Multi-partite quantum cryptographic protocols with noisy GHZ states
Kai Chen, Hoi-Kwong Lo
备注：
Revised Version, 21 pages, 7 figures. We have added a third-man cryptography protocol. Our protocols for the three cryptographic tasks can be implemented with the generation of only a single entangled pair at a time. This gives significant simplification over previous experimental implementations which require two entangled pairs generated simultaneously
发表信息：
Quantum Information & Computation Vol.7, No.8 (2007) 689-715
We propose a wide class of distillation schemes for multi-partite entangled states that are CSS-states. Our proposal provides not only superior efficiency, but also new insights on the connection between CSS-states and bipartite graph states. We then apply our distillation schemes to the tri-partite case for three cryptographic tasks--namely, (a) conference key agreement, (b) quantum sharing of classical secrets and (c) third-man cryptography. Moreover, we construct ``prepare-and-measure'' protocols for the above three cryptographic tasks which can be implemented with the generation of only a single entangled pair at a time. This gives significant simplification over previous experimental implementations which require two entangled pairs generated simultaneously. We also study the yields of those protocols and the threshold values of the fidelity above which the protocols can function securely. Rather surprisingly, our protocols will function securely even when the initial state does not violate the standard Bell-inequalities for GHZ states.
We propose two novel schemes to engineer four-partite entangled Greenberger-Horne-Zeilinger (GHZ) and W states in a deterministic way by using chains of (two-level) Rydberg atoms within the framework of cavity QED. These schemes are based on the resonant interaction of the atoms with a bimodal cavity that simultaneously supports, in contrast to a single-mode cavity, two independent modes of the photon field. In addition, we suggest the schemes to reveal the non-classical correlations for the engineered GHZ and W states. It is shown how these schemes can be extended in order to produce general …
We propose concurrence classes for general pure multipartite states based on an orthogonal complement of a positive operator valued measure on quantum phase. In particular, we construct $W^{m}$ class, $GHZ^{m}$, and $GHZ^{m-1}$ class concurrences for general pure $m$-partite states. We give explicit expressions for $W^{3}$ and $GHZ^{3}$ class concurrences for general pure three-partite states and for $W^{4}$, $GHZ^{4}$, and $GHZ^{3}$ class concurrences for general pure four-partite states.
The problem of noise incidence on qubits taking part of bipartite entanglement-based protocols is addressed. It is shown that the use of a three-partite GHZ state and measurements instead of their EPR counterparts allows the experimenter to detect $2/3$ of the times whenever one of the qubits involved in the measurement is affected by bit-flip noise through the mere observation of unexpected outcomes in the teleportation and superdense coding protocols when compared to the ideal case. It is shown that the use of post-selection after the detection of noise leads to an enhancement in the efficie…
We propose two classes of the generalized concurrence vectors of the multipartite systems consisting of qubits. Making use of them, we are able to, respectively, describe and quantify GHZ-class and W-class entanglement both in total and between arbitrary two partite in the multipartite system consisting of qubits. In the case of pure state of three qubits that one partite is separable, it is shown to exactly back to the usual Wootters' concurrence after introduce a whole concurrence vector. In principle, our method is applicable to any $N$-partite systems consisting of $N$ qubits.
The simulation of quantum correlations with alternative nonlocal resources, such as classical communication, gives a natural way to quantify their nonlocality. While multipartite nonlocal correlations appear to be useful resources, very little is known on how to simulate multipartite quantum correlations. We present the first known protocol that reproduces 3-partite GHZ correlations with bounded communication: 3 bits in total turn out to be sufficient to simulate all equatorial Von Neumann measurements on the 3-partite GHZ state.
We study the quantum coherence and its distribution of N-partite GHZ and W states of bosonic fields in the noninertial frames with arbitrary number of acceleration observers. We find that the coherence of both GHZ and W state reduces with accelerations and freezes in the limit of infinite accelerations. The freezing value of coherence depends on the number of accelerated observers. The coherence of N-partite GHZ state is genuinely global and no coherence exists in any subsystems. For the N-partite W state, however, the coherence is essentially bipartite types, and the total coherence is equal …
We investigate the generalized braid relation ($d-$level $N-$body braid relation) and its application to quantum entanglement. By means of finite-dimensional representations of Heisenberg-Weyl algebra, a set of $d^{N}\times d^{N}$ unitary matrix representations satisfying the generalized braid relation can be constructed. Such generalized braid matrices can entangle $d-$level $N-$partite quantum states. Acting the generalized braid matrices on the standard basis, one can obtain a set of maximally entangled basis. Further study shows that such entangled basis can be viewed as the $d-$level $N-$…
We develop separability criteria to identify non-$k$-separability $(k = 2,3,\ldots,n)$ and genuine multipartite entanglement in different classes of mixed $n$-partite quantum states using elements of density matrices. With the help of these criteria, we detect non-$k$-separability in $n$-qudit GHZ and W states respectively added with white noise. We also discuss the experimental implementation of our criteria by means of local observables.
