EPJ manuscript No. (will be inserted by the editor) Resource state structure for controlled quantum key distribution Arpan Das1;2;a, Sumit Nandi1;2;b, Sk Sazim3;c, and Pankaj Agrawal1;2;d 1 Institute of Physics, Sachivalaya Marg, Bhubaneswar 751005, Odisha, India. 2 Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400085, India. 3 Harish-Chandra Research Institute, HBNI, Allahabad 211019, India. Received: date / Revised version: date Abstract. Quantum entanglement plays a pivotal role in many communication protocols, like secret sharing and quantum cryptography. We consider a scenario where more than two parties are involved in a protocol and share a multipartite entangled state. In particular, we considered the protocol of Controlled Quantum Key Distribution (CoQKD), introduced in the Ref. Chin. Phys. Lett. 20, 183-185 (2003), where, two parties, Alice and Bob establish a key with the cooperation of other parties. Other parties control/supervise whether Alice and Bob can establish the key, its security and key rate. We discuss the case of three parties in detail and find suitable resource states. We discuss the controlling power of the third party, Charlie. We also examine the usefulness of the new resource states for generating conference key and for cooperative teleportation. We find that recently introduced Bell inequalities can be useful to establish the security of the conference key. We also generalize the scenario to more than three parties. PACS. XX.XX.XX No PACS code given 1 Introduction the two parties may be dishonest, ii) one of the two par- ties may be compromised by some eavesdropper, iii) the Quantum entanglement is an exquisite resource behind communication may be done only under some supervision. many quantum information processing protocols. In par- In this protocol, the controller/supervisor determines the ticular, usefulness of entanglement as a resource in cryp- state that Alice and Bob can share. This state can be a tography was demonstrated by Ekert [1] in a protocol product state, a partially entangled state, or a maximally which was an extension of the seminal BB84 scheme [2]. entangled state. The nature of this state will determine if Since then, many variants of this protocol have been pro- a key can be established or not, and the key rate and secu- posed using bipartite and multipartite entanglement [3]. rity. This protocol is different from MDI-QKD protocols In the Ekert's protocol, Alice and Bob share a set of Bell [6,7], where the third party (say Charlie) is an untrusted states and make measurement in three bases each. Of party. Alice and Bob send two photons to Charlie and he these, two bases are common. There are nine combina- projects the joint state onto a Bell state. Whereas, in Con- tions of bases, among which two combinations give corre- trolled QKD, Charlie's measurement is the first step of the lated results and are used to establish the secret key. Four protocol. The collapsed state between Alice and Bob after combinations of bases are used to establish the security the measurement of Charlie is used by them to establish using the violation of Bell-CHSH inequality [4] and the the secure key. In the next section, we illustrate various rest are discarded. We consider a variant of this protocol features of this protocol using three-qubit GHZ states. We arXiv:1903.10163v2 [quant-ph] 5 May 2020 using multipartite states. then find all three-qubit resource states that would allow In a multiparty case, we shall consider two scenarios. In the generation of a perfect key with optimal key rate, and the first scenario, a key is established between two parties, then generalize to more than three parties. We also discuss say Alice and Bob, with other parties controlling this key the security of the protocol using Bell-CHSH inequality. generation. This protocol of Controlled QKD (CoQKD) has also been discussed in the Ref. ([5]). In the second scenario, a secret key is established among all the parties. In the second scenario, the key, that is established For the first scenario, need for control may arise for a num- among all parties, is known as conference key. In a con- ber of different reasons. Some of them could be: i) one of ference key, all parties share a common key that they can use for secret communication with one another. Using the a Email:[email protected] new resource state (called NMM-state), introduced for Co- b Email:[email protected] QKD, we give a protocol to establish a conference key. We c Email:[email protected] use our recently introduced Bell inequalities [8,9] to estab- d Email:[email protected] lish the security of this conference key protocol. 