Quantum information processing in phase space: A modular variables approach

A. Ketterer, A. Keller, S. P. Walborn, T. Coudreau, and P. Milman

Phys. Rev. A 94, 022325 – Published 22 August 2016

Abstract

Binary quantum information can be fault-tolerantly encoded in states defined in infinite-dimensional Hilbert spaces. Such states define a computational basis, and permit a perfect equivalence between continuous and discrete universal operations. The drawback of this encoding is that the corresponding logical states are unphysical, meaning infinitely localized in phase space. We use the modular variables formalism to show that, in a number of protocols relevant for quantum information and for the realization of fundamental tests of quantum mechanics, it is possible to loosen the requirements on the logical subspace without jeopardizing their usefulness or their successful implementation. Such protocols involve measurements of appropriately chosen modular variables that permit the readout of the encoded discrete quantum information from the corresponding logical states. Finally, we demonstrate the experimental feasibility of our approach by applying it to the transverse degrees of freedom of single photons.

Received 9 December 2015

DOI:https://doi.org/10.1103/PhysRevA.94.022325

Physics Subject Headings (PhySH)

Quantum formalism Quantum information processing with continuous variables

General PhysicsQuantum Information, Science & Technology

Authors & Affiliations

A. Ketterer^1,*, A. Keller², S. P. Walborn³, T. Coudreau¹, and P. Milman^1,†

¹Laboratoire Matériaux et Phénomènes Quantiques, Sorbonne Paris Cité, Université Paris Diderot, CNRS UMR 7162, 75013 Paris, France
²Institut de Sciences Moléculaires d'Orsay, Bâtiment 350, Université Paris-Sud, Université Paris-Saclay, CNRS UMR 8214, 91405 Orsay, France
³Instituto de Física, Universidade Federal do Rio de Janeiro, Caixa Postal 68528, 21941-972 Rio de Janeiro, RJ, Brazil

^*andreas.ketterer@univ-paris-diderot.fr
^†perola.milman@univ-paris-diderot.fr

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Vol. 94, Iss. 2 — August 2016

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Images

Figure 1
Schematic representation of (a) the unbounded position and momentum eigenspectrum together with their discretization into boxes of length $ℓ$ and $2 π / ℓ$ , respectively, and of (b) the bounded spectra of the modular position and momentum operator. The red arrows indicate the displacements implementing the logical Pauli operators $\hat{X}, \hat{Y}$ , and $\hat{Z}$ . (c) Representation of the transverse distribution of a single photon created in a source S with a Gaussian wave function, which is transformed into a periodic diffraction pattern by passing through a grating with slit distance $L$ . In experiments, such a diffraction grating can be implemented easily using a spatial light modulator (SLM).
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Figure 2
(a) Density plots of the probability distributions of the logical states $| 0_{L} 〉$ and $| 1_{L} 〉$ , respectively, in the modular representation [see Eq. (14)] with $f (\bar{x}, \bar{p}), θ (\bar{x}, \bar{p})$ and $ϕ (\bar{x}, \bar{p})$ chosen as explained in the text and $ℓ = 2 \sqrt{π}$ . The full widths at half maximum (FWHM) of the distributions are indicated by $\tilde{Δ} = 2 \sqrt{2 ln 2} Δ$ and $\tilde{κ} = 2 \sqrt{2 ln 2} κ$ . (b) Plot of the wave function $ψ_{0, 1} (x)$ of the same two logical states in the position representation. In momentum space the same functions with switched roles of $\tilde{Δ}$ and $\tilde{κ}$ represent the logical states $| \pm_{L} 〉 = \frac{1}{\sqrt{2}} (| 0_{L} 〉 \pm | 1_{L} 〉)$ .
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Figure 3
Scheme showing the production, processing and detection of the transverse degrees of single photons. The photons are produced in a source (S) and then sent through a lens systems (LS) in order to prepare them in approximate plane waves. Further on, a periodic refraction grating prepares the photon's transverse wave function in a periodic state. The photons can be manipulated using a spatial light modulator (SLM) implementing the unitary operation ${\hat{U}}_{SLM}$ , or optionally $\hat{F} {\hat{U}}_{SLM} {\hat{F}}^{†}$ if placing lenses (L1, L2) before and after the SLM. Finally, the photons are measured with a spatially resolving detector (D).
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Figure 4
(a) Quantum circuit allowing for the measurement of the observables (28). $\hat{H}$ depict Hadamard gates and a controlled unitary gate $\hat{U}$ is applied if the ancilla is in the state $| 0 〉$ . The expectation value of (28) is given by $p_{0} - p_{1}$ , where $p_{0}$ ( $p_{1}$ ) are the probabilities of detecting the ancilla in the state $| 0 〉$ ( $| 1 〉$ ). In the case of the specific example mentioned in the text we choose $\hat{U} = \hat{X}, \hat{Y}, \hat{Z}$ . (b) Proposal of an experimental implementation of circuit (a) using the spatial field of single photons passing through a Mach-Zehnder interferometer. Controlled unitaries are realized by linear optical transformations inserted in one arm of the interferometer. Unitaries of the form $e^{i h (\hat{x})}$ or $e^{i h (\hat{p})}$ can be implemented using a SLM and lenses $(L)$ allowing us to switch from the position to the momentum space.
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