We develop protocols for preparing a GHZ state and, in general,a pure multi-partite maximally entangled state in a distributed network with apriori quantum entanglement between agents using classical communication and local operations. We investigate and characterize the minimal combinatorics of the sharing of EPR pairs required amongst agents in a network for the creation of multi-partite entanglement. We also characterize the minimal combinatorics of agents in the creation of pure maximal multi-partite entanglement amongst the set $N$ of $n$ agents in a network using apriori multi-partite en…
We generalize Greenberger-Horne-Zeilinger (GHZ) theorem to an arbitrary number of D-dimensional systems. Contrary to conventional approaches using compatible composite observables, we employ incompatible and concurrent observables, whose common eigenstate is still a generalized GHZ state. It is these concurrent observables which enable to prove a genuinely N-partite and D-dimensional GHZ theorem. Our principal idea is illustrated for a four-partite system with D which is an arbitrary multiple of 3. By extending to N qudits, we show that GHZ theorem holds as long as N is not divisible by all no…
For a tripartite pure state of three qubits, it is well known that there are two inequivalent classes of genuine tripartite entanglement, namely the GHZ-class and the W-class. Any two states within the same class can be transformed into each other with stochastic local operations and classical communication (SLOCC) with a non-zero probability. The optimal conversion probability, however, is only known for special cases. Here, we derive new lower and upper bounds for the optimal probability of transformation from a GHZ-state to other states of the GHZ-class. A key idea in the derivation of the …
The familiar Greenberger-Horne-Zeilinger (GHZ) states can be rewritten by entangling the Bell states for two qubits with a state of the third qubit, which is dubbed entangled entanglement. We show that in this way we obtain all 8 independent GHZ states that form the simplex of entangled entanglement, the magic simplex. The construction procedure allows a generalization to higher dimensions both, in the degrees of freedom (considering qudits) as well as in the number of particles (considering n-partite states). Such bases of GHZ-type states exhibit a certain geometry that is relevant for experi…
Quantum cryptographic conferencing (QCC) holds promise for distributing information-theoretic secure keys among multiple users over long distance. Limited by the fragility of Greenberger-Horne-Zeilinger (GHZ) state, QCC networks based on directly distributing GHZ states at long distance still face big challenge. Another two potential approaches are measurement device independent QCC and conference key agreement with single-photon interference, which was proposed based on the post-selection of GHZ states and the post-selection of W state, respectively. However, implementations of the former pro…
We experimentally prepare a new type of continuous variable genuine four-partite entangled states, the quantum correlation property of which is different from that of the four-mode GHZ and cluster states, and which has not any qubit counterpart to be proposed at present. In the criterion inequalities for the full inseparability of the genuine four-partite entangled states, the amplitude and phase quadrature correlation variances totally consisting of three-party combination from the four entangled modes are involved. The measured correlation variances among the quadratures of the prepared enta…
The theory of the asymptotic manipulation of pure bipartite quantum systems can be considered completely understood: The rates at which bipartite entangled states can be asymptotically transformed into each other are fully determined by a single number each, the respective entanglement entropy. In the multi-partite setting, similar questions of the optimally achievable rates of transforming one pure state into another are notoriously open. This seems particularly unfortunate in the light of the revived interest in such questions due to the perspective of experimentally realizing multi-partite …
We show that there is a unique maximal decomposition of a pure multi-partite (N>2) quantum state into a sum of states which are "locally orthogonal" in the sense that the local reduced state for a term in the sum lives in its own orthogonal subspace for each subsystem. Observers can make local measurements on any subsystem and determine which "branch" they are on. The Shannon entropy of the resulting branch weights defines a new measure of global, GHZ-like entanglement, which is insensitive to local pairwise entangling operations and vanishes when there is no piece of information recorded at e…
We show how to quantify tri-partite entanglement using entropies derived from experimental correlations. We use a multi-partite generalization of the entanglement of formation that is greater than zero if and only if the state is genuinely multi-partite entangled. We develop an entropic witness for tripartite entanglement, and show that the degree of violation of this witness places a lower limit on the tripartite entanglement of formation. We test our results in the three-qubit regime using the GHZ-Werner state and the W-Werner state, and in the high-dimensional pure-state regime using the tr…
The communication cost of simulating probability distributions obtained by measuring quantum states is a natural way to quantify quantum non-locality. While much is known in the case of bipartite entanglement, little has been done in the multipartite setting. In this paper, we focus on the GHZ state. Specifically, equatorial measurements lead to correlations similar to the ones obtained with Bell states. We give a protocol to simulate these measurements on the n-partite GHZ state using O(n^2) bits of communication on average.
