the outcome k takes d^{2} values with probabilities p_{k}=d^{−2}.
More precisely, the Bell detection is a positiveoperator valued measure with operators
where is the Bell state as in equation (68) and U_{k} is one of d^{2} teleportation unitaries, corresponding to generalized Pauli operators (described below). For any state ρ of the input system a, and outcome k of the Bell detection, the other EPR system B (Bob’s part) is projected onto U_{k}ρ. Once Alice has communicated k to Bob (feedforward), he applies the correction unitary U_{k}^{−1} to retrieve the original state ρ on its system B. Note that this process also teleports all correlations that the input system a may have with ancillary systems.
For CV systems (d→+∞), the ideal EPR source Φ_{AB} can be expressed as a TMSV state Φ^{μ} in the limit of infiniteenergy μ →+ ∞. The unitaries U_{k} are phasespace displacements D(k) with complex amplitude k (ref. 9). The CV Bell detection is also energyunbounded, corresponding to a projection onto the asymptotic EPR state up to phasespace displacements D(k). To deal with this, we need to consider a finiteenergy version of the measurement, defined as a quasiprojection onto displaced versions of the TMSV state Φ^{μ} with finite parameter μ. This defines a positiveoperator valued measure with operators
Optically, this can be interpreted as applying a balanced beamsplitter followed by two projections, one onto a positionsqueezed state and the other onto a momentumsqueezed state (both with finite squeezing). The ideal CV Bell detection is reproduced by taking the limit of μ→+∞ in equation (70). Thus, CV teleportation must always be interpreted a la Braunstein and Kimble^{25}, so that we first consider finite resources to compute the μdependent output and then we take the limit of large μ.
Teleportation unitaries
Let us characterize the set of teleportation unitaries for a qudit of dimension d. First, let us write k as a multiindex k=(a, b) with . The teleportation set is therefore composed of d^{2} generalized Pauli operators , where U_{ab} :=X^{a}Z^{b}. These are defined by introducing unitary (nonHermitian) operators
where ⊕ is the modulo d addition and
so that they satisfy the generalized commutation relation
Note that any qudit unitary can be expanded in terms of these generalized Pauli operators. We may construct the set of finitedimensional displacement operators D(j, a, b):=ω^{j}X^{a}Z^{b} with which form the finitedimensional Weyl–Heisenberg group (or Pauli group). For instance, for a qubit (d=2), we have and the group ±1 × {I, X, XZ, Z}. For a CV system (d=+∞), the teleportation set is composed of infinite displacement operators, that is, we have , where D(k) is a phasespace displacement operator^{2} with complex amplitude k. This set is the infinitedimensional Weyl–Heisenberg group.
It is important to note that, at any dimension (finite or infinite), the teleportation unitaries satisfy
where U_{f} is another teleportation unitary and is a phase. In fact, for finite d, let us write k and as multiindices, that is, k=(a, b) and . From , we see that . Then, for infinite d, we know that the displacement operators satisfy , for any two complex amplitudes u and v.
Now, let us represent a teleportation unitary as
It is clear that we have for DV systems, and for CV systems. Therefore satisfies the group structure
where G is a product of two groups of addition modulo d for DVs, while G is the translation group for CVs. Thus, the (multi)index of the teleportation unitaries can be taken from the abelian group G.
Teleportationcovariant channels
Let us a give a group representation to the property of teleportation covariance specified by equation (9). Following equation (75), we may express an arbitrary teleportation unitary as where g∈G. Calling , we see that equation (9) implies
so that and are generally different unitary representations of the same abelian group G. Thus, equation (9) can also be written as
for all h∈G, where .
The property of equation (9) is certainly satisfied if the channel is covariant with respect to the Weyl–Heisenberg group, describing the teleportation unitaries in both finite and infinitedimensional Hilbert spaces. This happens when the channel is dimensionpreserving and we may set V_{k}=U_{k′} for some k′ in equation (9). Equivalently, this also means that and are exactly the same unitary representation in equation (78). We call ‘Weylcovariant’ these specific types of telecovariant channels.
