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arXiv:1012.3474 · Physical Review A

Linear-optics realization of channels for single-photon multimode qudits

Marco Piani, David Pitkanen, Rainer Kaltenbaek, Norbert Lütkenhaus

Published Sep 6, 2011• 19 citations
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Abstract

We propose and theoretically study a method for the stochastic realization of arbitrary quantum channels on multimode single-photon qudits. In order for our method to be undemanding in its implementation, we restrict our analysis to linear-optical techniques, vacuum ancillary states, and nonadaptive schemes, but we allow for random switching between different optical networks. With our method it is possible to deterministically implement random-unitary channels and to stochastically implement general channels. We provide an expression for the optimal probability of success of our scheme and calculate this quantity for specific examples such as the qubit amplitude-damping channel. The success probability is shown to be related to the entanglement properties of the Choi-Jamio\l{}kowski state isomorphic to the channel.

DOI
10.1103/physreva.84.032304

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We propose and theoretically study a method for the stochastic realization of arbitrary quantum channels on multimode single-photon qudits. In order for our method to be undemanding in its implementation, we restrict our analysis to linear-optical techniques, vacuum ancillary states, and nonadaptive schemes, but we allow for random switching between different optical networks. With our method it is possible to deterministically implement random-unitary channels and to stochastically implement general channels. We provide an expression for the optimal probability of success of our scheme and calculate this quantity for specific examples such as the qubit amplitude-damping channel. The success probability is shown to be related to the entanglement properties of the Choi-Jamio\l{}kowski state isomorphic to the channel.
In a recent paper, have developed a scheme for the stochastic implementation of arbitrary quantum operations on multimode single-photon qudit states by using reconfigurable linear-optic systems. Based on this idea, we explore the use of phase modulation for the realization of qubit channels in the frequency basis. Single-photon states belonging to two different frequency modes differing by the modulator's driving frequency represent the input dual-rail qubit states. The channel is implemented by a phase modulator followed by a fiber Bragg grating, taking advantage of the high degree of reconfigurability and microwave bandwidth shown by electrooptic modulation technology. The channels are realized by a combination of three techniques: 1) suitably designed driving waveforms, which are probabilistically addressed to the modulator; 2) the corresponding addressing probabilities; and 3) the grating transmittance at the values of the frequency basis. The proposed scheme results in nonoptimal success probabilities but is shown to allow for a compact implementation of the conventional qubit random unitary channels and the qubit amplitude-damping channel.
We experimentally demonstrate a single-qubit decohering quantum channel using linear optics. We implement the channel, whose special cases include the familiar amplitude-damping channel and the bit-flip channel, using a single, static optical setup. Following a recent theoretical result (Piani et al 2011 Phys. Rev. A 84 032304), we realize the channel in an optimal way, maximizing the probability of success, i.e. the probability for the photonic qubit to remain in its encoding. Using a two-photon entangled resource, we characterize the channel using ancilla-assisted process tomography and find average process fidelities of 0.98 ± 0.01 and 0.976 ± 0.009 for amplitude-damping and the bit-flip case, respectively.
This thesis deals with the implementation of quantum channels using linear optics. We begin with overviews of some important concepts in both quantum information and quantum optics. First, we discuss the quantum bit and describe the evolution of the states via quantum channels. We then discuss both quantum state and process tomography, methods for how to determine which states and operations we are experimentally implementing in the lab. Second, we discuss topics in quantum optics such the generation of single photons, polarization entanglement, and the construction of an entangling gate. \n \nThe first experiment is the implementation of a quantum damping channel, which intentionally can add a specific type and amount of decohering noise to a photonic qubit. Specifically, we realized a class of quantum channels which contains both the amplitude-damping channel and the bit-flip channel, and did so with a single, static, optical setup. Many quantum channels, and some gates, can only be implemented probabilistically when using linear optics and