Multi-qubit quantum Rabi model, which is a fundamental model describing light-matter interaction, plays an important role in various physical systems. In this paper, we propose a theoretical method to simulate multi-qubit quantum Rabi model in a circuit quantum electrodynamics system. By means of external transversal and longitudinal driving fields, an effective Hamiltonian describing the multi-qubit quantum Rabi model is derived. The effective frequency of the resonator and the effective splitting of the qubits depend on the external driving fields. By adjusting the frequencies and the amplit…
Recently, several aspects of controlled quantum communication (e.g., bidirectional controlled state teleportation, controlled quantum secure direct communication, controlled quantum dialogue, etc.) have been studied using $n$-qubit ($n\geq3$) entanglement. Specially, a large number of schemes for bidirectional controlled state teleportation are proposed using $m$-qubit entanglement ($m\in\{5,6,7\}$). Here, we propose a set of protocols to illustrate that it is possible to realize all these tasks related to controlled quantum communication using only Bell states and permutation of particles (Po…
Multipartite reference-frame-independent quantum cryptographic communication
Korea Institute of Science and Technology (KIST)（韩国科学技术院）；Hyundai Motor Company（现代汽车公司）；Korea University of Science and Technology（韩国科学技术大学）；Korea University（高丽大学）
Secret sharing is a multi-party cryptographic primitive that can be applied to a network of partially distrustful parties for encrypting data that is both sensitive (it must remain secure) and important (it must not be lost or destroyed). When sharing classical secrets (as opposed to quantum states), one can distinguish between protocols that leverage bi-partite quantum key distribution (QKD) and those that exploit multi-partite entanglement. The latter class are known to be vulnerable to so-called participant attacks and, while progress has been made recently, there is currently no analysis t…
Secure cryptographic protocols are indispensable for modern communication systems. It is realized through an encryption process in cryptography. In quantum cryptography, Quantum Key Distribution (QKD) is a widely popular quantum communication scheme that enables two parties to establish a shared secret key that can be used to encrypt and decrypt messages. But security loopholes still exist in this cryptographic protocol, as an eavesdropper can in principle still intercept all the ciphertext to perform cryptanalysis and the key may get leaked to the eavesdropper, although it happens very rarely…
Reproducing with elementary resources the correlations that arise when a quantum system is measured (quantum state simulation), allows one to get insight on the operational and computational power of quantum correlations. We propose a family of models that can simulate von Neumann measurements in the x-y plane of the Bloch sphere on n-partite GHZ states using only bipartite nonlocal boxes. For the tripartite and fourpartite states, the models use only bipartite nonlocal boxes; they can be translated into classical communication schemes with finite average communication cost.