In finitedimension, a Weylcovariant channel must necessarily be a Pauli channel. In infinitedimension, a Weylcovariant channel commutes with displacements, which is certainly a property of the bosonic Gaussian channels. A simple channel that is telecovariant but not Weylcovariant is the erasure channel. This is in fact dimensionaltering (since it adds an orthogonal state to the input Hilbert space) and the output correction unitaries to be used in equation (9) have the augmented form V_{k}=U_{k}⊕I. Hybrid channels, mapping DVs into CVs or vice versa, cannot be Weylcovariant but they may be telecovariant. Finally, the amplitude damping channel is an example of a channel, which is not telecovariant.
Note that, for a quantum channel in finite dimension, we may easily rewrite equation (9) in terms of an equivalent condition for the Choi matrix. In fact, by evaluating the equality in equation (9) on the EPR state and using the property that , one finds
Thus, a finitedimensional is telecovariant if and only if, for any teleportation unitary U_{k}, we may write
for another generally different unitary V_{k}. There are finitedimensional channels satisfying conditions stronger than equation (80). For Pauli channels, we may write for any k, that is, the Choi matrix is invariant under twirling operations restricted to the generators of the Pauli group {U_{k}}. For depolarising channels, we may even write for an arbitrary unitary U. This means that the Choi matrix of a depolarising channel is an isotropic state.
LOCCaveraging in teleportation stretching
Consider an arbitrary adaptive protocol described by some fundamental preparation of the local registers ρ_{a}^{0}⊗ρ_{b}^{0} and a sequence of adaptive LOCCs . In general, these LOs may involve measurements. Call u_{i} the (vectorial) outcome of Alice’s and Bob’s local measurements performed within the ith adaptive LOCC, so that . It is clear that will be conditioned by measurements and outcomes of all the previous LOCCs, so that a more precise notation will be where the output u_{i} is achieved with a conditional probability . After n transmissions, we have a sequence of outcomes u=u_{0} … u_{n} with joint probability
and a sequence of LOCCs
The mean rate of the protocol is achieved by averaging the output state over all possible outcomes u, which is equivalent to considering the output state generated by the tracepreserving LOCCsequence .
In fact, suppose that the (normalized) output state (u) generated by the conditional is epsilonclose to a corresponding target state φ_{n}(u) with rate R_{n}(u). This means that we have in trace distance. The mean rate of the protocol is associated with the average target state . It is easy to show that φ_{n} is approximated by the mean output state generated by . In fact, by using the joint convexity of the trace distance^{1}, we may write
Now we show that the LOCCsimulation of a channel does not change the average output state and this state can be reorganized in a block form. The ith (normalized) conditional output can be expressed in terms of the i−1th output as follows
where is meant as with a_{i} being the system transmitted. Thus, after n transmissions, the conditional output state is , where
and the average output state is given by
where .
For some LOCC and resource state σ, let us write the simulation
where is Alice and Bob’s conditional LOCC with probability p(k). For simplicity we omit other technical labels that may describe independent local measurements or classical channels, because they will also be averaged at the end of the procedure. Let us introduce the vector k=k_{1} … k_{n} where k_{i} identifies a conditional LOCC associated with the ith transmission. Because the LOCCsimulation of the channel is fixed, we have the factorized probability p(k)=p(k_{1}) … p(k_{n}).
By replacing the simulation in equation (84), we obtain
By iteration, the latter equation yields
where
Therefore, the average output state of the original protocol may be equivalently expressed in the form
Finally, we may include the preparation ρ_{a}^{0}⊗ρ_{b}^{0} in the LOCC, so that we may write
To extend this technical proof to CV systems, we perform the replacement → with the probabilities becoming probability densities. Then, and σ may be both asymptotic, that is, defined as infiniteenergy limits and from corresponding finiteversions and σ^{μ}. In this case, we repeat the previous procedure for some μ and then we take the limit on the output state .
