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arXiv:0908.1604 · Physical Review A

Reconstruction of spatial qutrit states based on realistic measurement operators

Gen Taguchi, Tatsuo Dougakiuchi, M. Iinuma, Holger Friedrich Hofmann, Yutaka Kadoya

Published Dec 2, 2009• 17 citations

Abstract

Spatial qudit states can be realized by using multislits to discretize the transverse momentum of a photon. The merit of this kind of spatial qudit states is that the implementation of higher-dimensional qudits is relatively easy. As we have recently shown, the quantum states of these spatial qudits can be analyzed by scanning a single interference pattern. This method of single scan tomography can also be applied at higher dimensions, but the reconstruction becomes more sensitive to smaller details of the scanned patterns as the dimensions increase. In this paper, we investigate the effect of finite measurement resolution on the single scan tomography of spatial qutrits. Realistic measurement operators describing the spatial resolution of the measurement are introduced and the corresponding pattern functions for quantum state reconstruction are derived. We use the pattern functions to analyze experimental results for entangled pairs of spatial qutrits generated by spontaneous parametric downconversion. It is shown that a reliable reconstruction of the quantum state can be achieved with finite measurement resolution if this limitation of the measurement is included in the pattern functions of single scan tomography.

DOI: 10.1103/physreva.80.062102

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Spatial qudit states can be realized by using multislits to discretize the transverse momentum of a photon. The merit of this kind of spatial qudit states is that the implementation of higher-dimensional qudits is relatively easy. As we have recently shown, the quantum states of these spatial qudits can be analyzed by scanning a single interference pattern. This method of single scan tomography can also be applied at higher dimensions, but the reconstruction becomes more sensitive to smaller details of the scanned patterns as the dimensions increase. In this paper, we investigate the effect of finite measurement resolution on the single scan tomography of spatial qutrits. Realistic measurement operators describing the spatial resolution of the measurement are introduced and the corresponding pattern functions for quantum state reconstruction are derived. We use the pattern functions to analyze experimental results for entangled pairs of spatial qutrits generated by spontaneous parametric downconversion. It is shown that a reliable reconstruction of the quantum state can be achieved with finite measurement resolution if this limitation of the measurement is included in the pattern functions of single scan tomography.

We present a quantum state tomography method that enables the reconstruction of \emph{arbitrary}

d-

dimensional quantum states encoded in the discretized transverse momentum of photons, by using \emph{only}

d+1

experimental settings. To this end, we identify a family of bases with the property that the outcomes of a projective measurement are \emph{spatially multiplexed} on the interference pattern of the projected state. Using the proposed scheme we performed, as a proof-of-principle, an experimental reconstruction of

d=6-

dimensional states, for which a complete set of mutually unbiased bases does not exist. We obtained fidelity values above 0.97 for both pure and mixed states, reducing the number of experimental settings from

42

to only

7

.

We present a quantum state tomography method that enables the reconstruction of arbitrary d−dimensional quantum states encoded in the discretized transverse momentum of photons, by using only d+1 experimental settings. To this end, we identify a family of bases with the property that the outcomes of a projective measurement are spatially multiplexed on the interference pattern of the projected state. Using the proposed scheme, we performed, as a proof-of-principle, an experimental reconstruction of d=6−dimensional states, for which a complete set of mutually unbiased bases does not exist. We obtained fidelity values above 0.97 for both pure and mixed states, reducing the number of experimental settings from 42 to only 7.

We present a proof-of-principle demonstration of a method to characterize any pure spatial qudit of arbitrary dimension

d

, which is based on the classic phase-shift interferometry technique. In the proposed scheme a total of only

4d

measurement outcomes are needed, implying a significant reduction with respect to the standard schemes for quantum-state tomography which require on the order of

{d}^{2}

. By using this technique, we have experimentally reconstructed a large number of states ranging from

d=2

up to 14 with mean fidelity values higher than 0.97. For that purpose the qudits were codified in the discretized transverse-momentum position of single photons, once they are sent through an aperture with

d

slits. We provide an experimental implementation of the method based in a Mach-Zehnder interferometer, which allows one to reduce the number of measurement settings to four since the

d

slits can be measured simultaneously. Furthermore, it can be adapted to consider the reconstruction of the unknown state from the outcome frequencies of $4d\ensuremath{-}3$ fixed projectors independently of the encoding or the nature of the quantum system, allowing one to implement the reconstruction method in a general experiment.

An experimental demonstration of two-qutrit state tomography via one-qutrit symmetric informationally complete positive operator-valued measure (SIC-POVM) is presented. A two-qutrit state is encoded in the transversal paths of a spontaneous parametric down-converted photon pair. A spatial light modulator allows us to implement the necessary 81 POVM elements to reconstruct the state. The quality of the reconstruction is evaluated by comparing independent measurements with the predicted results calculated with the reconstructed state. Entanglement in the system is calculated via negativity and generalized robustness from the two-qutrit reconstructed density operator.

We produce and holographically measure entangled qudits encoded in transverse spatial modes of single photons. With the novel use of a quantum state tomography method that only requires two-state superpositions, we achieve the most complete characterization of entangled qutrits to date. Ideally, entangled qutrits provide better security than qubits in quantum bit commitment: we model the sensitivity of this to mixture and show experimentally and theoretically that qutrits with even a small amount of decoherence cannot offer increased security over qubits.