Elsevier

Optics Communications

Volume 419, 15 July 2018, Pages 30-35
Optics Communications

Experimental preparation and characterization of four-dimensional quantum states using polarization and time-bin modes of a single photon

https://doi.org/10.1016/j.optcom.2018.02.071Get rights and content

Abstract

We present a detailed method to prepare and characterize four-dimensional pure quantum states or ququarts using polarization and time-bin modes of a single-photon. In particular, we provide a simple method to generate an arbitrary pure ququart and fully characterize the state with quantum state tomography. We also verify the reliability of the recipe by showing experimental preparation and characterization of 20 ququart states in mutually unbiased bases. As qudits provide superior properties over qubits in many fundamental tests of quantum physics and applications in quantum information processing, the presented method will be useful for photonic quantum information science.

Introduction

A two dimensional quantum state or a qubit is a fundamental information carrier in quantum information science. Most quantum information processing protocols including quantum communications and quantum computing are based on preparation, manipulation and measurement of a number of qubits [1]. Although qubits are sufficient for most quantum information processing implementations, higher d-dimensional quantum states or qudits have attracted a lot due to their superior properties over qubits. Fundamentally, it is known that the violation of Bell-type inequality can be enhanced by increasing the dimension of the quantum states [[2], [3], [4], [5], [6]]. Qudits also have several advantages over qubits in practical applications of quantum information. For instance, with linear optical elements, the efficiency of Bell state measurement of qudits is higher than that of qubits, and thus, the efficiency of quantum teleportation can be increased [[7], [8], [9]]. In quantum key distribution, encoding the secrete keys in qudits rather than qubits increases the security against the eavesdropping attacks and the noise threshold during the communication [[10], [11], [12]]. It is also known that applying qudits may simplify the implementation of quantum computing [[13], [14]].

In photonic system, various degrees of freedom of a single photon, such as polarization, spatial mode, and photon arrival time can be a resource to encode the quantum information. For a qubit, polarization is the most widely chosen degree of freedom since it is easy to manipulate and measure [[15], [16]]. In order to implement a qudit with a single-photon, however, using only the polarization degree of freedom is no longer available since it is inherently two-dimensional system. Instead, it is necessary to bring multi-dimensional degrees of freedom such as spatial modes, time-bin modes, and frequency modes [[4], [5], [17], [18], [19], [20], [21], [22], [23], [24]].

An alternative approach to realize a qudit is combining two or more degrees of freedom in a single photon [[25], [26], [27], [28]]. A remarkable feature of this hybrid approach is that one can interpret each mode of a single particle as a qubit. This interpretation provides a concept of mode entanglement. Since the preparation, manipulation, and measurement of mode entanglement in a single particle are usually easier than those of two-particle entanglement, it is a good candidate for implementing many fundamental tests of quantum information processing such as quantum information transfer and teleportation [[29], [30], [31]]. In practice, mode entanglement can provide quantum supremacy in some quantum information applications including quantum communication and quantum computation [[32], [33], [34]].

In this paper, we provide detailed methods to prepare and characterize four-dimensional quantum states or ququarts using polarization and time-bin modes of a single photon. In particular, we present a method to prepare an arbitrary pure quantum states as well as detailed recipe to prepare quantum states in mutually unbiased bases (MUB). We also prove the reliability of our recipe by showing experimental preparation and measurement.

Section snippets

Preparation of four-dimensional quantum states

Let us start with defining the computational bases of time-bin and polarization modes as fast and slow time-bin (|F〉t, |S〉t), and horizontal and vertical polarization (|H〉p, |V〉p) states. Here, the subscripts t and p denote the time-bin and polarization modes, respectively. Note that each mode can be considered as an independent qubit, and thus, one can interpret the quantum states as two-qubit states.

The two-qubit states also can be considered as single ququart states. Let us define the

Characterization of four-dimensional quantum states

The density matrix of an experimentally implemented quantum state can be reconstructed by means of quantum state tomography and maximum likelihood estimation [[38], [39], [40]]. In order to conduct the quantum state tomography of a d-dimensional quantum state, one needs at least N=2d projection measurement results. The projectors are not uniquely defined but can have many different forms. We present a typical set of 16 projectors for ququart state tomography in Table 2.

It is remarkable that all

Experimental verification

We verify the reliability of the proposed recipe by showing the experimental preparation and characterization of all the 20 quantum states in MUB, listed in Table 1. Single photon state is generated via spontaneous parametric down-conversion with a 20 mm Type-II PPKTP crystal pumped by a 405 nm continuous-wave laser. The photon pairs are divided by a PBS. By measuring a photon count at one of the outputs of the PBS, one can conditionally prepare a single-photon state at the other output [[41],

Conclusion

We have provided a step-by-step procedure to generate four-dimensional pure photonic quantum states or ququarts with polarization and time-bin modes. We have also presented a detailed recipe to characterize the quantum states by means of quantum state tomography. The reliability of our recipes has been verified with the experimental implementation of 20 quantum states in MUB. Considering higher dimensional quantum states have superior properties over qubits in many quantum information

Acknowledgments

This work was supported by the ICT R&D program of MSIP/IITP (B0101-16-1355), and the KIST research program (2E27231).

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