1. Introduction

As a striking example of quantum algorithm, teleportation was discovered early on in the development of the field of quantum information processing. With either qubit [1] and continuous variable flavor [2], experiments were soon to follow [3, 4]. Quantum teleportation turns out to be an especially interesting primitive for prototyping experimental setups of quantum information processing. As was discovered in [5], it is possible to merge any Gaussian quantum gate together with the teleportation copying operation. An idea which has now evolved into the paradigm of one-way quantum computing and cluster state computation. Although Gaussian operations alone are not sufficient to achieve universal quantum information processing[6], several proposals have been made to mix together these one-way operations with non-Gaussian input states to achieve such universal operations [7-9]. These hybrid paradigm techniques are currently one of the most active area of research in quantum optics. With this long-term objectives of hybrid quantum operations, the research work of this thesis has been focused on the quantum teleportation of Schroedinger's cat states of light. Until now the continuous variable version of teleportation has indeed only been performed with Gaussian states, in a monomode sideband regime. There was a lack of results and understanding in this particular area, both experimentally and theoretically, and a lack of experimental research for relevant applications in quantum information processing. In the duration of this thesis, we have been able to demonstrate several results linked directly to these topics[10-12].

2. Non-Gaussian state generation and teleportation

We have demonstrated the experimental preparation of non-classical states of light closely approximating Schroedinger's cat states, with experimentally measured negative Wigner functions. We have also demonstrated the quantum manipulation of this fragile states and performed quantum teleportation of Schroedinger's cat states while preserving the quantum nature of these states at the output of teleportation[10]. To accommodate with the required time-resolving photon detection techniques and handle the wave-packet nature of these optical Schroedinger's cat states, we have developed a hybrid teleporter built with continuous-wave light yet able to directly operate in the time domain (see Fig.1). Both setups are individually evolved from the experimental setups of [13] and [14]. As a prototype of these new techniques in a fully quantum regime, this experiment constitutes a first step towards more advanced QIP protocols and future non-classical state manipulation experiments. Indeed although Schr¨odinger's cat states are often quoted for their potential applications[15, 16], the major experimental challenge of actually using and manipulating these fragile states has remained mostly unaddressed until this experimental demonstration.

Successfullness of continuous variable teleportation is a non-trivial problem as it is closely related to the kind of input states and entanglement used. For the Gaussian case, the fidelity F = <ψin|Pout|ψin> is the usual figure of merit, though F losses much of its meaning for non-Gaussian mixed states. Because non-classicality itself is an ambiguous propertie, the problem is even more complex when the input state is a Schroedinger's cat state. To verify the success of Schroedinger's cat states teleportation we perform experimental quantum tomography of the input and output states independently (see Fig. 2). We consider the input Win and output Wout Wigner functions and adopt the criteria of negativity teleportation in Wout given Win. The reconstructed input Wigner functionWin shows the two positive Gaussians of |α>and |-α> together with a central negative dip (Win(0, 0) = -0.171 ± 0.003) caused by the interferences of the|α>and |-α> superposition. The output Wigner function Wout retains the characteristic non-Gaussian shape as well as the negative dip (Wout(0, 0) = -0.022 ± 0.003) to a lesser degree. The degradation of the central negative dip and the full evolution of Win towards Wout can be fully understood as we explain in Sec. 3. We calculate that the fidelity Fcat is as high as 0.750 ± 0.005 for the input Wigner function Win with the nearest Schr¨odinger's cat having an amplitude equal to |αin|2 = 0.98. However, after the teleportation Wout fidelity is reduced to 0:46 ± 0.01 with the nearest Schr¨odinger's cat state having an amplitude |αout|2 = 0.66.

