doi: 10.1038/ncomms15043.
Fundamental limits of repeaterless quantum communications
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- PMID:28443624
- PMCID: PMC5414096
- DOI: 10.1038/ncomms15043
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Fundamental limits of repeaterless quantum communications
Stefano Pirandola et al. Nat Commun..
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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 point-to-point 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 two-way. These two-way 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 dimension-independent technique dubbed 'teleportation stretching', we establish these capacities for many fundamental channels, namely bosonic lossy channels, quantum-limited amplifiers, dephasing and erasure channels in arbitrary dimension. In particular, we exactly determine the fundamental rate-loss tradeoff affecting any protocol of quantum key distribution. Our findings set the limits of point-to-point quantum communications and provide precise and general benchmarks for quantum repeaters.
Conflict of interest statement
The authors declare no competing financial interests.
Figures
The first step is the preparation of the initial separable state
ofa andb by some adaptive LOCC Λ0. After the preparation of the local registers, there is the first transmission: Alice picks a system from her local registera1∈a, so that the register is updated asa→aa1; systema1 is sent through the channel
, with Bob getting the outputb1; Bob includes the output in his local register, which is updated asb1b→b; finally, Alice and Bob apply another adaptive LOCC Λ1 to their registersa,b. In the second transmission, Alice picks and sends another systema2∈a through channel
with outputb2 for Bob. The parties apply a further adaptive LOCC Λ2 to their registers and so on. This procedure is repeatedn times, with output state
for the Alice's and Bob's local registers.
ofa andb by some adaptive LOCC Λ0. After the preparation of the local registers, there is the first transmission: Alice picks a system from her local registera1∈a, so that the register is updated asa→aa1; systema1 is sent through the channel, with Bob getting the outputb1; Bob includes the output in his local register, which is updated asb1b→b; finally, Alice and Bob apply another adaptive LOCC Λ1 to their registersa,b. In the second transmission, Alice picks and sends another systema2∈a through channel with outputb2 for Bob. The parties apply a further adaptive LOCC Λ2 to their registers and so on. This procedure is repeatedn times, with output state for the Alice's and Bob's local registers.
(a) Consider the generalized teleportation of an input stateρ of ad-dimensional systema by using a resource stateσ of two systems,A andB, with corresponding dimensionsd andd′ (finite or infinite). Systemsa andA are subject to a Bell detection (triangle) with random outcomek. This outcome is associated with a projection onto a maximally entangled state up to an associated teleportation unitaryUk which is a Pauli operator ford<+∞ and a phase-displacement ford=+∞ (see Methods for the basics of quantum teleportation and the characterization of the teleportation unitaries). The classical outcomek is communicated to Bob, who applies a correction unitary
to his systemB with outputb. In general,Vk does not necessarily belong to the set {Uk}. On average, this teleportation LOCC defines a teleportation channel
froma tob. It is clear that this construction also teleports parta of an input state involving ancillary systems. (b) In general we may replace the teleportation LOCC (Bell detection and unitary corrections) with an arbitrary LOCC
: Alice performs a quantum operation
on her systemsa andA, communicates the classical variablek to Bob, who then applies another quantum operation
on his systemB. By averaging over the variablek, so that
is certainly trace-preserving, we achieve the simulation
for any input stateρ. We say that a channel
is ‘σ-stretchable' if it can be simulated by a resource stateσ for some LOCC
. Note that Alice's and Bob's LOs
and
are arbitrary quantum operations; they may involve other local ancillas and also have extra labels (due to additional local measurements), in which case
is assumed to be averaged over all these labels. (c) The most important case is when channel
can be simulated by a trace-preserving LOCC
applied to its Choi matrix
, with Φ being an EPR state. In this case, we say that the channel is ‘Choi-stretchable'. These definitions are suitably extended to bosonic channels.
to his systemB with outputb. In general,Vk does not necessarily belong to the set {Uk}. On average, this teleportation LOCC defines a teleportation channel froma tob. It is clear that this construction also teleports parta of an input state involving ancillary systems. (b) In general we may replace the teleportation LOCC (Bell detection and unitary corrections) with an arbitrary LOCC: Alice performs a quantum operation on her systemsa andA, communicates the classical variablek to Bob, who then applies another quantum operation on his systemB. By averaging over the variablek, so that is certainly trace-preserving, we achieve the simulation for any input stateρ. We say that a channel is ‘σ-stretchable' if it can be simulated by a resource stateσ for some LOCC. Note that Alice's and Bob's LOs and are arbitrary quantum operations; they may involve other local ancillas and also have extra labels (due to additional local measurements), in which case is assumed to be averaged over all these labels. (c) The most important case is when channel can be simulated by a trace-preserving LOCC applied to its Choi matrix, with Φ being an EPR state. In this case, we say that the channel is ‘Choi-stretchable'. These definitions are suitably extended to bosonic channels.
