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arXiv:1011.3504 · Physical Review A

Measurement strategy for spatially encoded photonic qubits

M. A. Solís-Prosser, Leonardo Neves

Published Nov 11, 2010• 10 citations
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Abstract

We propose a measurement strategy which can, probabilistically, reproduce the statistics of any observable for spatially encoded photonic qubits. It comprises the implementation of a two-outcome positive operator-valued measure followed by a detection in a fixed transverse position, making the displacement of the detection system unnecessary, unlike previous methods. This strategy generalizes a scheme recently demonstrated by one of us and co-workers, restricted to measurement of observables with equatorial eigenvectors only. The method presented here can be implemented with the current technology of programmable multipixel liquid-crystal displays. In addition, it can be straightforwardly extended to high-dimensional qudits and may be a valuable tool in optical implementations of quantum information protocols with spatial qubits and qudits.

DOI
10.1103/physreva.82.055801

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We propose a measurement strategy which can, probabilistically, reproduce the statistics of any observable for spatially encoded photonic qubits. It comprises the implementation of a two-outcome positive operator-valued measure followed by a detection in a fixed transverse position, making the displacement of the detection system unnecessary, unlike previous methods. This strategy generalizes a scheme recently demonstrated by one of us and co-workers, restricted to measurement of observables with equatorial eigenvectors only. The method presented here can be implemented with the current technology of programmable multipixel liquid-crystal displays. In addition, it can be straightforwardly extended to high-dimensional qudits and may be a valuable tool in optical implementations of quantum information protocols with spatial qubits and qudits.
We study the quantum communication protocol of remote state preparation (RSP) for pure states of qubits encoded in single photons transmitted through a double slit, the so-called spatial qubits. Two measurement strategies that one can adopt to remotely prepare the states are discussed. The first strategy is the well-known spatial postselection, where a single-pixel detector measures the transverse position of the photon between the focal and the image plane of a lens. The second strategy, proposed by ourselves, is a generalized measurement divided into two steps: the implementation of a two-outcome positive operator-valued measurement (POVM) followed by the spatial postselection at the focal plane of the lens by a two-pixel detector in each output of the POVM. In both cases we analyze the effects of the finite spatial resolution of the detectors over three figures of merit of the protocol, namely, the probability of preparation, the fidelity, and purity of the remotely prepared states. It is shown that our strategy improves these figures compared with spatial postselection, at the expense of increasing the classical communication cost as well as the required experimental resources. In addition, we present a modified version of our strategy for RSP of spatial qudits which is able to prepare arbitrary pure states, unlike spatial postselection alone. We expect that our study may also be extended for RSP of the angular spectrum of a single-photon field as an alternative for quantum teleportation which requires very inefficient nonlinear interactions.
One of the challenges of photon-based quantum teleportation is that both a source of entangled photons and an entangled basis measurement are required. For qubits, one can perform a probabilistic entangled basis measurement using linear optics, making the scheme efficient. However, for photonic qudits, an equivalent scheme remains difficult to devise. In this paper, we generalize the probabilistic photonic qubit teleportation protocol to qudits. The method relies on producing permutation entangled states nondeterministically which are superpositions of permutations of the spatial and qudit states of the photons. Our scheme nondeterministically teleports a photonic qudit using only entangled photon sources, linear optics, and photon detectors, and should be experimentally realizable for small qudit dimensions.
Quantum state tomography is an essential tool for characterizing quantum systems and underpins nearly every experimental realization of quantum technologies. Conventional tomography relies on performing a sequence of projective measurements on many identical copies of a quantum state, requiring the measurement apparatus to be reconfigured between successive acquisitions. As the Hilbert-space dimension increases, the number of required measurements grows rapidly; in practice, additional overcomplete measurements are often performed to improve robustness to experimental imperfections. Here, we introduce a tomography platform based on structured anisotropic media that performs informationally complete measurements of photonic polarization qubits within a single acquisition. The approach employs three liquid-crystal metasurfaces with spatially varying optic-axis orientations that transform the input polarization into a far-field distribution of discrete transverse-momentum modes. Each diffraction pattern uniquely determines the polarization state, enabling its reconstruction without sequential changes to the measurement apparatus. Unlike previous implementations, our scheme is intrinsically photon-number independent: the same optical device operates identically for arbitrary photon numbers, while the desired photon-number sector can be selected afterwards through post-selection of the corresponding nnn-fold coincidence events. We experimentally demonstrate single-frame quantum state tomography of both single- and two-photon polarization states, providing a simple and scalable route toward efficient quantum-state characterization.
We report an experimental implementation of a single-qubit generalized measurement scenario, the positive-operator valued measure (POVM), based on a quantum walk model. The qubit is encoded in a single-photon polarization. The photon performs a quantum walk on an array of optical elements, where the polarization-dependent translation is performed via birefringent beam displacers and a change of the polarization is implemented with the help of wave plates. We implement: (i) trine POVM, i.e., the POVM elements uniformly distributed on an equatorial plane of the Bloch sphere; (ii) symmetric-informationally-complete (SIC) POVM; and (iii) unambiguous discrimination of two nonorthogonal qubit states.
It is crucial for various quantum information processing tasks that the state of a quantum system can be determined reliably and efficiently from general quantum measurements. One important class of measurements for this purpose is symmetric informationally complete positive operator-valued measurements (SIC-POVMs). SIC-POVMs have the advantage of providing an unbiased estimator for the quantum state with the minimal number of outcomes needed for full tomography. By virtue of Naimark's dilation theorem, any POVM can always be realized with a suitable coupling between the system and an auxiliary system and by performing a projective measurement on the joint system. In practice, finding the appropriate coupling is rather nontrivial. Here we propose an experimental design for directly implementing SIC-POVMs using multiport devices and path-encoded qubits and qutrits, the utility of which has recently been demonstrated by several experimental groups around the world. Furthermore, we describe how these multiports can be attained in practice with an integrated photonic system composed of nested linear optical elements.
We have devised an optical scheme for the recently proposed protocol for encoding two qubits into one qutrit. In this protocol, Alice encodes an arbitrary pure product state of two qubits into a state of one qutrit. Bob can then restore either of the two encoded qubit states error-free but not both of them simultaneously. We have successfully realized this scheme experimentally using spatial-mode encoding. Each qubit (qutrit) was represented by a single photon that could propagate through two (three) separate fibers. We theoretically propose two generalizations of the original protocol. We have found a probabilistic operation that enables us to retrieve both qubits simultaneously with average fidelity above 90% and we have proposed an extension of the original encoding transformation to encode NNN qubits into one (N+1)(N+1)(N+1)-dimensional system.