3. Multimode model of teleportation

Although Gaussian states teleportation has been amply studied, due to the complex nature of non-Gaussian states and especially mixed non-Gaussian states, only few general results exist for this particular case. On top of these difficulties, to accommodate with the transient nature of our non-Gaussian input state used, the teleporter we used operates over a broad range of frequencies. Our main objective was to answer both these issues with a theoretical model as simple and efficient as possible[11]. In the Braunstein-Kimble teleportation scheme described in [17], the teleportation is expressed in phase space by the convolution Wout = Win 。Ge-r with r the EPR correlation parameter and Gα(q, p) a normalized Gaussian of standard deviation α. In this case teleportation of non-classical features such as negativity requiers 3 dB (r = ln √2) of squeezing[18], or equivalently a vacuum fidelity of F >= 2/3, which is also called the no-cloning limit. Various monomode and multimode models exist for the protocol of photon subtraction that we use to generate our experimental non-classical state [19, 20]. In the limit of small s and R, they are essentially equivalent and therefore we will assume that an APD trigger projects our input state onto a squeezed photon Ss|1>. TheWigner functionWref of this state is written Wref(q, p) = 2(e2sq2+ e-2sp2 - 1/2)G1/ √2(esq, e-sp). Although not pure our experimental input state happens to fit extremely well a simple loss model where the experimental Wigner function Win can be deduced from Wref by applying "beam splitter losses" 1 - η equivalent in phase space to Win(q, p) = (Wref * Gλ) (q/√η, p/√η)/η with λ = √1-η/2η . Wout is then written in the same form as Win and reads Wout(q, p) = (Wref * Gλ) (q/√η, p/√η)/η where λ has been changed to λ1 = √λ2 + (e-r)2 /η. With the equations above, we can express Win(0, 0) and Wout(0, 0) the negativity values at the origin of phase space at the input and output of the teleporter. Furthermore, both Wigner function can be calculated exactely and used to predict the output of the teleporter. As expected for unity gain teleportation, the Wout(0, 0) = 0 threshold is independant of s and can be expressed as a function of the two parameters η and r in our model by r = ln √2/(2η - 1) at threshold. APD triggered non-Gaussian statse have been shown to have complex multimode properties. In the limit of small s and small R a simple multimode picture describes the input state as e-s/2 (Ay+2-A2 )A+|0> with the wave packet mode operator A = ∫f(ω)aωdω, and f(ω) the bandwidth spectrum of the optical parametric oscillator used to produce the squeezed vacuum state Ss|0>. We investigate how this multimode aspect translates in term of teleportation and express in the Heisenberg picture the relation between input (xin, pin) and output (xout, pout) quadrature operators for unity gain teleportation. Thanks to this linear transformation and the linear model of input state in the Heisenberg picture, multimode teleportation reduces to monomode teleportation with an effective EPR parameter reff given by the relation e-reff = ∫ f(ω)e-r(ω)dω where as a matter of fact r(ω) is the EPR correlation spectrum of the teleporter, with unit 20/ln(10) dB. After we estimate precisely the relevant experimental parameters, we can now predict the shape of the input and output Wigner functions Win and Wout. We find fidelities of 0:987 and 0:988 between the experimental and predicted Wigner functions for Win and Wout. Furthermore, the predicted value of output negativity is Wout(0. 0) = -0.0243, to compare with the experimental results -0.022 ± 0.003.

4. Conditional Teleportation

The continuous variable teleportation protocol as it was proposed in [17] is a deterministic protocol, which always succeeds but also always adds a minimal amount of noise to the output teleported state. We present and demonstrate how to experimentally implement conditional teleportation to further enhance teleportation of nonclassical features of theWigner function as was originally proposed in [21]. In the Kimble-Braunstein teleportation protocol [17], homodyne detection is used by Alice to perform joint quadrature measurements and can be used in principle for conditioning: only when Alice's quadrature measurement results meet chosen requisite conditions that teleportation will be considered successful as an operation on its own. The conditioning scheme is based on a threshold mechanism. If Alice measurement ξ = (xu + ipv)/ √2 falls inside a circle of radius L, then the output teleported state is accepted. If not, the output teleported state is rejected. The conditioning algorithm exploits the fact that a smaller and smaller measured value of ξ on Alice side means smaller and smaller displacements on Bob side. In other words, if for instance ξ = 0, Bob does not need to perform any displacement and the output teleported state is naturally found to be identical to the input state. Experimentally, the output negativity W(0, 0|L) is estimated using the inverse Radon transform with the detection events satisfying the condition x2u + p2u <- 2L2 The evolution with the control parameter L of both the negativityW(0, 0jL) and the fraction of selected events can be evaluated for many values of L with the same experimental data set after the experimental measurement phase. Fig.4 shows experimental results of the probability of success P(L) of conditional teleportation with the value of the parameter L and the evolution of W(0; 0jL) with the value of the parameter L. Conditional teleportation can be seen as an advanced technique of noise filtering where information at the output of the teleporter which is judged too noisy is progressively removed from the final experimental data set.

5. Conclusion

In Sec.2, we have demonstrated an experimental quantum teleporter able to teleport full wave-packets of light while at the same time preserving the quantum characteristic of strongly non-classical superposition states, manifested in the negativity of the Wigner function. This experiment thus constitutes an important breakthrough and opens the door to universal QIP and further hybridization schemes between discrete and continuous variables techniques [7]. In Sec.3, we have shown that a careful multimode model of both the non-classical input state and the broadband teleportation apparatus were necessary to understand the exact behavior of the experimental apparatus and predict precisely the value of the negativity and the shape of theWigner function at the output of the teleporter. Our multimode models of non-classical wave-packet of light and broadband teleportation is precise enough to explain the measured experimental results. Finally in Sec.4, we have introduced the idea of Gaussian conditional teleportation to balance a continuous variable teleportation apparatus between determinism and amount of errors. We have shown experimentally that a simple conditioning done with Alice's measurements can be used to increase the negativity and the purity of the output teleported quantum state.

Figure 1: Quantum teleportation of photon subtracted squeezed vacuum states, experimental setup.