Details on Lemma 3 in relation to teleportation stretching with bosonic channels
For a bosonic channel, the Choi matrix and the ideal Bell detection are both energyunbounded. Therefore, any Choibased LOCC simulation of these channels must necessarily be asymptotic. Here we discuss in more detail how an asymptotic channel simulation leads to an asymptotic form of stretching as described in Lemma 3. Any operation or functional applied to is implicitly meant to be applied to the finiteenergy simulation , whose output then undergoes the μlimit.
Consider a bosonic channel with asymptotic simulation . As depicted in Fig. 9, this means that there is a channel generated by such that in the sense that
In other words, for any (energybounded) bipartite state ρ_{aa′}, whose a′part is propagated, the original channel output and the simulated channel output satisfy the limit
By teleportation stretching, we may equivalently decompose the output state ρ_{ab}^{μ} into the form
where _{μ} is a tracepreserving LOCC, which is includes both and the preparation of ρ_{aa′} (it is tracepreserving because we implicitly assume that we average over all possible measurements present in the simulation LOCC ). By taking the limit of μ→+∞ in equation (95), the state becomes the channel output state ρ_{ab} according to equation (94). Therefore, we have the limit
that we may compactly write as
Note that we may express equation (93) in a different form. In fact, consider the set of energyconstrained bipartite states , where is the total number operator. Then, for two bosonic channels, and , we may define the energybounded diamond norm
Using the latter definition and the fact that is a compact set, we have that the pointwise limit in equation (93) implies the following uniform limit
The latter expression is useful to generalize the reasoning to the adaptive protocol, with LOCCs applied before and after transmission. Consider the output after n adaptive uses of the channel , and the simulated output , which is generated by replacing with the imperfect channel . Explicitly, we may write
with its approximate version
where it is understood that and are applied to system a_{i} in the ith transmission, that is, .
Assume that the mean photon number of the total register states and is bounded by some large but yet finite value N(n). For instance, we may consider a sequence N(n)=N(0)+nt, where N(0) is the initial photon contribution and t is the channel contribution, which may be negative for energydecreasing channels (like the thermalloss channel) or positive for energyincreasing channels (like the quantum amplifier). We then prove
In fact, for n=2, we may write
where: (1) we use monotonicity under Λ_{2}; (2) we use the triangle inequality; (3) we use monotonicity with respect to ; and (4) we use the definition of equation (98) assuming a′=a_{i} and the energy bound N(n). Generalization to arbitrary n is just a technicality.
By using equation (99) we may write that, for any bound N(n) and ɛ≥0, there is a sufficiently large μ such that , so that equation (102) becomes
By applying teleportation stretching we derive , where _{μ} includes the original LOCCs Λ_{i} and the teleportation LOCCs . Thus, equation (104) implies
or, equivalently, .
Therefore, given an adaptive protocol with arbitrary register energy, and performed n times through a bosonic channel with asymptotic simulation, we may write its output state as the (tracenorm) limit
This means that we may formally write the asymptotic stretching for an asymptotic channel simulation .
More details on the oneshot REE bound given in Theorem 5
The main steps for proving equation (15) are already given in the main text. Here we provide more details of the formalism for the specific case of bosonic channels, involving asymptotic simulations . Given the asymptotic stretching of the output state as in equation (106), the simplification of the REE bound E_{R} explicitly goes as follows
where: (1) is a generic sequence of separable states that converges in trace norm, that is, such that there is a separable state so that ; (2) we use the lower semicontinuity of the relative entropy^{3}; (3) we use that _{μ} are specific types of converging separable sequences within the set of all such sequences; (4) we use the monotonicity of the relative entropy under tracepreserving LOCCs; and (5) we use the definition of REE for asymptotic states given in equation (4).