Teleporting an unknown qubit state is a paradigmatic quantum information processing task revealing the advantage of quantum communication protocols over their classical counterpart. For a teleportation protocol using a Bell state as quantum channel, the resource has been identified to be the concurrence. However, for mixed multi-partite states the lack of computable entanglement measures has made the identification of the quantum resource responsible for this advantage more challenging. Here, by building on previous results showing that localizable concurrence is the necessary resource for con…
Greenberger-Horne-Zeilinger (GHZ) paradox provides an all-versus-nothing test for the quantum nonlocality. In all the GHZ paradoxes known so far each observer is allowed to measure only two alternative observables. Here we shall present a general construction for GHZ paradoxes in which each observer measuring more than two observables given that the system is prepared in the $n$-qudit GHZ state. By doing so we are able to construct a multi-setting GHZ paradox for the $n$-qubit GHZ state, with $n$ being arbitrary, that is genuine $n$-partite, i.e., no GHZ paradox exists when restrict to a subse…
We consider the problem of teleporting an unknown information state within a quantum network by a sender, say, Alice to any given receiver out of several receivers, say, Bob(1), Bob(2), ...., Bob(n). For this task, we suggest two schemes using (n+1)-partite GHZ state as a quantum channel. The first scheme involves C-NOT operations on qubit of Bobs with the control as qubit of Alice, while the second scheme involves measurements in the Hadamard basis by Bobs on their qubits. We have generalized both schemes for quantum teleportation of arbitrary m-qubit information state within the quantum netw…
We show that for all $n\ge3$, an example of an $n$-partite quantum correlation that is not genuinely multipartite nonlocal but rather exhibiting anonymous nonlocality, that is, nonlocal but biseparable with respect to all bipartitions, can be obtained by locally measuring the $n$-partite Greenberger-Horne-Zeilinger (GHZ) state. This anonymity is a manifestation of the impossibility to attribute unambiguously the underlying multipartite nonlocality to any definite subset(s) of the parties, even though the correlation can indeed be produced by nonlocal collaboration involving only such subsets. …
Experimental demonstration of asynchronous measurement-device-independent quantum cryptographic conferencing
Nanyang Technological University（南洋理工大学）；China Telecom Research Institute（中国电信研究院）；Hangzhou Dianzi University（杭州电子科技大学）；CNRS（法国国家科学研究中心）；Université Côte d’Azur（蔚蓝海岸大学）；Sorbonne Université（索邦大学）；National University of Singapore（新加坡国立大学）；Majulab（马朱实验室）；National Centre for Advanced Integrated Photonics（国家先进集成光子学中心）；China Telecom Quantum Information Technology Group Corporation Limited（中国电信量子信息技术集团有限公司）
The spatial correlation with classical lights, which has some similar aspects as that with entangled lights, is an interesting and fundamentally important topic. But the features of high-order spatial correlation with classical lights are not well known, and the types of high-order correlations produced are of limit. Here, we propose a scheme to produce third-order spatial correlated states by modulating the phases of three laser beams. With the scheme we can produce Greenberger-Horne-Zeilinger-type (GHZ-type) and W-type spatial correlations with different phase modulations. Our scheme can be …
We present an explicit protocol ${\cal E}_0$ for faithfully teleporting an arbitrary two-qubit state via a genunie four-qubit entangled state. By construction, our four-partite state is not reducible to a pair of Bell states. Its properties are compared and contrasted with those of the four-party GHZ and W states. We also give a dense coding scheme ${\cal D}_0$ involving our state as a shared resource of entanglement. Both ${\cal D}_0$ and ${\cal E}_0$ indicate that our four-qubit state is a likely candidate for the genunine four-partite analogue to a Bell state.
In this paper we consider the quantum phase transition in the Ising model in the presence of a transverse field in one, two and three dimensions from a multi-partite entanglement point of view. Using \emph{exact} numerical solutions, we are able to study such systems up to 25 qubits. The Meyer-Wallach measure of global entanglement is used to study the critical behavior of this model. The transition we consider is between a symmetric GHZ-like state to a paramagnetic product-state. We find that global entanglement serves as a good indicator of quantum phase transition with interesting scaling b…
Self-testing is a device-independent examination of quantum devices based on correlations of observed statistics. Motivated by elegant progresses on self-testing strategies for measurements [Phys. Rev. Lett. 121, 250507 (2018)] and for states [New J. Phys. 20, 083041 (2018)], we develop a general self-testing procedure for multipartite generalized GHZ-state measurements. The key step is self-testing all measurement eigenstates for a general N-qubit multipartite GHZ-state measurement. Following our procedure, one only needs to perform local measurements on N-2 chosen parties of the N-partite ei…
We present the new exact upper bounds on the maximal Bell violation for the generalized N-qubit GHZ state, the N-qudit GHZ state and, in general, for an arbitrary N-partite quantum state, possibly infinite-dimensional. Our results indicate that, for an N-partite quantum state of any Hilbert space dimension, violation of any Bell-type inequality (either on correlation functions or on joint probabilities) with S settings and any number of outcomes at each site cannot exceed (2S-1)^{N-1}.