Figure 2: Quantum teleportation of photon subtracted squeezed vacuum states, experimental results. From left to right, input state electrical field, input state Wigner function, output state electrical field, output state Wigner function.

Figure 3: Threshold relation between EPR correlations r and input equivalent losses 1 - η.

Figure 4: Left: probability of success of conditional teleportation, experimental results and theoretical prediction. Right: improvement of output Wigner function negativity Wout with conditional teleportation.

量子アルゴリズムの例として、量子テレポーテーションは量子情報処理研究の初期に発見・提案された。そして、離散変数(量子ビット)の場合も連続変数の場合もその原理検証実験が行われた。ただし、当時は入出力が等しい「恒等変換」の量子アルゴリズムであり、それが量子情報処理においてどのように使われるかは余り明らかではなかった。ところがその後、量子テレポーテーションを基礎とした一方向(one-way)量子情報処理や補助入力誘起(ancilla-driven)量子情報処理が提案され、量子テレポーテーションは一躍ユニバーサル量子情報処理の中心的存在となった。これらのユニバーサル量子情報処理のためには、非古典の非ガウス型状態の生成および操作がキーとなるが、中でも非古典非ガウス型状態の量子テレポーテーションは極めて重要である。

論文提出者であるBenichi氏は、非古典非ガウス状態として「シュレーディンガーの猫状態」を生成し、その量子テレポーテーションを行った。また、実験の理論的解析のためのモデルや新たな実験解析法(量子トモグラフィーアルゴリズム)を考案し、実験結果をうまく解析・説明することに成功した。特に、新たな量子トモグラフィーアルゴリズムを考案し、純粋度の落ちる量子テレポーテーション出力において精度の高い解析を可能にした。さらに、新たな実験として、量子テレポーテーションの送信者側で条件を付けることを行い、量子テレポーテーションフィデリティ(正確に言うと、位相空間原点におけるウイグナー関数の負の値の程度)を上げることに成功した。

以下に本論文の各章の内容を要約する。

第1章では、本論文の基礎となる量子光学の基礎について述べている。特に、本論文では時間的に局在した状態を扱うため、基本的に周波数的にはマルチモードになる。そのため、周波数マルチモードのモデルについて記述している。

第2章では、非古典非ガウス型状態の生成について述べている。特に、スクイーズド光からの光子引き去り(Photon subtraction)によるシュレーディンガーの猫状態生成について述べている。スクイーズド光は偶数個の光子の重ね合わせ状態であるが、1つだけ光子を引き去るとシュレーディンガーの猫状態に極めて近い状態となる。Benichi氏はこの様子を氏の考案したモデルにより解析している。また、実際にシュレーディンガーの猫状態生成実験を行い、このモデルで解析を行っている。

第3章では、量子状態推定のための量子トモグラフィーについて述べている。前述したように、量子トモグラフィーはシュレーディンガーの猫状態量子テレポーテーション実験の成否を判定する重要な手法である。Benichi氏の考案した量子トモグラフィーアルゴリズムでは、特に状態の純粋度の低い場合の解析精度が上がるが、一般的に量子テレポーテーション出力では状態の純粋度が下がるので、氏のアルゴリズムは量子テレポーテーション実験の成否を解析するには理想的なアルゴリズムになっている。

第4章では、シュレーディンガーの猫状態量子テレポーテーションについて述べている。まずBenichi氏が考案したモデルを用い量子テレポーテーション過程の理論的解析を行っている。次に実際に行った実験について述べている。実験には2種類あり、1つは従来通り無条件で量子テレポーテーションを行うもの、もう1つは送信者側で条件を付け、条件をクリアした場合のみ量子テレポーテーションを行うものである。この実験の動機は、入力状態が単一光子状態の場合、送信者側で条件付けを行えば(具体的には測定結果がゼロに近ければ)、量子テレポーテーションフィデリティが格段に上がることが示唆されているからである。Benichi氏はこの理論をシュレーディンガーの猫状態の場合に拡張し、実際に実験を行った。条件付けによるイベントレート低下のため、条件付けと実験時間はトレードオフの関係になり、条件をきつくすることは困難であったが、ほぼ理論的予想通りの結果を得、位相空間原点におけるウイグナー関数の負の値の程度を上げることに成功している。

以上のように、Benichi氏は、非古典非ガウス状態として「シュレーディンガーの猫状態」を生成し、その量子テレポーテーションを行い、実験の理論的解析のためのモデルや新たな実験解析法(量子トモグラフィーアルゴリズム)を考案し、実験結果をうまく解析・説明することに成功した。さらに、送信者側で条件を付ける条件付き量子テレポーテーション実験を行い、位相空間原点におけるウイグナー関数の負の値の程度を上げることに成功した。本研究の成果はユニバーサルな量子情報処理を実現する上で重要な意義があるものと認められる。

よって、本論文は博士(工学)の学位論文として合格と認める。