Quantum Communication Complexity Protocol with Two Entangled Qutrits

Abstract

We formulate a two-party communication complexity problem and present its quantum solution that exploits the entanglement between two qutrits. We prove that for a broad class of protocols the entangled state can enhance the efficiency of solving the problem in the quantum protocol over any classical one if and only if the state violates Bell's inequality for two qutrits. To date only very few tasks in quantum communication and quantum computation require higher-dimensional systems than qubits as recourses. Quantum-key distribution based on higher alphabets was shown to be more secure than that based on qubits [9]. A certain quantum solution of the coin-flipping problem uses qutrits (3-dimensional quantum states) [10] and the quantum solution of the Byzantine agreement problem utilizes the entanglement between three qutrits [11].Here we formulate a two-party communication complexity problem and present its quantum solution which makes use of the entanglement between two qutrits. We prove that for a broad class of protocols the entangled state of two qutrits can enhance the efficiency of solving the problem in the quantum protocol, over any classical one if and only if the state violates Bell's inequality for two qutrits as derived by Collins et al. [12].

Correspondence between continuous-variable and discrete quantum systems of arbitrary dimensions

“…Such a mapping is also necessary if one wants to implement those quantum information tasks developed for discrete systems to CV systems, which exclusively requires higher-dimensional Hilbert spaces. These are, for example, the quantum key distribution based on higher alphabets [17] and the quantum solutions of the coin-flipping problem [18], of the Byzantine agreement problem [19], and of a certain communication complexity problem [20].In this paper we establish a mapping between a CV and a discrete system of arbitrary dimension. Mathematically, for an infinite-dimensional Hilbert space we construct the generators of SU(n) algebra for finite n, which build up the structure of a n-dimensional Hilbert space.…”

“…Such a mapping is also necessary if one wants to implement those quantum information tasks developed for discrete systems to CV systems, which exclusively requires higher-dimensional Hilbert spaces. These are, for example, the quantum key distribution based on higher alphabets [17] and the quantum solutions of the coin-flipping problem [18], of the Byzantine agreement problem [19], and of a certain communication complexity problem [20].…”

Experimental quantum communication complexity

“…Two types of CCP's are distinguished: the first minimizes the amount of information exchange necessary to solve the task with certainty [2,3,4]. The second maximizes the probability of successfully solving the task for restricted amount of information [4,5,6]. Such studies aim, e.g., at a speed up of distributed computations by increasing the communication efficiency, or at an optimization of VLSI circuits and data structures [7].…”

“…The second maximizes the probability of successfully solving the task for restricted amount of information [4,5,6]. Such studies aim, e.g., at a speed up of distributed computations by increasing the communication efficiency, or at an optimization of VLSI circuits and data structures [7].Quantum CCP protocols, using multi-particle entanglement, were proven to be clearly superior with respect to classical ones [2,3,4,5,6]. However, the technology of entanglement based multi-party quantum communication is still in a premature stage.…”

“…Quantum CCP protocols, using multi-particle entanglement, were proven to be clearly superior with respect to classical ones [2,3,4,5,6]. However, the technology of entanglement based multi-party quantum communication is still in a premature stage.…”