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Zhongguo kexue. Wulixue Lixue Tianwenxue

Optical realization of universal unitary operation of single partite polarization encoded qudit

Qing Lin

Published Feb 1, 2014• 1 citation
Publisher

Abstract

How to generate and manipulate high dimensional quantum state is a hot topic in the area of quantum information processing. In the optical system, based on the technique of linear optics and cross phase modulation, we consider the realization of universal unitary operation of single partite polarization encoded qudit. Firstly, we introduce the spatially encoded qudit as ancilla, and the entangler of polarization encoded qudit and spatially encoded qudit as core element. After that, the realization of a special unitary operation for single qutrit is discussed by one entangler and the Fourier transformation performed on the spatially encoded qutrit by a linear optical multi-port interferometer (LOMI). The corresponding success probability could be high, but this scheme is only available for a special type of unitary operations and cannot be developed to realize higher dimensional unitary operation. For general qudit universal unitary operation realization, we use qutrit unitary operation as example. The polarization encoded qutrit could be transformed to spatially encoded qutrit by one entangler and the measurement of polarization encoded qutrit, associated with the measurement results are used to control the transmission factors of mirrors placed on the spatial modes of spatially encoded qutrit. Then the desired universal unitary operation is performed on spatially encoded qutrit by a LOMI. Finally, one more entangler, a LOMI implementing Fourier transformation to spatially encoded qutrit, the measurement of spatially encoded qutrit and the phase shifts controlled by the measurement results through classical feed-forward will complete the realization of qutrit unitary operation. This approach could be developed for general case. By two entanglers, one LOMI for Fourier transformation of spatially encoded qudit and the measurement of polarization encoded qudit associated with its results are used to control the transmission factors of mirrors, the bi-directional mapping between polarization and spatially encoded qudits could be realized. Therefore, the universal unitary operation for general qudit could be realized by one more LOMI performed on the ancilla spatially encoded qutrit. In principle, our scheme is available for general single partite polarization encoded qudit and is feasible with the current experimental technology.