(a) Consider the teleportation of an input stateρa by using an EPR state ΦAA′ of systemsA andA′. The Bell detection
on systemsa andA teleports the input state ontoA′, up to a random teleportation unitary, that is,ρA′=Ukρa
. Because
is teleportation-covariant,Uk is mapped into an output unitaryVk and we may write
. Therefore, Bob just needs to receive the outcomek and apply
, so that
. Globally, the process describes the simulation of channel
by means of a generalized teleportation protocol over the Choi matrix
. (b) The procedure is also valid for CV systems. If the inputa is a bosonic mode, we need to consider finite-energy versions for the EPR state Φ and the Bell detection
, that is, we use a TMSV state Φμ and a corresponding quasi-projection
onto displaced TMSV states. At finite energyμ, the teleportation process froma toA′ is imperfect with some output
. However, for anyɛ>0 and input stateρa, there is a sufficiently large value ofμ such that
(refs 25, 26). Consider the transmitted state
. Because the trace distance decreases under channels, we have
. After the application of the correction unitary
, we have the output state
which satisfies
. Taking the asymptotic limit of largeμ, we achieve
→0 for any inputρa, therefore achieving the perfect asymptotic simulation of the channel. The asymptotic teleportation-LOCC is therefore
where
. The result is trivially extended to the presence of ancillas.
on systemsa andA teleports the input state ontoA′, up to a random teleportation unitary, that is,ρA′=Ukρa. Because is teleportation-covariant,Uk is mapped into an output unitaryVk and we may write. Therefore, Bob just needs to receive the outcomek and apply, so that. Globally, the process describes the simulation of channel by means of a generalized teleportation protocol over the Choi matrix. (b) The procedure is also valid for CV systems. If the inputa is a bosonic mode, we need to consider finite-energy versions for the EPR state Φ and the Bell detection, that is, we use a TMSV state Φμ and a corresponding quasi-projection onto displaced TMSV states. At finite energyμ, the teleportation process froma toA′ is imperfect with some output. However, for anyɛ>0 and input stateρa, there is a sufficiently large value ofμ such that (refs 25, 26). Consider the transmitted state. Because the trace distance decreases under channels, we have. After the application of the correction unitary, we have the output state which satisfies. Taking the asymptotic limit of largeμ, we achieve→0 for any inputρa, therefore achieving the perfect asymptotic simulation of the channel. The asymptotic teleportation-LOCC is therefore where. The result is trivially extended to the presence of ancillas.
(a) Consider theith transmission through channel
, where the input (i−1)th register state is given by
. After transmission through
and the adaptive LOCC Λi, the register state is updated to
. (b) Let us simulate the channel
by a LOCC
and a resource stateσ. (c) The simulation LOCC
can be combined with the adaptive LOCC Λi into a single ‘extended' LOCC Δi while the resource stateσ can be stretched back in time and out of the adaptive operations. We may therefore write
=Δi(
⊗σ). (d) We iterate the previous steps for all transmissions, so as to stretchn copiesσ⊗n and collapse all the extended LOCCs Δn o …o Δ1 into a single LOCC Λ. In other words, we may write
=Λ(
⊗σ⊗n). (e) Finally, we include the preparation of the separable state
into Λ and we also average over all local measurements present in Λ, so that we may write the output state as
=
(σ⊗n) for a trace-preserving LOCC
. The procedure is asymptotic in the presence of asymptotic channel simulations (bosonic channels).
, where the input (i−1)th register state is given by. After transmission through and the adaptive LOCC Λi, the register state is updated to. (b) Let us simulate the channel by a LOCC and a resource stateσ. (c) The simulation LOCC can be combined with the adaptive LOCC Λi into a single ‘extended' LOCC Δi while the resource stateσ can be stretched back in time and out of the adaptive operations. We may therefore write=Δi(⊗σ). (d) We iterate the previous steps for all transmissions, so as to stretchn copiesσ⊗n and collapse all the extended LOCCs Δn o …o Δ1 into a single LOCC Λ. In other words, we may write=Λ(⊗σ⊗n). (e) Finally, we include the preparation of the separable state into Λ and we also average over all local measurements present in Λ, so that we may write the output state as=(σ⊗n) for a trace-preserving LOCC. The procedure is asymptotic in the presence of asymptotic channel simulations (bosonic channels).
We depict the classes of channels that are considered in this work, together with the bounds for their two-way capacities.