Multiparty quantum communication provides delightful applications including quantum cryptographic communication and quantum secret sharing. Measurement-Device-Independent (MDI) quantum communication based on the Greenberg-Horne-Zeilinger (GHZ) state measurement provides a practical way to implement multiparty quantum communication. With the standard spatially localized GHZ state measurement, however, information can be imbalanced among the communication parties that can cause significant problems in multiparty cryptographic communication. Here, we propose an equitable multiparty quantum commun…
In this paper, we investigate the entanglement of multi-partite Grassmannian coherent states (GCSs) described by Grassmann numbers for $n>2$ degree of nilpotency. Choosing an appropriate weight function, we show that it is possible to construct some well-known entangled pure states, consisting of {\bf GHZ}, {\bf W}, Bell, cluster type and bi-separable states, which are obtained by integrating over tensor product of GCSs. It is shown that for three level systems, the Grassmann creation and annihilation operators $b$ and $b^†$ together with $b_{z}$ form a closed deformed algebra, i.e., $SU_{q}(2…
We consider the collision model of Ziman {\em et al.} and study the robustness of $N$-qubit Greenberger-Horne-Zeilinger (GHZ), W, and linear cluster states. Our results show that $N$-qubit entanglement of GHZ states would be extremely fragile under collisional decoherence, and that of W states could be more robust than of linear cluster states. We indicate that the collision model of Ziman {\em et al.} could provide a physical mechanism to some known results in this area of investigations. More importantly, we show that it could give a clue as to how $N$-partite distillable entanglement would …
In this article we present a general security strategy for quantum secret sharing (QSS) protocols based on the HBB scheme presented by Hillery, Bužek and Berthiaume [Phys. Rev A \textbf{59}, 1829 (1999)]. We focus on a generalization of the HBB protocol to $n$ communication parties thus including $n$-partite GHZ states. We show that the multipartite version of the HBB scheme is insecure in certain settings and impractical when going to large $n$. To provide security for such QSS schemes in general we use the framework presented by some of the authors [M. Huber, F. Minert, A. Gabriel, B. C. Hie…
We consider the class of (N+1)-partite states suitable for protocols where there is a powerful party, the authority, and the other N parties play the same role, namely the state of their system live in the symmetric Hilbert space. We show that, within this scenario, there is a "maximally entangled state" that can be transform by a LOCC protocol into any other state. In addition, we show how to make the protocol efficiently including the construction of the state and discuss security issues for possible applications to cryptographic protocols. As an immediate consequence we recover a sequential…
Quantum nonlocality of several four-qubit states is investigated by constructing a new Bell inequality. These include the Greenberger-Zeilinger-Horne (GHZ) state, W state, cluster state, and the state $|χ>$ that has been recently proposed in [PRL, {\bf 96}, 060502 (2006)]. The Bell inequality is optimally violated by $|χ>$ but not violated by the GHZ state. The cluster state also violates the Bell inequality though not optimally. The state $|χ>$ can thus be discriminated from the cluster state by using the inequality. Different aspects of four-partite entanglement are also studied by consideri…
Entanglement is a universal resource in quantum networks, yet entangled photon sources are typically custom-made for a specific use case. Versatility, both in terms of state modulation and tunability of the temporal properties of the photons, is the key to flexible network architectures and cryptographic primitives that go beyond quantum key distribution. Here, we report on a flexible source design that produces high-quality entanglement in continuous-wave and GHz-rate-pulsed operation modes. Utilizing off-the-shelf optical components, our approach uses a fiber-based Sagnac loop to generate po…
Mutually entangled multi-photon states are at the heart of all-optical quantum technologies. While impressive progresses have been reported in the generation of such quantum light states using free space apparatus, high-fidelity high-rate on-chip entanglement generation is crucial for future scalability. In this work, we use a bright quantum-dot based single-photon source to demonstrate the high fidelity generation of 4-photon Greenberg-Horne-Zeilinger (GHZ) states with a low-loss reconfigurable glass photonic circuit. We reconstruct the density matrix of the generated states using full quantu…
Quantun non-demolition (QND) variables are generlized to the nonlocal ones by proposing QND measurement networks of Bell states and multi-partite GHZ states, which means that we can generate and measure them without any destruction. One of its prospective applications in the quantum authentication system of the Quantum Security Automatic Teller Machine (QSATM) which is much more reliable than the classical ones is also presented.