Thus, from Theorem 1, we may write the following upper bound for the twoway capacity of a bosonic channel
The supremum over all adaptive protocols, which defines disappears in the right hand side of equation (108). The resulting bound applies to both energyconstrained protocols and the limit of energyunconstrained protocols. The proof of the further condition in equation (15) comes from the subadditivity of the REE over tensor product states. This subadditivity also holds for a tensor product of asymptotic states; it is proven by restricting the minimization on tensor product sequences in the corresponding definition of the REE.
Let us now prove equation (16). The two inequalities in equation (16) are simply obtained by using for a Choistretchable channel (where the Choi matrix is intended to be asymptotic for a bosonic channel). Then we show the equality . By restricting the optimization in to an input EPR state Φ, we get the direct part as already noticed in equation (6). For CVs, this means to choose an asymptotic EPR state , so that
and therefore
For the converse part, consider first DVs. By applying teleportation stretching to a single use of the channel , we may write for a tracepreserving LOCC . Then, the monotonicity of the REE leads to
For CVs, we have an asymptotic stretching where . Therefore, we may write
Since this is true for any ρ, it also applies to the supremum and, therefore, to the channel’s REE .
Proof of Proposition 6 on the oneshot REE bound for DV channels
At finite dimension, we may first use teleportation stretching to derive and then apply any upper bound to the distillable key , among which the REE bound has the best performance. Consider a key generation protocol described by a sequence of adaptive LOCCs (implicitly assumed to be averaged). If the protocol is implemented over a Choistretchable channel in finite dimension d, its stretching allows us to write the output as for a tracepreserving LOCC . Since any LOCCsequence is transformed into , any key generation protocol through becomes a key distillation protocol over copies of the Choi matrix . For large n, this means .
To derive the opposite inequality, consider Alice sending EPR states through the channel, so that the shared output will be . There exists an optimal LOCC on these states, which reaches the distillable key for large n. This is a specific key generation protocol over , so that we may write . Thus, for a ddimensional Choistretchable channel, we find
where we also exploit the fact that the distillable key of a DV state is bounded by its regularized REE^{27}. It is also clear that , where the latter equality is demonstrated in the proof of Theorem 5.
Note that cannot be directly written for a bosonic channel, because its Choi matrix is energyunbounded, so that its distillable key is not welldefined. By contrast, we know how to extend to bosonic channels and show at any dimension: This is the more general procedure of Theorem 5, which first exploits the general REE bound and then simplifies by means of teleportation stretching at any dimension.
Twoway quantum communication
Our method can be extended to more complex forms of quantum communication. In fact, our weak converse theorem can be applied to any scenario where two parties produce an output state by means of an adaptive protocol. All the details of the protocol are contained in the LOCCs which are collapsed into by teleportation stretching and then discarded using the REE.
Consider the scenario where Alice and Bob send systems to each other by choosing between two possible channels, (forward) or (backward), and performing adaptive LOCC after each single transmission (see also Fig. 10). The capacity is defined as the maximum number of target bits distributed per individual transmission, by using one of the two channels and , and assuming LOs assisted by unlimited twoway CCs.
In general, the feedback transmission may occur a fraction p of the rounds, with associated capacity
The lower bound is a convex combination of the individual capacities of the two channels, which is achievable by using independent LOCCsequences for the two channels.
Assume that are stretchable into the pair of resource states (σ, σ′). Then, we can stretch the protocol and decompose the output state as
where the tensor exponents n(1−p) and np are integers for suitably large n (it is implicitly understood that we consider suitable limits in the bosonic case). Using the monotonicity of the REE under tracepreserving LOCCs and its subadditivity over tensor products, we write
As previously said, our weak converse theorem can be applied to any adaptive protocol where two parties finally share a bipartite state . Thus, we may write
From equations (114) and (117), we find that must satisfy
For Choistretchable channels, this means
In particular, if the two channels are distillable, that is, and , then we may write
and the optimal strategy (value of p) corresponds to using the channel with maximum capacity.