We plot the secret-key rate (bits per channel use) versus Alice–Bob's distance (km) at the loss rate of 0.2 dB per km. The secret-key capacity of the channel (red line) sets the fundamental rate limit for point-to-point QKD in the presence of loss. Compare this capacity with a previous non-achievable upperbound (dotted line). We then show the maximum rates that are potentially achievable by current protocols, assuming infinitely long keys and ideal conditions, such as unit detector efficiencies, zero dark count rates, zero intrinsic error, unit error correction efficiency, zero excess noise (for CVs) and large modulation (for CVs). In the figure, we see that ideal implementations of CV protocols (purple lines) are not so far from the ultimate limit. In particular, we consider: (i) One-way no-switching protocol, coinciding with CV-MDI-QKD in the most asymmetric configuration (relay approaching Alice65). For high loss
, the rate scales asη/ln 4, which is just 1/2 of the capacity. Same scaling for the one-way switching protocol of ref. ; (ii) Two-way protocol with coherent states and homodyne detection which scales as
for high loss (thermal noise is needed for two-way to beat one-way QKD66). For the DV protocols (dashed lines), we consider: BB84 with single-photon sources with rateη/2; BB84 with weak coherent pulses and decoy states with rateη/(2e); and DV-MDI-QKD with rateη/(2e2). See Supplementary Note 6 for details on these ideal rates.
, the rate scales asη/ln 4, which is just 1/2 of the capacity. Same scaling for the one-way switching protocol of ref. ; (ii) Two-way protocol with coherent states and homodyne detection which scales as for high loss (thermal noise is needed for two-way to beat one-way QKD66). For the DV protocols (dashed lines), we consider: BB84 with single-photon sources with rateη/2; BB84 with weak coherent pulses and decoy states with rateη/(2e); and DV-MDI-QKD with rateη/(2e2). See Supplementary Note 6 for details on these ideal rates.
(a) Two-way capacity
of the thermal-loss channel as a function of transmissivityη for
=1 thermal photon. It is contained in the shadowed area identified by the lower bound (LB) and upper bound (UB) of equation (25). Our upper bound is clearly tighter than those based on the squashed entanglement, computed in ref. (dotted) and ref. (dashed). Note that
at high transmissivities. For
=0 (lossy channel) the shadowed region shrinks into a single line. (b) Two-way capacity
of the amplifier channel as a function of the gaing for
=1 thermal photon. It is contained in the shadowed specified by the bounds in equation (27). For small gains, we have
. For
=0 (quantum-limited amplifier) the shadowed region shrinks into a single line. (c) Two-way capacity
of the additive-noise Gaussian channel with added noiseξ. It is contained in the shadowed region specified by the bounds in equation (30). For small noise, we have
. Our upper bound is much tighter than those of ref. (dotted), ref. (dashed) and ref. (dot-dashed).
of the thermal-loss channel as a function of transmissivityη for=1 thermal photon. It is contained in the shadowed area identified by the lower bound (LB) and upper bound (UB) of equation (25). Our upper bound is clearly tighter than those based on the squashed entanglement, computed in ref. (dotted) and ref. (dashed). Note that at high transmissivities. For=0 (lossy channel) the shadowed region shrinks into a single line. (b) Two-way capacity of the amplifier channel as a function of the gaing for=1 thermal photon. It is contained in the shadowed specified by the bounds in equation (27). For small gains, we have. For=0 (quantum-limited amplifier) the shadowed region shrinks into a single line. (c) Two-way capacity of the additive-noise Gaussian channel with added noiseξ. It is contained in the shadowed region specified by the bounds in equation (30). For small noise, we have. Our upper bound is much tighter than those of ref. (dotted), ref. (dashed) and ref. (dot-dashed).
(a) Two-way capacity of the depolarizing channel
with arbitrary probabilityp. It is contained in the shadowed region specified by the bounds in equation (36). We also depict the best-known bound based on the squashed entanglement (dashed). (b) Two-way capacity of the amplitude damping channel
for arbitrary damping probabilityp. It is contained in the shadowed area identified by the lower bound (LB) of equation (48) and the upper bound (UB) of equation (49). We also depict the bound of equation (47) (upper solid line), which is good only at high dampings; and the bound
of ref. (dotted line), which is computed from the entanglement-assisted classical capacityCA. Finally, note the separation of the two-way capacity
from the unassisted quantum capacity
(dashed line).
with arbitrary probabilityp. It is contained in the shadowed region specified by the bounds in equation (36). We also depict the best-known bound based on the squashed entanglement (dashed). (b) Two-way capacity of the amplitude damping channel for arbitrary damping probabilityp. It is contained in the shadowed area identified by the lower bound (LB) of equation (48) and the upper bound (UB) of equation (49). We also depict the bound of equation (47) (upper solid line), which is good only at high dampings; and the bound of ref. (dotted line), which is computed from the entanglement-assisted classical capacityCA. Finally, note the separation of the two-way capacity from the unassisted quantum capacity (dashed line).
The finite-energy LOCC simulation
generates a teleportation channel
. Assume that
defines a target bosonic channel
according to the pointwise limit in equation (93). Then, we say that the bosonic channel
has asymptotic simulation
.
generates a teleportation channel. Assume that defines a target bosonic channel according to the pointwise limit in equation (93). Then, we say that the bosonic channel has asymptotic simulation.
The protocol employs a forward channel
and backward channel
. Transmissions are alternated with adaptive LOCCs
.
and backward channel. Transmissions are alternated with adaptive LOCCs.References
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