DOI
10.1360/n132013-00027

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How to generate and manipulate high dimensional quantum state is a hot topic in the area of quantum information processing. In the optical system, based on the technique of linear optics and cross phase modulation, we consider the realization of universal unitary operation of single partite polarization encoded qudit. Firstly, we introduce the spatially encoded qudit as ancilla, and the entangler of polarization encoded qudit and spatially encoded qudit as core element. After that, the realization of a special unitary operation for single qutrit is discussed by one entangler and the Fourier transformation performed on the spatially encoded qutrit by a linear optical multi-port interferometer (LOMI). The corresponding success probability could be high, but this scheme is only available for a special type of unitary operations and cannot be developed to realize higher dimensional unitary operation. For general qudit universal unitary operation realization, we use qutrit unitary operation as example. The polarization encoded qutrit could be transformed to spatially encoded qutrit by one entangler and the measurement of polarization encoded qutrit, associated with the measurement results are used to control the transmission factors of mirrors placed on the spatial modes of spatially encoded qutrit. Then the desired universal unitary operation is performed on spatially encoded qutrit by a LOMI. Finally, one more entangler, a LOMI implementing Fourier transformation to spatially encoded qutrit, the measurement of spatially encoded qutrit and the phase shifts controlled by the measurement results through classical feed-forward will complete the realization of qutrit unitary operation. This approach could be developed for general case. By two entanglers, one LOMI for Fourier transformation of spatially encoded qudit and the measurement of polarization encoded qudit associated with its results are used to control the transmission factors of mirrors, the bi-directional mapping between polarization and spatially encoded qudits could be realized. Therefore, the universal unitary operation for general qudit could be realized by one more LOMI performed on the ancilla spatially encoded qutrit. In principle, our scheme is available for general single partite polarization encoded qudit and is feasible with the current experimental technology.
Based on a special controlled-NOT gate, a multi-photon state encoded in polarizations of photons could be transformed into the corresponding single photon qudit encoded in spatial mode. It will make the processing on multi-photon change into the operation on a single-photon, if the inverse transformation from a single photon qudit back to a multi-photon state could be realized also. Associated with linear optical multi-port interferometer for single-photon unitary operation, the positive-operator value measurement and the universal unitary operation for multi-photon state are realized. This approach is more efficient than the previous one with decomposition into two-qubit gates in the circuit-based quantum computation, and it is feasible with using the current experimental technology.
Quantum entanglement plays a crucial role in the area of quantum information processing, which was widely used in various quantum information tasks. With the development of quantum information, more and more attentions are paid to high dimensional quantum state and entangled state, since higher dimension means stronger nonlocality, which enables securer and more powerful quantum information processing. Thereafter, how to create entangled qudits efficiently is worth discussed, which is the motivation of this paper. Based on the technique of cross phase modulation, we firstly introduce a single-photon qudit (n-dimension) encoded by spatial mode as ancilla, and then let it interact with n pairs of Bell states in the Kerr medias respectively. Finally, the arbitrary two-partite polarized entangled qudits could be generated after the multi-photon interference. This approach could be directly generalized to the generaion of arbitrary multi-partite polarized entangled qudits. The interactions between an ancilla single-photon qudit and n multi-photon GHZ states, associated with multi-photon interference, will yield the desired arbitrary multi-partite polarized entangled qudits. Except of the inevitable multi-photon interference, which is used to generate polarized qudit from single photons, all the generation processes are deterministic, therefore the generation efficiency is higher than the former proposals. In addition, using quantum nondemolition detection module to preserve two cases that all the single photons appear in the same spatial mode, the success probability of multi-photon interference, i.e., the generation of entangled qudits could be increased. Furthermore, the generated multi-partite polarized entangled qudits are flexible, whose coefficients could be adjusted flexibly by the single-photon unitary operations used in the generation of single-photon qudit. Thanks to this flexibility, it is convenient for the research of the property of entangled qudits and the many-body problem.
Qutrits, the triple level quantum systems in various forms, have been proposed for quantum information processing recently. By the methods presented in this paper a bi-photonic qutrit, which is encoded with the polarizations of two photons in the same spatial-temporal mode, can be mapped to a single photon qutrit in spatial modes. It will make arbitrary unitary operation on such bi-photonic qutrit possible if we can also realize the inverse map to polarization space. Among the two schemes proposed in this paper, the one based only on linear optics realizes an arbitrary U(3) operation with a very small success probability. However, if added with weak nonlinearity, the success probability can be greatly improved. These schemes are feasible with the current experimental technology.
Qutrits, the triple level quantum systems in various forms, have been proposed for quantum information processing recently. By the methods presented in this paper, a biphotonic qutrit, which is encoded with the polarizations of two photons in the same spatial-temporal mode, can be mapped to a single-photon qutrit in spatial modes. It will make arbitrary unitary operation on such biphotonic qutrit possible if we can also realize the inverse map to polarization space. Among the two schemes proposed in this paper, the one based only on linear optics realizes an arbitrary U(3) operation with a very small success probability. However, if added with weak nonlinearity, the success probability can be greatly improved. These schemes are feasible with the current experimental technology.
How to manipulate (operate or measure) single photons efficiently and simply is the basic problem in optical quantum information processing. We first present an efficient scheme to transform a biphoton polarization state to a corresponding single-photon state encoded by its polarization and spatial modes. This single-photon state carries both the information of the controlled and target photons. It will make the realization of bipartite positive-operator-valued measurements efficiently and simply. Moreover, the inverse transformation from the single-photon state back to the corresponding biphoton polarization state is also proposed. Using both the transformations, the realization of the arbitrary two-qubit unitary operation is simple with an M-Z interferometer. All the schemes are feasible with the current experimental technology.
Abstract Unitary operation is an essential step for quantum information processing. We first propose an iterative procedure for decomposing a general unitary operation without resorting to controlled-NOT gate and single-qubit rotation library. Based on the results of decomposition, we design two compact architectures to deterministically implement arbitrary two-qubit polarization-spatial and spatial-polarization collective unitary operations, respectively. The involved linear optical elements are reduced from 25 to 20 and 21 to 20, respectively. Moreover, the parameterized quantum computation can be flexibly manipulated by wave plates and phase shifters. As an application, we construct the specific quantum circuits to realize two-dimensional quantum walk and quantum Fourier transformation. Our schemes are simple and feasible with the current technology.