Note that we may also consider a twoway quantum communication protocol where the forward and backward transmissions occur simultaneously, and correspondingly define a capacity that quantifies the maximum number of target bits which are distributed in each double communication, forward and backward (instead of each single transmission, forward or backward). However, this case can be considered as a doubleband quantum channel.
Multiband quantum channel
Consider the communication scenario where Alice and Bob can exploit a multiband quantum channel, that is, a quantum channel whose single use involves the simultaneous transmission of m distinct systems. In practice, this channel is represented by a set of m independent channels or bands , that is, it can be written as
For instance, the bands may be bosonic Gaussian channels associated with difference frequencies.
In this case, the adaptive protocol is modified in such a way that each (multiband) transmission involves Alice simultaneously sending m quantum systems to Bob. These m input systems may be in a generally entangled state, which may also involve correlations with the remaining systems in Alice’s register. Before and after each multiband transmission, the parties perform adaptive LOCCs on their local registers a and b. The multiband protocol is therefore characterized by a LOCC sequence after n transmissions.
The definition of the generic twoway capacity is immediately extended to a multiband channel. This capacity quantifies the maximum number of target bits that are distributed (in parallel) for each multiband transmission by means of adaptive protocols. It must satisfy
where the lower bound is the sum of the twoway capacities of the single bands . This lower bound is obtained by using adaptive LOCCs that are independent between different , and considering an output state of the form where is the output associated with .
Now consider an adaptive protocol performed over a multiband channel, whose m bands are stretchable into m resources states {σ_{i}}. By teleportation stretching, we find that Alice and Bob’s output state can be decomposed in the form
(it is understood that the formulation is asymptotic for bosonic channels). This previous decomposition leads to
Using our weak converse theorem, we can then write
Combining equations (122) and (125) we may then write
For Choistretchable bands, this means
Finally, if the bands are distillable, that is, , then we find the additive result
Code availability
Source codes of the plots are available from the authors on request.
Data availability
No relevant research data were generated in this study.
Additional information
How to cite this article: Pirandola, S. et al. Fundamental limits of repeaterless quantum communications. Nat. Commun. 8, 15043 doi: 10.1038/ncomms15043 (2017).
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Acknowledgements
We acknowledge support from the EPSRC via the ‘UK Quantum Communications Hub’ (EP/M013472/1) and ‘qDATA’ (EP/L011298/1), and from the ERC (Starting Grant 308253 PACOMANEDIA). In alphabetic order we would like to thank G. Adesso, S. Bose, S.L. Braunstein, M. Christandl, T.P.W. Cope, R. DemkowiczDobrzanski, K. Goodenough, F. Grosshans, S. Guha, M. Horodecki, L. Jiang, J. Kolodynski, S. Lloyd, C. Lupo, L. Maccone, S. Mancini, A. MüllerHermes, B. Schumacher, G. Spedalieri, G. Vallone, M. Westmoreland, M.M. Wilde and A. Winter.
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Department of Computer Science and York Centre for Quantum Technologies, University of York, York YO10 5GH, UK
 Stefano Pirandola
 , Riccardo Laurenza
 & Carlo Ottaviani
Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK
 Leonardo Banchi
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Contributions
S.P. designed the general methodology, which reduces the study of adaptive protocols to singleletter bounds, by devising the technique of teleportation stretching at any dimension and showing its combination with the channel’s relative entropy of entanglement; S.P. proved the theorems, propositions and lemmas (with the exception of Theorem 7); S.P. also derived the analytical results for the bosonic Gaussian channels and wrote the manuscript. R.L. contributed to investigate bosonic channels and derived the analytical bounds for the discretevariable channels, besides studying the classical communication cost. C.O. analysed the ideal rates of current QKD protocols and performed numerical simulations. L.B. proved the general formula for the relative entropy of Gaussian states (Theorem 7), contributed to refine mathematical aspects of teleportation covariance and to improve the bounds for the discretevariable channels. All the authors checked the manuscript for accuracy.
Competing interests
The authors declare no competing financial interests.
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Correspondence to Stefano Pirandola.
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