2 Arpan Das et al.: Resource state structure for controlled quantum key distribution These resource states can be thought of as task-oriented p1 (j000i + j111i). If the controller Charlie performs mea- 2 maximally entangled states (TMESs) [10], where task is surement on his qubit in Hadamard basis, then the col- maximal Co-QKD, i.e. the generation of a perfect secret lapsed state between Alice and Bob is a Bell state: j ±i = key with optimal key rate. We also show how these states p1 (j00i±j11i). After the measurement, Charlie announces 2 can be generated from product states by using suitable his choice of basis publicly. Then following the Ekert's pro- multinary unitary operators. It turns out that these re- tocol, Alice and Bob can establish a perfect secret key with source states have the structure that makes them suitable optimal key rate between them, i. e. without any Quan- for cooperative quantum teleportation protocol also. We tum Bit Error rate (QBER) . After establishing the key, discuss some aspects of this protocol using new resource Alice and Bob both report the key rate to Charlie. If it states. is not optimal, then Charlie knows that there has been a The paper is organized as follows. In Sections2,3,4 cheating. Moreover, without the measurement performed and5, we cover the three-qubit scenario in detail. Specifi- by Charlie, Alice and Bob can not establish a key, as their cally, in the section2, we introduce the CoQKD protocol. qubits are in maximally mixed state. So, Charlie also su- In the section3, we propose the structure of three-qubit pervises the starting of the key generation process, which resource states suitable for maximal CoQKD. Then, we allows him to control the whole process. If Charlie makes discuss the CoQKD protocol using these resource states a measurement in a basis other than the Hadamard basis, in the section4. In the section5, we show the generation of shared state between Alice and Bob is not a Bell state, but the conference key by these resource states. In the section a partially entangled state. With this state, perfect key 6, we explore further for the suitable structure of states generation with optimal key rate would not be obtained. useful in CoQKD beyond three qubits. We discuss how If the key rate reported to Charlie is less than the opti- these new resource states are also suitable for cooperative mal w.r.t the shared state between Alice and Bob, Charlie teleportation in the section7. In the section8, we identify knows that there is a cheating involved and that run of these resource states as TMESs. Lastly, we conclude in key making is discarded. Therefore, for a particular basis the section9. 2 CoQKD Protocol In this section, we illustrate the CoQKD protocol using three-qubit GHZ state. The goal of this protocol is to es- tablish a secret key between two parties with the involve- ment of the other parties. As discusses earlier, there could be multiple reasons for the involvement of other parties. One of the parties with the final secret key is not trust- worthy, and wants to disrupt the key creation. Specifically, Fig. 1. CoQKD: Charlie supervises and controls the secret key in the key forming procedure, the dishonest party can de- between Alice and Bob. liberately make false statements in the public declaration rounds and affect the secret key. Moreover, he/she can be compromised by an external attacker, such that they two of Charlie, Alice and Bob can establish a perfect secret collaborate to act dishonestly. If there is a supervisor who key with optimal key rate, as the collapsed state between can control the shared entangled state between the two Alice and Bob is a maximally entangled state. The key is parties and supervises the key generation then a party can- made according to the Ekert's protocol where the secu- not cheat or be compromised by any external eavesdrop- rity is certified by the violation of CHSH [4] inequality. If per. Specifically, the supervisor knows exactly which state Charlie performs the measurement in a general basis, the the two parties are sharing and what the optimal key rate secret key rate is not optimal. In this case we introduce should be. If the parties report the controller/supervisor a a modified version of Ekert's protocol, where security is smaller key rate than the optimal, then there is a cheating guaranteed by the violation of CHSH inequality. Security involved and that run of the key generation is discarded. aspects and key rate are discussed in more detail in the The supervisor is also a controller. All the parties share section 4. a multipartite entangled state and the controller does a Advantage of this protocol is that Charlie can detect measurement on his/her system to initiate the key gener- if there is a cheating involved in the establishment of the ation process.