Versatile relative entropy bounds for quantum networks

Rigovacca, L.; Kato, Go; Bäuml, Stefan; Kim, M S; Munro, William J.; Azuma, Koji

doi:10.1088/1367-2630/aa9fcf

Cited by 41 publications

(55 citation statements)

References 65 publications

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“…For instance, our benchmarks may be used to evaluate performances in relay-assisted QKD protocols such as MDI-QKD and variants [56][57][58]. Related literature and other developments [59][60][61][62][63][64][65][66] are discussed in Supplementary Note 6.…”

Section: Discussionmentioning

confidence: 99%

End-to-end capacities of a quantum communication network

2019

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Section: Discussionmentioning

confidence: 99%

End-to-end capacities of a quantum communication network

2019

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“…Proof. For convenience of the reader, we give a complete proof, but we note that some of the essential steps are available in prior works [9,12,14]. From the assumption in (49), it follows that…”

Section: Max-rains Information As a Strong Converse Rate For Ppt-p-asmentioning

confidence: 99%

“…holds, which is known to occur generally for certain entanglement measures [9,12,16]or for certain channels with particular symmetries [9]. One of the main observations of [9], connected to earlier developments in [10][11][12][13][14], is that the amortized entanglement of a channel serves as an upper bound on the entanglement of the final state ω AB generated by an LOCC-or PPT-P-assisted quantum communication protocol that uses the channel n times:…”

Section: Introductionmentioning

confidence: 98%

“…Thus, if we find an upper bound on  « ( ) -Q PPT P, , then by (1), such an upper bound also bounds the physically relevant LOCC-assisted quantum capacity  « ( ) Q . A general approach for bounding these assisted capacities of a quantum channel has been developed recently in [9](see [10][11][12][13][14] for related notions). The starting point is to consider an entanglement measure E(A; B) ρ [15], which is evaluated for a bipartite state ρ AB .…”

Section: Introductionmentioning

confidence: 99%

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Amortization does not enhance the max-Rains information of a quantum channel

Berta

Wilde

2018

New J. Phys.

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Given an entanglement measure E, the entanglement of a quantum channel is defined as the largest amount of entanglement E that can be generated from the channel, if the sender and receiver are not allowed to share a quantum state before using the channel. The amortized entanglement of a quantum channel is defined as the largest net amount of entanglement Ethat can be generated from the channel, if the sender and receiver are allowed to share an arbitrary state before using the channel. Our main technical result is that amortization does not enhance the entanglement of an arbitrary quantum channel, when entanglement is quantified by the max-Rains relative entropy. We prove this statement by employing semi-definite programming (SDP)duality and SDPformulations for the max-Rains relative entropy and a channel's max-Rains information, found recently in Wang et al (arXiv:1709. 00200). The main application of our result is a single-letter, strong converse, and efficiently computable upper bound on the capacity of a quantum channel for transmitting qubits when assisted by positive-partial-transpose preserving (PPT-P) channels between every use of the channel. As the class of local operations and classical communication (LOCC) is contained in PPT-P, our result establishes a benchmark for the LOCC-assisted quantum capacity of an arbitrary quantum channel, which is relevant in the context of distributed quantum computation and quantum key distribution.In this paper, we are interested in placing upper bounds on the LOCC-assisted quantum capacity, and one way of simplifying the mathematics behind this task is to relax the class of free operations that the sender and receiver are allowed to perform between each channel use. With this in mind, we follow the approach of [7, 8]and relax the set LOCCto a larger class of operations known as PPT-preserving (PPT-P), standing for channels that are positive-partial transpose preserving. The resulting capacity is then known as the PPT-Passisted quantum capacity, and it is equal to the maximum rate at which qubits can be communicated reliably from a sender to a receiver, when they are allowed to use a PPT-P channel in between every use of the actual channel  . Figure 1 provides a visualization of such a PPT-P-assisted quantum communication protocol. Due to the containment LOCC⊂PPT-P [7, 8], the inequality    « « SB   ( ) Y J . 8 7 SB SB

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“…Given that, even in the case of point-to-point links, entanglement makes the characterization of this optimal value, the capacity, notably more complicated than its classical counterpart, with phenomena such as superactivation [9], it was unclear how much it would be possible to borrow from the theory of classical networks. However, in a series of fundamental works [10][11][12][13] it was established that the capacity of quantum networks for bipartite communication behaves similar to that of classical networks. Namely, given a network of quantum noisy channels and bounds on their capacities satisfying certain properties, one can conceptually construct a classical version of the quantum network where each quantum channel is replaced by a perfect classical channel with capacity given by the bound on the quantum channel capacity.…”

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confidence: 99%

Linear programs for entanglement and key distribution in the quantum internet

et al. 2020

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Quantum networks will allow to implement communication tasks beyond the reach of their classical counterparts. A pressing and necessary issue for the design of quantum network protocols is the quantification of the rates at which these tasks can be performed. Here, we propose a simple recipe that yields efficiently computable lower and upper bounds for network capacities. For this we make use of the max-flow min-cut theorem and its generalization to multi-commodity flows to obtain linear programs (LPs). We exemplify our recipe deriving the LPs for bipartite settings, settings where multi-pairs of users obtain entanglement in parallel as well as multipartite settings, covering almost all known situations. We also make use of a generalization of the concept of paths between user pairs in a network to Steiner trees spanning the group of users wishing to establish GHZ states. Contents 26A. Proof of Lemma 2 26 B. Proof of Lemma 3 27 C. Proof of Lemma 4 28 arXiv:1809.03120v2 [quant-ph] 4 Jan 2019 total/worst case network and P total/worst case networkas the maximum rate achievable by means of an adaptive protocol, as well as (3) the maximization over a weighted sum of rates over all user pairs. From [13,14], we can obtain upper bounds on the private capacities that involve (1) a minimization over multicuts, i.e. sets of edges the removal of which connects all pairs, or (2) the so-called minimum cut ratio [17]. Using the respective results of [17,19], we can, up to a factor of order O(log r), upper bound the capacities for both cases (1) and (2) by a maximization over concurrent multi-commodity flows, i.e. flows between several user pairs that can be achieved in parallel. In the first case we maximize the sum of flows for all user pairs, whereas in the second case we maximize the worst case flow that is guaranteed for every pair. Both multi-commodity flow maximizations can be cast into LPs. Applying the aggregated repeater protocol [12] to multiple user pairs we also obtain lower bounds in terms of the maximum concurrent multi-commodity flows, providing us with the following efficiently computable bounds: f total/worst case Q ↔ ≤ Q total/worst case network ≤ P total/worst case network ≤ O(log r)f total/worst case Esq/E

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Versatile relative entropy bounds for quantum networks

Cited by 41 publications

References 65 publications

End-to-end capacities of a quantum communication network

End-to-end capacities of a quantum communication network

Amortization does not enhance the max-Rains information of a quantum channel

Linear programs for entanglement and key distribution in the quantum internet

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