postselection. Our main result is that the optical setup achieves the optimal success probability for each channel. Using a novel ancilla-assisted tomography, we characterize each case of the channel, and find process fideilities of 0.98pm0.010.98 \\pm 0.010.98pm0.01 for the amplitude-damping channel and 0.976pm0.0090.976 \\pm 0.0090.976pm0.009 for the bit-flip. \n \nThe second experiment is an implementation of a protocol for quantum computing on encrypted data. The protocol provides the means for a client with very limited quantum power to use a server's quantum computer while maintaining privacy over the data. We perform a quantum process tomography for each gate in a universal set, showing that only when the proper decryption key is used on the output states, which is hidden from the server, then the action of the quantum gate is recovered. Otherwise, the gate acts as the completely depolarizing channel.
The main topic of this thesis is linear optics and the implementation of quantum operations (measurements, quantum channels, and unitary rotations) on optical systems. \nIn the opening chapter the basic notions needed to understand the rest of the thesis will be explained. \nThese notions include defining a quantum state, measurement, quantum channel and the linear optics tool set. \n \nThe work in this thesis takes both fundamental and practical approaches to studying linear optical networks. \nFor instance in the first chapter a proof is provided that shows that any unitary on a single mode Fock state can be realised with linear optics. \nThe proof is constructive, however the approach to realising the unitary is not suitable for experimental implementation \nbecause it requires complicated ancilla states. \nAs in the KLM proposal the procedure works only stochastically however by allowing the size of the ancilla to grow \nthe probability of failure can be made arbitrarly small. \n \nFurthermore we investigate the realisation of arbitrary channels in a specific encoding that we call a ddd-rail encoding. \nThe only ancilla state that we allow is a vacuum ancilliary state and further restrictions were considered (e.g. photon counting). \nA proof is provided that using these resources only random unitaries can be applied deterministically using linear optics. An expression for the optimal probability of success for realising \nmore general channels with these resources is also discussed. \n \nAs a final topic we also investigate the realisation of a quantum non-demolition measurement onto the dual rail qubit space. \nThe investigation is a blend of both fundamental and practical approaches. \nTo begin we employ a modified KLM-like procedure and show that the scheme can be realised perfectly but stochastically. \nThe probability that the proper measurement is made can be made arbitrarly close to one using a suitably large ancilla state. \nIn addition we consider an existing scheme \\cite{gisin10a} which uses practical sources (two single photon sources) to perform the measurement. \nThe scheme does not realise the true measurement but instead has a free parameter in it which is the transmittiviy of a beamsplitter. The measurement will project onto a space that has a vacuum component. By adjusting the transmittivity of this beamsplitter the vacuum component can be made arbitrarly small but only at the expense of the probability of success of the procedure. \nIn this thesis a modification that can be made to eliminate the vacuum component without changing the sources is introduced. The modification is surprisingly simple and only involves the addition of a single beamsplitter. In the proposal for the original amplifier it was used in simulations for DIQKD that included device imperfections. \nTo show the improvement of our modification these DIQKD simulations are reproduced using the modified amplifier and its results are compared to the results of the original amplifier.
A quantum channel describes the general evolution of quantum systems, which is therefore a critical issue for quantum simulation and a fundamental element of quantum information. To realize quantum channels, an efficient method is to randomly implement Kraus operators, which does not require any ancillary quantum system. For open systems, it is natural to extend the method from implementation of unitary Kraus operators to nonunitary ones. However, when some Kraus operators are not proportional to unitary ones, this method becomes probabilistic even if the channel is trace preserving. In this paper, to overcome this drawback, we propose an algorithm to deterministically realize arbitrary trace-preserving channels with only one ancillary qubit and finite iterations of evolutions of the combined states of the system and the ancilla. Moreover, to show the validity of the method, we experimentally realize conventional and modified Landau-Streater channels based on the algorithm. Our results shed light on quantum simulation of quantum channels for open systems.