Securing quantum networking tasks with multipartite Einstein-Podolsky-Rosen steering

Chien-Ying Huang, Neill Lambert, Che-Ming Li, Yen-Te Lu, and Franco Nori

Phys. Rev. A 99, 012302 – Published 2 January 2019

Abstract

Einstein-Podolsky-Rosen (EPR) steering is the explicit demonstration of the fact that the measurements of one party can influence the quantum state held by another distant party and do so even if the measurements themselves are untrusted. This has been shown to allow one-sided device-independent quantum information tasks between two remote parties. However, in general, advanced multiparty protocols for generic quantum technologies, such as quantum secret sharing and blind quantum computing for quantum networks, demand multipartite quantum correlations of graph states shared between more than two parties. Here, we show that when one part of a quantum multidimensional system composed of a two-colorable graph state (e.g., cluster and Greenberger-Horne-Zeilinger states) is attacked by an eavesdropper using a universal cloning machine, only one of the copy subsystems can exhibit multipartite EPR steering but not both. Such a no-sharing restriction secures both state sources and channels against cloning-based attacks for generic quantum networking tasks, such as distributed quantum information processing, in the presence of uncharacterized measurement apparatuses.

Received 15 March 2018

DOI:https://doi.org/10.1103/PhysRevA.99.012302

Physics Subject Headings (PhySH)

Measurement-based quantum computing Nonlocality Quantum communication Quantum cryptography Quantum entanglement Quantum networks

Quantum Information, Science & Technology

Authors & Affiliations

Chien-Ying Huang^1,2, Neill Lambert³, Che-Ming Li^1,*, Yen-Te Lu¹, and Franco Nori^3,4

¹Department of Engineering Science, National Cheng Kung University, Tainan 701, Taiwan
²Graduate Institute of Photonics and Optoelectronics, National Taiwan University, Taipei 10617, Taiwan
³Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan
⁴Department of Physics, University of Michigan, Ann Arbor, Michigan 48109-1040, USA

^*cmli@mail.ncku.edu.tw

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Vol. 99, Iss. 1 — January 2019

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Figure 1
Graph states for quantum networks. Graph states are created from a graph-state source, such as the star-graph states shown here, with the goal of using them for quantum information tasks. Each qudit is sent from the source to the corresponding quantum node through a quantum channel to implement said task, such as QSS [11, 12, 13, 14] or MBQC [15, 16, 17, 18, 19], whereas for BQC [23, 24], a specific initial state is sent from quantum nodes to the source for creating blind graph states. This construction is the essence of the modular and plug-and-play quantum network architecture [27].
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Figure 2
Sharing multipartite EPR steering with a quantum cloner. The party $C_{s}$ with an ancilla $C_{s}^{'}$ (not shown) wants to share the multipartite EPR steerability of $|G_{2}〉$ , between the groups $A_{s}$ and $B_{s}$ , by using a quantum cloning machine. See Appendix pp3 for detailed discussions.
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Figure 3
Quantum networks under both a quantum cloner attack and with untrusted measurements for quantum nodes. In addition to the cloning attack, the quantum-node holders may lose control over their measurement devices such that the system state may be in $ρ_{B_{s}}$ (3) in the worst case. For concreteness, it may happen in QSS where the untrusted parties in the group $B_{s}$ attempt to use their own measurements and the cloner to obtain knowledge of $A_{s}$ 's key without collaboration with the rest of the trusted parties in $B_{s}$ . The no-sharing of multipartite EPR steering can be used to secure QSS by excluding such a possibility with criteria (7) and (8).
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Figure 4
The lower bound of the secret key rate $R_{L}$ derived from noisy graph states. As demonstrated, the reduction in $R_{L}$ by white noise (with intensity $p$ ) on $|G_{2}〉$ is independent of $N$ . $R_{L}$ can be increased by increasing $d$ . This trend is comparable to the increase in the certified multipartite steering with $d$ , i.e., the noise tolerance of the criterion (7) (Appendix pp2-s3). Here the measurements used for creating secret keys, $A_{s m}$ and $B_{s m}$ , are extracted from Schmidt bases $A_{s m}^{(S)}$ and $B_{s m}^{(S)}$ [43] and satisfy the uncertainty relation under the two POVMs [48, 49] as introduced above, and each POVM is composed of two operator elements, respectively. See Appendix pp2-s1 for details. Note that for any cases where $I_{A_{s} B_{s}} - {log}_{2} d < 0$ , the corresponding key rates are set as zero.
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Figure 5
Two-colorable graph states: (a) five-qudit star-graph state and (b) five-qudit chain graph state. The vertices of these graphs can be divided into two sets. The vertices of each set can be given a color such that adjacent vertices have different colors, as shown in (c) and (d). An edge, $(i, j) \in E$ , corresponds to a unitary two-qudit transformation $U_{(i, j)}$ (A1) to cause a nonclassical correlation between qudits (e). The Schmidt form of a graph state, such as the star graph (e), with respect to the $A_{s}$ subsystem (golden) and $B_{s}$ subsystem (green) can be shown by first choosing a qudit in $A_{s}$ (the $i th$ qudit shown here) and then following the introduced procedure to find the state vector (A4) for the final Schmidt decomposition (A5).
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Figure 6
Noise tolerance of the steering criterion, $I_{A_{s} B_{s}} > {log}_{2} d$ , for arbitrarily two-colorable graph states.
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