## Abstract

We report on the experimental quantum teleportation of strongly nonclassical wave packets of light. To perform this full quantum operation while preserving and retrieving the fragile nonclassicality of the input state, we have developed a broadband, zero-dispersion teleportation apparatus that works in conjunction with time-resolved state preparation equipment. Our approach brings within experimental reach a whole new set of hybrid protocols involving discrete- and continuous-variable techniques in quantum information processing for optical sciences.

In the early development of quantum information processing (QIP), a communication protocol called quantum teleportation was discovered (*1*) that involves the transportation of an unknown arbitrary quantum state |ψ〉 by means of entanglement and classical information. Experimental realizations of quantum teleportation (*2*, *3*) and more advanced related operations (*4*) in the continuous-variable regime have been achieved by linear optics methods, although only for Gaussian states so far. However, at least third-order nonlinear operations are necessary for building a universal quantum computer (*5*)—something that Gaussian operations and Gaussian states alone cannot achieve. Photon subtraction techniques based on discrete-variable technology can provide useful nonlinearities and are used to generate Schrödinger’s-cat states and other optical non-Gaussian states (*6*). Schrödinger’s-cat states are of particular interest in this context, as they have been shown to be a useful resource for fault-tolerant QIP (*7*). It is therefore necessary to extend the continuous-variable technology to the technology used in the world of non-Gaussian states.

We have combined these two sets of technologies, and here we demonstrate such Gaussian operations on nonclassical non-Gaussian states by achieving experimental quantum teleportation of Schrödinger’s-cat states of light. Using the photon subtraction protocol, we generate quantum states closely approximating Schrödinger’s-cat states in a manner similar to (*8*–*11*). To accommodate the required time-resolving photon detection techniques and handle the wave-packet nature of these optical Schrödinger’s-cat states, we have developed a hybrid teleporter built with continuous-wave light yet able to directly operate in the time domain. For this purpose we constructed a time-gated source of Einstein-Podolsky-Rosen (EPR) correlations as well as a classical channel with zero phase dispersion (*12*). We were able to bring all the experimental parameters up to the quantum regime, and we performed successful quantum teleportation in the sense that both our input and output states are strongly nonclassical.

A superposition of the quasi-classical coherent state *6*). In these protocols, a squeezed vacuum state *s*. When a photon detection event occurs in the subtraction channel, *8*–*10*). The approximation is not perfect and can be quantified by means of the fidelity figure *13*).

To represent the superposition nature of these states, we use the Wigner formalism where for any quantum state *W*(*x*, *p*), where *x* and *p* are the phase-space position and momentum parameters. *W*(*x*, *p*) is called the Wigner function and holds information exactly equivalent to *14*). Although the position *W*(*x*, *p*) is always a definite real-valued function that can be numerically reconstructed if one performs a complete phase-resolved sequence of homodyne measurement *15*, *16*). *W*(*x*, *p*) is not a true probability distribution, however, as there exist quantum states whose Wigner functions are not positive. *W*(*x*, *p*) fails to be a positive distribution. Negativity in *W*(*x*, *p*) turns out to be an especially useful description of the nonclassicality of a Schrödinger’s-cat state *W*(*x*, *p*). In contrast, a statistical mixture of

In a quantum teleportation process, the input *W*_{in} and output *W*_{out} Wigner functions are related by the convolution (denoted ○)*r* is the EPR correlation parameter, *G*_{σ} is a normalized Gaussian of standard deviation σ, and *ħ* (Planck’s constant divided by 2π) has been set to 1 (*17*). When finite quantum entanglement *r* is used, *W*_{out} will be a thermalized copy of *W*_{in}. Only with infinite *r* will *G*_{σ} become a delta function so that *W*_{in} = *W*_{out}. The quality of quantum teleportation is usually evaluated according to the teleportation fidelity *F*_{tele} = 1/[1 + exp(–2*r*)] for Gaussian states (*18*). More important for our case, negative features of *W*_{in} (if any) can only be teleported and retrieved in *W*_{out} when *F*_{tele} ≥ 2/3 (*19*), a threshold also known as the no-cloning limit (*20*). However, the practical lower bound on *F*_{tele} will be higher because of decoherence and experimental imperfection of *W*_{in} (*21*). We have thus defined the success criterion of Schrödinger’s-cat–state teleportation as the successful transfer of its nonclassical features, or alternatively, successful teleportation of the Wigner function *W*_{in} negativity.

Our experimental quantum teleporter and Schrödinger’s-cat–state source (Fig. 1) upgrade the experiments described in (*3*) and (*10*), respectively. We use three optical parametric oscillators to generate the necessary squeezed vacua. One is used for the Schrödinger’s-cat–state preparation; the other two are combined together on a half beam splitter whose two exit ports are the resulting pair of EPR-correlated beams. The teleportation is conducted in three steps. Alice first receives both the input state and her EPR beam and performs two joint quadrature measurements, obtaining results *x*_{0} and *p*_{0}. Bob then receives Alice’s measurements

To benchmark our teleporter, we first evaluate the fidelity *F*_{tele} of teleportation of the vacuum state ^{2} = ½ + [exp(–2*r*)]* *(*ħ* = 1) from which we can deduce teleportation fidelity (Fig. 2). The blue traces are the shot-noise level, the noise spectrum of the input vacuum *r* = 0). We measure 4.8 dB of added noise above the shot noise, in agreement with the expected teleportation fidelity of 0.5. When Alice and Bob share entanglement, the added noise drops to that shown by the green traces: 1.4 dB above the shot noise around 1 MHz, corresponding to a fidelity of 0.83. This is in agreement with the experimental figure of −6.9 dB that we observe in direct measurement of the EPR correlations shared between Alice and Bob.

In contrast to quantum teleportation experiments conducted to date for narrow sidebands of light (*2*, *3*), our setup operates over a wide frequency bandwidth, as required by the nature of our input state. Because its generation relies on the detection of a single photon and the induced projection, a Schrödinger’s-cat state made via photon subtraction is a short wave packet of light. A phenomenological way to picture these wave packets is to consider them as the closed boxes containing the macroscopic superposition states as in Schrödinger’s original idea. This requires Alice and Bob to teleport every frequency component of these “box-like” wave packets for faithful teleportation to occur. In this way, Alice and Bob do not need to actually teleport the Schrödinger’s-cat states directly, but merely the potential boxes containing them. Consequently, Alice and Bob do not need to know when a detection event occurs; rather, they are only concerned with continuous and faithful “box” wave-packet teleportation, whichever state lies in the box. In fact, Alice and Bob actually teleport most of the time a squeezed vacuum state

In essence, our teleporter is a time-resolved apparatus that deconstructs the input wave packets into a stream of infinitely small time bins and reconstructs them at the output, within the extent of what we refer to as the teleportation bandwidth. This bandwidth is clearly visible in both of the green experimental traces where the added noise slowly increases with frequency (Fig. 2). This is a direct consequence of the finite bandwidth of squeezing used for entanglement. However, across the frequencies relevant to our input state, teleportation fidelity is always greater than the no-cloning limit of 2/3, a necessary regime for negativity teleportation. A very careful implementation of the classical channel has been required (*12*) to achieve experimental realization of this fidelity.

To verify the success of Schrödinger’s-cat–state teleportation, we perform experimental quantum tomography of the input and output states independently (Fig. 3). Both input and output marginal distributions exhibit the characteristic eye shape of photon-subtracted squeezed states, with a clear lack of detection events around the origin for any phase. Although necessary, this feature alone is not sufficient to confirm the presence of negativity in *W*_{in} or *W*_{out}. The reconstructed input Wigner function *W*_{in} shows the two positive Gaussians of *W*_{in}(0, 0) = −0.171 ± 0.003] caused by the interferences of the *W*_{out} retains the characteristic non-Gaussian shape as well as the negative dip [*W*_{out}(0, 0) = −0.022 ± 0.003] to a lesser degree. The degradation of the central negative dip and the full evolution of *W*_{in} toward *W*_{out} can be fully understood using Eq. 1 with a model of *W*_{in}, as was done in (*21*). Given the measured input state negativity of *W*_{in}(0, 0) = −0.171 and −6.9 dB of squeezing, the results of (*21*) predict an output negativity value of −0.02, in good agreement with measured output negativity. Although this figure does not take into account the input-state squeezing, a more detailed model shows that a squeezing parameter *s* = 0.28 affects output negativity in the third decimal place only (*12*). The experimental input and output states have an average photon number *12*). The increase in the output-state size is due to teleportation-induced thermalization. We calculate that the fidelity *F*_{cat} is as high as 0.750 ± 0.005 for the input Wigner function *W*_{in}, with the nearest Schrödinger’s-cat state having an amplitude |α_{in}|^{2} = 0.98 (*12*). However, after the teleportation *W*_{out}, fidelity is reduced to 0.46 ± 0.01, with the nearest Schrödinger’s-cat state having an amplitude |α_{out}|^{2} = 0.66. If *W*_{out} fidelity is calculated with |α_{in}|^{2} = 0.98, then *F*_{cat} = 0.45 ± 0.01.

We have demonstrated an experimental quantum teleporter able to teleport full wave packets of light up to a bandwidth of 10 MHz while at the same time preserving the quantum characteristic of strongly nonclassical superposition states, manifested in the negativity of the Wigner function. Although *F*_{cat} and *W*(0, 0) drop in the teleportation process, there is no theoretical limitation other than available squeezing, and stronger EPR correlations would achieve better fidelity and negativity transmission. The various more complex states generated as an application of photon subtraction so far (*22*, *23*) can be, in principle, readily sent through our broadband quantum teleporter. This opens the door to universal QIP and further hybridization schemes between discrete- and continuous-variable techniques (*24*).

## Supporting Online Material

www.sciencemag.org/cgi/content/full/332/6027/330/DC1

Materials and Methods

## References and Notes

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
See supporting material on
*Science*Online. - ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- Supported by the Strategic Information and Communications R&D Promotion (SCOPE) program of the Ministry of Internal Affairs and Communications of Japan, Special Coordination Funds for Promoting Science and Technology, Grants-in-Aid for Scientific Research, Global Center of Excellence, Advanced Photon Science Alliance, and Funding Program for World-Leading Innovative R&D on Science and Technology (FIRST) commissioned by the Ministry of Education, Culture, Sports, Science and Technology of Japan, Academy of Sciences of the Czech Republic, Japanese Society for the Promotion of Science, and the Australian Research Council, Center of Excellence (grant CE11E0096).